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A random map is a discrete-time dynamical system in which one of anumber of transformations is randomly selected and applied at eachiteration of the process. In this paper, we study random maps. Themain result provides a  necessary and sufficient condition for theexistence of absolutely continuous invariant measure for a randommap with constant probabilities and position-dependent probabilities

A generalization of Straube's theorem: existence of absolutely continuous invariant measures for random maps