Return times at periodic points in random dynamics

TL;DR

This paper establishes a quenched limit law for return times at periodic points in random dynamical systems, showing they follow a compound Poisson distribution under certain mixing and invariance conditions.

Contribution

It proves a new quenched limiting law for return times in random subshifts, extending understanding of statistical properties at periodic points.

Findings

Return times are compound Poisson distributed for almost every realization.

The distribution parameter is given by the escape rate at the periodic point.

Results apply to measures satisfying a generalized invariance and $\psi$-mixing conditions.

Abstract

We prove a quenched limiting law for random measures on subshifts at periodic points. We consider a family of measures $\{\mu_\omega\}_{\omega\in\Omega}$, where the `driving space' $\Omega$ is equipped with a probability measure which is invariant under a transformation $\theta$. We assume that the fibred measures $\mu_\omega$ satisfy a generalised invariance property and are $\psi$-mixing. We then show that for almost every $\omega$ the return times to cylinders $A_n$ at periodic points are in the limit compound Poisson distributed for a parameter $\vartheta$ which is given by the escape rate at the periodic point.