Hyperbolic and Ergodic Properties of DS

1. Hyperbolic and Ergodic Properties of DS By Anna Rapoport

2. 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. • And with understanding when and in which sense this behavior is robust under small modifications of the system.

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

4. 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; • {Ai OiAi O} holds (iAi )=∑i (Ai ) • A measure space is a triple (X,O,). If (X)=1, (X,O,) is called a probability space and  -a probability measure. • Consider (X,O) and (Y,S). Map f: X→Y is said to be measurable, if f -1(B)O for any BS.

5. Measure Preserving Map & Invariant Measure • 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 → Xif B O  f -1(B) O and (f -1(B) ) = (B).

6. Ergodicity & Unique Ergodicity • Considera 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. • 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. • Ex: Map x →x +a mod 1 with aR\Q, unique ergodic measure is Lebesgue measure.

7. 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 for µ-a.e.xX.

8. 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. • Thus, for an ergodic map, "space-averages = time-averages almost everywhere.“

9. Ergodic Hierarchy • 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

10. SRB-Measures • 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. • 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.

12. 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: Rn→Rn 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.

13. Types of Hyperbolic Fixed Points • Attracting – all the eigenvalues of Df(p) are inside 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. -1 0 1

14. Let f: M→M be a C1 Diff on a compact Riem. M. • Def : M – invariant set (f -1()= ) is a hyperbolic set if for every x there is a decomposition Tx(M)= Es(x)  Eu(x) such that: • (invariance) Df(x)E*(x)=E*(f(x)) , *=s,u; • (contraction) ║Df n(x)Es(x)║ Cnfor all n>0 • (expansion) ║Df -n(x)Eu(x)║ Cnfor all n>0 With C>0 and <1 independent on x. • Def: The diffeomorphism f is uniformlyhyperbolic or Axiom A if • The non-wandering set (f) is hyperbolic • The periodic points of f are dense in (f) • If M is hyperbolic – Anosov Diffeomorphism

15. Dynamical Decomposition • 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}=. • 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 .

16. 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. • Assuming Df isHölder, i is an attractor  it has a neighborhood U s.t. f(U) U and

17. 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 isHölder 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 :

18. 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 • 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.

19. Summary • 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.

20. 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. • 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”

21. Non-uniform hyperbolicity

22. Pails’ conjectures • Every system can be approximated by another having only finitely many attractors (appr in Cr 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.

23. Correlation decay • Another standard question concerns the correlation between  and  ◦ f n for large n. • If • 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 decayof correlations. • For mixingsystems (n) → 0 as n →

24. 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. • 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.

25. References • “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, 557-578 • “Introduction to the Ergodic Theory of chaotic billiards” N.Chernov, R.Markarian • “Introduction to the Modern Theory of DS ” A.Katok, B. Hasselblatt