Skip to main content Accessibility help
×
Home

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

  • Guotai Deng (a1), Chuntai Liu (a2) and Sze-Man Ngai (a3) (a4)

Abstract

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.

Copyright

Footnotes

Hide All

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.

Footnotes

References

Hide All
[1] Akiyama, S. and Gjini, N., On the connectedness of self-affine attractors . Arch. Math. (Basel) 82(2004), 153163. https://doi.org/10.1007/s00013-003-4820-z
[2] Bandt, C. and Wang, Y., Disk-like self-affine tiles in ℝ2 . Discrete Comput. Geom. 26(2001), 591601. https://doi.org/10.1007/s00454-001-0034-y
[3] Conner, G. R. and Thuswaldner, J. M., Self-affine manifolds . Adv. Math. 289(2016), 725783. https://doi.org/10.1016/j.aim.2015.11.022
[4] Deng, G., Liu, C., and Ngai, S.-M., Topological properties of a class of self-affine tiles in ℝ3 . Trans. Amer. Math. Soc. 370(2018), 13211350. https://doi.org/10.1090/tran/7055
[5] Deng, Q.-R. and Lau, K.-S., Connectedness of a class planar self-affine tiles . J. Math. Anal. Appl. 380(2011), 492500. https://doi.org/10.1016/j.jmaa.2011.03.043
[6] Hata, M., On the structure of self-similar sets . Japan J. Appl. Math. 2(1985), 381414. https://doi.org/10.1007/BF03167083
[7] He, X.-G., Kirat, I., and Lau, K.-S., Height reducing property of polynomials and self-affine tiles . Geom. Dedicata 152(2011), 153164. https://doi.org/10.1007/s10711-010-9550-3
[8] Hutchinson, J. E., Fractals and self-similarity . Indiana Univ. Math. J. 30(1981), 713747. https://doi.org/10.1512/iumj.1981.30.30055
[9] Kamae, T., Luo, J., and Tan, B., A gluing lemma for iterated function systems . Fractals 23(2015), 1550019, 10 pp. https://doi.org/10.1142/S0218348X1550019X
[10] Kirat, I. and Lau, K.-S., On the connectedness of self-affine tiles . J. London Math. Soc. (2) 62(2000), 291304. https://doi.org/10.1112/S002461070000106X
[11] Kirat, I., Lau, K.-S., and Rao, H., Expanding polynomials and connectedness of self-affine tiles . Discrete Comput. Geom. 31(2004), 275286. https://doi.org/10.1007/s00454-003-2879-8
[12] Lagarias, J. C. and Wang, Y., Self-affine tiles in ℝ n . Adv. Math. 121(1996), 2149. https://doi.org/10.1006/aima.1996.0045
[13] Leung, K.-S. and Lau, K.-S., Disklikeness of planar self-affine tiles . Trans. Amer. Math. Soc. 359(2007), 33373355. https://doi.org/10.1090/S0002-9947-07-04106-2
[14] Leung, K. S. and Luo, J. J., Connectedness of planar self-affine sets associated with non-consecutive collinear digit sets . J. Math. Anal. Appl. 395(2012), 208217. https://doi.org/10.1016/j.jmaa.2012.05.034
[15] Liu, J., Ngai, S.-M., and Tao, J., Connectedness of a class of two-dimensional self-affine tiles associated with triangular matrices . J. Math. Anal. Appl. 435(2016), 14991513. https://doi.org/10.1016/j.jmaa.2015.10.081
[16] Luo, J., Rao, H., and Tan, B., Topological structure of self-similar sets . Fractals 10(2002), 223227. https://doi.org/10.1142/S0218348X0200104X
[17] Ma, Y., Dong, X.-H., and Deng, Q.-R., The connectedness of some two-dimensional self-affine sets . J. Math. Anal. Appl. 420(2014), 16041616. https://doi.org/10.1016/j.jmaa.2014.06.054
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

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

  • Guotai Deng (a1), Chuntai Liu (a2) and Sze-Man Ngai (a3) (a4)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed