Absolutely continuous invariant measures for random maps with position dependent probabilities

By P. Góra, A. Boyarsky. 2003

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Abstract 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 τ k is chosen from a finite collection of maps with constant probability p k . In this note we allow the p k '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.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/76c3952a0621401cbecc09ee61b7576cc3d4ace1