Connectedness of a class of two-dimensional self-affine tiles associated with triangular matrices

Jingcheng Liu; Sze Man Ngai; Juan Tao

doi:10.1016/j.jmaa.2015.10.081

Connectedness of a class of two-dimensional self-affine tiles associated with triangular matrices

Jingcheng Liu, Sze Man Ngai, Juan Tao

Mathematical Sciences

Hunan Normal University

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

6 Scopus citations

Abstract

We study the connectedness of planar self-affine sets T(A,D) generated by a matrix of the form A = [p0-aq] together with nonconsecutive and noncollinear digit sets of the form D = l₀,l₁, . . . ,l_|p|-1 × m₀,m₁,. . .,m_|q|-1, where l₀, l₁, . . ., l_|p|-1 and m₀, m₁, . . ., m_|q|-1 are residue systems for |p| and |q| respectively. We give a necessary and sufficient condition for T(A,D) to be connected, and extend some results by Deng and Lau (2011) [5] to nonconsecutive digit sets.

Original language	English
Pages (from-to)	1499-1513
Number of pages	15
Journal	Journal of Mathematical Analysis and Applications
Volume	435
Issue number	2
DOIs	https://doi.org/10.1016/j.jmaa.2015.10.081
State	Published - 2016

Scopus Subject Areas

Analysis
Applied Mathematics

Keywords

Connectedness
Digit set
Self-affine tile

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title = "Connectedness of a class of two-dimensional self-affine tiles associated with triangular matrices",

abstract = "We study the connectedness of planar self-affine sets T(A,D) generated by a matrix of the form A = [p0-aq] together with nonconsecutive and noncollinear digit sets of the form D = l0,l1, . . . ,l|p|-1 × m0,m1,. . .,m|q|-1, where l0, l1, . . ., l|p|-1 and m0, m1, . . ., m|q|-1 are residue systems for |p| and |q| respectively. We give a necessary and sufficient condition for T(A,D) to be connected, and extend some results by Deng and Lau (2011) [5] to nonconsecutive digit sets.",

keywords = "Connectedness, Digit set, Self-affine tile",

author = "Jingcheng Liu and Ngai, \{Sze Man\} and Juan Tao",

note = "Publisher Copyright: {\textcopyright} 2015 Elsevier Inc.",

year = "2016",

doi = "10.1016/j.jmaa.2015.10.081",

language = "English",

volume = "435",

pages = "1499--1513",

journal = "Journal of Mathematical Analysis and Applications",

issn = "0022-247X",

publisher = "Academic Press Inc.",

number = "2",

}

TY - JOUR

T1 - Connectedness of a class of two-dimensional self-affine tiles associated with triangular matrices

AU - Liu, Jingcheng

AU - Ngai, Sze Man

AU - Tao, Juan

PY - 2016

Y1 - 2016

N2 - We study the connectedness of planar self-affine sets T(A,D) generated by a matrix of the form A = [p0-aq] together with nonconsecutive and noncollinear digit sets of the form D = l0,l1, . . . ,l|p|-1 × m0,m1,. . .,m|q|-1, where l0, l1, . . ., l|p|-1 and m0, m1, . . ., m|q|-1 are residue systems for |p| and |q| respectively. We give a necessary and sufficient condition for T(A,D) to be connected, and extend some results by Deng and Lau (2011) [5] to nonconsecutive digit sets.

AB - We study the connectedness of planar self-affine sets T(A,D) generated by a matrix of the form A = [p0-aq] together with nonconsecutive and noncollinear digit sets of the form D = l0,l1, . . . ,l|p|-1 × m0,m1,. . .,m|q|-1, where l0, l1, . . ., l|p|-1 and m0, m1, . . ., m|q|-1 are residue systems for |p| and |q| respectively. We give a necessary and sufficient condition for T(A,D) to be connected, and extend some results by Deng and Lau (2011) [5] to nonconsecutive digit sets.

KW - Connectedness

KW - Digit set

KW - Self-affine tile

U2 - 10.1016/j.jmaa.2015.10.081

DO - 10.1016/j.jmaa.2015.10.081

M3 - Article

SN - 0022-247X

VL - 435

SP - 1499

EP - 1513

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

ER -

Connectedness of a class of two-dimensional self-affine tiles associated with triangular matrices

Abstract

Scopus Subject Areas

Keywords

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