Invariant densities of random maps have lower bounds on their supports

Góra, Paweł; Boyarsky, Abraham; Islam, Md Shafiqul

doi:https://doi.org/10.1155/JAMSA/2006/79175

International Journal of Stochastic Analysis

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Volume 2006 | Article ID 079175 | https://doi.org/10.1155/JAMSA/2006/79175

Invariant densities of random maps have lower bounds on their supports

Paweł Góra,¹Abraham Boyarsky,¹and Md Shafiqul Islam²

Received08 Feb 2005

Revised02 Oct 2005

Accepted04 Oct 2005

Published25 May 2006

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.

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Copyright

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.

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