Spectral decomposition of surface diffeomorphisms Sylvain Crovisier - PowerPoint PPT Presentation

Spectral decomposition of 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 Ω p f q “ K 1 \ ¨ ¨ ¨ \ K ℓ , where K i 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 p U q Ă U . M “ R Y Ť R n P Z p f n p U qz f n ` 1 p U qq Y A § This allows to define the chain-recurrent set and § its decomposition into chain-recurrence classes . U A Renormalization domains. D Ă M topological disc satisfying: f τ p D q Ă D , f i p D q 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 H p O q : “ W s p O q | X W u p O q . § Transitive set. § Hyperbolic periodic orbits and basic subsets are dense.

Panorama of surface dynamics h top p f q “ 0 h top p f q ą 0 dissipative Plykin attractor Morse-Smale H´ enon conservative hamiltonian standard map . . . general 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 odometer Theorem (Franks-Handel, Le Calvez-Tal). If h top p f 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 h top p H p O qX H p O 1 qq“ 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 R zr´ χ, χ s . Theorem (Buzzi-C-Sarig). For any class H p 0 q and χ ą 0 , there exists: – a locally compact Markov shift on a countable alphabet p Σ , σ q , – a H¨ older map π : Σ Ñ H p O q satisfying π ˝ σ “ f ˝ π , (a) µ p π p Σ # qq“ 1 for any χ -hyperbolic measure µ „ H p O q , such that (b) π ´ 1 p y q X Σ # is finite for all y P π p H p O qq , (c) p Σ , σ q is transitive (irreductible).

Dynamics on homoclinic classes f : C 2 -diffeomorphism of surface µ is χ -hyperbolic if its Lyapunov exponents are in R zr´ χ, χ s . Theorem (Buzzi-C-Sarig). For any class H p 0 q and χ ą 0 , there exists: – a locally compact Markov shift on a countable alphabet p Σ , σ q , – a H¨ older map π : Σ Ñ H p O q satisfying π ˝ σ “ f ˝ π , (a) µ p π p Σ # qq“ 1 for any χ -hyperbolic measure µ „ H p O q , such that (b) π ´ 1 p y q X Σ # is finite for all y P π p H p O qq , (c) p Σ , σ q is transitive (irreductible). Corollary. (1) For C α -potentials, H p O q supports at most one hyperbolic equilibrium. Its support is H p O q . It is Bernoulli (up to a finite extension). ( Buzzi-C-Sarig) (2) H p O q supports at most one hyperbolic SRB. ( Hertz-Hertz-Tahzibi-Ures) Questions. Existence? Speed of mixing? Limit theorems?

Dynamics with infinitely many pieces

If H p O 1 q , H p O 2 q , . . . are distinct homoclinic classes, how does the hyperbolicity on H p O n q degenerate as n Ñ `8 ?

If H p O 1 q , H p O 2 q , . . . are distinct homoclinic classes, how does the hyperbolicity on H p O n q degenerate as n Ñ `8 ? Restatement 1. Does there exist a surface diffeomorphism f with infinitely many periodic orbits O 1 , O 2 , . . . such that: – H p O i q ‰ H p O j q 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 Λ p f q : “ log max p} Df } 8 , } Df ´ 1 } 8 q . ă r 1 , there exists a C r - Theorem (BCS). For any 1 ď r diffeomorphism f with infinitely many saddles O n such that ‚ H p O n q X H p O m q “ H for all n ‰ m, ‚ all Lyapunov exponents have their modulus equal to Λ p f q r 1 .

Low exponents: a counter-example Let Λ p f q : “ log max p} Df } 8 , } Df ´ 1 } 8 q . ă r 1 , there exists a C r - Theorem (BCS). For any 1 ď r diffeomorphism f with infinitely many saddles O n such that ‚ H p O n q X H p O m q “ H for all n ‰ m, ‚ all Lyapunov exponents have their modulus equal to Λ p f q r 1 . D f µ ´ n α 0 n µ ´ n { r f n 0 n µ ´ n µ angle α “ µ p 1 r ´ 1 q n λ 1 n expansion } Df n | D } “ µ r 0 ă λ ă µ ´ 1 ă 1

Large exponents and C 2 -topology Let Λ p f q : “ log max p} Df } 8 , } Df ´ 1 } 8 q . Theorem. If f is a C 2 -diffeomorphism and p O n q are saddles whose Lyapunov exponents have all their modulus larger than 15 16 Λ p f q , then there exists n ‰ m such that H p O n q “ H p O m q . Question. Does this hold for C r -diffeomorphisms and saddles whose Lyapunov exponents have their modulus larger than Λ p f q r ?

Large exponents and C 2 -topology: proof Direct approach. ρ P p 0 , 1 q such that r σ r ρ Theorem (C-Pujals) . Let σ, r σ, ρ, r σρ ą σ . Then the points x having a direction E Ă T x M satisfying σ n ď } Df n | E p x q} ď σ n , ρ n ď } Df n | E p x q} 2 | detDf n p x q| ď ρ n , @ n ě 0 , r r 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. ρ P p 0 , 1 q such that r σ r ρ Theorem (C-Pujals) . Let σ, r σ, ρ, r σρ ą σ . Then the points x having a direction E Ă T x M satisfying σ n ď } Df n | E p x q} ď σ n , ρ n ď } Df n | E p x q} 2 | detDf n p x q| ď ρ n , @ n ě 0 , r r 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 ? If lim 1 n log } Df n p z q| E } “ ´ λ , then the condition Pliss lemma. } Df n | E p x q} ď σ n , @ n ě 0 , λ ` log σ holds for a set of iterates x : “ f k p z q with density ě Λ p f q` log σ .

Large entropy and C r -topology Let Λ p f q : “ log max p} Df } 8 , } Df ´ 1 } 8 q . r 1 and any C r - Theorem (BCS) . For any 1 ă r ă diffeomorphism f , there exists at most finitely many different homoclinic classes H p O q with entropy h top p H p O qq ą Λ p f q r 1 .

Large entropy and C r -topology Let Λ p f q : “ log max p} Df } 8 , } Df ´ 1 } 8 q . r 1 and any C r - Theorem (BCS) . For any 1 ă r ă diffeomorphism f , there exists at most finitely many different homoclinic classes H p O q with entropy h top p H p O qq ą Λ p f q r 1 . Yomdin theory gives two ingredients: ‚ a sequence H p O n q with h top ą Λ p f q r 1 accumulates on a non-trivial class H p O 8 q .

Large entropy and C r -topology Let Λ p f q : “ log max p} Df } 8 , } Df ´ 1 } 8 q . r 1 and any C r - Theorem (BCS) . For any 1 ă r ă diffeomorphism f , there exists at most finitely many different homoclinic classes H p O q with entropy h top p H p O qq ą Λ p f q r 1 . Yomdin theory gives two ingredients: ‚ a sequence H p O n q with h top ą Λ p f q H p O 8 q r 1 accumulates on a non-trivial class H p O 8 q . H p O n q ‚ a class H p O n q with h top ą Λ p f q does not r 1 decompose into “components” with small diameter.

Large entropy and C r -topology Let Λ p f q : “ log max p} Df } 8 , } Df ´ 1 } 8 q . r 1 and any C r - Theorem (BCS) . For any 1 ă r ă diffeomorphism f , there exists at most finitely many different homoclinic classes H p O q with entropy h top p H p O qq ą Λ p f q r 1 . Yomdin theory gives two ingredients: ‚ a sequence H p O n q with h top ą Λ p f q H p O 8 q r 1 accumulates on a non-trivial class H p O 8 q . H p O n q ‚ a class H p O n q with h top ą Λ p f q does not r 1 decompose into “components” with small diameter.

Zero entropy and mild dissipation f : mildly dissipative diffeomorphism of the disc enon f b , c : p x , y q ÞÑ p x 2 ` c ` y , ´ bx q with | b | ă 1 (example H´ 4 ) Theorem (C-Pujals-Tresser). If h top p f 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.