Absolutely continuous invariant measures for random maps with position dependent probabilities ✩

A random map is discrete-time dynamical system in which one of a number of transformations is randomly selected and applied at each iteration of the process. Usually the map τk is chosen from a finite collection of maps with constant probability pk. In this note we allow the pk’s to be functions of position. In this case, the random map cannot be considered to be a skew product. The main result provides a sufficient condition for the existence of an absolutely continuous invariant measure for position dependent random maps on [0, 1]. Geometrical and topological properties of sets of absolutely continuous invariant measures, attainable by means of position dependent random maps, are studied theoretically and numerically.  2003 Elsevier Science (USA). All rights reserved.