Invariant measures for Markov operators with application to function systems

A new sufficient condition for the existence of an invariant measure for Markov operators defined on Polish spaces is presented. This criterion is applied to iterated function systems. 0. Introduction. The theory of Markov processes is a fast developing topic which has been extensively studied during the last few years. The reason for this study was the progress in the theory of fractals. Markov processes can be considered from two points of view. They can be investigated by purely probabilistic and purely analytic methods. In our paper we use the second approach. More precisely, let {Zn} be a homogeneous Markov chain taking values in a metric space (X, %) and let π be its transition kernel, i.e. Prob(Zn+1 ∈ A |Zn = xn, . . . , Z0 = x0) = π(xn, A) for all n ∈ N and Borel sets A. It is of great interest to give sufficient conditions for the existence of an invariant probability measure for the Markov chain {Zn}, i.e. a probability measure μ∗ satisfying μ∗(A) =