The cyclic behavior of the constrictive Markov operators

Let S be a Polish space, and let M_{varSigma} be the Banach space of finite signed measures on the Borel Sigma-algebra varSigma of S. Given a constrictive Markov operator T:M_{varSigma}rightarrow M_{varSigma}, we use the asymptotic periodic decomposition of T to determine the set of T-invariant distributions in M_{varSigma} and the set of T-ergodic distributions. We also give the relationship between the asymptotic periodic decomposition and the cycles of the process relative to the operator T.