Microlocal methods for dynamical systems

QMath 13 Georgia Tech

Maciej Zworski

UC Berkeley

October 10 2016

Dynamical systems

Dynamical systems

Dynamical systems

Dynamical systems

Dynamical systems

Dynamical systems a statistical approach

Dynamical systems a statistical approach

Linear Non-linear

Dynamical systems a statistical approach

Linear Non-linear

Dynamical systems a statistical approach

Completely integrable Chaotic

In the chaotic case positions and directions get uniformlydistributed

Typical questionsHow long do we have to wait to have uniform distribution

Are there periodic orbits and what information to they contain

In the chaotic case positions and directions get uniformlydistributed

Recent work on the rate of decay for billiards byBaladindashDemersndashLiverani rsquo15

A dynamical analogue of the Riemann zeta function

Ruelle zetafunction

Replace primes with prime closed orbits in ζ(s) =prod

p(1minus pminuss)minus1

A dynamical analogue of the Riemann zeta function Ruelle zetafunction

Replace primes with prime closed orbits in ζ(s) =prod

p(1minus pminuss)minus1

A dynamical analogue of the Riemann zeta function Ruelle zetafunction

Replace primes with prime closed orbits in ζ(s) =prod

p(1minus pminuss)minus1

A dynamical analogue of the Riemann zeta function Ruelle zetafunction

Replace primes with prime closed orbits in ζ(s) =prod

p(1minus pminuss)minus1

Replace p by log `γ where `γ is the length of a prime closed orbit

A dynamical analogue of the Riemann zeta function Ruelle zetafunction

Replace primes with prime closed orbits in ζ(s) =prod

p(1minus pminuss)minus1

A dynamical analogue of the Riemann zeta function Ruelle zetafunction

Replace primes with prime closed orbits in ζ(s) =prod

p(1minus pminuss)minus1

A dynamical analogue of the Riemann zeta function Ruelle zetafunction

Replace primes with prime closed orbits in ζ(s) =prod

p(1minus pminuss)minus1

Replace p by log `γ where `γ is the length of a prime closed orbit

A dynamical analogue of the Riemann zeta function Ruelle zetafunction

Replace primes with prime closed orbits in ζ(s) =prod

p(1minus pminuss)minus1

A dynamical analogue of the Riemann zeta function Ruelle zetafunction

Replace primes with prime closed orbits in ζ(s) =prod

p(1minus pminuss)minus1

Replace p by log `γ where `γ is the length of a prime closed orbit

A dynamical analogue of the Riemann zeta function Ruelle zetafunction

Replace primes with prime closed orbits in ζ(s) =prod

p(1minus pminuss)minus1

Replace p by log `γ where `γ is the length of a prime closed orbit

A dynamical analogue of the Riemann zeta function Ruelle zetafunction

Replace primes with prime closed orbits in ζ(s) =prod

p(1minus pminuss)minus1

ζD(s) =prodγ

(1minus eminuss`γ )

Replace p by e`γ where `γ is the length of a prime closed orbit

It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system

That includes the time at which we achieve uniform distribution

A dynamical analogue of the Riemann zeta function Ruelle zetafunction

Replace primes with prime closed orbits in ζ(s) =prod

p(1minus pminuss)minus1

ζD(s) =prodγ

(1minus eminuss`γ )

Replace p by e`γ where `γ is the length of a prime closed orbit

It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system

That includes the time at which we achieve uniform distribution

A dynamical analogue of the Riemann zeta function Ruelle zetafunction

Replace primes with prime closed orbits in ζ(s) =prod

p(1minus pminuss)minus1

ζD(s) =prodγ

(1minus eminuss`γ )

Replace p by e`γ where `γ is the length of a prime closed orbit

It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system

That includes the time at which we achieve uniform distribution

Dynamical zeta functions have been studied by many authors

Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15

Y (`1 `2 φ) for `1 = `2 = 7 φ = π2

Dynamical zeta functions have been studied by many authors

Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15

Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC

ldquoI must admit a positive answer would be a little shockingrdquo

Dynamical zeta functions have been studied by many authors

Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15

Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X )

eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

A dynamical analogue of the Riemann zeta function Ruelle zetafunction

Replace primes with prime closed orbits in ζ(s) =prod

p(1minus pminuss)minus1

ζD(s) =prodγ

(1minus eminuss`γ )

Replace p by e`γ where `γ is the length of a prime closed orbit

It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system

That includes the time at which we achieve uniform distribution

A dynamical analogue of the Riemann zeta function Ruelle zetafunction

Replace primes with prime closed orbits in ζ(s) =prod

p(1minus pminuss)minus1

ζD(s) =prodγ

(1minus eminuss`γ )

Replace p by e`γ where `γ is the length of a prime closed orbit

It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system

That includes the time at which we achieve uniform distribution

Dynamical zeta functions have been studied by many authors

Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15

Y (`1 `2 φ) for `1 = `2 = 7 φ = π2

Dynamical zeta functions have been studied by many authors

Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15

Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC

ldquoI must admit a positive answer would be a little shockingrdquo

Dynamical zeta functions have been studied by many authors

Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15

Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X )

eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

A dynamical analogue of the Riemann zeta function Ruelle zetafunction

Replace primes with prime closed orbits in ζ(s) =prod

p(1minus pminuss)minus1

ζD(s) =prodγ

(1minus eminuss`γ )

Replace p by e`γ where `γ is the length of a prime closed orbit

It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system

That includes the time at which we achieve uniform distribution

Dynamical zeta functions have been studied by many authors

Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15

Y (`1 `2 φ) for `1 = `2 = 7 φ = π2

Dynamical zeta functions have been studied by many authors

Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15

Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC

ldquoI must admit a positive answer would be a little shockingrdquo

Dynamical zeta functions have been studied by many authors

Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15

Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X )

eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Dynamical zeta functions have been studied by many authors

Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15

Y (`1 `2 φ) for `1 = `2 = 7 φ = π2

Dynamical zeta functions have been studied by many authors

Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15

Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC

ldquoI must admit a positive answer would be a little shockingrdquo

Dynamical zeta functions have been studied by many authors

Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15

Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X )

eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Dynamical zeta functions have been studied by many authors

Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15

Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC

ldquoI must admit a positive answer would be a little shockingrdquo

Dynamical zeta functions have been studied by many authors

Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15

Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X )

eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Dynamical zeta functions have been studied by many authors

Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15

Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X )

eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X )

eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X )

eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

What is an Anosov flow

TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous

dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0

|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0

()t

+

E

E

Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X )

eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X )

eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X )

eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)

For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C

Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13

The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X )

eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X )

eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

x

ξ

ξ2 + V (x) = E

T lowastRn

x

ξ T lowastMξ(Xx )

=0

Scattering theory FaurendashSjostrand

0

Q

partTlowastM

Elowastu

Elowasts

DyatlovndashZ DyatlovndashGuillarmou

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X )

eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X )

eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X )

eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X )

eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X )

eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Special case

Let (M g) be a smooth oriented compact Riemannian surface of

negative curvature

ζD(s) =prodγ

(1minus eminuss`γ ) Re s 1

Theorem (DyatlovndashZ rsquo16)

Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and

s2minus2gζD(s)|s=0 6= 0

In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X )

eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X )

eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X )

eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Pollicott-Ruelle Resonances

Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))

ρf g (t) =

intX

eminusitP f (x)g(x)dx

Power spectrum

ρf g (λ) =

int infin0

ρf g (t)e iλtdt

Resonances poles of ρf g (λ)

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

A ldquorealrdquo life example

Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

ρf g (t) =

intX

eminusitP f (x)g(x)dx

PollicottndashRuelle resonances are the poles of ρf g (λ)

m(λ) = dim

u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu

E lowastu = (Eu otimes E0)perp sub T lowastX

BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11

To obtain exponential decay of correlations one needs a gap

ν0 = supν no resonances for Imλ gt minusν

ν1 = supν finite n of resonances for Imλ gt minusν

ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

A ldquorealrdquo life investigation of the gap

Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Microlocal analysis (semiclassical version)

I Phase space (x ξ) isin T lowastX

I Semiclassical parameter hrarr 0 the effective wavelength

I Classical observables a(x ξ) isin Cinfin(T lowastX )

I Quantization Oph(a) = a(x hi partx

) Cinfin(X )rarr Cinfin(X )

semiclassical pseudodifferential operator

Basic examples

I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator

I a(x ξ) = ξj =rArr Oph(a) = hi partxj

Classical-quantum correspondence

I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)

I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow

I Example [Oph(ξk)Oph(xj)] = hi δjk

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u

Oph(b)u+ hminus1

f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Standard semiclassical estimates

General question

P = Oph(p) Pu = f =rArr u f

Control u microlocally

Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u

Elliptic estimate

p = 0 supp a

Propagation of singularities

supp ab 6= 0etHp

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Phase space description of singularities

u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0

(x ξ) isinWF(u)

mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)

Examples

I WF(δ0) = (0 ξ) ξ isin Rn 0

I WF((x minus i0)minus1) = (0 ξ) ξ gt 0

I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ )

=ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Pγ is the linearized Poincare map

Return to the Ruelle zeta function (with a slight change ofconvention)

ζD(λ) =prodγ

(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)

ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X

ζk(λ) =

(minusinfinsum

m=1

sumγ

e imλ`γ trandkPmγ

m| det(I minus Pmγ )|

) Imλ 1

Theorem (DyatlovndashZ rsquo16)

ζk(λ) extends to an entire function and the multiplicities of itszeros are given by

dimu isin Dprime(X Ωk

0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu

where Ωk

0 are k-forms satisfying ιVu = 0

The starting point for relating the zeta function to thedistributional spectrum of 1

i LV is is the AtiyahndashBottndashGuillemintrace formula

tr eminusitPh =infinsum

m=1

sumγ

`γδ(t minusm`γ)

|I minus Pmγ |

P = hV i

This is related to the first building block of the zeta function

ζ0(λ) = exp

(minusinfinsum

m=1

sumγ

e imλ`γ

m| det(I minus Pmγ )|

)Since

(P minus z)minus1 =i

h

int infin0

eminusitPhdt

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

The starting point for relating the zeta function to thedistributional spectrum of 1

i LV is is the AtiyahndashBottndashGuillemintrace formula

tr eminusitPh =infinsum

m=1

sumγ

`γδ(t minusm`γ)

|I minus Pmγ |

P = hV i

This is related to the first building block of the zeta function

ζ0(λ) = exp

(minusinfinsum

m=1

sumγ

e imλ`γ

m| det(I minus Pmγ )|

)Since

(P minus z)minus1 =i

h

int infin0

eminusitPhdt

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

The starting point for relating the zeta function to thedistributional spectrum of 1

i LV is is the AtiyahndashBottndashGuillemintrace formula

tr eminusitPh =infinsum

m=1

sumγ

`γδ(t minusm`γ)

|I minus Pmγ |

P = hV i

This is related to the first building block of the zeta function

ζ0(λ) = exp

(minusinfinsum

m=1

sumγ

e imλ`γ

m| det(I minus Pmγ )|

)

Since

(P minus z)minus1 =i

h

int infin0

eminusitPhdt

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

The starting point for relating the zeta function to thedistributional spectrum of 1

i LV is is the AtiyahndashBottndashGuillemintrace formula

tr eminusitPh =infinsum

m=1

sumγ

`γδ(t minusm`γ)

|I minus Pmγ |

P = hV i

This is related to the first building block of the zeta function

ζ0(λ) = exp

(minusinfinsum

m=1

sumγ

e imλ`γ

m| det(I minus Pmγ )|

)Since

(P minus z)minus1 =i

h

int infin0

eminusitPhdt

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

The starting point for relating the zeta function to thedistributional spectrum of 1

i LV is is the AtiyahndashBottndashGuillemintrace formula

tr eminusitPh =infinsum

m=1

sumγ

`γδ(t minusm`γ)

|I minus Pmγ |

P = hV i

This is related to the first building block of the zeta function

ζ0(λ) = exp

(minusinfinsum

m=1

sumγ

e imλ`γ

m| det(I minus Pmγ )|

)Since

(P minus z)minus1 =i

h

int infin0

eminusitPhdt

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0

= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R

KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

The starting point for relating the zeta function to thedistributional spectrum of 1

i LV is is the AtiyahndashBottndashGuillemintrace formula

tr eminusitPh =infinsum

m=1

sumγ

`γδ(t minusm`γ)

|I minus Pmγ |

P = hV i

This is related to the first building block of the zeta function

ζ0(λ) = exp

(minusinfinsum

m=1

sumγ

e imλ`γ

m| det(I minus Pmγ )|

)Since

(P minus z)minus1 =i

h

int infin0

eminusitPhdt

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

The starting point for relating the zeta function to thedistributional spectrum of 1

i LV is is the AtiyahndashBottndashGuillemintrace formula

tr eminusitPh =infinsum

m=1

sumγ

`γδ(t minusm`γ)

|I minus Pmγ |

P = hV i

This is related to the first building block of the zeta function

ζ0(λ) = exp

(minusinfinsum

m=1

sumγ

e imλ`γ

m| det(I minus Pmγ )|

)Since

(P minus z)minus1 =i

h

int infin0

eminusitPhdt

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

The starting point for relating the zeta function to thedistributional spectrum of 1

i LV is is the AtiyahndashBottndashGuillemintrace formula

tr eminusitPh =infinsum

m=1

sumγ

`γδ(t minusm`γ)

|I minus Pmγ |

P = hV i

This is related to the first building block of the zeta function

ζ0(λ) = exp

(minusinfinsum

m=1

sumγ

e imλ`γ

m| det(I minus Pmγ )|

)

Since

(P minus z)minus1 =i

h

int infin0

eminusitPhdt

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

The starting point for relating the zeta function to thedistributional spectrum of 1

i LV is is the AtiyahndashBottndashGuillemintrace formula

tr eminusitPh =infinsum

m=1

sumγ

`γδ(t minusm`γ)

|I minus Pmγ |

P = hV i

This is related to the first building block of the zeta function

ζ0(λ) = exp

(minusinfinsum

m=1

sumγ

e imλ`γ

m| det(I minus Pmγ )|

)Since

(P minus z)minus1 =i

h

int infin0

eminusitPhdt

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

The starting point for relating the zeta function to thedistributional spectrum of 1

i LV is is the AtiyahndashBottndashGuillemintrace formula

tr eminusitPh =infinsum

m=1

sumγ

`γδ(t minusm`γ)

|I minus Pmγ |

P = hV i

This is related to the first building block of the zeta function

ζ0(λ) = exp

(minusinfinsum

m=1

sumγ

e imλ`γ

m| det(I minus Pmγ )|

)Since

(P minus z)minus1 =i

h

int infin0

eminusitPhdt

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0

= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R

KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

The starting point for relating the zeta function to thedistributional spectrum of 1

i LV is is the AtiyahndashBottndashGuillemintrace formula

tr eminusitPh =infinsum

m=1

sumγ

`γδ(t minusm`γ)

|I minus Pmγ |

P = hV i

This is related to the first building block of the zeta function

ζ0(λ) = exp

(minusinfinsum

m=1

sumγ

e imλ`γ

m| det(I minus Pmγ )|

)Since

(P minus z)minus1 =i

h

int infin0

eminusitPhdt

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

The starting point for relating the zeta function to thedistributional spectrum of 1

i LV is is the AtiyahndashBottndashGuillemintrace formula

tr eminusitPh =infinsum

m=1

sumγ

`γδ(t minusm`γ)

|I minus Pmγ |

P = hV i

This is related to the first building block of the zeta function

ζ0(λ) = exp

(minusinfinsum

m=1

sumγ

e imλ`γ

m| det(I minus Pmγ )|

)

Since

(P minus z)minus1 =i

h

int infin0

eminusitPhdt

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

The starting point for relating the zeta function to thedistributional spectrum of 1

i LV is is the AtiyahndashBottndashGuillemintrace formula

tr eminusitPh =infinsum

m=1

sumγ

`γδ(t minusm`γ)

|I minus Pmγ |

P = hV i

This is related to the first building block of the zeta function

ζ0(λ) = exp

(minusinfinsum

m=1

sumγ

e imλ`γ

m| det(I minus Pmγ )|

)Since

(P minus z)minus1 =i

h

int infin0

eminusitPhdt

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

The starting point for relating the zeta function to thedistributional spectrum of 1

i LV is is the AtiyahndashBottndashGuillemintrace formula

tr eminusitPh =infinsum

m=1

sumγ

`γδ(t minusm`γ)

|I minus Pmγ |

P = hV i

This is related to the first building block of the zeta function

ζ0(λ) = exp

(minusinfinsum

m=1

sumγ

e imλ`γ

m| det(I minus Pmγ )|

)Since

(P minus z)minus1 =i

h

int infin0

eminusitPhdt

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0

= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R

KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

The starting point for relating the zeta function to thedistributional spectrum of 1

i LV is is the AtiyahndashBottndashGuillemintrace formula

tr eminusitPh =infinsum

m=1

sumγ

`γδ(t minusm`γ)

|I minus Pmγ |

P = hV i

This is related to the first building block of the zeta function

ζ0(λ) = exp

(minusinfinsum

m=1

sumγ

e imλ`γ

m| det(I minus Pmγ )|

)

Since

(P minus z)minus1 =i

h

int infin0

eminusitPhdt

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

The starting point for relating the zeta function to thedistributional spectrum of 1

i LV is is the AtiyahndashBottndashGuillemintrace formula

tr eminusitPh =infinsum

m=1

sumγ

`γδ(t minusm`γ)

|I minus Pmγ |

P = hV i

This is related to the first building block of the zeta function

ζ0(λ) = exp

(minusinfinsum

m=1

sumγ

e imλ`γ

m| det(I minus Pmγ )|

)Since

(P minus z)minus1 =i

h

int infin0

eminusitPhdt

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

The starting point for relating the zeta function to thedistributional spectrum of 1

i LV is is the AtiyahndashBottndashGuillemintrace formula

tr eminusitPh =infinsum

m=1

sumγ

`γδ(t minusm`γ)

|I minus Pmγ |

P = hV i

This is related to the first building block of the zeta function

ζ0(λ) = exp

(minusinfinsum

m=1

sumγ

e imλ`γ

m| det(I minus Pmγ )|

)Since

(P minus z)minus1 =i

h

int infin0

eminusitPhdt

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0

= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R

KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

The starting point for relating the zeta function to thedistributional spectrum of 1

i LV is is the AtiyahndashBottndashGuillemintrace formula

tr eminusitPh =infinsum

m=1

sumγ

`γδ(t minusm`γ)

|I minus Pmγ |

P = hV i

This is related to the first building block of the zeta function

ζ0(λ) = exp

(minusinfinsum

m=1

sumγ

e imλ`γ

m| det(I minus Pmγ )|

)Since

(P minus z)minus1 =i

h

int infin0

eminusitPhdt

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

The starting point for relating the zeta function to thedistributional spectrum of 1

i LV is is the AtiyahndashBottndashGuillemintrace formula

tr eminusitPh =infinsum

m=1

sumγ

`γδ(t minusm`γ)

|I minus Pmγ |

P = hV i

This is related to the first building block of the zeta function

ζ0(λ) = exp

(minusinfinsum

m=1

sumγ

e imλ`γ

m| det(I minus Pmγ )|

)Since

(P minus z)minus1 =i

h

int infin0

eminusitPhdt

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0

= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R

KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

The starting point for relating the zeta function to thedistributional spectrum of 1

i LV is is the AtiyahndashBottndashGuillemintrace formula

tr eminusitPh =infinsum

m=1

sumγ

`γδ(t minusm`γ)

|I minus Pmγ |

P = hV i

This is related to the first building block of the zeta function

ζ0(λ) = exp

(minusinfinsum

m=1

sumγ

e imλ`γ

m| det(I minus Pmγ )|

)Since

(P minus z)minus1 =i

h

int infin0

eminusitPhdt

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0

= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R

KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0

= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R

KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0

= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R

KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0

= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R

KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0

= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R

KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R

KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R

KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Here the trace is a formal trace obtained by integration over thediagonal

Au(x) =

intX

KA(x y)u(y)dy

tr A =

intX

KA(x x)dx

Allowed under a wave front set condition

WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)

tr A =

intRδ((aminus 1)x) =

1

|aminus 1| a 6= 1

WF(KA) = (x ax minusaη η) η 6= 0

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

e it0λh trϕlowastminust0(P minus hλ)minus1 =part

partλlog ζ0(λ)

Need

WF(Kϕlowastminust0

(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty

Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this

0

Q

partTlowastM

Elowastu

Elowasts

The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature

What is the order of vanishing m of ζD(λ) at λ = 0

When M = ΓH2 (constant curvature)

Selberg trace formula =rArr m = 2g minus 2

In general

ζD(λ) =ζ1(λ)

ζ0(λ)ζ2(λ)

m = m1(0)minusm0(0)minusm2(0)

mj(0) = dimu isin Dprime(X Ωj

0) LrVu = 0 WF(u) sub E lowastu

where Ωj

0(X ) are j-forms satisfying ιVu = 0

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Regularity result

Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin

Here L2(X ) is defined using the smooth invariant measure on SlowastM

dvol = α and dα

where α is the contact form

α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1

α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof

If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )

Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)

Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1

Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence

〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0

t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)

It follows that du|EuoplusEs = 0 and that means that du = ϕα

0 = α and d(du) = ϕα and dα =rArr ϕ = 0

=rArr du = 0

=rArr u = const

If V 2u = 0 WF(u) sub E lowastu then Vu = const

ButintX Vudvol = 0 so const = 0

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m1(0) = dim Y1 = dim H1(X C)

Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu

1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have

cvol(X ) =

intdu and α =

intu and dα = 0

We conclude that du = 0

2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))

Y1 3 u 7rarr [uminus dϕ] isin H1(X C)

is an isomorphism

Showing semisimplity is a little bit tricky

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention

m0(0) = m2(0) = 1 m1(0) = dim H1(X C)

Since for surfaces of genus g ge 2

H1(SlowastMC) H1(MC)

it follows that the order of vanishing of ζD at 0 is

m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2

Thanks for your attention