Quantitative Recurrence and Dimension Theory

Abstract

Poincaré’s recurrence theorem (Theorem 2.2.2) is one of the basic but fundamental results of the theory of dynamical systems. Unfortunately it only provides information of a qualitative nature. In particular, it does not consider the following natural problems: with which frequency the orbit of a point visits a given set of positive measure; with which rate the orbit of a point returns to an arbitrarily small neighborhood of the initial point. Birkhoff’s ergodic theorem (Theorem 2.2.3) gives a complete answer to the first problem. The second problem of quantitative recurrence has experienced a growing interest during the last decade, also in connection with other fields, including for example compression algorithms. We describe in this chapter several results that provide partial answers to the problem.