Finite state automata and homeomorphism of self-similar sets

Abstract

The topological and metrical equivalence of fractals is an important topic in analysis. In this paper, we use a class of finite state automata, called $\Sigma$-automaton, to construct psuedo-metric spaces, and then apply them to the study of classification of self-similar sets. We first introduce a notion of topology automaton of a fractal gasket, which is a simplified version of neighbor automaton; we show that a fractal gasket is homeomorphic to the psuedo-metric space induced by the topology automaton. Then we construct a universal map to show that psuedo-metric spaces induced by different automata can be bi-Lipschitz equivalent. As an application, we obtain a rather general sufficient condition for two fractal gaskets to be homeomorphic or Lipschitz equivalent.