Journal of Applied Mathematics and Stochastic Analysis
Volume 2006 (2006), Article ID 79175, 13 pages
doi:10.1155/JAMSA/2006/79175
Invariant densities of random maps have lower bounds on their supports
1Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke Street West, Montreal, Quebec H4B 1R6, Canada
2Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta T1K 3M4, Canada
Received 8 February 2005; Revised 2 October 2005; Accepted 4 October 2005
Copyright © 2006 Paweł Góra 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.
Abstract
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. The asymptotic properties of a random map are described by its invariant densities. If Pelikan's average expanding condition is satisfied, then the random map has invariant densities. For individual maps, piecewise expanding is sufficient to establish many important properties of the invariant densities, in particular, the fact that the densities are bounded away from 0 on their supports. It is of interest to see if this property is transferred to random maps satisfying Pelikan's condition. We show that if all the maps constituting the random map are piecewise expanding, then the same result is true. However, if one or more of the maps are not expanding, this may not be true: we present an example where Pelikan's condition is satisfied, but not all the maps are piecewise expanding, and show that the invariant density is not separated from 0.