How inhomogeneous Cantor sets can pass a point

Wenxia Li; Zhiqiang Wang

doi:10.1007/s00209-022-03099-0

How inhomogeneous Cantor sets can pass a point

Wenxia Li^*
, Zhiqiang Wang

^*Corresponding author for this work

School of Mathematical Sciences

East China Normal University

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

1 Scopus citations

Abstract

For x> 0 , let Υ(x)={(a,b):x∈Ea,b,a>0,b>0,a+b≤1},where E_a_,_b is the unique nonempty compact invariant set generated by the inhomogeneous IFS Ψa,b={f0(x)=ax,f1(x)=b(x+1)}.We show that the set Υ (x) is a Lebesgue null set with full Hausdorff dimension and the intersection of the sets Υ (x₁) , … , Υ (x_ℓ) still has full Hausdorff dimension for any finite number of positive numbers x₁, … , x_ℓ.

Original language	English
Pages (from-to)	1429-1449
Number of pages	21
Journal	Mathematische Zeitschrift
Volume	302
Issue number	3
DOIs	https://doi.org/10.1007/s00209-022-03099-0
State	Published - Nov 2022

Keywords

Cantor set
Hausdorff dimension
Inhomogeneous
Intersection
Thickness

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title = "How inhomogeneous Cantor sets can pass a point",

abstract = "For x> 0 , let Υ(x)=\{(a,b):x∈Ea,b,a>0,b>0,a+b≤1\},where Ea,b is the unique nonempty compact invariant set generated by the inhomogeneous IFS Ψa,b=\{f0(x)=ax,f1(x)=b(x+1)\}.We show that the set Υ (x) is a Lebesgue null set with full Hausdorff dimension and the intersection of the sets Υ (x1) , … , Υ (xℓ) still has full Hausdorff dimension for any finite number of positive numbers x1, … , xℓ.",

keywords = "Cantor set, Hausdorff dimension, Inhomogeneous, Intersection, Thickness",

author = "Wenxia Li and Zhiqiang Wang",

note = "Publisher Copyright: {\textcopyright} 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.",

year = "2022",

month = nov,

doi = "10.1007/s00209-022-03099-0",

language = "英语",

volume = "302",

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TY - JOUR

T1 - How inhomogeneous Cantor sets can pass a point

AU - Li, Wenxia

AU - Wang, Zhiqiang

PY - 2022/11

Y1 - 2022/11

N2 - For x> 0 , let Υ(x)={(a,b):x∈Ea,b,a>0,b>0,a+b≤1},where Ea,b is the unique nonempty compact invariant set generated by the inhomogeneous IFS Ψa,b={f0(x)=ax,f1(x)=b(x+1)}.We show that the set Υ (x) is a Lebesgue null set with full Hausdorff dimension and the intersection of the sets Υ (x1) , … , Υ (xℓ) still has full Hausdorff dimension for any finite number of positive numbers x1, … , xℓ.

AB - For x> 0 , let Υ(x)={(a,b):x∈Ea,b,a>0,b>0,a+b≤1},where Ea,b is the unique nonempty compact invariant set generated by the inhomogeneous IFS Ψa,b={f0(x)=ax,f1(x)=b(x+1)}.We show that the set Υ (x) is a Lebesgue null set with full Hausdorff dimension and the intersection of the sets Υ (x1) , … , Υ (xℓ) still has full Hausdorff dimension for any finite number of positive numbers x1, … , xℓ.

KW - Cantor set

KW - Hausdorff dimension

KW - Inhomogeneous

KW - Intersection

KW - Thickness

UR - https://www.scopus.com/pages/publications/85136974135

U2 - 10.1007/s00209-022-03099-0

DO - 10.1007/s00209-022-03099-0

M3 - 文章

AN - SCOPUS:85136974135

SN - 0025-5874

VL - 302

SP - 1429

EP - 1449

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 3

ER -

How inhomogeneous Cantor sets can pass a point

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Keywords

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