Scaling Distances on Finitely Ramified Fractals

TL;DR

This paper establishes an equivalence between two types of distance properties on finitely ramified fractals and provides a matrix-based criterion for their existence.

Contribution

It proves the equivalence of Lipschitz and scaling diameter properties on finitely ramified fractals and introduces a matrix condition characterizing the existence of such distances.

Findings

The two properties are equivalent on a large class of fractals.

A necessary and sufficient matrix condition is provided.

The condition involves asymptotic behavior of matrix products.

Abstract

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.