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Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities. (English) Zbl 0653.60057

Consider a discrete-time Markov process on a locally compact metric space X obtained by randomly iterating Lipschitz maps \(w_ 1,...,w_ N\); the probability \(p_ i(x)\) of choosing map \(w_ i\) at each step is allowed to depend on the current position x. Assume sets of finite diameter in X are relatively compact. It is shown that if the maps are average- contractive, i.e., \[ \sum^{N}_{i=1}p_ i(x)\log (d(w_ ix,w_ iy)/d(x,y))<0 \] uniformly in x and y, and if the \(p_ i's\) are bounded away from zero and satisfy a Dini-type continuity condition (weaker than Hölder-continuity), then the process converges in distribution to a unique invariant measure.
Also discussed are Perron-Frobenius theory and primitive weakly almost- periodic Markov operators, discontinuous maps, Julia sets, and running dynamical systems backwards.

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J35 Transition functions, generators and resolvents
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