Limit Laws for Recurrence Times in Expanding Maps of an Interval

Abstract

The problem of entrance time dates back to the first developments of ergodic theory. A well known historical example being the raging discussions at the beginning of the century about the period of Poincaré recurrence. As for many questions in ergodic theory, the main step came much later with Birkhoff’s ergodic theorem. We recall that if we have a space X equipped with a σ-algebra A and a measure μ on this σ-algebra which is invariant and ergodic for a measurable transformation f of X, then for any measurable set A of positive measure, and for μ almost every initial condition x, the orbit of x spends (asymptotically) in A a fraction of time equal to μ(A). In other words, for μ almost every x, we have

$$ \mathop{{\lim }}\limits_{{n \to \infty }} \frac{1}{n}\sum\limits_{{j = 0}}^{{n - 1}} {\chi A} ({{f}^{j}}(x)) = \mu (A), $$

where as usual χA denotes the characteristic function of the set A, and f ^j is the j^th iterate of f. This means that roughly the orbit of x recurs to A with a period equal to 1/µ(A), and this is a natural time scale associated to any measurable set A with positive measure. At this level of generality one cannot say much more about the law of recurrence time to A ¹, but by analogy with the techniques of probability theory, one can hope to get interesting asymptotic results when the measure of A tends to zero. To be more precise we define the first entrance time to A as the random variable given by

$$ {{\tau }_{A}}(x) = \min \{ n0;{{f}^{n}}(x) \in A. $$

Note that it follows immediately from Birkhoff’s ergodic theorem that τ _A is almost surely finite.