Topological properties of Rauzy fractals

Topological properties of Rauzy fractals
[Propriétés topologiques des fractals de Rauzy]

Siegel, Anne ; Thuswaldner, Jörg M.

Mémoires de la Société Mathématique de France, no. 118 (2009) , 140 p.

Résumé
Abstract

Les fractals de Rauzy apparaissent dans diverses branches des mathématiques telles que la théorie des nombres, les systèmes dynamiques, la combinatoire et la théorie des quasi-cristaux. De nombreuses questions font alors intervenir la structure topologique des fractals. Cette monographie propose une étude systématique des propriétés topologiques des fractals de Rauzy. Les premiers chapitres de ce document rappellent les enjeux mathématiques relatifs aux fractals de Rauzy ainsi que les principaux résultats connus à leur sujet. Sont ensuite discutés des propriétés de pavages, de connexité, d’homéomorphisme à un disque, ainsi que le groupe fondamental de ces ensembles. Les méthodes s’appuient sur des résultats en topologie du plan et sur la construction de graphes pour décrire la structure des pavages associés aux fractals. De nombreux exemples caractéristiques sont présentés. Un chapitre final discute des principales perspectives de recherches liées à cette thématique.

Substitutions are combinatorial objects (one replaces a letter by a word) which produce sequences by iteration. They occur in many mathematical fields, roughly as soon as a repetitive process appears. In the present monograph we deal with topological and geometric properties of substitutions, in particular, we study properties of the Rauzy fractals associated to substitutions. To be more precise, let $σ$ be a substitution over the finite alphabet $𝒜$ . We assume that the incidence matrix of $σ$ is primitive and that its dominant eigenvalue is a unit Pisot number (i.e., an algebraic integer greater than one whose norm is equal to one and all of whose Galois conjugates are of modulus strictly smaller than one). It is well-known that one can attach to $σ$ a set $𝒯$ which is called central tile or Rauzy fractal of $σ$ . Such a central tile is a compact set that is the closure of its interior and decomposes in a natural way in $n = | 𝒜 |$ subtiles $𝒯 (1), ..., 𝒯 (n)$ . The central tile as well as its subtiles are graph directed self-affine sets that often have fractal boundary. Pisot substitutions and central tiles are of high relevance in several branches of mathematics like tiling theory, spectral theory, Diophantine approximation, the construction of discrete planes and quasicrystals as well as in connection with numeration like generalized continued fractions and radix representations. The questions coming up in all these domains can often be reformulated in terms of questions related to the topology and the geometry of the underlying central tile. After a thorough survey of important properties of unit Pisot substitutions and their associated Rauzy fractals the present monograph is devoted to the investigation of a variety of topological properties of $𝒯$ and its subtiles. Our approach is an algorithmic one. In particular, we dwell upon the question whether $𝒯$ and its subtiles induce a tiling, calculate the Hausdorff dimension of their boundary, give criteria for their connectivity and homeomorphy to a closed disk and derive properties of their fundamental group. The basic tools for our criteria are several classes of graphs built from the description of the tiles $𝒯 (i)$ ( $1 \leq i \leq n$ ) as the solution of a graph directed iterated function system and from the structure of the tilings induced by these tiles. These graphs are of interest in their own right. For instance, they can be used to construct the boundaries $\partial 𝒯$ as well as $\partial 𝒯 (i)$ ( $1 \leq i \leq n$ ) and all points where two, three or four different tiles of the induced tilings meet. When working with central tiles in one of the above mentioned contexts it is often useful to know such intersection properties of tiles. In this sense the present monograph also aims at providing tools for “everyday’s life” when dealing with topological and geometric properties of substitutions. Many examples are given throughout the text in order to illustrate our results. Moreover, we give perspectives for further directions of research related to the topics discussed in this monograph.

MR Zbl | 2 citations dans Numdam

DOI : 10.24033/msmf.430

Classification : 28A80, 11A63, 54F65
Keywords: Rauzy fractal, tiling, beta-numeration, connectivity, homeomorphy to a disk, fundamental group
Mot clés : Fractal de Rauzy, Pavage, beta-numération, connexité, homéomorphisme à un disque, groupe fondamental

@book{MSMF_2009_2_118__1_0,
     author = {Siegel, Anne and Thuswaldner, J\"org M.},
     title = {Topological properties of {Rauzy} fractals},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {118},
     year = {2009},
     doi = {10.24033/msmf.430},
     mrnumber = {2721985},
     zbl = {1229.28021},
     language = {en},
     url = {http://archive.numdam.org/item/MSMF_2009_2_118__1_0/}
}

TY  - BOOK
AU  - Siegel, Anne
AU  - Thuswaldner, Jörg M.
TI  - Topological properties of Rauzy fractals
T3  - Mémoires de la Société Mathématique de France
PY  - 2009
IS  - 118
PB  - Société mathématique de France
UR  - http://archive.numdam.org/item/MSMF_2009_2_118__1_0/
DO  - 10.24033/msmf.430
LA  - en
ID  - MSMF_2009_2_118__1_0
ER  -

%0 Book
%A Siegel, Anne
%A Thuswaldner, Jörg M.
%T Topological properties of Rauzy fractals
%S Mémoires de la Société Mathématique de France
%D 2009
%N 118
%I Société mathématique de France
%U http://archive.numdam.org/item/MSMF_2009_2_118__1_0/
%R 10.24033/msmf.430
%G en
%F MSMF_2009_2_118__1_0

Siegel, Anne; Thuswaldner, Jörg M. Topological properties of Rauzy fractals. Mémoires de la Société Mathématique de France, Série 2, no. 118 (2009), 140 p. doi : 10.24033/msmf.430. http://numdam.org/item/MSMF_2009_2_118__1_0/

Bibliographie
Cité par

[1] B. Adamczewski & Y. Bugeaud – « On the complexity of algebraic numbers. I. Expansions in integer bases », Ann. of Math. 165 (2007), p. 547–565. | MR | Zbl

[2] B. Adamczewski, Y. Bugeaud & L. Davison – « Continued fractions and transcendental numbers », Ann. Inst. Fourier (Grenoble) 56 (2006), p. 2093–2113. | MR | EuDML | Zbl | Numdam

[3] B. Adamczewski, C. Frougny, A. Siegel & W. Steiner – « Rational numbers with purely periodic beta-expansion », J. London Math. Soc. 42 (2010), p. 538–552. | MR | Zbl

[4] R. L. Adler – « Symbolic dynamics and Markov partitions », Bull. Amer. Math. Soc. (N.S.) 35 (1998), p. 1–56. | Zbl

[5] R. L. Adler & B. Weiss – Similarity of automorphisms of the torus, Memoirs of the American Mathematical Society, No. 98, Amer. Math. Soc., 1970. | MR | Zbl

[6] S. Akiyama – « Pisot numbers and greedy algorithm », in Number theory (Eger, 1996), de Gruyter, 1998, p. 9–21. | MR | Zbl

[7] —, « Self affine tiling and Pisot numeration system », in Number theory and its applications (Kyoto, 1997), Dev. Math., vol. 2, Kluwer Acad. Publ., 1999, p. 7–17. | MR | Zbl

[8] —, « Cubic Pisot units with finite beta expansions », in Algebraic number theory and Diophantine analysis (Graz, 1998), de Gruyter, 2000, p. 11–26. | Zbl

[9] —, « On the boundary of self affine tilings generated by Pisot numbers », J. Math. Soc. Japan 54 (2002), p. 283–308. | MR | Zbl

[10] —, « Pisot number system and its dual tiling », in Physics and Theoretical Computer Science (Cargese, 2006), IOS Press, 2007, p. 133–154.

[11] S. Akiyama, G. Barat, V. Berthé & A. Siegel – « Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions », Monatsh. Math. 155 (2008), p. 377–419. | MR | Zbl

[12] S. Akiyama, T. Borbély, H. Brunotte, A. Pethő & J. M. Thuswaldner – « Generalized radix representations and dynamical systems. I », Acta Math. Hungar. 108 (2005), p. 207–238. | MR | Zbl

[13] S. Akiyama, H. Brunotte, A. Pethő & J. M. Thuswaldner – « Generalized radix representations and dynamical systems. II », Acta Arith. 121 (2006), p. 21–61. | MR | EuDML | Zbl

[14] —, « Generalized radix representations and dynamical systems. III », Osaka J. Math. 45 (2008), p. 347–374. | MR | Zbl

[15] —, « Generalized radix representations and dynamical systems. IV », Indag. Math. (N.S.) 19 (2008), p. 333–348. | MR | Zbl

[16] S. Akiyama, G. Dorfer, J. M. Thuswaldner & R. Winkler – « On the fundamental group of the Sierpiński-gasket », Topology Appl. 156 (2009), p. 1655–1672. | MR | Zbl

[17] S. Akiyama & G. Nertila – « On the connectedness of self-affine tilings », Arch. Math. 82 (2004), p. 153–163. | Zbl

[18] S. Akiyama, H. Rao & W. Steiner – « A certain finiteness property of Pisot number systems », J. Number Theory 107 (2004), p. 135–160. | MR | Zbl

[19] S. Akiyama & K. Scheicher – « Intersecting two-dimensional fractals with lines », Acta Sci. Math. (Szeged) 71 (2005), p. 555–580. | MR | Zbl

[20] C. Allauzen – « Une caractérisation simple des nombres de Sturm », J. Théor. Nombres Bordeaux 10 (1998), p. 237–241. | MR | EuDML | Numdam

[21] J.-P. Allouche & J. O. Shallit – Automatic sequences: Theory and applications, Cambridge Univ. Press, 2002. | MR

[22] J. Anderson & I. Putnam – « Topological invariants for substitution tilings and their associated $C^{*}$ -algebras », Ergodic Theory Dynam. Systems 18 (1998), p. 509–537. | MR | Zbl

[23] P. Arnoux – « Un exemple de semi-conjugaison entre un échange d’intervalles et une translation sur le tore », Bull. Soc. Math. France 116 (1988), p. 489–500. | MR | EuDML | Zbl | Numdam

[24] P. Arnoux, J. Bernat & X. Bressaud – « Geometrical models for substitutions », Experiment. Math. (2010), to appear. | MR | Zbl

[25] P. Arnoux, V. Berthé, H. Ei & S. Ito – « 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, p. 059–078. | MR | Zbl

[26] P. Arnoux, V. Berthé, T. Fernique & D. Jamet – « Functional stepped surfaces, flips, and generalized substitutions », Theoret. Comput. Sci. 380 (2007), p. 251–265. | MR | Zbl

[27] P. Arnoux, V. Berthé, A. Hilion & A. Siegel – « Fractal representation of the attractive lamination of an automorphism of the free group », Ann. Inst. Fourier (Grenoble) 56 (2006), p. 2161–2212. | MR | EuDML | Zbl | Numdam

[28] P. Arnoux, V. Berthé & S. Ito – « Discrete planes, $ℤ^{2}$ -actions, Jacobi-Perron algorithm and substitutions », Ann. Inst. Fourier (Grenoble) 52 (2002), p. 305–349. | MR | EuDML | Zbl | Numdam

[29] P. Arnoux, M. Furukado, E. Harriss & S. Ito – « Algebraic numbers, group automorphisms and substitution rules on the plane », Trans. Amer. Math. Soc. (2010), in press.

[30] P. Arnoux & S. Ito – « Pisot substitutions and Rauzy fractals », Bull. Belg. Math. Soc. Simon Stevin 8 (2001), p. 181–207. | MR | Zbl

[31] V. Baker, M. Barge & J. Kwapisz – « Geometric realization and coincidence for reducible non-unimodular pisot tiling spaces with an application to beta-shifts », Ann. Inst. Fourier 56 (2006), p. 2213–2248. | MR | EuDML | Zbl | Numdam

[32] C. Bandt & G. Gelbrich – « Classification of self-affine lattice tilings », J. London Math. Soc. 50 (1994), p. 581–593. | MR | Zbl

[33] G. Barat, V. Berthé, , P. Liardet & J. M. Thuswaldner – « Dynamical directions in numeration », Ann. Inst. Fourier (Grenoble) 56 (2006), p. 1987–2092. | MR | EuDML | Numdam

[34] M. Barge & B. Diamond – « Coincidence for substitutions of Pisot type », Bull. Soc. Math. France 130 (2002), p. 619–626. | MR | EuDML | Zbl | Numdam

[35] M. Barge, B. Diamond & R. Swanson – « The branch locus for one-dimensional Pisot tiling spaces », Fund. Math. 204 (2009), p. 215–240. | MR | EuDML | Zbl

[36] M. Barge & J. Kwapisz – « Geometric theory of unimodular Pisot substitutions », Amer. J. Math. 128 (2006), p. 1219–1282. | MR | Zbl

[37] F. Bassino – « Beta-expansions for cubic Pisot numbers », in LATIN 2002: Theoretical informatics (Cancun), Lecture Notes in Comput. Sci., vol. 2286, Springer, 2002, p. 141–152. | MR | Zbl

[38] L. E. Baum & M. M. Sweet – « Continued fractions of algebraic power series in characteristic $2$ », Ann. of Math. 103 (1976), p. 593–610. | MR | Zbl

[39] M.-P. Béal & D. Perrin – « Symbolic dynamics and finite automata », in Handbook of Formal Languages (G. Rozenberg & A. Salomaa, éds.), vol. 2, Springer, 1997, p. 463–503. | MR

[40] J. Bernat – « Arithmetics in $β$ -numeration », Discrete Math. Theor. Comput. Sci. 9 (2007), p. 85–106. | MR | Zbl

[41] —, « Computation of $L_{\oplus}$ for several cubic Pisot numbers », Discrete Math. Theor. Comput. Sci. 9 (2007), p. 175–193. | MR | Zbl

[42] J. Berstel & D. Perrin – « The origins of combinatorics on words », European J. Combin. 28 (2007), p. 996–1022. | MR | Zbl

[43] V. Berthé, S. Ferenczi & L. Q. Zamboni – « Interactions between dynamics, arithmetics and combinatorics: the good, the bad, and the ugly », in Algebraic and topological dynamics, Contemp. Math., vol. 385, Amer. Math. Soc., 2005, p. 333–364. | MR | Zbl

[44] V. Berthé & T. Fernique – « Brun expansions of stepped surfaces », Preprint (2010). | MR | Zbl

[45] V. Berthé & A. Siegel – « Tilings associated with beta-numeration and substitutions », INTEGERS (Electronic Journal of Combinatorial Number Theory) 5 (2005). | MR | EuDML | Zbl

[46] —, « Purely periodic $β$ -expansions in the Pisot non-unit case », J. Number Theory 127 (2007), p. 153–172. | MR | Zbl

[47] V. Berthé, A. Siegel, W. Steiner, P. Surer & J. M. Thuswaldner – « Fractal tiles associated with shift radix systems », Advances in Mathematics (2010), in press. | MR | Zbl

[48] V. Berthé, A. Siegel & J. M. Thuswaldner – « Substitutions, Rauzy fractals, and tilings », in Combinatorics, Automata, and Number Theory (V. Berthé & M. Rigo, éds.), Encyclopedia of Mathematics and its Applications, Cambridge Univ. Press, to appear. | MR | Zbl

[49] A. Bertrand-Mathis – « Développement en base $θ$ ; répartition modulo un de la suite ${(x θ^{n})}_{n \geq 0}$ ; langages codés et $θ$ -shift », Bull. Soc. Math. France 114 (1986), p. 271–323. | MR | EuDML | Zbl | Numdam

[50] M. Bestvina, M. Feighn & M. Handel – « Laminations, trees, and irreducible automorphisms of free groups », Geom. Funct. Anal. 7 (1997), p. 215–244. | Zbl

[51] —, « Laminations, trees, and irreducible automorphisms of free groups », Geom. Funct. Anal. 7 (1997), p. 215–244. | MR | Zbl

[52] M. Bestvina & M. Handel – « Train tracks and automorphisms of free groups », Ann. of Math. 135 (1992), p. 1–51. | MR | Zbl

[53] F. Blanchard – « $β$ -expansions and symbolic dynamics », Theoret. Comput. Sci. 65 (1989), p. 131–141. | MR | Zbl

[54] E. Bombieri & J. E. Taylor – « Which distributions of matter diffract? An initial investigation », J. Physique 47 (1986), p. C3–19–C3–28. | MR | Zbl

[55] R. Bowen – Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math., vol. 470, Springer, 1975. | MR | Zbl

[56] —, « Markov partitions are not smooth », Proc. Amer. Math. Soc. 71 (1978), p. 130–132. | MR | Zbl

[57] c. Burdik, C. Frougny, J.-P. Gazeau & R. Krejcar – « Beta-integers as natural counting systems for quasicrystals », J. of Physics A: Math. Gen. 31 (1998), p. 6449–6472. | MR

[58] J. W. Cannon & G. R. Conner – « The combinatorial structure of the Hawaiian earring group », Topology Appl. 106 (2000), p. 225–271. | MR | Zbl

[59] V. Canterini – « Connectedness of geometric representation of substitutions of Pisot type », Bull. Belg. Math. Soc. Simon Stevin 10 (2003), p. 77–89. | MR | Zbl

[60] V. Canterini & A. Siegel – « Geometric representation of substitutions of Pisot type », Trans. Amer. Math. Soc. 353 (2001), p. 5121–5144. | MR | Zbl

[61] J. Cassaigne, S. Ferenczi & L. Q. Zamboni – « Imbalances in Arnoux-Rauzy sequences », Ann. Inst. Fourier (Grenoble) 50 (2000), p. 1265–1276. | MR | EuDML | Zbl

[62] E. Cawley – « Smooth Markov partitions and toral automorphisms », Ergodic Theory Dynam. Systems 11 (1991), p. 633–651. | MR | Zbl

[63] N. Chekhova, P. Hubert & A. Messaoudi – « Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci », J. Théor. Nombres Bordeaux 13 (2001), p. 371–394. | MR | EuDML | Numdam

[64] A. Cobham – « Uniform tag sequences », Math. Systems Theory 6 (1972), p. 164–192. | MR | Zbl

[65] G. R. Conner & J. W. Lamoreaux – « On the existence of universal covering spaces for metric spaces and subsets of the Euclidean plane », Fund. Math. 187 (2005), p. 95–110. | EuDML | Zbl

[66] D. Cooper – « Automorphisms of free groups have finitely generated fixed point sets », J. Algebra 111 (1987), p. 453–456. | MR | Zbl

[67] T. Coulbois, A. Hilion & M. Lustig – « $ℝ$ -trees and laminations for free groups. I. Algebraic laminations », J. Lond. Math. Soc. 78 (2008), p. 723–736. | MR | Zbl

[68] —, « $ℝ$ -trees and laminations for free groups. II. The dual lamination of an $ℝ$ -tree », J. Lond. Math. Soc. 78 (2008), p. 737–754. | Zbl

[69] —, « $ℝ$ -trees and laminations for free groups. III. Currents and dual $ℝ$ -tree metrics », J. Lond. Math. Soc. 78 (2008), p. 755–766. | Zbl

[70] —, « $ℝ$ -trees, dual laminations and compact systems of partial isometries », Math. Proc. Cambridge Philos. Soc. 147 (2009), p. 345–368. | MR | Zbl

[71] D. Crisp, W. Moran, A. Pollington & P. Shiue – « Substitution invariant cutting sequences », J. Théor. Nombres Bordeaux 5 (1993), p. 123–137. | MR | EuDML | Zbl | Numdam

[72] F. M. Dekking – « The spectrum of dynamical systems arising from substitutions of constant length », Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41 (1977/78), p. 221–239. | MR | Zbl

[73] O. Delgrange & E. Rivals – « Star: an algorithm to search for tandem approximate repeats », Bioinformatics 20 (2004), p. 2812–20.

[74] J.-M. Dumont & A. Thomas – « Systemes de numeration et fonctions fractales relatifs aux substitutions », Theoret. Comput. Sci. 65 (1989), p. 153–169. | MR | Zbl

[75] —, « Digital sum moments and substitutions », Acta Arith. 64 (1993), p. 205–225. | MR | EuDML | Zbl

[76] —, « Gaussian asymptotic properties of the sum-of-digits function », J. Number Theory 62 (1997), p. 19–38. | MR | Zbl

[77] F. Durand – « A generalization of Cobham’s theorem », Theory Comput. Syst. 31 (1998), p. 169–185. | MR | Zbl

[78] F. Durand & A. Messaoudi – « Boundary of the rauzy fractal set in $ℝ \times ℂ$ generated by $p (x) = x^{4} - x^{3} - x^{2} - x - 1$ », Osaka J. of Math. (2010), in press.

[79] K. Eda & K. Kawamura – « The fundamental groups of one-dimensional spaces », Topology Appl. 87 (1998), p. 163–172. | MR | Zbl

[80] H. Ei & S. Ito – « Tilings from some non-irreducible Pisot substitutions », Discrete Math. Theor. Comput. Sci. 7 (2005), p. 81–122. | MR | Zbl

[81] H. Ei, S. Ito & H. Rao – « Atomic surfaces, tilings and coincidences II. reducible case », Ann. Inst. Fourier 56 (2006), p. 2285–2313. | MR | EuDML | Zbl | Numdam

[82] M. Einsiedler & K. Schmidt – « Markov partitions and homoclinic points of algebraic $𝐙^{d}$ -actions », Tr. Mat. Inst. Steklova 216 (1997), p. 265–284. | MR | Zbl

[83] K. Falconer – Fractal geometry, Mathematical foundations and applications, John Wiley & Sons Ltd., 1990. | MR

[84] D.-J. Feng, M. Furukado, S. Ito & J. Wu – « Pisot substitutions and the Hausdorff dimension of boundaries of atomic surfaces », Tsukuba J. Math. 30 (2006), p. 195–223. | MR | Zbl

[85] T. Fernique – « Generation and recognition of digital planes using multi-dimensional continued fractions », in Discrete geometry for computer imagery, Lecture Notes in Comput. Sci., vol. 4992, Springer, 2008, p. 33–44. | MR | Zbl

[86] N. P. Fogg – Substitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Math., vol. 1794, Springer, 2002. | MR | Zbl

[87] C. Frougny & B. Solomyak – « Finite beta-expansions », Ergodic Theory Dynam. Systems 12 (1992), p. 45–82. | MR

[88] C. Fuchs & R. Tijdeman – « Substitutions, abstract number systems and the space filling property », Ann. Inst. Fourier (Grenoble) 56 (2006), p. 2345–2389. | MR | EuDML | Zbl | Numdam

[89] B. Gaujal, A. Hordijk & D. V. Der Laan – « On the optimal open-loop control policy for deterministic and exponential polling systems », Probability in Engineering and Informational Sciences 21 (2007), p. 157–187. | MR | Zbl

[90] J.-P. Gazeau & J.-L. Verger-Gaugry – « Geometric study of the beta-integers for a Perron number and mathematical quasicrystals », J. Théor. Nombres Bordeaux 16 (2004), p. 125–149. | MR | EuDML | Zbl | Numdam

[91] M. Hata – « On the structure of self-similar sets », Japan J. Appl. Math. 2 (1985), p. 381–414. | MR | Zbl

[92] G. A. Hedlund – « Remarks on the work of Axel Thue on sequences », Nordisk Mat. Tidskr. 15 (1967), p. 148–150. | MR | Zbl

[93] M. Hollander – « Linear numeration systems, finite beta expansions, and discrete spectrum of substitution dynamical systems », Thèse, University of Washington, 1996. | MR

[94] P. Hubert & A. Messaoudi – « Best simultaneous Diophantine approximations of Pisot numbers and Rauzy fractals », Acta Arith. 124 (2006), p. 1–15. | MR | EuDML | Zbl

[95] S. Ito – « Simultaneous approximations and dynamical systems (on the simultaneous approximation of $(α, α^{2})$ satisfying $α^{3} + k α - 1 = 0$ ) », Sūrikaisekikenkyūsho Kōkyūroku 958 (1996), p. 59–61. | MR | Zbl

[96] S. Ito, J. Fujii, H. Higashinoand & S.-I. Yasutomi – « On simultaneous approximation to $(α, α^{2})$ with $α^{3} + k α - 1 = 0$ », J. Number Theory 99 (2003), p. 255–283. | MR | Zbl

[97] S. Ito & M. Kimura – « On Rauzy fractal », Japan J. Indust. Appl. Math. 8 (1991), p. 461–486. | MR | Zbl

[98] S. Ito & M. Ohtsuki – « Modified Jacobi-Perron algorithm and generating Markov partitions for special hyperbolic toral automorphisms », Tokyo J. Math. 16 (1993), p. 441–472. | MR | Zbl

[99] —, « Parallelogram tilings and Jacobi-Perron algorithm », Tokyo J. Math. 17 (1994), p. 33–58. | MR | Zbl

[100] S. Ito & H. Rao – « Purely periodic $β$ -expansion with Pisot base », Proc. Amer. Math. Soc. 133 (2005), p. 953–964. | MR | Zbl

[101] —, « Atomic surfaces, tilings and coincidences I. Irreducible case », Israel J. Math. 153 (2006), p. 129–155. | MR | Zbl

[102] C. Kalle & W. Steiner – « Beta-expansions, natural extensions and multiple tilings », Trans. Amer. Math. Soc. (2010), in press. | MR

[103] E. R. Van Kampen – « On some characterizations of $2$ -dimensional manifolds », Duke Math. J. 1 (1935), p. 74–93. | MR | JFM

[104] M. Keane – « Interval exchange transformations », Math. Z. 141 (1975), p. 25–31. | MR | EuDML | Zbl

[105] J. Kellendonk & I. Putnam – « Tilings, $C^{*}$ -algebras, and $K$ -theory », in Directions in mathematical quasicrystals (M. Baake et al., éds.), AMS CRM Monogr. Ser., vol. 13, 2000, p. 177–206. | MR | Zbl

[106] R. Kenyon & A. Vershik – « Arithmetic construction of sofic partitions of hyperbolic toral automorphisms », Ergodic Theory Dynam. Systems 18 (1998), p. 357–372. | MR | Zbl

[107] K. Kuratowski – Topology. Vol. II, New edition, revised and augmented. Translated from the French by A. Kirkor, Academic Press, 1968. | MR | Zbl

[108] J. C. Lagarias & Y. Wang – « Self affine tiles in $ℝ^{n}$ », Adv. Math. 121 (1996), p. 21–49. | MR | Zbl

[109] —, « Substitution Delone sets », Discrete Comput. Geom. 29 (2003), p. 175–209. | MR | Zbl

[110] S. Le Borgne – « Un codage sofique des automorphismes hyperboliques du tore », in Séminaires de Probabilités de Rennes (1995), Publ. Inst. Rech. Math. Rennes, vol. 1995, Univ. Rennes I, 1995, p. 35. | MR | EuDML

[111] —, « Un codage sofique des automorphismes hyperboliques du tore », C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), p. 1123–1128. | MR

[112] —, « Un codage sofique des automorphismes hyperboliques du tore », Bol. Soc. Brasil. Mat. (N.S.) 30 (1999), p. 61–93. | MR

[113] J.-Y. Lee, R. V. Moody & B. Solomyak – « Pure point dynamical and diffraction spectra », Ann. Henri Poincaré 3 (2002), p. 1003–1018. | MR | Zbl

[114] D. Lind & B. Marcus – An introduction to symbolic dynamics and coding, Cambridge Univ. Press, 1995. | MR | Zbl

[115] A. N. Livshits – « On the spectra of adic transformations of markov compacta », Uspekhi Mat. Nauk 42 (1987), p. 189–190. | MR | Zbl

[116] —, « Some examples of adic transformations and automorphisms of substitutions », Selecta Math. Soviet. 11 (1992), p. 83–104. | MR

[117] B. Loridant & J. M. Thuswaldner – « Interior components of a tile associated to a quadratic canonical number system », Topology Appl. 155 (2008), p. 667–695. | MR | Zbl

[118] M. Lothaire – Applied combinatorics on words, Encyclopedia of Mathematics and its Applications, vol. 105, Cambridge Univ. Press, 2005. | MR | Zbl

[119] J. M. Luck, C. Godrèche, T. A. Janner & Janssen – « The nature of the atomic surfaces of quasiperiodic self-similar structures », J. Phys. A 26 (1993), p. 1951–1999. | MR

[120] J. Luo – « A note on a self-similar tiling generated by the minimal Pisot number », Fractals 10 (2002), p. 335–339. | MR | Zbl

[121] J. Luo, S. Akiyama & J. M. Thuswaldner – « On the boundary connectedness of connected tiles », Math. Proc. Cambridge Philos. Soc. 137 (2004), p. 397–410. | MR | Zbl

[122] J. Luo, H. Rao & B. Tan – « Topological structure of self-similar sets », Fractals 10 (2002), p. 223–227. | MR | Zbl

[123] J. Luo & J. M. Thuswaldner – « On the fundamental group of self-affine plane tiles », Ann. Inst. Fourier (Grenoble) 56 (2006), p. 2493–2524. | MR | EuDML | Zbl | Numdam

[124] J. Luo & Z.-L. Zhou – « Disk-like tiles derived from complex bases », Acta Math. Sin. (Engl. Ser.) 20 (2004), p. 731–738. | MR | Zbl

[125] R. D. Mauldin & S. C. Williams – « Hausdorff dimension in graph directed constructions », Trans. Amer. Math. Soc. 309 (1988), p. 811–829. | MR | Zbl

[126] A. Messaoudi – « Propriétés arithmétiques et dynamiques du fractal de Rauzy », J. Théor. Nombres Bordeaux 10 (1998), p. 135–162. | MR | EuDML | Numdam

[127] —, « Frontière du fractal de Rauzy et système de numération complexe », Acta Arith. 95 (2000), p. 195–224. | MR | EuDML | Zbl

[128] —, « Propriétés arithmétiques et topologiques d’une classe d’ensembles fractales », Acta Arith. 121 (2006), p. 341–366. | MR | EuDML

[129] R. V. Moody – « Model sets: a survey », in From Quasicrystals to More Complex Systems (F. Axel & J.-P. Gazeau, éds.), Les Editions de Physique, Springer, Berlin, 2000, p. 145–166.

[130] H. M. Morse – « Recurrent geodesics on a surface of negative curvature », Trans. Amer. Math. Soc. 22 (1921), p. 84–100. | MR | JFM

[131] B. Mossé – « Recognizability of substitutions and complexity of automatic sequences », Bull. Soc. Math. Fr. 124 (1996), p. 329–346. | MR | EuDML | Zbl | Numdam

[132] S.-M. Ngai & N. Nguyen – « The Heighway dragon revisited », Discrete Comput. Geom. 29 (2003), p. 603–623. | MR | Zbl

[133] S.-M. Ngai & T.-M. Tang – « A technique in the topology of connected self-similar tiles », Fractals 12 (2004), p. 389–403. | MR | Zbl

[134] —, « Topology of connected self-similar tiles in the plane with disconnected interiors », Topology Appl. 150 (2005), p. 139–155. | MR | Zbl

[135] W. Parry – « On the $β$ -expansion of real numbers », Acta Math. Acad. Sci. Hungar. 11 (1960), p. 401–416. | MR | Zbl

[136] B. Praggastis – « Numeration systems and Markov partitions from self-similar tilings », Trans. Amer. Math. Soc. 351 (1999), p. 3315–3349. | MR | Zbl

[137] N. Priebe-Franck – « A primer of substitution tilings of the Euclidean plane », Expo. Math. 26 (2008), p. 295–326. | MR | Zbl

[138] Y.-H. Qu, H. Rao & Y.-M. Yang – « Periods of $β$ -expansions and linear recurrent sequences », Acta Arith. 120 (2005), p. 27–37. | MR | EuDML | Zbl

[139] M. Queffélec – Substitution dynamical systems—spectral analysis, Lecture Notes in Mathematics, 1294. Springer, 1987. | MR | Zbl

[140] C. Radin – « Space tilings and substitutions », Geom. Dedicata 55 (1995), p. 257–264. | MR | Zbl

[141] G. Rauzy – « Nombres algébriques et substitutions », Bull. Soc. Math. France 110 (1982), p. 147–178. | MR | EuDML | Zbl | Numdam

[142] J.-P. Reveillès – « Géométrie discrète, calcul en nombres entiers et algorithmique », Thèse de Doctorat, Université Louis Pasteur, Strasbourg, 1991.

[143] M. Rigo & W. Steiner – « Abstract $β$ -expansions and ultimately periodic representations », J. Number Theory 17 (2005), p. 283–299. | MR | EuDML | Zbl | Numdam

[144] E. A. J. Robinson – « Symbolic dynamics and tilings of $ℝ^{d}$ », in Symbolic dynamics and its applications, Proc. Sympos. Appl. Math., Amer. Math. Soc. Providence, RI, vol. 60, 2004, p. 81–119. | Zbl

[145] D. Roy – « Approximation to real numbers by cubic algebraic integers. II », Ann. of Math. 158 (2003), p. 1081–1087. | MR | Zbl

[146] W. Rudin – « Some theorems on Fourier coefficients », Proc. Amer. Math. Soc. 10 (1959), p. 855–859. | MR | Zbl

[147] T. Sadahiro – « Multiple points of tilings associated with Pisot numeration systems », Theoret. Comput. Sci. 359 (2006), p. 133–147. | MR | Zbl

[148] Y. Sano, P. Arnoux & S. Ito – « Higher dimensional extensions of substitutions and their dual maps », J. Anal. Math. 83 (2001), p. 183–206. | MR | Zbl

[149] K. Scheicher & J. M. Thuswaldner – « Canonical number systems, counting automata and fractals », Math. Proc. Cambridge Philos. Soc. 133 (2002), p. 163–182. | MR | Zbl

[150] K. Schmidt – « On periodic expansions of Pisot numbers and Salem numbers », Bull. London Math. Soc. 12 (1980), p. 269–278. | MR | Zbl

[151] —, « Algebraic coding of expansive group automorphisms and two-sided beta-shifts », Monatsh. Math. 129 (2000), p. 37–61. | Zbl

[152] —, « Algebraic coding of expansive group automorphisms and two-sided beta-shifts », Monatsh. Math. 129 (2000), p. 37–61. | MR | Zbl

[153] M. Senechal – « What is $...$ a quasicrystal? », Notices Amer. Math. Soc. 53 (2006), p. 886–887. | MR | Zbl

[154] A. Siegel – « Représentation des systèmes dynamiques substitutifs non unimodulaires », Ergodic Theory Dynam. Systems 23 (2003), p. 1247–1273. | MR

[155] —, « Pure discrete spectrum dynamical system and periodic tiling associated with a substitution », Ann. Inst. Fourier (Grenoble) 54 (2004), p. 341–381. | MR | Zbl | Numdam

[156] V. F. Sirvent – « Geodesic laminations as geometric realizations of Pisot substitutions », Ergodic Theory Dynam. Systems 20 (2000), p. 1253–1266. | MR | Zbl

[157] V. F. Sirvent & B. Solomyak – « Pure discrete spectrum for one-dimensional substitution systems of Pisot type », Canad. Math. Bull. 45 (2002), p. 697–710. | MR | Zbl

[158] V. F. Sirvent & Y. Wang – « Self-affine tiling via substitution dynamical systems and Rauzy fractals », Pacific J. Math. 206 (2002), p. 465–485. | MR | Zbl

[159] S. Smale – « Differentiable dynamical systems », Bull. Amer. Math. Soc. 73 (1967), p. 747–817. | MR | Zbl

[160] B. De Smit – « The fundamental group of the Hawaiian earring is not free », Internat. J. Algebra Comput. 2 (1992), p. 33–37. | MR | Zbl

[161] B. Solomyak – « Dynamics of self-similar tilings », Ergodic Theory Dynam. Systems 17 (1997), p. 695–738. | MR | Zbl

[162] —, « Tilings and dynamics », in EMS Summer School on Combinatorics, Automata and Number Theory, 2006.

[163] W. Steiner – « Digital expansions and the distribution of related functions », 2000, http://www.liafa.jussieu.fr/~steiner/.

[164] A. Thue – « Über unendliche Zeichenreihen », Norske Vid. Selsk. Skr. Mat. Nat. Kl. 37 (1906), p. 1–22. | JFM

[165] —, « Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen », Norske Vid. Selsk. Skr. Mat. Nat. Kl. 43 (1912), p. 1–67. | JFM

[166] W. P. Thurston – « Groups, tilings and finite state automata », Lectures notes distributed in conjunction with the Colloquium Series, in AMS Colloquium lectures, 1989.

[167] J. M. Thuswaldner – « Unimodular Pisot substitutions and their associated tiles », J. Théor. Nombres Bordeaux 18 (2006), p. 487–536. | MR | EuDML | Zbl | Numdam

[168] W. A. Veech – « Interval exchange transformations », J. Anal. Math. 33 (1978), p. 222–272. | MR | Zbl

[169] R. F. Williams – « Classification of one dimensional attractors », in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., 1970, p. 341–361. | MR

[170] S.-I. Yasutomi – « On Sturmian sequences which are invariant under some substitutions », in Number theory and its applications (Kyoto, 1997), Dev. Math., vol. 2, Kluwer Acad. Publ., 1999, p. 347–373. | MR | Zbl

Cité par Sources :