Hyperbolic and Hyperbolic and Ergodic Properties Ergodic Properties

of DSof DSBy Anna RapoportBy Anna Rapoport

IntroductionIntroduction In general terms, DYNAMICS is In general terms, DYNAMICS is

concerned with describing for the concerned with describing for the majority of systems how the majority majority of systems how the majority of orbits behave as time goes to of orbits behave as time goes to infinity.infinity.

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

Algebra and Algebra and -algebra-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 A Measure space & a Probability SpaceProbability Space

If O is an algebra of subsets of X, the function :O →[0,] is called a measure on O if ()=0; {Ai OiAi 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 f -1-1(B)(B) O for any BS.S.

Measure Preserving Map & Measure Preserving Map & Invariant MeasureInvariant Measure

If If (X,O,) and (Y,S,) are measure spaces, the map f: X → Y is called measure preserving if BS 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 & Unique ErgodicityErgodicity ConsiderConsider a measure-preserving map a measure-preserving map ff of a of a

probability space probability space (X,O,). A set A set AAO is called f-invariant if f -1(A) =A.

Def:Def: ff is said to be is said to be ergodicergodic if every if every ff-invariant -invariant set has measure 0 or 1.set has measure 0 or 1.

Ex: Ex: The map The map x →x → 2x mod 1 2x mod 1 on [0,1] with on [0,1] with Lebesgue measure is ergodic.Lebesgue measure is ergodic.

Given a map Given a map ff and a and a -algebra, there may be -algebra, there may be many ergodic measures. If there is only one many ergodic measures. If there is only one ergodic measure, then ergodic measure, then ff is called is called uniquely uniquely ergodicergodic. .

Ex: Ex: Map Map x →x → x +a mod 1 x +a mod 1 with with aaR\Q, R\Q, unique unique ergodic measure is Lebesgue measure.ergodic measure is Lebesgue measure.

Birkhoff Ergodic Birkhoff Ergodic TheoremTheorem

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. xxXX..

To illustrate this, take To illustrate this, take to be the to be the characteristic function of some subset characteristic function of some subset AA XX so that so that

The left-hand side just says how often The left-hand side just says how often the orbit of the orbit of xx ( (x , f(x) , f x , f(x) , f 22(x),…) (x),…) lies in lies in AA, , and the right-hand side is just the and the right-hand side is just the Lebesgue measure of Lebesgue measure of AA. .

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

Ergodic HierarchyErgodic Hierarchy A measure-preserving mapA measure-preserving map f f of a probability of a probability

space space (X,O,) is ergodic for every A,BO

Mixing:

Bernoulli Kolmogorov (K-mixing) Mixing Ergodic

SRB-Measures SRB-Measures SRB-measure is called after Sinai, Ruelle, SRB-measure is called after Sinai, Ruelle,

Bowen, who first constructed them.Bowen, who first constructed them. Def:Def: An ergodic measure An ergodic measure is an is an SRB-measure SRB-measure if if

there exists a subset there exists a subset UU X X with with m(U)>0m(U)>0 and and such that for each continuous function such that for each continuous function    

for for xxU.U.

Recall, it always holds for Recall, it always holds for -a.e. -a.e. xxXX by the by the Birkhoff Ergodic Theorem. Birkhoff Ergodic Theorem.

The difference for an SRB-measure is that the The difference for an SRB-measure is that the “time average=space average” for a set of initial “time average=space average” for a set of initial points with positive Lebesgue measure. That is points with positive Lebesgue measure. That is why this measure is also referred to as the why this measure is also referred to as the physical (natural)physical (natural) invariant measure. invariant measure.

DefinitionsDefinitions 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.

Types of Hyperbolic Types of Hyperbolic Fixed PointsFixed Points

Attracting Attracting – all the – all the eigenvalues of eigenvalues of Df(p) Df(p) are inside the unit are inside the unit circle.circle.

RepellingRepelling – all the – all the eigenvalues of eigenvalues of Df(p) Df(p) are outside the unit are outside the unit circle. circle.

Saddle typeSaddle type – some of – some of the eigenvalues of the eigenvalues of Df(p) Df(p) are outside the are outside the unit circle and some unit circle and some are inside.are inside.

1-1 0

Let Let f: M→M f: M→M be a Cbe a C1 1 Diff on a compact Riem. Diff on a compact Riem. M.M.

Def : Def : MM – invariant set ( – invariant set (f f -1-1(()= )= )) is a is a hyperbolic set hyperbolic set if for every xif for every x there is a there is a decomposition decomposition TTxx(M)= E(M)= Ess(x) (x) E Euu(x)(x) such such that:that: (invariance)(invariance) Df(x)E*(x)=E*(f(x)) , Df(x)E*(x)=E*(f(x)) , *=s,u; *=s,u; (contraction) ║(contraction) ║Df Df nn(x)E(x)Ess(x)(x)║║ C Cnn for all n>0for all n>0 (expansion) ║(expansion) ║Df Df -n-n(x)E(x)Euu(x)(x)║║ C Cnn for all n>0for all n>0With C>0 and With C>0 and <1 independent on <1 independent on xx..

Def: Def: The diffeomorphism The diffeomorphism f f is is uniformlyuniformly hyperbolichyperbolic or or Axiom AAxiom A if if The non-wandering set The non-wandering set (f)(f) is hyperbolic is hyperbolic The periodic points of The periodic points of ff are dense in are dense in (f)(f)

If M is hyperbolic – If M is hyperbolic – Anosov DiffeomorphismAnosov Diffeomorphism

Dynamical Dynamical DecompositionDecomposition

Def Def : An invariant set : An invariant set is is transitivetransitive if it if it contains some dense orbit {contains some dense orbit {f f nn(x)(x): n : n 0}. 0}.

DefDef: An invariant set : An invariant set is is isolatedisolated if it admits if it admits a neighborhood U s.t. {a neighborhood U s.t. {x x : : f f nn(x)(x) U U n}=n}=..

Theorem (Smale): Theorem (Smale): If If f: M→M f: M→M is uniformly is uniformly hyperbolic hyperbolic (f)(f) = = 11 … … N N – finite disjoint – finite disjoint union of compact invariant sets union of compact invariant sets i i transitive transitive and isolated. The and isolated. The -limit and -limit and -limit sets of -limit sets of every orbit is contained in some every orbit is contained in some i i ..

Attractor and a Basin of Attractor and a Basin of AttractionAttraction

Def Def : : i i is a (hyperbolic) is a (hyperbolic) attractorattractor if if the the basin of attractionbasin of attraction B( B(i i )) = {x = {x M: M: (x)(x) i i }} has a positive has a positive Lebesgue measure.Lebesgue measure.

Assuming Assuming DfDf is is Hölder, Hölder, i i is an is an attractor attractor it has a neighborhood it has a neighborhood UU s.t. s.t. f(U)f(U) U U andand

Dynamics near Dynamics near elementary pieceselementary pieces

Let Let f: M→M f: M→M be uniformly hyperbolic be uniformly hyperbolic and and = =ii be any of the elementary be any of the elementary

pieces of the dynamics and assume pieces of the dynamics and assume DfDf isis Hölder continuous.Hölder continuous.

Theorem (Sinai, Ruelle, Bowen): Theorem (Sinai, Ruelle, Bowen): Every Every attractor of attractor of ff has a unique invariant has a unique invariant probability measure probability measure s.t. Lebesgue s.t. Lebesgue a.e. a.e. xx B( B() ) for any continuous for any continuous function function ::

The concept of SRB-measures in the The concept of SRB-measures in the context of Anosov systems has been context of Anosov systems has been introduced by Y.G. Sinai in the 1960's introduced by Y.G. Sinai in the 1960's

Later the existence of SRB-measures has Later the existence of SRB-measures has been shown for Axiom A systems by R. been shown for Axiom A systems by R. Bowen and D. Ruelle Bowen and D. Ruelle

More recently M. Benedicks and L.-S. More recently M. Benedicks and L.-S. Young have shown that the Henon-map Young have shown that the Henon-map has an SRB-measure for a ``large'' set of has an SRB-measure for a ``large'' set of parameter values. parameter values.

However, it is still one of the major However, it is still one of the major problems in Ergodic Theory to establish problems in Ergodic Theory to establish the existence of SRB-measures for a more the existence of SRB-measures for a more general class of dynamical systems. general class of dynamical systems.

Summary Summary

Hyperbolic systems admit a Hyperbolic systems admit a decomposition into finitely many decomposition into finitely many invariant and indecomposable invariant and indecomposable (transitive) pieces.(transitive) pieces.

The dynamics on each elementary The dynamics on each elementary piece and the statistics of orbits in piece and the statistics of orbits in the basins are well-understood.the basins are well-understood.

Uniform Hyperbolicity is Uniform Hyperbolicity is not enough!not enough!

““Strange” attractors of Lorenz and Henon Strange” attractors of Lorenz and Henon showed that uniform hyperbolicity is too showed that uniform hyperbolicity is too strong condition for a general description strong condition for a general description of dynamics.of dynamics.

A version of hyperbolicity with A version of hyperbolicity with considerably weaker assumptions emerged considerably weaker assumptions emerged following the works of Oseledec and Pesin.following the works of Oseledec and Pesin.

““expansions and contractions everywhere” expansions and contractions everywhere” on a compact set is replaced by on a compact set is replaced by ““asymptoticasymptotic expansions and contractions expansions and contractions almost everywherealmost everywhere””

Non-uniform Non-uniform hyperbolicityhyperbolicity

Pails’ conjecturesPails’ conjectures

Every system can be approximated by Every system can be approximated by another having only finitely many another having only finitely many attractors (appr in Cattractors (appr in Crr topology) topology) supporting SRB measures whose basins supporting SRB measures whose basins cover a full Lebesgue measure subset of cover a full Lebesgue measure subset of the manifold (Axiom A systems are dense)the manifold (Axiom A systems are dense)

Time averages should not be much Time averages should not be much affected if small random errors in affected if small random errors in parameter space are introduced at each parameter space are introduced at each iteration: stochastic stability. iteration: stochastic stability.

Correlation decayCorrelation decay Another standard question concerns the Another standard question concerns the

correlation between correlation between and and ◦ ◦ f f nn for large for large n.n.

IfIf

Then one could ask if Then one could ask if (n)(n) → → 0 0 as n →as n → and at what speed.and at what speed.

E.g. if E.g. if (n)(n) ~ e ~ e--nn for some for some >0 >0 independent of independent of , then this is a property , then this is a property of the dynamical system of the dynamical system (f, (f, )) and we say and we say that that (f, (f, )) has has exponential decay of exponential decay of correlationscorrelations. If . If (n)(n) ~ n ~ n-- for some for some >0, >0, polynomial decaypolynomial decay of correlations.of correlations.

For mixingFor mixing systems systems (n)(n) → → 0 0 as n →as n →

SummarySummary

Since for chaotic systems orbits are Since for chaotic systems orbits are sensitive to initial conditions, and so sensitive to initial conditions, and so essentially unpredictable over long essentially unpredictable over long periods of time, one focus on statistical periods of time, one focus on statistical properties of large sets of trajectories.properties of large sets of trajectories.

For Anosov diffeomorphisms and Axiom For Anosov diffeomorphisms and Axiom A attractors, SRB measures always A attractors, SRB measures always exist, correlation decay is exponential.exist, correlation decay is exponential.

Outside the Axiom A category there are Outside the Axiom A category there are no general results.no general results.

ReferencesReferences ““Developments in Chaotic Dynamics” Developments in Chaotic Dynamics” Lai-Lai-

Sang YoungSang Young, Notices of the AMS, Volume , Notices of the AMS, Volume 45, N. 1045, N. 10

““Dynamics: A Probabilistic and Geometric Dynamics: A Probabilistic and Geometric Perspective” Perspective” Marcelo VianaMarcelo Viana, Documenta , Documenta Mathematica, Extra Volume ICM 1998, I, Mathematica, Extra Volume ICM 1998, I, 557-578557-578

““Introduction to the Ergodic Theory of Introduction to the Ergodic Theory of chaotic billiards” chaotic billiards” N.Chernov, R.MarkarianN.Chernov, R.Markarian

““Introduction to the Modern Theory of DS Introduction to the Modern Theory of DS ” ” A.Katok, B. HasselblattA.Katok, B. Hasselblatt