Recurrence for random dynamical systems

TL;DR

This paper investigates recurrence behavior in random dynamical systems, introducing concepts of quenched and annealed return times, and establishes a relationship between recurrence rate and local measure dimension for highly mixing systems.

Contribution

It introduces new recurrence concepts for random systems and proves the equivalence of recurrence rate and local dimension in super-polynomially mixing cases.

Findings

Recurrence rate equals local dimension in highly mixing systems

Defined quenched and annealed return times for random maps

Established foundational concepts for recurrence in random dynamical systems

Abstract

This paper is a first step in the study of the recurrence behavior in random dynamical systems and randomly perturbed dynamical systems. In particular we define a concept of quenched and annealed return times for systems generated by the composition of random maps. We moreover prove that for super-polynomially mixing systems, the random recurrence rate is equal to the local dimension of the stationary measure.