Topology automaton of self-similar sets and its applications to metrical classifications

Abstract The topological and metrical classifications of fractal sets are important topics in analysis. The goal of the present paper is to carry out such studies by using a finite state automaton. Firstly, we introduce Σ-automaton for self-similar sets, and we define topology automaton for fractal gaskets. Next, we show that a fractal gasket is always bi-Hölder equivalent to the pseudo-metric space induced by its topology automaton. Thirdly, we investigate when the pseudo-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 bi-Hölder or bi-Lipschitz equivalent.