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Statistical properties of chaotic systems. (English) Zbl 0718.58038

This one hundred pages paper presents a unified approach to numerous recent results in the theory of abstract and smooth dynamical systems, related to the notion of chaotic behavior. It is mostly concerned with the classification problem of such systems in order to answer the following question: “How the knowledge of the past of a given stochastic process allows to predict its future?” This work is essentially two- fold: The first part is an overview of the classical theory of both smooth and abstract systems (definitions and essential concepts). The second part is a detailed discussion of the isomorphism theory illustrated by a lot of examples and most of the announced new results are then proved.
It is quite impossible to give a complete account of this encyclopedic paper but the central facts are the notions of \(\alpha\)-congruence and \(\alpha\)-entropy. We list below the most important studied topics:
\(\bullet\) Isomorphisms and \(\alpha\)-congruence of Bernoulli flows and Markov processes.
\(\bullet\) Stochastic stability and instability. Random perturbations.
\(\bullet\) Smooth dynamical systems and hyperbolic structures. Geodesic, Anosov and A flows.
\(\bullet\) Lyapunov exponents and Oseledec’s theorem
\(\bullet\) Smooth hyperbolic systems are Bernoulli.
\(\bullet\).......

MSC:

37D99 Dynamical systems with hyperbolic behavior
37C10 Dynamics induced by flows and semiflows
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37A99 Ergodic theory
60J99 Markov processes
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References:

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