Algebraic criteria for Lipschitz equivalence of dust‐like self‐similar sets

Xi, Lifeng; Xiong, Yan

doi:10.1112/jlms.12392

Cited by 7 publications

(2 citation statements)

References 34 publications

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“…Falconer and Marsh [8] proved that all Cantor sets are Hölder equivalent. The study of Lipschitz equivalence of fractal sets derives from 1990s and it has become a very active topic in recent years [6,8,9,13,17,19,21,25]. Most of the studies focus on self-similar sets which are totally disconnected.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2023

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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.

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Section: Introductionmentioning

confidence: 99%

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

et al. 2023

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“…For studies of quasi-symmetric equivalence of fractal sets, see [4,19]. The study of Lipschitz equivalence of fractal sets derives from 1990's and it becomes a very active topic in recent years [5,6,9,11,14,15,18,22], where most of the studies focus on self-similar sets which are totally disconnected.…”

Section: Introductionmentioning

confidence: 99%

Finite state automata and homeomorphism of self-similar sets

Huang¹,

Wen²,

Yang³

et al. 2021

Preprint

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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 Σ-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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Lipschitz images and dimensions

Balka,

Keleti

2024

Advances in Mathematics

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Algebraic criteria for Lipschitz equivalence of dust‐like self‐similar sets

Cited by 7 publications

References 34 publications

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

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

Finite state automata and homeomorphism of self-similar sets

Lipschitz images and dimensions

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