Return times at periodic points in random dynamics

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.