The global statistics of return times: return time dimensions versus generalized measure dimensions

TL;DR

This paper explores the relationship between measure-based and return time-based dimensions in dynamical systems, establishing inequalities and conjecturing typical behaviors with minimal assumptions.

Contribution

It derives inequalities linking measure and return time dimensions and discusses their optimality, providing new insights into the statistical properties of dynamical systems.

Findings

Established inequalities between measure and return time dimensions

Analyzed optimality using von Neumann--Kakutani and Gaspard--Wang maps

Conjectured typical behavior of return time dimensions

Abstract

We investigate the relations holding among generalized dimensions of invariant measures in dynamical systems and similar quantities defined by the scaling of global averages of powers of return times. Because of a heuristic use of Kac theorem, these latter have been used in place of the former in numerical and experimental investigations; to mark this distinction, we call them return time dimensions. We derive a full set of inequalities linking measure and return time dimensions and we comment on their optimality with the aid of two maps due to von Neumann -- Kakutani and to Gaspard -- Wang. We conjecture the behavior of return time dimensions in a typical system. We only assume ergodicity of the dynamical system under investigation.