Entropy and Poincar\'e recurrence from a geometrical viewpoint

TL;DR

This paper explores Poincaré recurrence using geometric methods, establishing relationships between entropy, return times, and dynamical properties, and providing new insights into the growth rates and connections among these concepts.

Contribution

It offers a geometric proof linking metric entropy to return times and reveals linear growth of minimal return times, extending classical recurrence results.

Findings

Metric entropy equals exponential growth rate of return times

Minimal return times grow linearly with the length of dynamical balls

Relations between recurrence, dimension, entropy, and Lyapunov exponents are established

Abstract

We study Poincar\'e recurrence from a purely geometrical viewpoint. We prove that the metric entropy is given by the exponential growth rate of return times to dynamical balls. This is the geometrical counterpart of Ornstein-Weiss theorem. Moreover, we show that minimal return times to dynamical balls grow linearly with respect to its length. Finally, some interesting relations between recurrence, dimension, entropy and Lyapunov exponents of ergodic measures are given.