Journal of Applied Mathematics and Stochastic Analysis
Volume 2005 (2005), Issue 2, Pages 133-141

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

Md. Shafiqul Islam, Pawel Góra, and Abraham Boyarsky

Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke Street West , Quebec, Montreal H4B 1R6, Canada

Received 29 February 2004; Revised 1 September 2004

Copyright © 2005 Md. Shafiqul Islam et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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