HAUSDORFF MEASURES of A CLASS of MORAN SETS

Xiaofang Jiang; Qinghui Liu; Guizhen Wang; Zhiying Wen

doi:10.1142/S0218348X2050053X

HAUSDORFF MEASURES of A CLASS of MORAN SETS

Xiaofang Jiang, Qinghui Liu^*, Guizhen Wang, Zhiying Wen

^*Corresponding author for this work

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

1 Citation (Scopus)

Abstract

Let (n,c) be the class of Moran sets with integer n ≥ 2 and real c satisfying nc < 1. It is well known that the Hausdorff dimension of any set in this class is s = -logcn. We show that for any E (n,c), 2s 2 (n - 1)c 1 - cs ≤s(E) ≤ 1, where s(E) denotes s-dimensional Hausdorff measure of E. For any a with 2s 2 (n - 1)c 1 - cs ≤ a ≤ 1, there exists a self-similar set E (n,c) such that s(E) = a.

Original language	English
Article number	2050053
Journal	Fractals
Volume	28
Issue number	3
DOIs	https://doi.org/10.1142/S0218348X2050053X
Publication status	Published - 1 May 2020

Keywords

Hausdorff Dimension
Hausdorff Measure
Moran Sets

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title = "HAUSDORFF MEASURES of A CLASS of MORAN SETS",

abstract = "Let (n,c) be the class of Moran sets with integer n ≥ 2 and real c satisfying nc < 1. It is well known that the Hausdorff dimension of any set in this class is s = -logcn. We show that for any E (n,c), 2s 2 (n - 1)c 1 - cs ≤s(E) ≤ 1, where s(E) denotes s-dimensional Hausdorff measure of E. For any a with 2s 2 (n - 1)c 1 - cs ≤ a ≤ 1, there exists a self-similar set E (n,c) such that s(E) = a.",

keywords = "Hausdorff Dimension, Hausdorff Measure, Moran Sets",

author = "Xiaofang Jiang and Qinghui Liu and Guizhen Wang and Zhiying Wen",

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AU - Jiang, Xiaofang

AU - Liu, Qinghui

AU - Wang, Guizhen

AU - Wen, Zhiying

PY - 2020/5/1

Y1 - 2020/5/1

N2 - Let (n,c) be the class of Moran sets with integer n ≥ 2 and real c satisfying nc < 1. It is well known that the Hausdorff dimension of any set in this class is s = -logcn. We show that for any E (n,c), 2s 2 (n - 1)c 1 - cs ≤s(E) ≤ 1, where s(E) denotes s-dimensional Hausdorff measure of E. For any a with 2s 2 (n - 1)c 1 - cs ≤ a ≤ 1, there exists a self-similar set E (n,c) such that s(E) = a.

AB - Let (n,c) be the class of Moran sets with integer n ≥ 2 and real c satisfying nc < 1. It is well known that the Hausdorff dimension of any set in this class is s = -logcn. We show that for any E (n,c), 2s 2 (n - 1)c 1 - cs ≤s(E) ≤ 1, where s(E) denotes s-dimensional Hausdorff measure of E. For any a with 2s 2 (n - 1)c 1 - cs ≤ a ≤ 1, there exists a self-similar set E (n,c) such that s(E) = a.

KW - Hausdorff Dimension

KW - Hausdorff Measure

KW - Moran Sets

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VL - 28

JO - Fractals

JF - Fractals

IS - 3

M1 - 2050053

ER -

HAUSDORFF MEASURES of A CLASS of MORAN SETS

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