On the statistical distribution of first--return times of balls and cylinders in chaotic systems

TL;DR

This paper investigates the statistical distribution of first-return times in chaotic dynamical systems across different settings, linking these distributions to entropy measures and Lyapunov exponents.

Contribution

It provides a comprehensive analysis of first-return time distributions in symbolic and geometric chaotic systems, establishing new relations with entropy and stability metrics.

Findings

Distribution of return times relates to Renyi entropies.

Results apply to Bernoulli shifts and toral automorphisms.

Derived formulas connect return times with Lyapunov exponents.

Abstract

We study returns in dynamical systems: when a set of points, initially populating a prescribed region, swarms around phase space according to a deterministic rule of motion, we say that the return of the set occurs at the earliest moment when one of these points comes back to the original region. We describe the statistical distribution of these "first--return times" in various settings: when phase space is composed of sequences of symbols from a finite alphabet (with application for instance to biological problems) and when phase space is a one and a two-dimensional manifold. Specifically, we consider Bernoulli shifts, expanding maps of the interval and linear automorphisms of the two dimensional torus. We derive relations linking these statistics with Renyi entropies and Lyapunov exponents.