Stacey, On Devaney's definition of chaos, Amer.
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Colonius, R. Fabbri, and R. Johnson, Chain recurrence, growth rates and ergodic limits, Ergodic Theory and Dynamical Systems, to appear. Colonius and W.
Hewitt and K. Hurley, Chain recurrence, semiflows, and gradients, J. Equations 7, pp. Katok and B. Russian original Nemytskii and V. However, for nonuniformly hyperbolic dynamical systems there is an effective decomposition:.
Lyapunov exponents of cocycles over non-uniformly hyperbolic systems
A central ingredient in this proof is the Hopf argument, which rests on the fact that even for nonuniformly hyperbolic systems the stable and unstable foliations have the absolute continuity property discussed above. Examples show that there may indeed be countably many ergodic components. This is referred to as local ergodicity. Another approach to proving local ergodicity due to Burns-Gerber and Katok-Burns is based on Lyapunov function techniques and is closely related to the approach of Wojtkowski on computing Lyapunov exponents via an infinitesimally eventually strict cone family.
One can also establish local ergodicity for partially hyperbolic systems with negative central exponents either volume-preserving or with partially hyperbolic attractors.
As in the uniformly hyperbolic case, hyperbolic invariant measures tend to have much stronger ergodic properties than ergodicity. In particular, if this decomposition is trivial, that is, for a weakly mixing smooth invariant measure, the diffeomorphism is a Bernoulli automorphism. There are intimate connections between Lyapunov exponents and entropy. The central pertinent result is the Pesin Entropy Formula , which has been developed rather extensively and is therefore the subject of a separate entry.
The content of this formula and related ones is that the entropy of a measure is given exactly by the total expansion in the system, and these also establish close relations to fractal dimensions of measures and their stable or unstable conditionals.
As a direct consequence of the definition one obtains. This says that Sinai-Ruelle-Bowen measures are "physical measures" in that there is a positive probability that a point chosen at random with respect to Lebesgue measure will display asymptotics that reflect the Sinai-Ruelle-Bowen measure. Accordingly, establishing the existence of a Sinai-Ruelle-Bowen measure on an attractor is a major step that immediately provides a detailed probabilistic picture of the dynamics on the attractor.
Benedicks-Young Theorem. Geodesic flows of manifolds of nonpositive curvature are amenable to the techniques of Pesin theory as well and provide important examples.
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This result extends to multidimensional manifolds with no focal points. The Anosov Closing Lemma generalizes to orbit segments of nonuniformly hyperbolic diffeomorphisms that almost close up. The required closeness of the beginning and end points depend on the Pesin set in which they lie, as does the degree of approximation by the resulting hyperbolic periodic orbit. As a result, hyperbolic periodic points are dense in the support of every hyperbolic measure. Moreover, if there is any hyperbolic measure then the measures supported on hyperbolic periodic orbits are weakly dense in the set of hyperbolic measures.
Moreover, the Livshitz Theorem generalizes to this setting as well, but with lower regularity of the solution of the cohomological equation:. Livshitz Theorem. There is also a spectral decomposition for continuous nonatomic hyperbolic invariant Borel probability measures: Each Pesin set is contained in a finite union of orbit closures. Indeed, whenever there is a continuous nonatomic hyperbolic invariant Borel probability measure then there is a horseshoe, that is, a compact locally maximal uniformly hyperbolic set on which the diffeomorphism corresponds to a topological Markov chain via a Markov partition.
In particular, the diffeomorphism has positive topological entropy. Yakov Pesin and Boris Hasselblatt , Scholarpedia, 3 1 Sinai , Ergodic Theory and Dynamical Systems, v. The paper extends the famous Sinai-Ruelle-Bowen measures for classical hyperbolic attractors to partially hyperbolic ones and thus provides the basis for ergodic theory of partially hyperbolic attractors. Jiang , Communications in Mathematical Physics, v. This paper provides a complete description of equilibrium measures for the class of coupled map lattices -- an infinite chain of weakly interacting hyperbolic attractors - that are famous discrete models in mathematical physics , applied in physics, biology, etc.
It reveals a persistent chaotic behavior in dynamical networks obtained by discretization of some famous evolution-type PDS. Barreira and J. Schmeling , Annals of Mathematics.
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The paper gives a complete solution of the Eckmann-Ruelle conjecture in dimension theory of smooth dynamical systems.