On the Absolutely Continuous Invariant Measures for Random Maps

Abstract

We study the dynamics of a new family of transformations. We find a general formula for the invariant density of any transformation in our family. The properties of the family allow us to prove that the invariant density function [Formula: see text] of random map constructed from our family maps [Formula: see text] is the combination [Formula: see text] where [Formula: see text] are the invariant density functions of [Formula: see text] respectively. We also consider another family of transformations, and prove that the invariant density for any transformation of the family is [Formula: see text].