First return maps of random maps and invariant measures

We consider a family of transformations and study a random dynamical system such that one transformation is randomly selected from the family and then applied on each iteration. To study a random dynamical system with indifferent fixed points, we define the first return random map. In general, the selection of transformations in the first return random map depends on the position in the state space, even if the selection of transformations in the original random map is independent of the position. We give a theorem which describe how to make an invariant measure of the original random map from that of the first return random map. We apply our theorem developed in this paper to one-dimensional random maps with indifferent fixed points and show the existence of absolutely continuous -finite invariant measures under some conditions. Further, we research whether the invariant measures are finite or not under some conditions.