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 {µω}_ω∈Ω, 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 ϑ which is given by the escape rate at the periodic point.