Entry and return times for semi-flows

TL;DR

This paper explores the relationship between entry and return times in dynamical systems, generalizing previous results to non-ergodic maps and continuous-time systems through stationary point processes.

Contribution

It introduces a new interpretation of entry and return times using stationary point processes, extending existing theorems to broader classes of dynamical systems.

Findings

Generalization of entry and return time results to non-ergodic maps

Extension of the theory to continuous-time dynamical systems

Interpretation of times via Palm distributions

Abstract

Haydn, Lacroix and Vaienti [Ann. Probab. 33 (2005)] proved that, for a given ergodic map, the entry time distribution converges in the small target limit, if and only if the corresponding return time distribution converges. The present note explains how entry and return times can be interpreted in terms of stationary point processes and their Palm distribution. This permits a generalization of the results by Haydn et al. to non-ergodic maps and continuous-time dynamical systems.