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The Structure ofHyperbolic Sets
Todd Fisher
Department of Mathematics
University of Maryland, College Park
The Structure of Hyperbolic Sets – p. 1/35
Outline of TalkHistory and Examples
The Structure of Hyperbolic Sets – p. 2/35
Outline of TalkHistory and Examples
Properties
The Structure of Hyperbolic Sets – p. 2/35
Outline of TalkHistory and Examples
Properties
Structure
The Structure of Hyperbolic Sets – p. 2/35
3 ways dynamicsstudied
Measurable: Probabilistic and statisticalproperties.
The Structure of Hyperbolic Sets – p. 3/35
3 ways dynamicsstudied
Measurable: Probabilistic and statisticalproperties.
Topological: This studies functions that areonly assumed to be continuous.
The Structure of Hyperbolic Sets – p. 3/35
3 ways dynamicsstudied
Measurable: Probabilistic and statisticalproperties.
Topological: This studies functions that areonly assumed to be continuous.
Smooth: Assume there is a derivative at everypoint.
The Structure of Hyperbolic Sets – p. 3/35
Advantages toSmooth
The local picture given by derivative
The Structure of Hyperbolic Sets – p. 4/35
Advantages toSmooth
The local picture given by derivative
Very useful in hyperbolic case. Tangent spaceTΛM splits into expanding Eu and contractingdirections Es.
The Structure of Hyperbolic Sets – p. 4/35
Advantages toSmooth
The local picture given by derivative
Very useful in hyperbolic case. Tangent spaceTΛM splits into expanding Eu and contractingdirections Es.
For instance, say
f(x, y) =
[
1/2 0
0 2
] [
x
y
]
The Structure of Hyperbolic Sets – p. 4/35
Stable set andUnstable set
The stable set of a point x ∈ M is
W s(x) = {y ∈ M |d(fn(x), fn(y)) → 0 as n → ∞}.
The Structure of Hyperbolic Sets – p. 5/35
Stable set andUnstable set
The stable set of a point x ∈ M is
W s(x) = {y ∈ M |d(fn(x), fn(y)) → 0 as n → ∞}.
The unstable set of a point x ∈ M is
W u(x) = {y ∈ M |d(f−n(x), f−n(y)) → 0
as n → ∞}.
The Structure of Hyperbolic Sets – p. 5/35
One Origin - CelestialMechanics
In 19th century Poincaré began to study stabilityof solar system.
The Structure of Hyperbolic Sets – p. 6/35
One Origin - CelestialMechanics
In 19th century Poincaré began to study stabilityof solar system.
For a flow from a differential equation with fixedhyperbolic saddle point p and pointx ∈ W s(p) ∩ W u(p).
The Structure of Hyperbolic Sets – p. 6/35
One Origin - CelestialMechanics
In 19th century Poincaré began to study stabilityof solar system.
For a flow from a differential equation with fixedhyperbolic saddle point p and pointx ∈ W s(p) ∩ W u(p).
P
The Structure of Hyperbolic Sets – p. 6/35
TransverseHomoclinic Point
If we look at a function f picture can becomemore complicated. This was in some sense thestart of chaotic dynamics.
A point x ∈ W s(p) ⋔ W u(p) is called a transversehomoclinic point.
The Structure of Hyperbolic Sets – p. 7/35
Homoclinic Tangle
p
x
The Structure of Hyperbolic Sets – p. 8/35
Homoclinic Tangle
f(x)
x
p
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Homoclinic Tangle
f(x)
x
p
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Homoclinic Tangle
p
x
f(x)
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Further Results onHomoclinic Points
In 1930’s Birkhoff showed that near a transversehomoclinic point ∃ pn → x such that pn periodic
The Structure of Hyperbolic Sets – p. 9/35
Further Results onHomoclinic Points
In 1930’s Birkhoff showed that near a transversehomoclinic point ∃ pn → x such that pn periodic
In 1960’s Smale showed the following:
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Further Results onHomoclinic Points
In 1930’s Birkhoff showed that near a transversehomoclinic point ∃ pn → x such that pn periodic
In 1960’s Smale showed the following:
nf (D)
D
p x
The Structure of Hyperbolic Sets – p. 9/35
Smale’s HorseshoeSmale generalized picture as follows:Take the unit square R = [0, 1] × [0, 1] map thesquare as shown below.
BA
f(R)
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There exist two region A and B in R such that
f |A and f |B looks like
[
1/3 0
0 3
]
The Structure of Hyperbolic Sets – p. 10/35
Invariant Set forHorseshoe - 1
We want points that never leave R.
Λ =⋂
n∈Z
fn(R)
The Structure of Hyperbolic Sets – p. 11/35
Invariant Set forHorseshoe -3
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Invariant Set forHorseshoe -3
The Structure of Hyperbolic Sets – p. 12/35
Invariant Set forHorseshoe -3
The Structure of Hyperbolic Sets – p. 12/35
Dynamics ofHorseshoe
Then Λ is Middle Thirds Cantor × Middle ThirdsCantor
The Structure of Hyperbolic Sets – p. 13/35
Dynamics ofHorseshoe
Then Λ is Middle Thirds Cantor × Middle ThirdsCantor
The set Λ is chaotic in the sense of Devaney.
periodic points of Λ are dense
there is a point with a dense orbit (transitive)
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Dynamics ofHorseshoe
Then Λ is Middle Thirds Cantor × Middle ThirdsCantor
The set Λ is chaotic in the sense of Devaney.
periodic points of Λ are dense
there is a point with a dense orbit (transitive)
So Horseshoe is very interesting dynamically.
The Structure of Hyperbolic Sets – p. 13/35
Hyperbolic Set
Hyperbolic A compact set Λ is hyperbolic if it isinvariant (f(Λ) = Λ) and the tangent space has acontinuous invariant splitting TΛM = E
s ⊕ Eu
where Es is uniformly contracting and E
u isuniformly expanding.
So ∃ C > 0 and λ ∈ (0, 1) such that:‖Dfn
x v‖ ≤ Cλn‖v‖ ∀ v ∈ Esx, n ∈ N and
‖Df−nx v‖ ≤ Cλn‖v‖ ∀ v ∈ E
ux, n ∈ N
The Structure of Hyperbolic Sets – p. 14/35
HyperbolicProperties -1
For a point of a hyperbolic set x ∈ Λ the stableand unstable sets are immersed copies of R
m
and Rn where m = dim(Es) and n = dim(Eu).
TxWs(x) = Es
x and TxWu(x) = Eu
x
Closed + Bounded + Hyperbolic = InterestingDynamics
The Structure of Hyperbolic Sets – p. 15/35
Morse-SmaleDiffeormophisms
A diffeo. f is Morse-Smale if the only recurrentpoints are a finite number hyperbolic periodicpoints and the stable and unstable manifolds ofeach periodic point is transverse.
so dynamics are gradient like.
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Weak PalisConjecture
Note: Recent Theorem says horseshoes are verycommon for diffeomorphisms.
Theorem 1 (Weak Palis Conjecture) For anysmooth manifold the set of there is an open anddense set of C1 diffeomorphisms that are eitherMorse-Smale or contain a horseshoe.
Proof announced by Crovisier, based on work ofBonatti, Gan, and Wen.
The Structure of Hyperbolic Sets – p. 17/35
Hyperbolic ToralAutomorphisms
Take the Matrix A =
[
2 1
1 1
]
.
This matrix has det(A) = 1, one eigenvalueλs ∈ (0, 1), and one eigenvalue λu ∈ (1,∞). Soone contracting direction and one expandingdirection.
The Structure of Hyperbolic Sets – p. 18/35
AnosovDiffeomorphisms
Since A has determinant 1 it preserves Z2 there
is induced map on torus fA from A. At everypoint x ∈ T
2 there is a contacting and expandingdirection.
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AnosovDiffeomorphisms
Since A has determinant 1 it preserves Z2 there
is induced map on torus fA from A. At everypoint x ∈ T
2 there is a contacting and expandingdirection.
A diffeomorphism is Anosov if the entire manifoldis a hyperbolic set. So fA is Anosov.
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Anosov Diffeos in2-dimensions
In two dimensions only the torus supportsAnosov diffeomorphisms and all are topologicallyconjugate to hyperbolic toral automorphisms.
Two maps f : X → X and g : Y → Y areconjugate if there is a continuoushomeomorphism h : X → Y such that hf = gh.
The Structure of Hyperbolic Sets – p. 20/35
AttractorsDefinition 2 A set X has an attractingneighborhood if ∃ neighborhood U of X such thatX =
⋂
n∈Nfn(U).
The Structure of Hyperbolic Sets – p. 21/35
AttractorsDefinition 3 A set X has an attractingneighborhood if ∃ neighborhood U of X such thatX =
⋂
n∈Nfn(U).
A hyperbolic set Λ is a hyperbolic attractor if Λ istransitive (contains a point with a dense orbit)and has an attracting neighborhood.
The Structure of Hyperbolic Sets – p. 21/35
AttractorsDefinition 4 A set X has an attractingneighborhood if ∃ neighborhood U of X such thatX =
⋂
n∈Nfn(U).
A hyperbolic set Λ is a hyperbolic attractor if Λ istransitive (contains a point with a dense orbit)and has an attracting neighborhood.
For a compact surface result of Plykin says theremust be at least 3 holes for a hyperbolic attractor.
The Structure of Hyperbolic Sets – p. 21/35
Plykin AttractorV
The Structure of Hyperbolic Sets – p. 22/35
Plykin AttractorV
f(V)
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Dynamics ofAttractors
V is an attracting neighborhood and
Λ =⋂
n∈N
fn(V ).
In two dimensions a hyperbolic attractor lookslocally like a Cantor set × interval.
The interval is the unstable direction the Cantorset is the stable direction.
Hyperbolic attractors have dense periodic pointsand a point with a dense orbit.
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Locally Maximal
A hyperbolic set Λ is locally maximal (or isolated ) if∃ open set U such that
Λ =⋂
n∈Z
fn(U)
All the examples we looked at are locallymaximal
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Properties of LocallyMaximal Sets
Locally maximal transitive hyperbolic sets havenice properties including:
1. Shadowing
2. Structural Stability
3. Markov Partitions
4. SRB measures (for attractors)
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Shadowing
A sequence x1, x2, ..., xn is an ǫ pseudo-orbit if
d(f(xi), xi+1) < ǫ for all 1 ≤ i < n.
A point y δ-shadows an ǫ pseudo-orbit ifd(f i(y), xi) < δ for all 1 ≤ i ≤ n.
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Shadowing Diagram
1
x1
f(x )
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Shadowing Diagram
x
1
2
x1
f(x )
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Shadowing Diagram
3
2f(x )
xf(x )
1x
2
1
x
The Structure of Hyperbolic Sets – p. 27/35
Shadowing Diagram
y3
2f(x )
xf(x )
1x
2
1
x
The Structure of Hyperbolic Sets – p. 27/35
Shadowing Diagram
f(y)y3
2f(x )
xf(x )
1x
2
1
x
The Structure of Hyperbolic Sets – p. 27/35
Shadowing Diagram
2f (y)
f(y)y3
2f(x )
xf(x )
1x
2
1
x
The Structure of Hyperbolic Sets – p. 27/35
Shadowing Theorem
Theorem 5 (Shadowing Theorem) Let Λ be acompact hyperbolic invariant set. Given δ > 0 ∃
ǫ, η > 0 such that if {xj}j2j=j1
is an ǫ pseudo-orbitfor f with d(xj,Λ) < η for j1 ≤ j ≤ j2, then ∃ ywhich δ-shadows {xj}. If j1 = −∞ and j2 = ∞,then y is unique. If Λ is locally maximal andj1 = −∞ and j2 = ∞, then y ∈ Λ.
The Structure of Hyperbolic Sets – p. 28/35
Structural Stability
Theorem 6 (Structural Stability) If Λ is ahyperbolic set for f , then there exists a C1 openset U containing f such that for all g ∈ U thereexists a hyperbolic set Λg and homeomorphismh : Λ → Λg such that hf = gh.
The Structure of Hyperbolic Sets – p. 29/35
Markov PartitionDecomposition of Λ into finite number ofdynamical rectangles R1, ..., Rn such that foreach 1 ≤ i ≤ n
The Structure of Hyperbolic Sets – p. 30/35
Markov PartitionDecomposition of Λ into finite number ofdynamical rectangles R1, ..., Rn such that foreach 1 ≤ i ≤ nint(Ri) ∩ int(Rj) = ∅ if i 6= j
The Structure of Hyperbolic Sets – p. 30/35
Markov PartitionDecomposition of Λ into finite number ofdynamical rectangles R1, ..., Rn such that foreach 1 ≤ i ≤ nint(Ri) ∩ int(Rj) = ∅ if i 6= j
for some ǫ sufficiently small Ri is(W u
ǫ (x) ∩ Ri) × (W sǫ (x) ∩ Ri)
The Structure of Hyperbolic Sets – p. 30/35
Markov PartitionDecomposition of Λ into finite number ofdynamical rectangles R1, ..., Rn such that foreach 1 ≤ i ≤ nint(Ri) ∩ int(Rj) = ∅ if i 6= j
for some ǫ sufficiently small Ri is(W u
ǫ (x) ∩ Ri) × (W sǫ (x) ∩ Ri)
x ∈ Ri, f(x) ∈ Rj, and i → j is an allowedtransition in Σ, then
f(W s(x,Ri)) ⊂ Rj and f−1(W u(f(x), Rj)) ⊂ Ri.
The Structure of Hyperbolic Sets – p. 30/35
SRB MeasursIf Λ is a hyperbolic attractor ∃ measure µ on Λsuch that for a.e x in basin of attraction and anyobservable φ:
limn→∞
1
nΣn
i=1φ(f i(x)) =
∫
Λ
φ dµ
The Structure of Hyperbolic Sets – p. 31/35
Question 1Katok and Hasselblatt:Question 1: “Let Λ be a hyperbolic set... and V anopen neighborhood of Λ. Does there exist alocally maximal hyperbolic set Λ̃ such thatΛ ⊂ Λ̃ ⊂ V ?”
The Structure of Hyperbolic Sets – p. 32/35
Question 1Katok and Hasselblatt:Question 1: “Let Λ be a hyperbolic set... and V anopen neighborhood of Λ. Does there exist alocally maximal hyperbolic set Λ̃ such thatΛ ⊂ Λ̃ ⊂ V ?”Crovisier (2001) answers no for specific exampleon four torus.
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Question 1Katok and Hasselblatt:Question 1: “Let Λ be a hyperbolic set... and V anopen neighborhood of Λ. Does there exist alocally maximal hyperbolic set Λ̃ such thatΛ ⊂ Λ̃ ⊂ V ?”Crovisier (2001) answers no for specific exampleon four torus.Related questions:
1. Can this be robust?
2. Can this happen on other manifolds, in lowerdimension, on all manifolds?
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Robust and MarkovTheorems
Theorem 7 (F.) On any compact manifold M ,where dim(M) ≥ 2, there exists a C1 open set ofdiffeomorphisms, U , such that any f ∈ U has ahyperbolic set that is not contained in a locallymaximal hyperbolic set.
Theorem 8 (F.) If Λ is a hyperbolic set and V isa neighborhood of Λ, then there exists ahyperbolic set Λ̃ with a Markov partition such thatΛ ⊂ Λ̃ ⊂ V .
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Hyperbolic Sets withInterior
Theorem 9 (F.) If Λ is a hyperbolic set withnonempty interior, then f is Anosov if
1. Λ is transitive
2. Λ is locally maximal and M is a surface
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Hyperbolic Attractorson Surfaces
Theorem 10 (F.) If M is a compact smoothsurface, Λ is a hyperbolic attractor for f , andhyperbolic for g, then Λ is either a hyperbolicattractor or repeller for g.
So if we know the topology of the set and weknow that it is hyperbolic we know it is anattractor.A set Λ is a repeller if there exists neighborhoodV such that Λ =
⋂
n∈Nf−n(V ).
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