Statistical Properties of Dynamical Systems with Some Hyperbolicity

Abstract

This paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts (though not necessarily all) of their phase spaces.The main results are for discrete time systems.To put this work into context, recall that for Axiom A attractors the picture has been fairly complete since the 1970's (see [S1], [B], [R2]).Since then much progress has been made on two fronts: there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero Lyapunov exponents ([O], [P1], [Ka], [LY]), and there are detailed analyses of specific kinds of dynamical systems including, for example, billiards, 1-dimensional and Hénon-type maps (Statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero Lyapunov exponents.The goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than Axiom A. This class will not be defined explicitly, but it includes some of the much studied examples.By looking at regular returns to sets with good hyperbolic properties, one could give systems in this class a simple dynamical representation.Conditions for the existence of natural invariant measures, exponential mixing and central limit theorems are given in terms of the return times.These conditions can be checked in concrete situations, giving a unified way of proving a number of results, some new and some old.Among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of Hénon-type maps.The dynamical picture we wish to focus on is the following.Let f be the map in question, and suppose that f admits a "horseshoe" Λ with infinitely many branches and variable return times.More precisely, Λ has a product structure in the sense