Topological properties of a class of self-affine tiles in $\mathbb {R}^3$
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- by Guotai Deng, Chuntai Liu and Sze-Man Ngai PDF
- Trans. Amer. Math. Soc. 370 (2018), 1321-1350 Request permission
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.References
- Shigeki Akiyama and Nertila Gjini, On the connectedness of self-affine attractors, Arch. Math. (Basel) 82 (2004), no. 2, 153–163. MR 2047669, DOI 10.1007/s00013-003-4820-z
- Shigeki Akiyama and Jörg M. Thuswaldner, A survey on topological properties of tiles related to number systems, Geom. Dedicata 109 (2004), 89–105. MR 2113188, DOI 10.1007/s10711-004-1774-7
- Christoph Bandt, Self-similar sets. V. Integer matrices and fractal tilings of $\textbf {R}^n$, Proc. Amer. Math. Soc. 112 (1991), no. 2, 549–562. MR 1036982, DOI 10.1090/S0002-9939-1991-1036982-1
- C. Bandt, Combinatorial topology of three-dimensional self-affine tiles, http://arxiv.org /abs/1002.0710.
- C. Bandt and Y. Wang, Disk-like self-affine tiles in $\Bbb R^2$, Discrete Comput. Geom. 26 (2001), no. 4, 591–601. MR 1863811, DOI 10.1007/s00454-001-0034-y
- R. H. Bing, The geometric topology of 3-manifolds, American Mathematical Society Colloquium Publications, vol. 40, American Mathematical Society, Providence, RI, 1983. MR 728227, DOI 10.1090/coll/040
- Gregory R. Conner and Jörg M. Thuswaldner, Self-affine manifolds, Adv. Math. 289 (2016), 725–783. MR 3439698, DOI 10.1016/j.aim.2015.11.022
- Qi-Rong Deng and Ka-sing Lau, Connectedness of a class of planar self-affine tiles, J. Math. Anal. Appl. 380 (2011), no. 2, 493–500. MR 2794408, DOI 10.1016/j.jmaa.2011.03.043
- Götz Gelbrich, Self-affine lattice reptiles with two pieces in $\textbf {R}^n$, Math. Nachr. 178 (1996), 129–134. MR 1380707, DOI 10.1002/mana.19961780107
- Masayoshi Hata, On the structure of self-similar sets, Japan J. Appl. Math. 2 (1985), no. 2, 381–414. MR 839336, DOI 10.1007/BF03167083
- John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
- Teturo Kamae, Jun Luo, and Bo Tan, A gluing lemma for iterated function systems, Fractals 23 (2015), no. 2, 1550019, 10. MR 3351926, DOI 10.1142/S0218348X1550019X
- Richard Kenyon, Self-replicating tilings, Symbolic dynamics and its applications (New Haven, CT, 1991) Contemp. Math., vol. 135, Amer. Math. Soc., Providence, RI, 1992, pp. 239–263. MR 1185093, DOI 10.1090/conm/135/1185093
- Ibrahim Kirat and Ka-Sing Lau, On the connectedness of self-affine tiles, J. London Math. Soc. (2) 62 (2000), no. 1, 291–304. MR 1772188, DOI 10.1112/S002461070000106X
- Ibrahim Kirat, Ka-Sing Lau, and Hui Rao, Expanding polynomials and connectedness of self-affine tiles, Discrete Comput. Geom. 31 (2004), no. 2, 275–286. MR 2060641, DOI 10.1007/s00454-003-2879-8
- Jeffrey C. Lagarias and Yang Wang, Self-affine tiles in $\textbf {R}^n$, Adv. Math. 121 (1996), no. 1, 21–49. MR 1399601, DOI 10.1006/aima.1996.0045
- King-Shun Leung and Ka-Sing Lau, Disklikeness of planar self-affine tiles, Trans. Amer. Math. Soc. 359 (2007), no. 7, 3337–3355. MR 2299458, DOI 10.1090/S0002-9947-07-04106-2
- King-Shun Leung and Jun Jason Luo, Connectedness of planar self-affine sets associated with non-consecutive collinear digit sets, J. Math. Anal. Appl. 395 (2012), no. 1, 208–217. MR 2943615, DOI 10.1016/j.jmaa.2012.05.034
- Jun Luo, Hui Rao, and Bo Tan, Topological structure of self-similar sets, Fractals 10 (2002), no. 2, 223–227. MR 1910665, DOI 10.1142/S0218348X0200104X
- Sze-Man Ngai and Tai-Man Tang, Topology of connected self-similar tiles in the plane with disconnected interiors, Topology Appl. 150 (2005), no. 1-3, 139–155. MR 2133675, DOI 10.1016/j.topol.2004.11.009
- W. P. Thurston, Groups, tilings, and finite state automata, AMS colloquium lecture notes, 1989.
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
- MR Author ID: 818471
- 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
- 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.
- © Copyright 2017 American Mathematical Society
- 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
- MathSciNet review: 3729502