Asymptotic properties of two interacting maps

In this paper we consider a system whose state x changes to sigma(x) if a perturbation occurs at the time t, for t > 0, t is not an element of N. Moreover, the state x changes to the new state eta(x) at time t, for t is an element of N. It is assumed that the number of perturbations in an interval (0, t) is a Poisson process. Here eta and sigma are measurable maps from a measure space (E, A, mu) into itself. We give conditions for the existence of a stationary distribution of the system when the maps eta and sigma commute, and we prove that any stationary distribution is an invariant measure of these maps