On statistical properties of chaotic dynamical systems

Abstract

This paper is devoted to the methods of investigating statistical properties of chaotic dynamical systems. By statistical properties we mean the rate of the decay of correlations, the central limit theorem and other probabilistic limit theorems. Surveys of known results in this area may be found in [3, 9, 8]. An effective method of proving these properties is based on Markov approximations to dynamical systems. This approach is an alternative to the conventional Perron-Frobenius operator techniques. It was Ya. Sinai and his school [14, 15, 4, 5, 6] who systematically developed the methods of Markov partitions, Markov symbolic dynamics and measuretheoretic Markov approximations to Anosov diffeomorphisms and billiards. Bowen [2] extended these methods to Smale’s Axiom A diffeomorphisms. A general construction of Markov chains approximating discrete-time dynamical systems was introduced by Bunimovich and Sinai in [4, 5] and later studied in [6, 8]. It was shown that the chaotic behavior of the dynamical system ensures special conditions on transition probabilities of the approximating Markov chain. In turn, under those conditions one can prove probabilistic limit theorems and establish strong bounds on correlation functions for the original dynamical system – some general results are displayed in [8]. Here we continue studying the techniques of Markov approximations to dynamical systems. We explain how approximating Markov chains can be constructed for dynamical systems with discrete and continuous time. We also introduce a brand new condition on the transition probabilities of Markov chains that implies strong bounds on the correlations. Our new condition is weaker than all those studied earlier in [4, 6, 8].