Contracting on average iterated function systems by metric change

Abstract

We study contraction conditions for an iterated function system (IFS) of continuous maps on a metric space which are chosen randomly, identically and independently. We investigate metric changes, preserving the topological structure of the space, which turn the IFS into one which is contracting on average. For the particular case of a system of C 1-diffeomorphisms of the circle which is proximal and does not have a probability measure simultaneously invariant by every map, we derive an equivalent metric which contracts on average and conclude that it carries a unique stationary probability measure. Here we strongly use local contraction properties established by Malicet in 2017, which then imply that the IFS has a negative random Lyapunov exponent and generalizes a result by Baxendale.