All iterated function systems are Lipschitz up to an equivalent metric

A finite family F=₁,…,fₙ\ 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 {Hu}. However, in {LS} authors studied highly non-contractive IFSs, i.e. such families F=₁,…,fₙ\ of continuous selfmaps that for any remetrization of X each function fᵢ has Lipschitz constant >1, i=1,…,n. They asked when one can remetrize X that F is Lipschitz IFS, i.e. all fᵢ's are Lipschitz (not necessarily contractive), i=1,…,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 F=₁,…,fₙ\ of continuous selfmaps of X. However, our construction is valid even for some specific infinite families of continuous functions.