Remetrization results for possibly infinite self-similar systems

Abstract

In this paper we introduce a concept of possibly infinite self-similar system which generalizes the attractor of a possibly infinite iterated function system whose constitutive functions are $\varphi$-contractions. We prove that for a uniformly possibly infinite self-similar system there exists a remetrization which makes contractive all its constitutive functions. Then, based on this result, we show that for such a system there exist a comparison function $\varphi $ and a remetrization of the system which makes $\varphi $-contractive all its constitutive functions. Finally we point out that in the case of a finite set of constitutive functions our concept of a possibly infinite self-similar system coincides with Kameyama's concept of a topological self-similar system.