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Topological Properties of a Class of Higher-dimensional Self-affine Tiles

Part of: Real and complex geometry Designs and configurations Classical measure theory Discrete geometry

Published online by Cambridge University Press:  03 May 2019

Guotai Deng ,
Chuntai Liu  and
Sze-Man Ngai
Guotai Deng
Affiliation:
School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, P. R. China Email: [email protected]
Chuntai Liu
Affiliation:
School of Mathematics and Computer Science, Wuhan Polytechnic University, Wuhan 430023, P. R. China Email: [email protected]
Sze-Man Ngai
Affiliation:
Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP) (Ministry of Education of China), College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460-8093, USA Email: [email protected]
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Abstract

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We construct a family of self-affine tiles in $\mathbb{R}^{d}$ ($d\geqslant 2$) with noncollinear digit sets, which naturally generalizes a class studied originally by Q.-R. Deng and K.-S. Lau in $\mathbb{R}^{2}$, and its extension to $\mathbb{R}^{3}$ by the authors. We obtain necessary and sufficient conditions for the tiles to be connected and for their interiors to be contractible.

Keywords

self-affine tileconnectednessball-like tile

MSC classification

Primary: 28A80: Fractals
Secondary: 52C22: Tilings in $n$ dimensions 05B45: Tessellation and tiling problems 51M20: Polyhedra and polytopes; regular figures, division of spaces
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 4 , December 2019 , pp. 727 - 740
Copyright
© Canadian Mathematical Society 2019 

Footnotes

Author G. T. D. was supported by the Fundamental Research Funds for the Central Universities CCNU19TS071. Author C. T. L. was supported in part by the National Natural Science Foundation of China grant 11601403, China Scholarship Council and Research and Innovation Initiatives of WHPU 2018Y18. Author S. M. N. was supported in part by the National Natural Science Foundation of China grants 11771136 and 11271122, the Hunan Province Hundred Talents Program, Construct Program of the Key Discipline in Hunan Province, and a Faculty Research Scholarly Pursuit Funding from Georgia Southern University.

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