×

zbMATH — the first resource for mathematics

Unimodular Pisot substitutions and their associated tiles. (English) Zbl 1161.37016
Let \(\sigma\) be a unimodular Pisot substitution over a \(d\)-letter alphabet and let \(X_1,\dots,X_d\) be the associated Rauzy fractals. In this paper the author investigates the boundaries \(\partial X_i\) (\(1\leq i\leq d\)) of these fractals. To this matter, he defines the contact graph \(\mathcal{C}\) of \(\sigma\). If \(\sigma\) satisfies the super coincidence condition [see S. Ito and H. Rao, Isr. J. Math. 153, 129–155 (2006; Zbl 1143.37013)] – also sometimes called the geometric coincidence condition [see M. M. Barge and B. Diamond, Bull. Soc. Math. Fr. 130, 619–626 (2002; Zbl 1028.37008)] – the contact graph can be used to set up a self-affine graph directed system whose attractors are certain pieces of the boundaries \(\partial X_i\) (\(1\leq i\leq d\)).
From this graph directed system the author derives a formula for the fractal dimension of \(\partial X_i\) in which eigenvalues of the adjacency matrix of \(\mathcal{C}\) occurs. The main advantage of the contact graph is its relatively easy shape. The author calculates the contact graph for the substitutions \(\sigma(1)=1^b 2\), \(\sigma(2)=1^a 3\) and \(\sigma(3)=1\) with \({b\geq a\geq 1}\). It turns out that it has roughly the same shape for each substitution from this class. The knowledge of the contact graphs of these substitutions enables us to establish an explicit formula for the Hausdorff dimension of the boundary of the associated Rauzy fractals.

MSC:
37B10 Symbolic dynamics
11B85 Automata sequences
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
28A80 Fractals
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] Akiyama, S.Self affine tilings and Pisot numeration systems. In Number Theory and its Applications (1999), K. Győry and S. Kanemitsu, Eds., Kluwer, pp. 1-17. · Zbl 0999.11065
[2] Akiyama, S.On the boundary of self affine tilings generated by Pisot numbers. J. Math. Soc. Japan 54, 2 (2002), 283-308. · Zbl 1032.11033
[3] Akiyama, S., Sadahiro, T.A self-similar tiling generated by the minimal Pisot number. Acta Math. Info. Univ. Ostraviensis 6 (1998), 9-26. · Zbl 1024.11066
[4] Arnoux, P., Berthé, V., Ei, H., Ito, S.Tilings, quasicrystals, discrete planes, generalized substitutions, and multidimensional continued fractions. In Discrete models: combinatorics, computation, and geometry (Paris, 2001), Discrete Math. Theor. Comput. Sci. Proc., AA. Maison Inform. Math. Discrèt. (MIMD), Paris, 2001, pp. 059-078 (electronic). · Zbl 1017.68147
[5] Arnoux, P., Berthé, V. Siegel, A.Two-dimensional iterated morphisms and discrete planes. Theoret. Comput. Sci. 319, 1-3 (2004), 145-176. · Zbl 1068.37004
[6] Arnoux, P. Ito, S.Pisot substitutions and Rauzy fractals. Bull. Belg. Math. Soc. Simon Stevin 8, 2 (2001), 181-207. Journées Montoises d’Informatique Théorique (Marne-la-Vallée, 2000). · Zbl 1007.37001
[7] Barge, M. Diamond, B.Coincidence for substitutions of Pisot type. Bull. Soc. Math. France 130, 4 (2002), 619-626. · Zbl 1028.37008
[8] Barge, M. Kwapisz, J.Geometric theory of unimodular Pisot substitutions. Preprint. · Zbl 1152.37011
[9] Berthé, V. Siegel, A.Purely periodic beta-expansions in the Pisot non-unit case. Preprint.
[10] Berthé, V. Siegel, A.Tilings associated with beta-numeration and substitutions. Integers 5 (2005), no. 3, A2, (electronic). · Zbl 1139.37008
[11] Canterini, V. Siegel, A.Automate des préfixes-suffixes associé à une substitution primitive. J. Théor. Nombres Bordeaux 13, 2 (2001), 353-369. · Zbl 1071.37011
[12] Canterini, V. Siegel, A.Geometric representation of substitutions of Pisot type. Trans. Amer. Math. Soc. 353, 12 (2001), 5121-5144 (electronic). · Zbl 1142.37302
[13] Deliu, A., Geronimo, J., Shonkwiler, R. Hardin, D.Dimension associated with recurrent self similar sets. Math. Proc. Camb. Philos. Soc. 110 (1991), 327-336. · Zbl 0742.28002
[14] Dumont, J. M. Thomas, A.Digital sum moments and substitutions. Acta Arith. 64 (1993), 205-225. · Zbl 0774.11041
[15] Falconer, K. J.The Hausdorff dimension of self-affine fractals. Math. Proc. Camb. Phil. Soc. 103 (1988), 339-350. · Zbl 0642.28005
[16] Falconer, K. J.Fractal Geometry. John Wiley and Sons, Chichester, 1990. · Zbl 0689.28003
[17] Falconer, K. J.The dimension of self-affine fractals II. Math. Proc. Camb. Phil. Soc. 111 (1992), 169-179. · Zbl 0797.28004
[18] Falconer, K. J.Techniques in Fractal Geometry. John Wiley and Sons, Chichester, New York, Weinheim, Brisbane, Singapore, Toronto, 1997. · Zbl 0869.28003
[19] Feng, D.-J., Furukado, M., Ito, S. Wu, J.Pisot substitutions and the Hausdorff dimension of boundaries of atomic surfaces. Tsukuba J. Math. 30 (2006), 195-224. · Zbl 1130.37318
[20] Fogg, N. P.Substitutions in dynamics, arithmetics and combinatorics. Vol. 1794 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2002. Edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel. · Zbl 1014.11015
[21] Frougny, C. Solomyak, B.Finite beta-expansions. Ergodic Theory Dynam. Systems 12, 4 (1992), 713-723. · Zbl 0814.68065
[22] Gröchenig, K. Haas, A.Self-similar lattice tilings. J. Fourier Anal. Appl. 1 (1994), 131-170. · Zbl 0978.28500
[23] Holton, C. Zamboni, L. Q.Geometric realizations of substitutions. Bull. Soc. Math. France 126, 2 (1998), 149-179. · Zbl 0931.11004
[24] Ito, S. Kimura, M.On Rauzy fractal. Japan J. Indust. Appl. Math. 8, 3 (1991), 461-486. · Zbl 0734.28010
[25] Ito, S. Rao, H.Atomic surfaces, Tilings and coincidence I. Irreducible case. Israel J. Math., to appear. · Zbl 1143.37013
[26] Ito, S. Sano, Y.Substitutions, atomic surfaces, and periodic beta expansions. In Analytic number theory (Beijing/Kyoto, 1999), vol. 6 of Dev. Math. Kluwer Acad. Publ., Dordrecht, 2002, pp. 183-194. · Zbl 1022.11002
[27] Lind, D. Marcus, B.An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge, 1995. · Zbl 1106.37301
[28] Mauldin, R. D. Williams, S. C.Hausdorff dimension in graph directed constructions. Trans. Amer. Math. Soc. 309 (1988), 811-829. · Zbl 0706.28007
[29] Messaoudi, A.Propriétés arithmétiques et dynamiques du fractal de Rauzy. J. Théor. Nombres Bordeaux 10, 1 (1998), 135-162. · Zbl 0918.11048
[30] Messaoudi, A.Frontière du fractal de Rauzy et système de numération complexe. Acta Arith. 95, 3 (2000), 195-224. · Zbl 0968.28005
[31] Mossé, B.Recognizability of substitutions and complexity of automatic sequences. Bull. Soc. Math. Fr. 124, 2 (1996), 329-346. · Zbl 0855.68072
[32] Parry, W.On the \(β \)-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401-416. · Zbl 0099.28103
[33] Queffélec, M.Substitution dynamical systems—spectral analysis. Vol. 1294 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1987. · Zbl 0642.28013
[34] Rauzy, G.Nombres algébriques et substitutions. Bull. Soc. Math. France 110, 2 (1982), 147-178. · Zbl 0522.10032
[35] Sano, Y., Arnoux, P. Ito, S.Higher dimensional extensions of substitutions and their dual maps. J. Anal. Math. 83 (2001), 183-206. · Zbl 0987.11013
[36] Scheicher, K. Thuswaldner, J. M.Canonical number systems, counting automata and fractals. Math. Proc. Cambridge Philos. Soc. 133, 1 (2002), 163-182. · Zbl 1001.68070
[37] Scheicher, K. Thuswaldner, J. M.Neighbours of self-affine tiles in lattice tilings. In Proceedings of the Conference “Fractals in Graz” (2002), P. Grabner and W. Woess, Eds., pp. 241-262. · Zbl 1040.52013
[38] Siegel, A.Représentation des systèmes dynamiques substitutifs non unimodulaires. Ergodic Theory Dynam. Systems 23, 4 (2003), 1247-1273. · Zbl 1052.37009
[39] Siegel, A.Pure discrete spectrum dynamical system and periodic tiling associated with a substitution. Ann. Inst. Fourier (Grenoble) 54, 2 (2004), 341-381. · Zbl 1083.37009
[40] Sirvent, V. F. Wang, Y.Self-affine tiling via substitution dynamical systems and Rauzy fractals. Pacific J. Math. 206, 2 (2002), 465-485. · Zbl 1048.37015
[41] Solomyak, B.Substitutions, adic transformations, and beta-expansions. In Symbolic dynamics and its applications (New Haven, CT, 1991). Vol. 135 of Contemp. Math. Amer. Math. Soc., Providence, RI, 1992, pp. 361-372. · Zbl 0771.28013
[42] Thurston, W.Groups, tilings and finite state automata. AMS Colloquium Lecture Notes, 1989.
[43] Vince, A.Digit tiling of euclidean space. In Directions in Mathematical Quasicrystals (Providence, RI, 2000), Amer. Math. Soc., pp. 329-370. · Zbl 0972.52012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.