All iterated function systems are Lipschitz up to an equivalent metric

TL;DR

This paper proves that any finite family of continuous selfmaps on a metric space can be re-metrized with an equivalent metric so that all maps become Lipschitz, extending the concept beyond contractive systems.

Contribution

The authors construct a new equivalent metric under which any finite family of continuous maps becomes Lipschitz, addressing a question about remetrization of non-contractive iterated function systems.

Findings

Any finite family of continuous selfmaps can be made Lipschitz via an equivalent metric.

The construction applies even to some infinite families of continuous functions.

The result generalizes the understanding of iterated function systems beyond contractive cases.

Abstract

A finite family $\mathcal{F}=\{f_1,\ldots,f_n\}$ of continuous selfmaps of a given metric space $X$ is called an iterated function system (shortly IFS). In a case of contractive selfmaps of a complete metric space is well-known that IFS has an unique attractor \cite{Hu}. However, in \cite{LS} authors studied highly non-contractive IFSs, i.e. such families $\mathcal{F}=\{f_1,\ldots,f_n\}$ of continuous selfmaps that for any remetrization of $X$ each function $f_i$ has Lipschitz constant $>1, i=1,\ldots,n.$ They asked when one can remetrize $X$ that $\mathcal{F}$ is Lipschitz IFS, i.e. all $f_i's$ are Lipschitz (not necessarily contractive), $ i=1,\ldots,n$. We give a general positive answer for this problem by constructing respective new metric (equivalent to the original one) on $X$, determined by a given family $\mathcal{F}=\{f_1,\ldots,f_n\}$ of continuous selfmaps of $X$. However, our construction is valid even for some specific infinite families of continuous functions.