Return times at periodic points in random dynamics

We prove a quenched limiting law for random measures on subshifts at periodic points. We consider a family of measures $\{μ_ω\}_{ω\inΩ}$, where the `driving space' $Ω$ is equipped with a probability measure which is invariant under a transformation $θ$. We assume that the fibred measures $μ_ω$ satisfy a generalised invariance property and are $ψ$-mixing. We then show that for almost every $ω$ 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.

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