Dimensions of the Boundary of a Graph-Directed Self-Similar Set With Overlaps

Guotai Deng; Chuntai Liu; Sze Man Ngai

Dimensions of the Boundary of a Graph-Directed Self-Similar Set With Overlaps

Guotai Deng, Chuntai Liu, Sze Man Ngai

Mathematical Sciences

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

2 Scopus citations

Abstract

We set up a framework for computing the Hausdorff and box dimensions of the boundary of each component F_i of a graph self-similar family defined by a graph-directed iterated function system satisfying the so-called graph finite boundary type condition. We show that ∂F_i has the same Hausdorff and box dimension, with the corresponding Hausdorff measure being positive and σ-finite. These results are natural extensions of existing ones concerning the dimensions of the boundary of a self-similar tile.

Original language	American English
Journal	Houston Journal of Mathematics
Volume	42
State	Published - Jan 1 2016

Keywords

Boundary
Dimensions
Graph-directed Self-similar Set
Overlaps

DC Disciplines

Education
Mathematics

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https://www.math.uh.edu/~hjm/Vol42-1.html

Cite this

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title = "Dimensions of the Boundary of a Graph-Directed Self-Similar Set With Overlaps",

abstract = "We set up a framework for computing the Hausdorff and box dimensions of the boundary of each component Fi of a graph self-similar family defined by a graph-directed iterated function system satisfying the so-called graph finite boundary type condition. We show that ∂Fi has the same Hausdorff and box dimension, with the corresponding Hausdorff measure being positive and σ-finite. These results are natural extensions of existing ones concerning the dimensions of the boundary of a self-similar tile.",

keywords = "Boundary, Dimensions, Graph-directed Self-similar Set, Overlaps",

author = "Guotai Deng and Chuntai Liu and Ngai, {Sze Man}",

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year = "2016",

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language = "American English",

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T1 - Dimensions of the Boundary of a Graph-Directed Self-Similar Set With Overlaps

AU - Deng, Guotai

AU - Liu, Chuntai

AU - Ngai, Sze Man

PY - 2016/1/1

Y1 - 2016/1/1

N2 - We set up a framework for computing the Hausdorff and box dimensions of the boundary of each component Fi of a graph self-similar family defined by a graph-directed iterated function system satisfying the so-called graph finite boundary type condition. We show that ∂Fi has the same Hausdorff and box dimension, with the corresponding Hausdorff measure being positive and σ-finite. These results are natural extensions of existing ones concerning the dimensions of the boundary of a self-similar tile.

AB - We set up a framework for computing the Hausdorff and box dimensions of the boundary of each component Fi of a graph self-similar family defined by a graph-directed iterated function system satisfying the so-called graph finite boundary type condition. We show that ∂Fi has the same Hausdorff and box dimension, with the corresponding Hausdorff measure being positive and σ-finite. These results are natural extensions of existing ones concerning the dimensions of the boundary of a self-similar tile.

KW - Boundary

KW - Dimensions

KW - Graph-directed Self-similar Set

KW - Overlaps

UR - https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/402

UR - https://www.math.uh.edu/~hjm/Vol42-1.html

M3 - Article

SN - 0362-1588

VL - 42

JO - Houston Journal of Mathematics

JF - Houston Journal of Mathematics

ER -

Dimensions of the Boundary of a Graph-Directed Self-Similar Set With Overlaps

Abstract

Keywords

DC Disciplines

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