Poincar\'e recurrence for observations

TL;DR

This paper investigates how the recurrence times of observations in high-dimensional dynamical systems relate to the Hausdorff dimension of the observed measure, especially under super-polynomial decay of correlations.

Contribution

It extends Boshernitzan's work by establishing a link between recurrence rates of observations and Hausdorff dimensions in systems with super-polynomial decay of correlations.

Findings

Recurrence rates for observations equal pointwise dimensions under certain conditions.

Decay of correlations influences the relationship between recurrence times and measure dimensions.

Provides theoretical insights into the behavior of observed dynamical systems.

Abstract

A high dimensional dynamical system is often studied by experimentalists through the measurement of a relatively low number of different quantities, called an observation. Following this idea and in the continuity of Boshernitzan's work, for a measure preserving system, we study Poincar\'e recurrence for the observation. The link between the return time for the observation and the Hausdorff dimension of the image of the invariant measure is considered. We prove that when the decay of correlations is super polynomial, the recurrence rates for the observations and the pointwise dimensions relatively to the push-forward are equal.