Recurrence times and Rates of Mixing

This paper is part of an attempt to understand the speed of mixing and related statistical properties for chaotic dynamical systems. More precisely, we are interested in systems that are expanding or hyperbolic on large parts (though not necessarily all) of their phase spaces. A natural approach to this problem is to pick a suitable reference set, and to regard a part of the system as having “renewed” itself when it makes a “full” return to this set. We obtain in this way a representation of the dynamical system in question, described in terms of a reference set and return times. We propose to study this object abstractly, that is to say, to set aside the specific characteristics of the original system and to understand its statistical properties purely in terms of these recurrence times. Needless to say, if we are to claim that this approach is valid, we must also show that it is implementable, and that it gives reasonable results in interesting, concrete situations. The ideas described above were put forth in [Y]; they continue to be the underlying theme of the present paper. In [Y] we focused on mixing at exponential speeds. One of the aims of this paper is to extend the abstract part of this study to all speeds of mixing. Of particular interest is when the recurrence is polynomial, i.e. when the probability of not returning in the first n iterates is of order n−α. We will show in this case that the speed of mixing is of order n−α+1. More generally, let R