Hyperbolic and Ergodic Properties of DS By Anna Rapoport

Introduction In general terms, DYNAMICS is concerned with describing for the majority of systems how the majority of orbits behave as time goes to infinity. In general terms, DYNAMICS is concerned with describing for the majority of systems how the majority of orbits behave as time goes to infinity. And with understanding when and in which sense this behavior is robust under small modifications of the system. And with understanding when and in which sense this behavior is robust under small modifications of the system.

Algebra and  -algebra For any topological space X, the Borel  -algebra of X is the  -algebra B generated by the open sets of X.

A Measure space & a Probability Space If O is an algebra of subsets of X, the function  :O →[0,  ] is called a measure on O if  (  )=0;  {A i  O  i A i  O} holds  (  i A i )=∑ i  (A i ) A measure space is a triple (X,O,  ). If  (X)=1, (X,O,  ) is called a probability space and  -a probability measure. f -1 (B)  S. Consider (X,O) and (Y,S). Map f: X→Y is said to be measurable, if f -1 (B)  O for any B  S.

Measure Preserving Map & Invariant Measure If If (X,O,  ) and (Y,S, ) are measure spaces, the map f: X → Y is called measure preserving if B  S  f -1 (B)  O and  (f -1 (B) ) = (B). The measure  is invariant under f : X → X if B  O  f -1 (B)  O and  (f -1 (B) ) =  (B).

Ergodicity & Unique Ergodicity Consider a measure-preserving map f of a probability space A set A  Consider a measure-preserving map f of a probability space (X,O,  ). A set A  O is called f- invariant if f -1 (A) =A. Def: f is said to be ergodic if every f-invariant set has measure 0 or 1. Def: f is said to be ergodic if every f-invariant set has measure 0 or 1. Ex: The map x → 2x mod 1 on [0,1] with Lebesgue measure is ergodic. Ex: The map x → 2x mod 1 on [0,1] with Lebesgue measure is ergodic. Given a map f and a  -algebra, there may be many ergodic measures. If there is only one ergodic measure, then f is called uniquely ergodic. Given a map f and a  -algebra, there may be many ergodic measures. If there is only one ergodic measure, then f is called uniquely ergodic. Ex: Map x → x +a mod 1 with a  R\Q, unique ergodic measure is Lebesgue measure. Ex: Map x → x +a mod 1 with a  R\Q, unique ergodic measure is Lebesgue measure.

Birkhoff Ergodic Theorem  Let  be an integrable function on a measure space with probability measure µ, and let f be an ergodic transformation (i.e. f -1 (A)=A implies µ(A)=0 or 1), then x  X. for µ-a.e. x  X.

To illustrate this, take  to be the characteristic function of some subset A  X so that To illustrate this, take  to be the characteristic function of some subset A  X so that The left-hand side just says how often the orbit of x (x, f(x), f 2 (x),…) lies in A, and the right- hand side is just the Lebesgue measure of A. The left-hand side just says how often the orbit of x (x, f(x), f 2 (x),…) lies in A, and the right- hand side is just the Lebesgue measure of A. Thus, for an ergodic map, "space-averages = time-averages almost everywhere.“ Thus, for an ergodic map, "space-averages = time-averages almost everywhere.“

Ergodic Hierarchy A measure-preserving map f of a probability space A measure-preserving map f of a probability space (X,O,  ) is ergodic  for every A,B  O Mixing: Bernoulli  Kolmogorov (K-mixing)  Mixing  Ergodic

SRB-Measures SRB-measure is called after Sinai, Ruelle, Bowen, who first constructed them. SRB-measure is called after Sinai, Ruelle, Bowen, who first constructed them. Def: An ergodic measure  is an SRB-measure if there exists a subset U  X with m(U)>0 and such that for each continuous function  for  x  U. Def: An ergodic measure  is an SRB-measure if there exists a subset U  X with m(U)>0 and such that for each continuous function  for  x  U. Recall, it always holds for  -a.e. x  X by the Birkhoff Ergodic Theorem. Recall, it always holds for  -a.e. x  X by the Birkhoff Ergodic Theorem. The difference for an SRB-measure is that the “time average=space average” for a set of initial points with positive Lebesgue measure. That is why this measure is also referred to as the physical (natural) invariant measure. The difference for an SRB-measure is that the “time average=space average” for a set of initial points with positive Lebesgue measure. That is why this measure is also referred to as the physical (natural) invariant measure.

Definitions For definiteness let us confine ourselves to discrete time dynamical systems. Manifolds are smooth, compact, without boundary. Measures are probabilities on the Borel  -algebra. Def : The linear map T: R n →R n is called hyperbolic if none of its eigenvalues lies on the unit circle. Def : A nonlinear map f is said to have a hyperbolic fixed point at p if f(p)=p and Df(p) is a hyperbolic linear map.

Types of Hyperbolic Fixed Points Attracting – all the eigenvalues of Df(p) are inside the unit circle. Attracting – all the eigenvalues of Df(p) are inside the unit circle. Repelling – all the eigenvalues of Df(p) are outside the unit circle. Repelling – all the eigenvalues of Df(p) are outside the unit circle. Saddle type – some of the eigenvalues of Df(p) are outside the unit circle and some are inside. Saddle type – some of the eigenvalues of Df(p) are outside the unit circle and some are inside. 10

Let f: M→M be a C 1 Diff on a compact Riem. M. Let f: M→M be a C 1 Diff on a compact Riem. M. Def :  M – invariant set (f -1 (  )=  ) is a hyperbolic set if for every x  there is a decomposition T x (M)= E s (x)  E u (x) such that: Def :  M – invariant set (f -1 (  )=  ) is a hyperbolic set if for every x  there is a decomposition T x (M)= E s (x)  E u (x) such that: (invariance) Df(x)E*(x)=E*(f(x)), *=s,u; (invariance) Df(x)E*(x)=E*(f(x)), *=s,u; (contraction) ║Df n (x)E s (x)║  C n for all n>0 (contraction) ║Df n (x)E s (x)║  C n for all n>0 (expansion) ║Df -n (x)E u (x)║  C n for all n>0 (expansion) ║Df -n (x)E u (x)║  C n for all n>0 With C>0 and 0 and <1 independent on x. Def: The diffeomorphism f is uniformly hyperbolic or Axiom A if Def: The diffeomorphism f is uniformly hyperbolic or Axiom A if The non-wandering set  (f) is hyperbolic The non-wandering set  (f) is hyperbolic The periodic points of f are dense in  (f) The periodic points of f are dense in  (f) If M is hyperbolic – Anosov Diffeomorphism If M is hyperbolic – Anosov Diffeomorphism

Dynamical Decomposition Def : An invariant set  is transitive if it contains some dense orbit {f n (x): n  0}. Def : An invariant set  is transitive if it contains some dense orbit {f n (x): n  0}. Def: An invariant set  is isolated if it admits a neighborhood U s.t. {x : f n (x)  U  n}= . Def: An invariant set  is isolated if it admits a neighborhood U s.t. {x : f n (x)  U  n}= . Theorem (Smale): If f: M→M is uniformly hyperbolic  (f) =  1  …   N – finite disjoint union of compact invariant sets  i transitive and isolated. The  -limit and  -limit sets of every orbit is contained in some  i. Theorem (Smale): If f: M→M is uniformly hyperbolic  (f) =  1  …   N – finite disjoint union of compact invariant sets  i transitive and isolated. The  -limit and  -limit sets of every orbit is contained in some  i.

Attractor and a Basin of Attraction Def :  i is a (hyperbolic) attractor if the basin of attraction B(  i ) = {x  M:  (x)   i } has a positive Lebesgue measure. Def :  i is a (hyperbolic) attractor if the basin of attraction B(  i ) = {x  M:  (x)   i } has a positive Lebesgue measure. Assuming Df is Hölder,  i is an attractor  it has a neighborhood U s.t. f(U)  U and Assuming Df is Hölder,  i is an attractor  it has a neighborhood U s.t. f(U)  U and

Dynamics near elementary pieces Let f: M→M be uniformly hyperbolic and  =  i be any of the elementary pieces of the dynamics and assume Df is Hölder continuous. Let f: M→M be uniformly hyperbolic and  =  i be any of the elementary pieces of the dynamics and assume Df is Hölder continuous. Theorem (Sinai, Ruelle, Bowen): Every attractor of f has a unique invariant probability measure  s.t. Lebesgue a.e. x  B(  ) for any continuous function  : Theorem (Sinai, Ruelle, Bowen): Every attractor of f has a unique invariant probability measure  s.t. Lebesgue a.e. x  B(  ) for any continuous function  :

The concept of SRB-measures in the context of Anosov systems has been introduced by Y.G. Sinai in the 1960's The concept of SRB-measures in the context of Anosov systems has been introduced by Y.G. Sinai in the 1960's Later the existence of SRB-measures has been shown for Axiom A systems by R. Bowen and D. Ruelle Later the existence of SRB-measures has been shown for Axiom A systems by R. Bowen and D. Ruelle More recently M. Benedicks and L.-S. Young have shown that the Henon-map has an SRB- measure for a ``large'' set of parameter values. More recently M. Benedicks and L.-S. Young have shown that the Henon-map has an SRB- measure for a ``large'' set of parameter values. However, it is still one of the major problems in Ergodic Theory to establish the existence of SRB- measures for a more general class of dynamical systems. However, it is still one of the major problems in Ergodic Theory to establish the existence of SRB- measures for a more general class of dynamical systems.

Summary Hyperbolic systems admit a decomposition into finitely many invariant and indecomposable (transitive) pieces. Hyperbolic systems admit a decomposition into finitely many invariant and indecomposable (transitive) pieces. The dynamics on each elementary piece and the statistics of orbits in the basins are well- understood. The dynamics on each elementary piece and the statistics of orbits in the basins are well- understood.

Uniform Hyperbolicity is not enough! “Strange” attractors of Lorenz and Henon showed that uniform hyperbolicity is too strong condition for a general description of dynamics. “Strange” attractors of Lorenz and Henon showed that uniform hyperbolicity is too strong condition for a general description of dynamics. A version of hyperbolicity with considerably weaker assumptions emerged following the works of Oseledec and Pesin. A version of hyperbolicity with considerably weaker assumptions emerged following the works of Oseledec and Pesin. “expansions and contractions everywhere” on a compact set is replaced by “asymptotic expansions and contractions almost everywhere” “expansions and contractions everywhere” on a compact set is replaced by “asymptotic expansions and contractions almost everywhere”

Non-uniform hyperbolicity

Pails’ conjectures Every system can be approximated by another having only finitely many attractors (appr in C r topology) supporting SRB measures whose basins cover a full Lebesgue measure subset of the manifold (Axiom A systems are dense) Every system can be approximated by another having only finitely many attractors (appr in C r topology) supporting SRB measures whose basins cover a full Lebesgue measure subset of the manifold (Axiom A systems are dense) Time averages should not be much affected if small random errors in parameter space are introduced at each iteration: stochastic stability. Time averages should not be much affected if small random errors in parameter space are introduced at each iteration: stochastic stability.

Correlation decay Another standard question concerns the correlation between  and  ◦ f n for large n. Another standard question concerns the correlation between  and  ◦ f n for large n. If If Then one could ask if  (n) → 0 as n →  and at what speed. Then one could ask if  (n) → 0 as n →  and at what speed. E.g. if  (n) ~ e -  n for some  >0 independent of , then this is a property of the dynamical system (f, ) and we say that (f, ) has exponential decay of correlations. If  (n) ~ n -  for some  >0, polynomial decay of correlations. E.g. if  (n) ~ e -  n for some  >0 independent of , then this is a property of the dynamical system (f,  ) and we say that (f,  ) has exponential decay of correlations. If  (n) ~ n -  for some  >0, polynomial decay of correlations. For mixing systems  (n) → 0 as n →  For mixing systems  (n) → 0 as n → 

Summary Since for chaotic systems orbits are sensitive to initial conditions, and so essentially unpredictable over long periods of time, one focus on statistical properties of large sets of trajectories. Since for chaotic systems orbits are sensitive to initial conditions, and so essentially unpredictable over long periods of time, one focus on statistical properties of large sets of trajectories. For Anosov diffeomorphisms and Axiom A attractors, SRB measures always exist, correlation decay is exponential. For Anosov diffeomorphisms and Axiom A attractors, SRB measures always exist, correlation decay is exponential. Outside the Axiom A category there are no general results. Outside the Axiom A category there are no general results.

References “Developments in Chaotic Dynamics” Lai-Sang Young, Notices of the AMS, Volume 45, N. 10 “Developments in Chaotic Dynamics” Lai-Sang Young, Notices of the AMS, Volume 45, N. 10 “Dynamics: A Probabilistic and Geometric Perspective” Marcelo Viana, Documenta Mathematica, Extra Volume ICM 1998, I, “Dynamics: A Probabilistic and Geometric Perspective” Marcelo Viana, Documenta Mathematica, Extra Volume ICM 1998, I, “Introduction to the Ergodic Theory of chaotic billiards” N.Chernov, R.Markarian “Introduction to the Ergodic Theory of chaotic billiards” N.Chernov, R.Markarian “Introduction to the Modern Theory of DS ” A.Katok, B. Hasselblatt “Introduction to the Modern Theory of DS ” A.Katok, B. Hasselblatt