Stationary Random Iterations of Lipschitz Maps: A Generalized Displacement Condition for the Existence of Invariant Structures
We study random iterations generated by infinitely many Lipschitz maps on non-compact complete separable metric spaces. We generalize classical displacement conditions ensuring the existence and uniqueness of invariant structures. We prove two main results. The first uses the graph transform operator to extend Elton's condition [A multiplicative ergodic theorem for Lipschitz maps. Stochastic Process. Appl. 34(1) (1990)] for the existence and uniqueness of invariant graphs. The second uses the Markov operator to generalize a displacement condition for stationary measures, introduced by Diaconis and Freedman [Iterated random functions. SIAM Rev. 41(1) (1999)] and by Kloeckner [Optimal transportation and stationary measures for iterated function systems. Math. Proc. Cambridge Philos. Soc. 173(1) (2022)]. We present several examples illustrating the applicability of our results.