Asymptotic properties of two interacting maps

Abstract.

In this paper we consider a system whose state x changes to σ(x) if a perturbation occurs at the time t, for $t > 0,\;t \notin \mathbb{N}$. Moreover, the state x changes to the new state η(x) at time t, for $t \in \mathbb{N}$. It is assumed that the number of perturbations in an interval (0, t) is a Poisson process. Here η and σ are measurable maps from a measure space ${\left( {E,{\cal A},\mu } \right)}$ into itself. We give conditions for the existence of a stationary distribution of the system when the maps η and σ commute, and we prove that any stationary distribution is an invariant measure of these maps.