Multiple codings of self-similar sets with overlaps

Karma Dajani; Kan Jiang; Derong Kong; Wenxia Li

doi:10.1016/j.aam.2020.102146

Multiple codings of self-similar sets with overlaps

Karma Dajani, Kan Jiang^*, Derong Kong, Wenxia Li

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

In this paper we consider a general class E of self-similar sets with complete overlaps. Given a self-similar iterated function system Φ=(E,{f_i}_i=1^m)∈E on the real line, for each point x∈E we can find a sequence (i_k)=i₁i₂…∈{1,…,m}^N, called a coding of x, such that x=limn→∞⁡f_i₁∘f_i₂∘⋯∘f_{i_n}(0). For k=1,2,…,ℵ₀ or 2^ℵ₀ we investigate the subset U_k(Φ) which consists of all x∈E having precisely k different codings. Among several equivalent characterizations we show that U₁(Φ) is closed if and only if U_ℵ₀(Φ) is an empty set. Furthermore, we give explicit formulae for the Hausdorff dimension of U_k(Φ), and show that the corresponding Hausdorff measure of U_k(Φ) is always infinite for any k≥2. Finally, we explicitly calculate the local dimension of the self-similar measure at each point in U_k(Φ) and U_ℵ₀(Φ).

Original language	English
Article number	102146
Number of pages	49
Journal	Advances in Applied Mathematics
Volume	124
DOIs	https://doi.org/10.1016/j.aam.2020.102146
Publication status	Published - Mar 2021

Bibliographical note

DBLP License: DBLP's bibliographic metadata records provided through http://dblp.org/ are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.

Funding

The authors wish to thank the anonymous referee for many useful comments, which greatly improved the presentation of the paper. They also thank Junjie Miao for many discussions on the earlier version of Theorem 3 . Jiang was supported by NSFC No. 11701302 , Zhejiang Provincial Natural Science Foundation of China with No. LY20A010009 , and the K.C. Wong Magna Fund in Ningbo University . Kong was supported by NSFC No. 11971079 and the Fundamental and Frontier Research Project of Chongqing No. cstc2019jcyj-msxmX0338 and No. cx2019067 . Li was supported by NSFC No. 11671147 , 11571144 and Science and Technology Commission of Shanghai Municipality (STCSM) No. 18dz2271000 . Xi was supported by NSFC No. 11831007 .

Keywords

Countable expansions
Hausdorff dimension
Multiple expansions
Unique expansion

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Multiple codings of self-similar sets with overlapsFinal published version, 614 KB

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title = "Multiple codings of self-similar sets with overlaps",

abstract = "In this paper we consider a general class E of self-similar sets with complete overlaps. Given a self-similar iterated function system Φ=(E,{fi}i=1m)∈E on the real line, for each point x∈E we can find a sequence (ik)=i1i2…∈{1,…,m}N, called a coding of x, such that x=limn→∞⁡fi1∘fi2∘⋯∘fin(0). For k=1,2,…,ℵ0 or 2ℵ0 we investigate the subset Uk(Φ) which consists of all x∈E having precisely k different codings. Among several equivalent characterizations we show that U1(Φ) is closed if and only if Uℵ0(Φ) is an empty set. Furthermore, we give explicit formulae for the Hausdorff dimension of Uk(Φ), and show that the corresponding Hausdorff measure of Uk(Φ) is always infinite for any k≥2. Finally, we explicitly calculate the local dimension of the self-similar measure at each point in Uk(Φ) and Uℵ0(Φ).",

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AB - In this paper we consider a general class E of self-similar sets with complete overlaps. Given a self-similar iterated function system Φ=(E,{fi}i=1m)∈E on the real line, for each point x∈E we can find a sequence (ik)=i1i2…∈{1,…,m}N, called a coding of x, such that x=limn→∞⁡fi1∘fi2∘⋯∘fin(0). For k=1,2,…,ℵ0 or 2ℵ0 we investigate the subset Uk(Φ) which consists of all x∈E having precisely k different codings. Among several equivalent characterizations we show that U1(Φ) is closed if and only if Uℵ0(Φ) is an empty set. Furthermore, we give explicit formulae for the Hausdorff dimension of Uk(Φ), and show that the corresponding Hausdorff measure of Uk(Φ) is always infinite for any k≥2. Finally, we explicitly calculate the local dimension of the self-similar measure at each point in Uk(Φ) and Uℵ0(Φ).

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Multiple codings of self-similar sets with overlaps

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