Contracting on average iterated function systems by metric change

TL;DR

This paper explores how changing the metric on a space can make a randomly chosen iterated function system contract on average, especially for certain circle diffeomorphisms, without altering the topological structure.

Contribution

It introduces a method to find metric changes that induce average contraction in IFSs, including for specific circle diffeomorphisms, expanding the understanding of contraction conditions.

Findings

Identifies metric changes that induce average contraction in IFSs

Derives a strongly equivalent metric for circle diffeomorphisms that contracts on average

Shows the applicability to systems without a common invariant measure

Abstract

We study contraction conditions for an iterated function system 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 a strongly equivalent metric which contracts on average.