Invariant densities of random maps have lower bounds on their supports

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.