Finite state automata and homeomorphism of self-similar sets

Liangyi Huang; Zhiying Wen; Yamin Yang; Yunjie Zhu

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arXiv · 2112.02322

Finite state automata and homeomorphism of self-similar sets

Liangyi Huang·Zhiying Wen·Yamin Yang·Yunjie Zhu

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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.

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BibTeX RIS

Liangyi Huang, Zhiying Wen, Yamin Yang, Yunjie Zhu. 2021-12-04. Finite state automata and homeomorphism of self-similar sets. https://arxiv.org/abs/2112.02322

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