Scaling Distances on Finitely Ramified Fractals

In previous papers by A. Kameyama and by J. Kigami distances on fractals have been discussed having two different but similar properties. One property is that the maps defining the fractal are Lipschitz of prescribed constants less than 1, the other is that the diameters of the copies of the fractal are asymptotic to prescribed scaling factors. In this paper, 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 distances. Such a condition is expressed in terms of asymptotic behavior of the product of certain matrices associated to the fractal.