Spectral decomposition of surface diffeomorphisms Sylvain Crovisier - - PowerPoint PPT Presentation

Spectral decomposition

• f surface diffeomorphisms

Sylvain Crovisier

(CNRS - University Paris-Sud) New trends in Lyapunov Exponents – Lisbon July 7th, 2020

f : a diffeomorphism of a surface M.

How does the system break down into elementary pieces?

f : a diffeomorphism of a surface M.

How does the system break down into elementary pieces?

Questions. (local) Describe the dynamics on each piece. (global) How many pieces? how are they organized? If the number is infinite, does their “size” go to zero?

Example: hyperbolic dynamics

f : Axiom A diffeomorphism Theorem (Smale’s spectral decomposition). There exists a partition of the non-wandering set Ωpf q “ K1 \ ¨ ¨ ¨ \ Kℓ, where Ki is a transitive locally maximal hyperbolic set (basic set).

Morse-Smale Anosov horseshoe Plykin attractor

Each piece can be coded, admits a thermodynamical formalism (unique equilibrium measure for C α-potential), satisfies standard limit theorems,...

Decomposition: several candidates

Trapping regions. U Ă M open set satisfying f pUq Ă U.

§ M “ R Y Ť

nPZpf npUqzf n`1pUqq Y A

§ This allows to define the chain-recurrent set and its decomposition into chain-recurrence classes.

U R A

Renormalization domains. D Ă M topological disc satisfying: f τpDq Ă D, f ipDq X D “ H for 0 ă i ă τ.

§ This induces a new diffeomorphism f τ on D.

D

Elementary pieces: several candidates

Maximal transitive sets. Invariant compact sets that are transitive and maximal for the inclusion. Chain-recurrence classes. Invariant compact sets that are chain-transitive (ε-dense periodic ε-pseudo-orbits, @ε) and maximal for the inclusion. Homoclinic classes. For O hyperbolic periodic orbit HpOq :“ W spOq | X W upOq. § Transitive set. § Hyperbolic periodic orbits and basic subsets are dense.

Panorama of surface dynamics

htoppf q “ 0 htoppf q ą 0

dissipative conservative general

. . .

Morse-Smale hamiltonian standard map H´ enon Plykin attractor

Theorem (Newhouse). There exists an abundant set of surface diffeomorphisms with infinitely many attractors.

Dynamics on pieces: the zero-entropy case

f : orientation-preserving homeomorphism of the sphere. Models with zero entropy.

Morse-Smale irrational attractor

• dometer

Theorem (Franks-Handel, Le Calvez-Tal). If htoppf q “ 0, then any f -invariant transitive compact set – either is a periodic orbit, – or factorizes on an odometer, – or has irrational type.

Dynamics on homoclinic classes

f : C 8-diffeomorphism of surface µ (ergodic) is hyperbolic if its Lyapunov exponents are ‰ 0. Spectral decomposition for non-uniformly hyperbolic dynamics. – Any ergodic hyperbolic µ is supported on a homoclinic class. (Katok) – For any distinct homoclinic classes htoppHpOqXHpO1qq“0. A transitive set contains at most one (non-trivial) homoclinic class.

(Buzzi - C - Sarig)

Dynamics on homoclinic classes

f : C 2-diffeomorphism of surface µ is χ-hyperbolic if its Lyapunov exponents are in Rzr´χ, χs. Theorem (Buzzi-C-Sarig). For any class Hp0q and χ ą 0, there exists: – a locally compact Markov shift on a countable alphabet pΣ, σq, – a H¨

• lder map π: Σ Ñ HpOq satisfying π ˝ σ “ f ˝ π,

such that (a) µpπpΣ#qq“1 for any χ-hyperbolic measure µ„HpOq, (b) π´1pyq X Σ# is finite for all y P πpHpOqq, (c) pΣ, σq is transitive (irreductible).

Dynamics on homoclinic classes

f : C 2-diffeomorphism of surface µ is χ-hyperbolic if its Lyapunov exponents are in Rzr´χ, χs. Theorem (Buzzi-C-Sarig). For any class Hp0q and χ ą 0, there exists: – a locally compact Markov shift on a countable alphabet pΣ, σq, – a H¨

• lder map π: Σ Ñ HpOq satisfying π ˝ σ “ f ˝ π,

such that (a) µpπpΣ#qq“1 for any χ-hyperbolic measure µ„HpOq, (b) π´1pyq X Σ# is finite for all y P πpHpOqq, (c) pΣ, σq is transitive (irreductible).

• Corollary. (1) For C α-potentials, HpOq supports at most one hyperbolic equilibrium.

Its support is HpOq. It is Bernoulli (up to a finite extension). (Buzzi-C-Sarig) (2) HpOq supports at most one hyperbolic SRB. (Hertz-Hertz-Tahzibi-Ures)

• Questions. Existence? Speed of mixing? Limit theorems?

Dynamics with infinitely many pieces

If HpO1q, HpO2q, . . . are distinct homoclinic classes, how does the hyperbolicity on HpOnq degenerate as n Ñ `8?

If HpO1q, HpO2q, . . . are distinct homoclinic classes, how does the hyperbolicity on HpOnq degenerate as n Ñ `8?

Restatement 1. Does there exist a surface diffeomorphism f with infinitely many periodic orbits O1, O2, . . . such that: – HpOiq ‰ HpOjq for i ‰ j, – their Lyapunov exponents are uniformly bounded away from 0? Restatement 2. How uniform is the Pesin theory wrt the measure?

Low exponents: a counter-example

Let Λpf q :“ log maxp}Df }8, }Df ´1}8q. Theorem (BCS). For any 1 ď r ă r1, there exists a C r- diffeomorphism f with infinitely many saddles On such that ‚ HpOnq X HpOmq “ H for all n ‰ m, ‚ all Lyapunov exponents have their modulus equal to Λpf q

r1 .

Low exponents: a counter-example

Let Λpf q :“ log maxp}Df }8, }Df ´1}8q. Theorem (BCS). For any 1 ď r ă r1, there exists a C r- diffeomorphism f with infinitely many saddles On such that ‚ HpOnq X HpOmq “ H for all n ‰ m, ‚ all Lyapunov exponents have their modulus equal to Λpf q

r1 . λ µ µ´n 1

0n

f n f

0 ă λ ă µ´1 ă 1

µ´n µ´n{r 0n α D

angle α “ µp 1

r ´1qn

expansion }Df n|D} “ µ

n r

Large exponents and C 2-topology

Let Λpf q :“ log maxp}Df }8, }Df ´1}8q.

• Theorem. If f is a C 2-diffeomorphism and pOnq are saddles whose

Lyapunov exponents have all their modulus larger than 15

16Λpf q,

then there exists n ‰ m such that HpOnq “ HpOmq.

• Question. Does this hold for C r-diffeomorphisms and saddles

whose Lyapunov exponents have their modulus larger than Λpf q

r ?

Large exponents and C 2-topology: proof

Direct approach.

Theorem (C-Pujals). Let σ, r σ, ρ, r ρ P p0, 1q such that r

σr ρ σρ ą σ.

Then the points x having a direction E Ă TxM satisfying @n ě 0, r σn ď }Df n|Epxq} ďσn, r ρn ď }Df n|E pxq}2

|detDf npxq| ďρn,

have a one-dimensional stable manifold varying continuously for the C 1-topology w.r.t. x and f .

Large exponents and C 2-topology: proof

Direct approach.

Theorem (C-Pujals). Let σ, r σ, ρ, r ρ P p0, 1q such that r

σr ρ σρ ą σ.

Then the points x having a direction E Ă TxM satisfying @n ě 0, r σn ď }Df n|Epxq} ďσn, r ρn ď }Df n|E pxq}2

|detDf npxq| ďρn,

have a one-dimensional stable manifold varying continuously for the C 1-topology w.r.t. x and f . How to satisfy these conditions simultaneously on E s and E u? Pliss lemma. If lim 1

n log }Df npzq|E} “ ´λ, then the condition

@n ě 0, }Df n|Epxq} ď σn, holds for a set of iterates x :“ f kpzq with density ě

λ `log σ Λpf q`log σ.

Large entropy and C r-topology

Let Λpf q :“ log maxp}Df }8, }Df ´1}8q. Theorem (BCS). For any 1 ă r ă r1 and any C r- diffeomorphism f , there exists at most finitely many different homoclinic classes HpOq with entropy htoppHpOqq ą Λpf q

r1 .

Large entropy and C r-topology

Let Λpf q :“ log maxp}Df }8, }Df ´1}8q. Theorem (BCS). For any 1 ă r ă r1 and any C r- diffeomorphism f , there exists at most finitely many different homoclinic classes HpOq with entropy htoppHpOqq ą Λpf q

r1 .

Yomdin theory gives two ingredients: ‚ a sequence HpOnq with htop ą Λpf q

r 1

accumulates on a non-trivial class HpO8q.

Large entropy and C r-topology

Let Λpf q :“ log maxp}Df }8, }Df ´1}8q. Theorem (BCS). For any 1 ă r ă r1 and any C r- diffeomorphism f , there exists at most finitely many different homoclinic classes HpOq with entropy htoppHpOqq ą Λpf q

r1 .

Yomdin theory gives two ingredients: ‚ a sequence HpOnq with htop ą Λpf q

r 1

accumulates on a non-trivial class HpO8q. ‚ a class HpOnq with htop ą Λpf q

r 1

does not decompose into “components” with small diameter.

HpO8q HpOnq

Large entropy and C r-topology

Let Λpf q :“ log maxp}Df }8, }Df ´1}8q. Theorem (BCS). For any 1 ă r ă r1 and any C r- diffeomorphism f , there exists at most finitely many different homoclinic classes HpOq with entropy htoppHpOqq ą Λpf q

r1 .

Yomdin theory gives two ingredients: ‚ a sequence HpOnq with htop ą Λpf q

r 1

accumulates on a non-trivial class HpO8q. ‚ a class HpOnq with htop ą Λpf q

r 1

does not decompose into “components” with small diameter.

HpO8q HpOnq

Zero entropy and mild dissipation

f : mildly dissipative diffeomorphism of the disc (example H´ enon fb,c : px, yq ÞÑ px2 ` c ` y, ´bxq with |b| ă 1

4)

Theorem (C-Pujals-Tresser). If htoppf q “ 0 and mild disspation then ‚ either the set of periods is bounded, ‚ or f is renormalizable: a finite collection of renormalization domains contain all the periodic orbits with large period.

Zero entropy and mild dissipation

f : mildly dissipative diffeomorphism of the disc (example H´ enon fb,c : px, yq ÞÑ px2 ` c ` y, ´bxq with |b| ă 1

4)

Theorem (C-Pujals-Tresser). If htoppf q “ 0 and mild disspation then ‚ either the set of periods is bounded, ‚ or f is renormalizable: a finite collection of renormalization domains contain all the periodic orbits with large period.

Corollary.

• Each chain-recurrence class is either

an odometer or a set of periodic points.

• Hierarchical structure on the periodic set.
• Any sequence pOnq (with increasing periods)

‚ accumulates on an odometer, and ‚ its upper Lyapunov exponents Ý Ñ

nÑ8 0.

Wiman theorem gives: |b|ă1{4 ñ each stable branch is unbounded.

To go further...

f : mildly dissipative C 8-diffeomorphism of the disc with htoppf qą0

How are homoclinic classes globally organized?

To go further...

f : mildly dissipative C 8-diffeomorphism of the disc with htoppf qą0

How are homoclinic classes globally organized?

Each class HpOq has a period

(= gcd of the orbits O1 „ HpOq)

Questions. If htoppf q ! 1, do the homoclinic classes share a hierarchical structure up to some period? Is there only finitely many (non-trivial) homoclinic classes of a given period? Are homoclinic classes separated by renormalization domains?

Thank you!

(take care and try to keep the exponent negative...)