Scaling distances on finitely ramified fractals

Abstract

In this article we study two problems about the existence of a distance $d$ on a given fractal having certain properties. In the first problem, we require that the maps ${\psi }_i$ defining the fractal be Lipschitz of prescribed constants less than $1$ with respect to the distance $d$, and in the second one, we require that arbitrary compositions of the maps ${\psi }_i$ be uniformly bi-Lipschitz of related constants. Both problems have been investigated previously by other authors. In this article, on a large class of finitely ramified fractals, we prove that these two problems are equivalent and give a necessary and sufficient condition for the existence of such a distance. Such a condition is expressed in terms of asymptotic behavior of the product of certain matrices associated to the fractal.