Geodesic
Topological Properties of a Class of Higher-dimensional Self-affine Tiles
Canadian mathematical bulletin, Tome 62 (2019) no. 4, pp. 727-740

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/S0008439519000237
Mots-clés : self-affine tile, connectedness, ball-like tile 
Deng, Guotai; Liu, Chuntai; Ngai, Sze-Man. Topological Properties of a Class of Higher-dimensional Self-affine Tiles. Canadian mathematical bulletin, Tome 62 (2019) no. 4, pp. 727-740. doi: 10.4153/S0008439519000237
@article{10_4153_S0008439519000237,
     author = {Deng, Guotai and Liu, Chuntai and Ngai, Sze-Man},
     title = {Topological {Properties} of a {Class} of {Higher-dimensional} {Self-affine} {Tiles}},
     journal = {Canadian mathematical bulletin},
     pages = {727--740},
     year = {2019},
     volume = {62},
     number = {4},
     doi = {10.4153/S0008439519000237},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439519000237/}
}
TY  - JOUR
AU  - Deng, Guotai
AU  - Liu, Chuntai
AU  - Ngai, Sze-Man
TI  - Topological Properties of a Class of Higher-dimensional Self-affine Tiles
JO  - Canadian mathematical bulletin
PY  - 2019
SP  - 727
EP  - 740
VL  - 62
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/S0008439519000237/
DO  - 10.4153/S0008439519000237
ID  - 10_4153_S0008439519000237
ER  - 
%0 Journal Article
%A Deng, Guotai
%A Liu, Chuntai
%A Ngai, Sze-Man
%T Topological Properties of a Class of Higher-dimensional Self-affine Tiles
%J Canadian mathematical bulletin
%D 2019
%P 727-740
%V 62
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/S0008439519000237/
%R 10.4153/S0008439519000237
%F 10_4153_S0008439519000237

[1] Akiyama, S. and Gjini, N., On the connectedness of self-affine attractors . Arch. Math. (Basel) 82(2004), 153–163. Google Scholar | DOI

[2] Bandt, C. and Wang, Y., Disk-like self-affine tiles in ℝ2 . Discrete Comput. Geom. 26(2001), 591–601. Google Scholar | DOI

[3] Conner, G. R. and Thuswaldner, J. M., Self-affine manifolds . Adv. Math. 289(2016), 725–783. Google Scholar | DOI

[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), 1321–1350. Google Scholar | DOI

[5] Deng, Q.-R. and Lau, K.-S., Connectedness of a class planar self-affine tiles . J. Math. Anal. Appl. 380(2011), 492–500. Google Scholar | DOI

[6] Hata, M., On the structure of self-similar sets . Japan J. Appl. Math. 2(1985), 381–414. Google Scholar | DOI

[7] He, X.-G., Kirat, I., and Lau, K.-S., Height reducing property of polynomials and self-affine tiles . Geom. Dedicata 152(2011), 153–164. Google Scholar | DOI

[8] Hutchinson, J. E., Fractals and self-similarity . Indiana Univ. Math. J. 30(1981), 713–747. Google Scholar | DOI

[9] Kamae, T., Luo, J., and Tan, B., A gluing lemma for iterated function systems . Fractals 23(2015), 1550019, 10 pp. Google Scholar | DOI

[10] Kirat, I. and Lau, K.-S., On the connectedness of self-affine tiles . J. London Math. Soc. (2) 62(2000), 291–304. Google Scholar | DOI

[11] Kirat, I., Lau, K.-S., and Rao, H., Expanding polynomials and connectedness of self-affine tiles . Discrete Comput. Geom. 31(2004), 275–286. Google Scholar | DOI

[12] Lagarias, J. C. and Wang, Y., Self-affine tiles in ℝ n . Adv. Math. 121(1996), 21–49. Google Scholar | DOI

[13] Leung, K.-S. and Lau, K.-S., Disklikeness of planar self-affine tiles . Trans. Amer. Math. Soc. 359(2007), 3337–3355. Google Scholar | DOI

[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), 208–217. Google Scholar | DOI

[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), 1499–1513. Google Scholar | DOI

[16] Luo, J., Rao, H., and Tan, B., Topological structure of self-similar sets . Fractals 10(2002), 223–227. Google Scholar | DOI

[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), 1604–1616. Google Scholar | DOI

Cité par Sources :