All iterated function systems are Lipschitz up to an equivalent metric

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.