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Topological properties of a class of self-affine tiles in $ \mathbb{R}^3$


Authors: Guotai Deng, Chuntai Liu and Sze-Man Ngai
Journal: Trans. Amer. Math. Soc. 370 (2018), 1321-1350
MSC (2010): Primary 28A80, 52C22; Secondary 05B45, 51M20
DOI: https://doi.org/10.1090/tran/7055
Published electronically: September 25, 2017
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Abstract: We construct a class of connected self-affine tiles in $ \mathbb{R}^3$ and prove that it contains a subclass of tiles that are homeomorphic to a unit ball in $ \mathbb{R}^3$. Our construction is obtained by generalizing a two-dimensional one by Deng and Lau. The proof of ball-likeness is inspired by the construction of a homeomorphism from Alexander's horned ball to a 3-ball.


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Additional Information

Guotai Deng
Affiliation: School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, People’s Republic of China
Email: hilltower@163.com

Chuntai Liu
Affiliation: School of Mathematics and Computer Science, Wuhan Polytechnic University, Wuhan 430023, People’s Republic of China
Email: lct984@163.com

Sze-Man Ngai
Affiliation: College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China – and – Department of Mathematical Sciences, Georgia Southern University, Statesboro, Georgia 30460-8093
Email: smngai@georgiasouthern.edu

DOI: https://doi.org/10.1090/tran/7055
Keywords: Self-affine tile, connectedness, ball-like tiles
Received by editor(s): November 8, 2014
Received by editor(s) in revised form: March 4, 2016, and July 19, 2016
Published electronically: September 25, 2017
Additional Notes: The first author was supported by the China Scholarship Council. The second author was supported by the National Natural Science Foundation of China grant 11601403. The third author was supported in part by the National Natural Science Foundation of China grant 11271122, the Hunan Province Hundred Talents Program, and a Faculty Research Scholarly Pursuit Funding from Georgia Southern University.
Article copyright: © Copyright 2017 American Mathematical Society

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