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Self-Affine Tiles Generated by a Finite Number of Matrices

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Abstract

We study self-affine tiles generated by iterated function systems consisting of affine mappings whose linear parts are defined by different matrices. We obtain an interior theorem for these tiles. We prove a tiling theorem by showing that for such a self-affine tile, there always exists a tiling set. We also obtain a more complete interior theorem for reptiles, which are tiles obtained when the matrices in the iterated function system are similarities. Our results extend some of the classical ones by Lagarias and Wang (Adv. Math. 121(1), 21–49 (1996)), where the IFS maps are defined by a single matrix.

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Acknowledgements

Part of this work was carried out when the first two authors were visiting the Department of Mathematical Sciences of Georgia Southern University. They thank the Department for its hospitality and support. The authors also thank the anonymous reviewers for some helpful comments and suggestions, especially the use of the joint spectral radius as a basic assumption in our main results.

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Authors and Affiliations

  1. School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, People’s Republic of China

    Guotai Deng

  2. School of Mathematics and Computer Science, Wuhan Polytechnic University, Wuhan, 430023, People’s Republic of China

    Chuntai Liu

  3. 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, 410081, Hunan, People’s Republic of China

    Sze-Man Ngai

  4. Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, 30460-8093, USA

    Sze-Man Ngai

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  1. Guotai Deng
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  2. Chuntai Liu
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  3. Sze-Man Ngai
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Correspondence to Sze-Man Ngai.

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The first author is supported by the Fundamental Research Funds for the Central Universities CCNU19TS071, the National Natural Science Foundation of China Grant (12071171) and Hubei Provincial Natural Science Foundation of China (2020CFB833). The second author is supported in part by the National Natural Science Foundation of China Grant (11601403), China Scholarship Council and Hubei Provincial Natural Science Foundation of China (2019CFB602). The third author is supported in part by the National Natural Science Foundation of China Grants 12271156 and 11771136, the Hunan Province’ Hundred Talents Program, Construct Program of the Key Discipline in Hunan Province, and a Faculty Research Scholarly Pursuit Award from Georgia Southern University.

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Deng, G., Liu, C. & Ngai, SM. Self-Affine Tiles Generated by a Finite Number of Matrices. Discrete Comput Geom 70, 620–644 (2023). https://doi.org/10.1007/s00454-023-00529-6

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  • DOI: https://doi.org/10.1007/s00454-023-00529-6

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