Note on limit distribution of normalized return times and escape rate

TL;DR

This paper investigates the conditions under which normalized return times to shrinking targets follow an exponential distribution and establishes a proportional relationship between escape rate and set size in certain mixing systems.

Contribution

It provides a necessary and sufficient condition for exponential limit distribution of normalized return times and links escape rate to set size in $\psi$-mixing systems.

Findings

Normalized return times follow exponential distribution under specific conditions.

Escape rate is proportional to the size of the set in $\psi$-mixing systems.

Identifies sweep-out sequence as key to understanding return time distributions.

Abstract

In this note we discuss limit distribution of normalized return times for shrinking targets and draw a necessary and sufficient condition using sweep-out sequence in order for the limit distribution to be exponential with parameter $1$. The normalizing coefficients are the same as sizes of the targets. Moreover we study escape rate, namely the exponential decay rate of sweep-out sequence and prove that in $\psi$-mixing systems for a certain class of sets the escape rate is in limit proportional to the size of the set.