Scaling distances on finitely ramified fractals.

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 ψi 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 ψi 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. [ABSTRACT FROM AUTHOR]