On construction of asymptotically stable iterated function system with probabilities

Abstract

ABSTRACTWe consider a Markov chain generated by random iterations of a family of mappings indexed by elements of an arbitrary measurable space. Under sufficiently weak assumptions we construct a family of place-dependent probability measures such that considered Markov chain converges to a stationary distribution. We also prove some sufficient condition for asymptotic stability of a family of i.i.d. mappings and we apply obtained result for discrete white noise random dynamical systems showing analogous probabilistic long-time behavior.