FROBENIUS-PERRON OPERATORS AND APPROXIMATION OF INVARIANT-MEASURES FOR SET-VALUED DYNAMICAL-SYSTEMS

A set-valued dynamical system F on a Borel space X induces a set-valued operator F on M(X) - the set of probability measures on X. We define a representation of F, each of which induces an explicitly defined selection of F; and use this to extend the notions of invariant measure and Frobenius-Perron operators to set-valued maps. We also extend a method of S. Ulam to Markov finite approximations of invariant measures to the set-valued case and show how this leads to the approximation of T-invariant measures for transformations tau, where T corresponds to the closure of the graph of tau.