Isomorphism and bi-Lipschitz equivalence between the univoque sets

Kan Jiang; Lifeng Xi; Shengnan Xu; Jinjin Yang

doi:10.3934/dcds.2020271

Article Contents

2020, Volume 40, Issue 11: 6089-6114. Doi: 10.3934/dcds.2020271

This issue Previous Article Next Article Weak solutions to the continuous coagulation model with collisional breakage

Isomorphism and bi-Lipschitz equivalence between the univoque sets

Department of Mathematics, Ningbo University, Ningbo, Zhejiang, China

Received: April 2018

Revised: May 2020

Published online: November 2020

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Abstract

In this paper, we consider a class of self-similar sets, denoted by $ \mathcal{A} $, and investigate the set of points in the self-similar sets having unique codings. We call such set the univoque set and denote it by $ U_1 $. We analyze the isomorphism and bi-Lipschitz equivalence between the univoque sets. The main result of this paper, in terms of the dimension of $ U_1 $, is to give several equivalent conditions which describe that the closure of two univoque sets, under the lazy maps, are measure theoretically isomorphic with respect to the unique measure of maximal entropy. Moreover, we prove, under the condition $ U_1 $ is closed, that isomorphism and bi-Lipschitz equivalence between the univoque sets have resonant phenomenon.

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Mathematics Subject Classification: 37B10, 28A78, 28A80.

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Figure 1. First iteration of $ K $

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Figure 3. First iteration of $ K_1 $

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Figure 4. First iteration of $ K_2 $

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Figure 2. First iteration of $ K $

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