A version of Kac's lemma on first return times for suspension flows

TL;DR

This paper extends Kac's lemma to suspension flows, providing formulas for mean return times that account for flow reparametrizations and escape times, thus generalizing classical discrete-time results.

Contribution

It introduces a formula for mean return times in suspension flows, incorporating flow reparametrizations and escape times, thus generalizing Kac's lemma.

Findings

Mean return time varies linearly with flow reparametrizations.

Formulas account for mean escaping time from the set.

Extension of Kac's lemma to continuous-time suspension flows.

Abstract

In this article we study the mean return times to a given set for suspension flows. In the discrete time setting, this corresponds to the classical version of Kac's lemma \cite{K} that the mean of the first return time to a set with respect to the normalized probability measure is one. In the case of suspension flows we provide formulas to compute the mean return time. In particular, this varies linearly with continuous reparametrizatons of the flow and takes into account the mean escaping time from the original set.