Topology automaton of self-similar sets and its applications to metrical classifications. (1st May 2023)

• Record Type: Journal Article
• Authors: Huang, Liang-Yi; Wen, Zhi-Ying; Yang, Ya-Min; Zhu, Yun-Jie
• Is Part Of: Nonlinearity. Volume 36:Number 5(2023)
• Page Start: 2541
• Page End: 2566
• Publication Date: 2023-05-01
• Subjects: finite state automaton -- bi-Hölder equivalence -- Lipschitz equivalence -- self-similar set; 28A80 -- 26A16 -- 68Q45; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/acc304
• Languages: English
• ISSNs: 0951-7715

Abstract

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.