Self-similar structure on intersection of homogeneous symmetric Cantor sets

2010 ◽  
Vol 284 (2-3) ◽  
pp. 298-316 ◽  
Author(s):  
Wenxia Li ◽  
Yuanyuan Yao ◽  
Yunxiu Zhang
Keyword(s):  
Cantor Sets ◽  
Self Similar

Self-similar structure on the intersection of middle-(1 − 2β) Cantor sets with β ∊ (1/3, 1/2)

Nonlinearity ◽  
2008 ◽  
Vol 21 (12) ◽  
pp. 2899-2910 ◽  
Author(s):  
Yuru Zou ◽  
Jian Lu ◽  
Wenxia Li
Keyword(s):  
Cantor Sets ◽  
Self Similar

scholarly journals Resonance between Cantor sets

2009 ◽  
Vol 29 (1) ◽  
pp. 201-221 ◽  
Author(s):  
YUVAL PERES ◽  
PABLO SHMERKIN
Keyword(s):  
Hausdorff Dimension ◽  
Iterated Function System ◽  
Unit Interval ◽  
Irrational Rotation ◽  
Cantor Sets ◽  
Scaling Parameter ◽  
Iterated Function ◽  
Self Similar ◽  
Image Position ◽  
Geometric Resonance

AbstractLet Ca be the central Cantor set obtained by removing a central interval of length 1−2a from the unit interval, and then continuing this process inductively on each of the remaining two intervals. We prove that if log b/log a is irrational, then where dim is Hausdorff dimension. More generally, given two self-similar sets K,K′ in ℝ and a scaling parameter s>0, if the dimension of the arithmetic sum K+sK′ is strictly smaller than dim (K)+dim (K′)≤1 (‘geometric resonance’), then there exists r<1 such that all contraction ratios of the similitudes defining K and K′ are powers of r (‘algebraic resonance’). Our method also yields a new result on the projections of planar self-similar sets generated by an iterated function system that includes a scaled irrational rotation.

A REMARK ON SELF-SIMILAR SUBSETS OF UNIFORM CANTOR SETS WITH SMALL HAUSDORFF DIMENSION

Fractals ◽  
2020 ◽  
Vol 28 (04) ◽  
pp. 2050057
Author(s):  
HUI RAO ◽  
ZHI-YING WEN ◽  
YING ZENG
Keyword(s):  
Hausdorff Dimension ◽  
Difficult Problem ◽  
Cantor Set ◽  
Cantor Sets ◽  
Symbolic Space ◽  
Self Similar

Recently there are several works devoted to the study of self-similar subsets of a given self-similar set, which turns out to be a difficult problem. Let [Formula: see text] be an integer and let [Formula: see text]. Let [Formula: see text] be the uniform Cantor set defined by the following set equation: [Formula: see text] We show that for any [Formula: see text], [Formula: see text] and [Formula: see text] essentially have the same self-similar subsets. Precisely, [Formula: see text] is a self-similar subset of [Formula: see text] if and only if [Formula: see text] is a self-similar subset of [Formula: see text], where [Formula: see text] (similarly [Formula: see text]) is the coding map from the symbolic space [Formula: see text] to [Formula: see text].

SELF-SIMILARITY AND LIPSCHITZ EQUIVALENCE OF UNIONS OF CANTOR SETS

Fractals ◽  
2018 ◽  
Vol 26 (05) ◽  
pp. 1850061
Author(s):  
CHUNTAI LIU
Keyword(s):  
Cantor Set ◽  
Cantor Sets ◽  
Necessary And Sufficient Condition ◽  
Self Similarity ◽  
Lipschitz Equivalence ◽  
Fractal Sets ◽  
Strong Separation ◽  
Self Similar ◽  
Strong Separation Condition ◽  
Necessary And Sufficient

Self-similarity and Lipschitz equivalence are two basic and important properties of fractal sets. In this paper, we consider those properties of the union of Cantor set and its translate. We give a necessary and sufficient condition that the union is a self-similar set. Moreover, we show that the union satisfies the strong separation condition if it is of the self-similarity. By using the augment tree, we characterize the Lipschitz equivalence between Cantor set and the union of Cantor set and its translate.

scholarly journals Self-similar structure on intersections of triadic Cantor sets

2008 ◽  
Vol 337 (1) ◽  
pp. 617-631 ◽  
Author(s):  
Guo-Tai Deng ◽  
Xing-Gang He ◽  
Zhi-Xiong Wen
Keyword(s):  
Cantor Sets ◽  
Self Similar

scholarly journals Embeddings of self-similar ultrametric Cantor sets

2011 ◽  
Vol 158 (16) ◽  
pp. 2148-2157 ◽  
Author(s):  
Antoine Julien ◽  
Jean Savinien
Keyword(s):  
Cantor Sets ◽  
Self Similar

scholarly journals Self-similar measures and intersections of Cantor sets

1998 ◽  
Vol 350 (10) ◽  
pp. 4065-4087 ◽  
Author(s):  
Yuval Peres ◽  
Boris Solomyak
Keyword(s):  
Cantor Sets ◽  
Self Similar

SELF-SIMILAR SUBSETS OF SYMMETRIC CANTOR SETS

Fractals ◽  
2017 ◽  
Vol 25 (01) ◽  
pp. 1750003 ◽  
Author(s):  
YING ZENG
Keyword(s):  
Cantor Set ◽  
Cantor Sets ◽  
Self Similar ◽  
Affine Embeddings

This paper concerns the affine embeddings of general symmetric Cantor sets. Under certain condition, we show that if a self-similar set [Formula: see text] can be affinely embedded into a symmetric Cantor set [Formula: see text], then their contractions are rationally commensurable. Our result supports Conjecture 1.2 in [D. J. Feng, W. Huang and H. Rao, Affine embeddings and intersections of Cantor sets, J. Math. Pures Appl. 102 (2014) 1062–1079].

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