parablender episode ii - byu math · parablender episode ii pierre berger, ... and so one for every...

Parablender episode II

Pierre Berger, CNRS Universite Paris 13

Second part of the mini-course ”Blender and parablender”started by Christian Bonatti, CNRS-Dijon.

1/23

Recent of forthcoming results obtained thanks to parablenders

Definition (Arnold Topology)

A family (fa)a∈Rk is of class C r , if the map (a, x) 7→ fa(x) is ofclass C r .

Definition (Arnold-Kolmogorov Typicallity)

A property (P) is C r -k-typical if for generic C r -families (fa)a∈Rk ofC r -dynamics fa, for Lebesgue almost every a ∈ Rk sufficientlysmall, the map fa satisfies (P).

QuestionWhat are the pathological property which are typical in the senseof Arnold-Kolmogorov?

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Theorem (Berger 2015-2016)

For every ∞ > r ≥ 1, there exists an open set of U of C r -familiesof C r -dynamics, and a topologically generic set R ⊂ U so that forevery (fa)a ∈ R, for every ‖a‖ ≤ 1, the map fa displays infinitelymany sinks accumulating on a blender.

Theorem (Berger-Crovisier-Pujals in Preparation)

There exists an open set U of C r -surface maps dynamics in whicha typical dynamics displays infinitely many sinks accumulating on ablender.

3/23

Here is a negative answer to a Problem of Arnold (1989).

Theorem (Berger 2016)

For every ∞ > r ≥ 1, there exists an open set of U of C r -familiesof C r -dynamics, so that for every (an)n≥1 ∈ NN, there exists atopologically generic set R ⊂ U so that for every (fa)a ∈ R, forevery ‖a‖ ≤ 1, the number of saddle points of fa of period n is notbounded by (an)n.

4/23

Here is a negative answer to a question of Colli late (90’s):

Theorem (Juan David Rojas Gacha in Preparation, IMPA)

For every ∞ > r ≥ 2, there exists an open set of U of C r -familiesof C r -dynamics, and a topologically generic set R ⊂ U so that forevery (fa)a ∈ R, for Lebesgue a.e. ‖a‖ ≤ 1, the map fa displaysinfinitely many NUH-Henon like Attractors.

5/23

Here is a negative answer to a conjecture communicated byIlyashenko:

Theorem (Ivan Shilin in Preparation, Moscow)

For every ∞ > r ≥ 2, there exists an open set of U of C r -familiesof C r -dynamics, and a topologically generic set R ⊂ U so that forevery (fa)a ∈ R, for every ‖a‖ ≤ 1, the map fa displays a Lyapunovunstable Milnor attractor.

6/23

Make your own theorem:1) Take your favorite pathological generic properties,2) Show it by using a blender,3) Show its typicality by using a parablender

7/23

IFS Blender and Parablenders

Figure: From Blender to IFS

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An IFS of contracting maps (fi )i of Rn is a C r -blender if its limitset

Λ = limn→∞

⋃i1,...,in

{fin ◦ · · · ◦ fi1(0)}

contains robustly a non-empty open set U of Rn (amongC r -perturbation of (fi )i ).

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Sufficient (and necessary?) condition is the following:

(O) There exists a family of open sets (Ui )1≤i≤k so that:

cl(f −1i (Ui )) ⊂ ∪iUi , ∀1 ≤ i ≤ k .

Observe that ∪iUi ⊂ Λ.

Example

Let k = 2, n = 1, f1(x) = 23(x − 1) + 1, f−1(x) = 2

3(x + 1)− 1.U1 := (−0.1, 0.9), U−1 = (−0.9, 0.1).

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cl(f −1i (Ui )) ⊂ ∪iUi , ∀1 ≤ i ≤ k .


Example

Let k = 2× 2, n = 2.f1,1(x , y) = (23(x − 1) + 1, 23(y − 1) + 1),f−1,1(x) = (23(x + 1)− 1, 23(y − 1) + 1),f1,−1(x , y) = (23(x − 1) + 1, 23(y + 1)− 1),f−1,−1(x) = (23(x + 1)− 1, 23(y + 1)− 1),V1 := (−0.1, 0.9), V−1 = (−0.9, 0.1). Ui ,j := Vi × Vj .

And so one for every n ≥ 2: the open covering property is stableby product.

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cl(f −1i (Ui )) ⊂ ∪iUi , ∀1 ≤ i ≤ k .


Example (B.-Crovisier-Pujals, Hare-Sidorov.)

Let k = 2, n ≥ 2.Put:

U :=

1 1 0 · · · 00 1 1 · · · 0...

. . .. . .

. . ....

0 · · · 0 1 10 · · · · · · 0 1

Take λ < 1 be close to 1. f1(X ) = λUX + (0, . . . , 0, 1).f−1(X ) = λUX − (0, . . . , 0, 1).

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Parablender for IFS

Given a C r -family of contracting maps fi = (fi a)a∈Rk , a C r -familyof IFS (fi )i acts on C r -family of points (za)a as:

(za)a 7→ (fi a(zi ))a

Hence it acts on the space J r0M on C r -jet J r0 z of family z = (za)aof points (za)a at a = 0:

J r0 z 7→ J r0 fi ◦ J r0 z = J r0(fa i (za))

It is a C r -para-blender if the IFS (J r0 fi )i is a blender.

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cl(J r0f−1i (Ui )) ⊂ ∪iUi , ∀1 ≤ i ≤ k .

Observe that:

∪iUi ⊂ Λ = limn→∞

⋃i1,...,in

{J r0fin ◦ · · · ◦ J r0fi1(0)} .

Example

Let k = 4, n ≥ 1.f1,1(a, x) = 2

3(x − 1− a) + 1 + a,f−1,1(a, x) = 2

3(x + 1 + a)− 1− a,f1,−1(a, x) = 2

3(x − 1− a) + 1 + a,f−1,−1(a, x) = 2

3(x + 1 + a)− 1− a,V1 := (−0.1, 0.9), V−1 = (−0.9, 0.1).Ui ,j := {x0 + ay0 : (x0, y0) ∈ Vi × Vj}.

7/23

Example (B.-Crovisier-Pujals)

Let k = 2, r ≥ 1. Take λ < 1 be close to 1.Put X = (x0, x1, · · · , xr ). X (a) := x0 + x1a + · · ·+ xra

r .

U :=

1 0 · · · 0

1 1. . .

......

. . .. . . 0

0 · · · 1 1

f1a(X (a)) = λ(1 + a) · X (a) + 1 + o(ar )f1a(X (a)) = (λUX + (1, 0, . . . , 0))(a) + o(ar ).

f−1a(X (a)) = λ(1 + a) · X (a)− 1 + o(ar )f−1a(X (a)) = (λUX − (1, 0, . . . , 0))(a) + o(ar ).

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Blender and Parablenders for endomorphisms

Let K be a hyperbolic set of a surface endomorphism f and let(W u

loc(←−z ; f ))←−z ∈←−K

be a continuous family of local unstablemanifolds.

Here←−K := {(zi )i≤0 ∈ KZ−

: f (zi ) = zi+1} is the set of preorbits inK .

If the following union contains C 1-robustly a non-empty open setU of the surface, then K is a blender.⋃

←−z ∈K

W uloc(←−z ; f ) ⊃ U .

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For instance let (fi )1≤i≤k be a blender IFS, (Ii )1≤i≤k disjointintervals of I , and g : t Ik → I be an expanding, surjective map.Then the following endomorphism is a blender.

f : (x , y) ∈ tIi × Rd → (g(x), fi (y)) ∈ I × Rd

(O) There exists a family of open sets (Ui )1≤i≤k associated toinverse branches (gi )1≤i≤k so that:

cl(gi (Ui )) ⊂ ∪iUi , ∀1 ≤ i ≤ k .

Observe that ∪iUi ⊂⋃←−z ∈K W u

loc(←−z ; f ).

10/23

Parablender for families of surface endomorphisms

A family of blenders (Ka)a is a C r -parablender at a = 0, if thereexists an open set U of C r -families of points (za)a so that for every

C r -perturbation (fa)a of the dynamics (fa)a, there exists←−k ∈

←−K ,

and a C r -family of points (pa)a ∈ (W uloc(←−k ; fa))a so that:

d(pa, za) = o(|a|r )

Observe that U ⊂⋃←−z ∈K J r0W

uloc(←−z ; f0).

11/23

Example of parablenderFor instance let (fδa)δ∈{−1,1}d be the parablender IFS defined by:

fδa(y) =2

3(y −

∑0≤j<d

δjaj) +

∑0≤j<d

δjaj

(Ii )i∈{−1,1}d disjoint intervals of I , and g : t Ik → I be anexpanding, surjective map. Then the following defines aC r -parablender:

fa : (x , y) ∈ tIi × Rd → (g(x), fia(y)) ∈ I × Rd

(O) There exists a family of open sets (Uδ)δ∈{−1,1}d associated toinverse branches (gδa)δ so that:

cl(J r0gδa(Uδ)) ⊂ ∪δUδ , ∀δ ∈ {−1, 1}d

Observe that ∪δUδ ⊂⋃←−z ∈K J r0W

uloc(←−z ; fa).

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Example of parablenderFor λ < 1 close to 1, let (fia)i∈{−1,1} be the parablender IFSdefined by:

f1a(y) = λ(1 + a) · y + 1

f−1a(y) = λ(1 + a) · y − 1

(Ii )i=1,−1 two disjoint intervals of I , and g : t Ik → I be anexpanding, surjective map. Then the following define aC r -parablender:

fa : (x , y) ∈ tIi × Rd → (g(x), fia(y)) ∈ I × Rd

(O) There exists a family of open sets (Ui )i=1,−1 associated toinverse branches (gia)i=1,−1 so that:

cl(J r0gia(Ui )) ⊂ ∪iUi , ∀i = 1,−1

Observe that ∪iUi ⊂⋃←−z ∈K J r0W

uloc(←−z ; fa).

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Inclination and para-inclination Lemma

Lemma (Inclination)

Let K be a hyperbolic set for an endomorphism, with index(ds , du). Let x ∈ K .Let V be a du-dimensional sumanifold, which is transverse toW s

loc(x) at a point y .Then an open neighborhood of f n(y) in f n(V ) is C r -close toW s

loc(f n(x)) for n large.

Proof.Make a surgery such that y is a hyperbolic fixed point, with V aslocal unstable manifold. Then K ∪ {f n(y) : n ≥ 0} is a hypebrolicset. The Lemma follows from the continuity of the local unstablemanifolds.

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Lemma (para-inclination)

Let (fa)a be a family of dynamics which leaves invariant a family ofhyperbolic sets (Ka)a, with index (ds , du). Let (xa)a ∈ (Ka)a.Let (Va)a be a family of du-dimensional sumanifolds Va which aretransverse to W s

loc(xa) at a point ya.Then an open neighborhood of (f na (ya))a in (f na (Va))a is C r -closeto (W s

loc(f na (xa)))a.

Proof.From the previous argument it suffices to show that the continuityof the local unstable manifolds for parameter families. This is OKby HPS Theory.

15/23

Sinks created nearby a dissipativehomoclinic tangency

LemmaLet fa be a C∞-surface self-mapping which displays a saddle fixedpoint Pa with a homoclinic tangency at a point Ha, which isdissipative: |det DP fa| < 1. Then, for every N ≥ 1 large, thereexists a C r -perturbation fa of fa which displays a sink of period≥ N.

16/23

LemmaLet (fa)a be a C∞-family of surface mappings so that f0 = f andthe homoclinic tangency persists for the (dissipative) continuationof Pa at a point Ha which depends continuously of a. Then, forevery N ≥ 1 large, there exists a C∞-perturbation (fa)a of (fa)a, sothat fa displays a sink of period ≥ N for every ‖a‖ ≤ 1.

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Precise statement of the Theorem

PreliminaryA fixed point for a C r -map is a protectively hyperbolic source if itseigenvalues have different modulii.

LemmaThe immediate repulsion basin of S is C r−1-foliated by strongunstable manifolds. This foliation is denoted by Fuu.

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Let 1 ≤ r <∞. Let U be the set of C r -surface dynamics fdisplaying

• a projectively hyperbolic source S ,

• a blender K ,

• a dissipative saddle point P,

so that:

• S belongs to the covered domain of K ,

• W u(P) intersects transversally W s(K ),

• W s(P) is robustly tangent to Fuu(S).

Then a C r -generic map f in displays infinitely many sinks.

19/23

Let (fa)a be a C r -family in U, and assume that the projectivelyhyperbolic source Sa, the blender Ka, the dissipative saddle pointPa are well defined by continuation. We recall that:

• the source Sa belongs to the covered domain of Ka,

• W u(Pa) intersects transversally W s(Ka),

• W s(Pa) is robustly tangent to Fuu(Sa).

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Let U be the set of C r -families so that:

• (Ka)a is a C r -parablender ; (Sa)a belongs to its covereddomain defined by local unstable manifolds which are nottangent to weak unstable direction of Sa.

• Ka is included in the immediate repulsion basin B of Sa andthe stable direction of Ka is not tangent to Fuu(Sa), for everya.

Then a C r -generic family (fa)a ∈ U satisfies that fa displaysinfinitely many sinks for every ‖a‖ ≤ 1.

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LemmaFor every (fa)a ∈ U of class C∞, for every ε > 0, there existsα > 0 so that there exists a ε-C r -perturbation so that W u(Pa)contains Sa for every ‖a‖ ≤ α.

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LemmaFor every (fa)a ∈ U of class C∞, for every ε > 0, there existsα > 0 so that there exists a ε-C r -perturbation so that Pa displaysa persistent homoclinic tangency for every ‖a‖ ≤ α.

LemmaFor every (fa)a ∈ U of class C∞, for every ε > 0, there existsα > 0 so that for every N ≥ 1, there exists a ε-C r -perturbation sothat fa displays a sinks of period ≥ N for every ‖a‖ ≤ α.

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parablender episode ii - byu math · parablender episode ii pierre berger, ... and so one for every...

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