On the Existence of Geometrically Attracting Measures for Iterated Function Systems with Varying Sets of Transformations

Abstract

In this paper, we focus on the asymptotic behavior of random iterated functions systems, in which both the family of transformations and the distribution of selecting them vary at each step. We consider two types of dynamics: the time-inhomogeneous Markov chain arising from forward iterates and the non-Markovian process generated by backward iterates. For both settings, we provide certain criteria for the existence of a probability measure that is geometrically attracting in the bounded Lipschitz distance, independently of the initial distribution. Finally, we illustrate our results through applications to specific models involving affine transformations.