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Consider a proper metric space X and a sequence of i.i.d. random continuousmappings F_n from X to X. It induces the stochastic dynamical system (SDS)X_n^x = F_n(X_{n-1}^x) starting at x in X. In this paper, we study existenceand uniqueness of invariant measures, as well as recurrence and ergodicity ofthis process. In the first part, we elaborate, improve and complete theunpublished work of Martin Benda on local contractivity, which merits publicityand provides an important tool for studying stochastic iterations. We considerthe case when the F_n are contractions and, in particular, discuss recurrencecriteria and their sharpness for reflected random walk.  In the second part, we consider the case where the F_n are Lipschitzmappings. The main results concern the case when the associated Lipschitzconstants are log-centered. Prinicpal tools are the Chacon-Ornstein theorem anda hyperbolic extension of the space X as well as the process (X_n^x). Theresults are applied to the reflected affine stochastic recursion on thenon-negative half-line.

Stochastic dynamical systems with weak contractivity properties (with a chapter featuring results of Martin Benda)