Finite state automata and homeomorphism of self-similar sets

TL;DR

This paper introduces a novel automaton-based framework to classify self-similar fractals, establishing conditions for their topological and metric equivalence through pseudo-metric spaces and automata theory.

Contribution

It develops a new automaton-based approach to analyze and classify self-similar sets, providing criteria for their homeomorphism and Lipschitz equivalence.

Findings

Fractal gaskets are homeomorphic to pseudo-metric spaces from topology automata.

Different automata induce bi-Lipschitz equivalent pseudo-metric spaces.

Provides general sufficient conditions for homeomorphism and Lipschitz equivalence of fractal gaskets.

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.