Topological properties of Rauzy fractals  [ Propriétés topologiques des fractals de Rauzy ] (2009)


Siegel, Anne; Thuswaldner, Jörg M.
Mémoires de la Société Mathématique de France, Tome 118 (2009) 140 p
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consulter l'article sur le site de la revue
doi : 10.24033/msmf.430
URL stable : http://www.numdam.org/item?id=MSMF_2009_2_118__1_0

Bibliographie

[1] B. Adamczewski & Y. Bugeaud« On the complexity of algebraic numbers. I. Expansions in integer bases », Ann. of Math. 165 (2007), p. 547–565. Zbl 1195.11094 | MR 2299740

[2] B. Adamczewski, Y. Bugeaud & L. Davison« Continued fractions and transcendental numbers », Ann. Inst. Fourier (Grenoble) 56 (2006), p. 2093–2113. Numdam | Zbl 1152.11034 | | MR 2290775

[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. Zbl 1211.11010 | MR 2651949

[4] R. L. Adler« Symbolic dynamics and Markov partitions », Bull. Amer. Math. Soc. (N.S.) 35 (1998), p. 1–56. Zbl 0892.58019

[5] R. L. Adler & B. WeissSimilarity of automorphisms of the torus, Memoirs of the American Mathematical Society, No. 98, Amer. Math. Soc., 1970. Zbl 0195.06104 | MR 257315

[6] S. Akiyama« Pisot numbers and greedy algorithm », in Number theory (Eger, 1996), de Gruyter, 1998, p. 9–21. Zbl 0919.11063 | MR 1628829

[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. Zbl 0999.11065 | MR 1738803

[8] —, « Cubic Pisot units with finite beta expansions », in Algebraic number theory and Diophantine analysis (Graz, 1998), de Gruyter, 2000, p. 11–26. Zbl 1001.11038

[9] —, « On the boundary of self affine tilings generated by Pisot numbers », J. Math. Soc. Japan 54 (2002), p. 283–308. Zbl 1032.11033 | MR 1883519

[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. Zbl 1190.11005 | MR 2461585

[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. Zbl 1110.11003 | MR 2162561

[13] S. Akiyama, H. Brunotte, A. Pethő & J. M. Thuswaldner« Generalized radix representations and dynamical systems. II », Acta Arith. 121 (2006), p. 21–61. Zbl 1142.11055 | | MR 2216302

[14] —, « Generalized radix representations and dynamical systems. III », Osaka J. Math. 45 (2008), p. 347–374. Zbl 1217.11007 | MR 2441944

[15] —, « Generalized radix representations and dynamical systems. IV », Indag. Math. (N.S.) 19 (2008), p. 333–348. Zbl 1190.11041 | MR 2513054

[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. Zbl 1182.57001 | MR 2521702

[17] S. Akiyama & G. Nertila« On the connectedness of self-affine tilings », Arch. Math. 82 (2004), p. 153–163. Zbl 1063.37008

[18] S. Akiyama, H. Rao & W. Steiner« A certain finiteness property of Pisot number systems », J. Number Theory 107 (2004), p. 135–160. Zbl 1052.11055 | MR 2059954

[19] S. Akiyama & K. Scheicher« Intersecting two-dimensional fractals with lines », Acta Sci. Math. (Szeged) 71 (2005), p. 555–580. Zbl 1111.11006 | MR 2206596

[20] C. Allauzen« Une caractérisation simple des nombres de Sturm », J. Théor. Nombres Bordeaux 10 (1998), p. 237–241. Numdam | | MR 1828243

[21] J.-P. Allouche & J. O. ShallitAutomatic sequences: Theory and applications, Cambridge Univ. Press, 2002. MR 1997038

[22] J. Anderson & I. Putnam« Topological invariants for substitution tilings and their associated C * -algebras », Ergodic Theory Dynam. Systems 18 (1998), p. 509–537. Zbl 1053.46520 | MR 1631708

[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. Numdam | Zbl 0703.58045 | | MR 1005392

[24] P. Arnoux, J. Bernat & X. Bressaud« Geometrical models for substitutions », Experiment. Math. (2010), to appear. Zbl 1266.37008 | MR 2802726

[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. Zbl 1017.68147 | MR 1888763

[26] P. Arnoux, V. Berthé, T. Fernique & D. Jamet« Functional stepped surfaces, flips, and generalized substitutions », Theoret. Comput. Sci. 380 (2007), p. 251–265. Zbl 1119.68136 | MR 2330996

[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. Numdam | Zbl 1146.20020 | | MR 2290778

[28] P. Arnoux, V. Berthé & S. Ito« Discrete planes, 2 -actions, Jacobi-Perron algorithm and substitutions », Ann. Inst. Fourier (Grenoble) 52 (2002), p. 305–349. Zbl 1017.11006 | | MR 1906478

[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. Zbl 1007.37001 | MR 1838930

[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. Numdam | Zbl 1138.37008 | | MR 2290779

[32] C. Bandt & G. Gelbrich« Classification of self-affine lattice tilings », J. London Math. Soc. 50 (1994), p. 581–593. Zbl 0820.52012 | MR 1299459

[33] G. Barat, V. Berthé, , P. Liardet & J. M. Thuswaldner – « Dynamical directions in numeration », Ann. Inst. Fourier (Grenoble) 56 (2006), p. 1987–2092. Numdam | | MR 2290774

[34] M. Barge & B. Diamond« Coincidence for substitutions of Pisot type », Bull. Soc. Math. France 130 (2002), p. 619–626. Numdam | Zbl 1028.37008 | | MR 1947456

[35] M. Barge, B. Diamond & R. Swanson« The branch locus for one-dimensional Pisot tiling spaces », Fund. Math. 204 (2009), p. 215–240. Zbl 1185.37013 | | MR 2520153

[36] M. Barge & J. Kwapisz« Geometric theory of unimodular Pisot substitutions », Amer. J. Math. 128 (2006), p. 1219–1282. Zbl 1152.37011 | MR 2262174

[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. Zbl 1152.11342 | MR 1966122

[38] L. E. Baum & M. M. Sweet« Continued fractions of algebraic power series in characteristic 2 », Ann. of Math. 103 (1976), p. 593–610. Zbl 0312.10024 | MR 409372

[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 1470015

[40] J. Bernat« Arithmetics in β-numeration », Discrete Math. Theor. Comput. Sci. 9 (2007), p. 85–106. Zbl 1152.68456 | MR 2318443

[41] —, « Computation of L for several cubic Pisot numbers », Discrete Math. Theor. Comput. Sci. 9 (2007), p. 175–193. Zbl 1165.11061 | MR 2306527

[42] J. Berstel & D. Perrin« The origins of combinatorics on words », European J. Combin. 28 (2007), p. 996–1022. Zbl 1111.68092 | MR 2300777

[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. Zbl 1156.37301 | MR 2180244

[44] V. Berthé & T. Fernique« Brun expansions of stepped surfaces », Preprint (2010). Zbl 1236.11011 | MR 2765621

[45] V. Berthé & A. Siegel« Tilings associated with beta-numeration and substitutions », INTEGERS (Electronic Journal of Combinatorial Number Theory) 5 (2005). Zbl 1139.37008 | | MR 2191748

[46] —, « Purely periodic β-expansions in the Pisot non-unit case », J. Number Theory 127 (2007), p. 153–172. Zbl 1197.11139 | MR 2362431

[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. Zbl 1221.11018 | MR 2735753

[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. Zbl 1247.37015 | MR 2759108

[49] A. Bertrand-Mathis« Développement en base θ; répartition modulo un de la suite (xθ n ) n0 ; langages codés et θ-shift », Bull. Soc. Math. France 114 (1986), p. 271–323. Numdam | Zbl 0628.58024 | | MR 878240

[50] M. Bestvina, M. Feighn & M. Handel« Laminations, trees, and irreducible automorphisms of free groups », Geom. Funct. Anal. 7 (1997), p. 215–244. Zbl 0884.57002

[51] —, « Laminations, trees, and irreducible automorphisms of free groups », Geom. Funct. Anal. 7 (1997), p. 215–244. Zbl 0884.57002 | MR 1445386

[52] M. Bestvina & M. Handel« Train tracks and automorphisms of free groups », Ann. of Math. 135 (1992), p. 1–51. Zbl 0757.57004 | MR 1147956

[53] F. Blanchard« β-expansions and symbolic dynamics », Theoret. Comput. Sci. 65 (1989), p. 131–141. Zbl 0682.68081 | MR 1020481

[54] E. Bombieri & J. E. Taylor« Which distributions of matter diffract? An initial investigation », J. Physique 47 (1986), p. C3–19–C3–28. Zbl 0693.52002 | MR 866320

[55] R. BowenEquilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math., vol. 470, Springer, 1975. Zbl 0308.28010 | MR 442989

[56] —, « Markov partitions are not smooth », Proc. Amer. Math. Soc. 71 (1978), p. 130–132. Zbl 0417.58011 | MR 474415

[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 1644115

[58] J. W. Cannon & G. R. Conner« The combinatorial structure of the Hawaiian earring group », Topology Appl. 106 (2000), p. 225–271. Zbl 0955.57002 | MR 1775709

[59] V. Canterini« Connectedness of geometric representation of substitutions of Pisot type », Bull. Belg. Math. Soc. Simon Stevin 10 (2003), p. 77–89. Zbl 1031.37015 | MR 2032327

[60] V. Canterini & A. Siegel« Geometric representation of substitutions of Pisot type », Trans. Amer. Math. Soc. 353 (2001), p. 5121–5144. Zbl 1142.37302 | MR 1852097

[61] J. Cassaigne, S. Ferenczi & L. Q. Zamboni« Imbalances in Arnoux-Rauzy sequences », Ann. Inst. Fourier (Grenoble) 50 (2000), p. 1265–1276. Zbl 1004.37008 | | MR 1799745

[62] E. Cawley« Smooth Markov partitions and toral automorphisms », Ergodic Theory Dynam. Systems 11 (1991), p. 633–651. Zbl 0754.58028 | MR 1145614

[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. Numdam | | MR 1879664

[64] A. Cobham« Uniform tag sequences », Math. Systems Theory 6 (1972), p. 164–192. Zbl 0253.02029 | MR 457011

[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. Zbl 1092.57001 |

[66] D. Cooper« Automorphisms of free groups have finitely generated fixed point sets », J. Algebra 111 (1987), p. 453–456. Zbl 0628.20029 | MR 916179

[67] T. Coulbois, A. Hilion & M. Lustig« -trees and laminations for free groups. I. Algebraic laminations », J. Lond. Math. Soc. 78 (2008), p. 723–736. Zbl 1197.20019 | MR 2456901

[68] —, « -trees and laminations for free groups. II. The dual lamination of an -tree », J. Lond. Math. Soc. 78 (2008), p. 737–754. Zbl 1198.20023

[69] —, « -trees and laminations for free groups. III. Currents and dual -tree metrics », J. Lond. Math. Soc. 78 (2008), p. 755–766. Zbl 1200.20018

[70] —, « -trees, dual laminations and compact systems of partial isometries », Math. Proc. Cambridge Philos. Soc. 147 (2009), p. 345–368. Zbl 1239.20030 | MR 2525931

[71] D. Crisp, W. Moran, A. Pollington & P. Shiue« Substitution invariant cutting sequences », J. Théor. Nombres Bordeaux 5 (1993), p. 123–137. Numdam | Zbl 0786.11041 | | MR 1251232

[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. Zbl 0348.54034 | MR 461470

[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. Zbl 0679.10010 | MR 1020484

[75] —, « Digital sum moments and substitutions », Acta Arith. 64 (1993), p. 205–225. Zbl 0774.11041 | | MR 1225425

[76] —, « Gaussian asymptotic properties of the sum-of-digits function », J. Number Theory 62 (1997), p. 19–38. Zbl 0869.11009 | MR 1430000

[77] F. Durand« A generalization of Cobham’s theorem », Theory Comput. Syst. 31 (1998), p. 169–185. Zbl 0895.68081 | MR 1491657

[78] F. Durand & A. Messaoudi« Boundary of the rauzy fractal set in × 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. Zbl 0922.55008 | MR 1624308

[80] H. Ei & S. Ito« Tilings from some non-irreducible Pisot substitutions », Discrete Math. Theor. Comput. Sci. 7 (2005), p. 81–122. Zbl 1153.37323 | MR 2164061

[81] H. Ei, S. Ito & H. Rao« Atomic surfaces, tilings and coincidences II. reducible case », Ann. Inst. Fourier 56 (2006), p. 2285–2313. Numdam | Zbl 1119.52013 | | MR 2290782

[82] M. Einsiedler & K. Schmidt« Markov partitions and homoclinic points of algebraic 𝐙 d -actions », Tr. Mat. Inst. Steklova 216 (1997), p. 265–284. Zbl 0954.37008 | MR 1632169

[83] K. FalconerFractal geometry, Mathematical foundations and applications, John Wiley & Sons Ltd., 1990. MR 1102677

[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. Zbl 1130.37318 | MR 2248292

[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. Zbl 1138.68592 | MR 2503454

[86] N. P. FoggSubstitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Math., vol. 1794, Springer, 2002. Zbl 1014.11015 | MR 1970385

[87] C. Frougny & B. Solomyak« Finite beta-expansions », Ergodic Theory Dynam. Systems 12 (1992), p. 45–82. MR 1200339

[88] C. Fuchs & R. Tijdeman« Substitutions, abstract number systems and the space filling property », Ann. Inst. Fourier (Grenoble) 56 (2006), p. 2345–2389. Numdam | Zbl 1194.11023 | | MR 2290784

[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. Zbl 1128.90018 | MR 2350992

[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. Numdam | Zbl 1075.11007 | | MR 2145576

[91] M. Hata« On the structure of self-similar sets », Japan J. Appl. Math. 2 (1985), p. 381–414. Zbl 0608.28003 | MR 839336

[92] G. A. Hedlund« Remarks on the work of Axel Thue on sequences », Nordisk Mat. Tidskr. 15 (1967), p. 148–150. Zbl 0153.33101 | MR 228875

[93] M. Hollander« Linear numeration systems, finite beta expansions, and discrete spectrum of substitution dynamical systems », Thèse, University of Washington, 1996. MR 2694876

[94] P. Hubert & A. Messaoudi« Best simultaneous Diophantine approximations of Pisot numbers and Rauzy fractals », Acta Arith. 124 (2006), p. 1–15. Zbl 1116.28009 | | MR 2262136

[95] S. Ito« Simultaneous approximations and dynamical systems (on the simultaneous approximation of (α,α 2 ) satisfying α 3 +kα-1=0) », Sūrikaisekikenkyūsho Kōkyūroku 958 (1996), p. 59–61. Zbl 0917.11029 | MR 1468000

[96] S. Ito, J. Fujii, H. Higashinoand & S.-I. Yasutomi« On simultaneous approximation to (α,α 2 ) with α 3 +kα-1=0 », J. Number Theory 99 (2003), p. 255–283. Zbl 1135.11326 | MR 1968452

[97] S. Ito & M. Kimura« On Rauzy fractal », Japan J. Indust. Appl. Math. 8 (1991), p. 461–486. Zbl 0734.28010 | MR 1137652

[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. Zbl 0805.11056 | MR 1247666

[99] —, « Parallelogram tilings and Jacobi-Perron algorithm », Tokyo J. Math. 17 (1994), p. 33–58. Zbl 0805.52011 | MR 1279568

[100] S. Ito & H. Rao« Purely periodic β-expansion with Pisot base », Proc. Amer. Math. Soc. 133 (2005), p. 953–964. Zbl 1099.11062 | MR 2117194

[101] —, « Atomic surfaces, tilings and coincidences I. Irreducible case », Israel J. Math. 153 (2006), p. 129–155. Zbl 1143.37013 | MR 2254640

[102] C. Kalle & W. Steiner« Beta-expansions, natural extensions and multiple tilings », Trans. Amer. Math. Soc. (2010), in press. MR 2888207

[103] E. R. Van Kampen« On some characterizations of 2-dimensional manifolds », Duke Math. J. 1 (1935), p. 74–93. Zbl 61.0638.03 | MR 1545866

[104] M. Keane« Interval exchange transformations », Math. Z. 141 (1975), p. 25–31. Zbl 0278.28010 | | MR 357739

[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. Zbl 0972.52015 | MR 1798993

[106] R. Kenyon & A. Vershik« Arithmetic construction of sofic partitions of hyperbolic toral automorphisms », Ergodic Theory Dynam. Systems 18 (1998), p. 357–372. Zbl 0915.58077 | MR 1619562

[107] K. KuratowskiTopology. Vol. II, New edition, revised and augmented. Translated from the French by A. Kirkor, Academic Press, 1968. Zbl 0158.40901 | MR 259835

[108] J. C. Lagarias & Y. Wang« Self affine tiles in n », Adv. Math. 121 (1996), p. 21–49. Zbl 0893.52013 | MR 1399601

[109] —, « Substitution Delone sets », Discrete Comput. Geom. 29 (2003), p. 175–209. Zbl 1037.52017 | MR 1957227

[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 1396814

[111] —, « Un codage sofique des automorphismes hyperboliques du tore », C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), p. 1123–1128. MR 1423437

[112] —, « Un codage sofique des automorphismes hyperboliques du tore », Bol. Soc. Brasil. Mat. (N.S.) 30 (1999), p. 61–93. MR 1686988

[113] J.-Y. Lee, R. V. Moody & B. Solomyak« Pure point dynamical and diffraction spectra », Ann. Henri Poincaré 3 (2002), p. 1003–1018. Zbl 1025.37004 | MR 1937612

[114] D. Lind & B. MarcusAn introduction to symbolic dynamics and coding, Cambridge Univ. Press, 1995. Zbl 1106.37301 | MR 1369092

[115] A. N. Livshits« On the spectra of adic transformations of markov compacta », Uspekhi Mat. Nauk 42 (1987), p. 189–190. Zbl 0635.47007 | MR 896889

[116] —, « Some examples of adic transformations and automorphisms of substitutions », Selecta Math. Soviet. 11 (1992), p. 83–104. MR 1155902

[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. Zbl 1148.28009 | MR 2395584

[118] M. LothaireApplied combinatorics on words, Encyclopedia of Mathematics and its Applications, vol. 105, Cambridge Univ. Press, 2005. Zbl 1133.68067 | MR 2165687

[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 1220802

[120] J. Luo« A note on a self-similar tiling generated by the minimal Pisot number », Fractals 10 (2002), p. 335–339. Zbl 1077.37500 | MR 1932445

[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. Zbl 1070.37010 | MR 2092067

[122] J. Luo, H. Rao & B. Tan« Topological structure of self-similar sets », Fractals 10 (2002), p. 223–227. Zbl 1075.28005 | MR 1910665

[123] J. Luo & J. M. Thuswaldner« On the fundamental group of self-affine plane tiles », Ann. Inst. Fourier (Grenoble) 56 (2006), p. 2493–2524. Numdam | Zbl 1119.52012 | | MR 2290788

[124] J. Luo & Z.-L. Zhou« Disk-like tiles derived from complex bases », Acta Math. Sin. (Engl. Ser.) 20 (2004), p. 731–738. Zbl 1063.11006 | MR 2096785

[125] R. D. Mauldin & S. C. Williams« Hausdorff dimension in graph directed constructions », Trans. Amer. Math. Soc. 309 (1988), p. 811–829. Zbl 0706.28007 | MR 961615

[126] A. Messaoudi« Propriétés arithmétiques et dynamiques du fractal de Rauzy », J. Théor. Nombres Bordeaux 10 (1998), p. 135–162. Numdam | | MR 1827290

[127] —, « Frontière du fractal de Rauzy et système de numération complexe », Acta Arith. 95 (2000), p. 195–224. Zbl 0968.28005 | | MR 1793161

[128] —, « Propriétés arithmétiques et topologiques d’une classe d’ensembles fractales », Acta Arith. 121 (2006), p. 341–366. | MR 2224401

[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. Zbl 48.0786.06 | MR 1501161

[131] B. Mossé« Recognizability of substitutions and complexity of automatic sequences », Bull. Soc. Math. Fr. 124 (1996), p. 329–346. Numdam | Zbl 0855.68072 | | MR 1414542

[132] S.-M. Ngai & N. Nguyen« The Heighway dragon revisited », Discrete Comput. Geom. 29 (2003), p. 603–623. Zbl 1028.28010 | MR 1976609

[133] S.-M. Ngai & T.-M. Tang« A technique in the topology of connected self-similar tiles », Fractals 12 (2004), p. 389–403. Zbl 1304.28009 | MR 2109984

[134] —, « Topology of connected self-similar tiles in the plane with disconnected interiors », Topology Appl. 150 (2005), p. 139–155. Zbl 1077.37019 | MR 2133675

[135] W. Parry« On the β-expansion of real numbers », Acta Math. Acad. Sci. Hungar. 11 (1960), p. 401–416. Zbl 0099.28103 | MR 142719

[136] B. Praggastis« Numeration systems and Markov partitions from self-similar tilings », Trans. Amer. Math. Soc. 351 (1999), p. 3315–3349. Zbl 0984.11008 | MR 1615950

[137] N. Priebe-Franck« A primer of substitution tilings of the Euclidean plane », Expo. Math. 26 (2008), p. 295–326. Zbl 1151.52016 | MR 2462439

[138] Y.-H. Qu, H. Rao & Y.-M. Yang« Periods of β-expansions and linear recurrent sequences », Acta Arith. 120 (2005), p. 27–37. Zbl 1155.11337 | | MR 2189716

[139] M. QueffélecSubstitution dynamical systems—spectral analysis, Lecture Notes in Mathematics, 1294. Springer, 1987. Zbl 0642.28013 | MR 924156

[140] C. Radin« Space tilings and substitutions », Geom. Dedicata 55 (1995), p. 257–264. Zbl 0835.52018 | MR 1334449

[141] G. Rauzy« Nombres algébriques et substitutions », Bull. Soc. Math. France 110 (1982), p. 147–178. Numdam | Zbl 0522.10032 | | MR 667748

[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. Numdam | Zbl 1084.11059 | | MR 2152225

[144] E. A. J. Robinson« Symbolic dynamics and tilings of d », in Symbolic dynamics and its applications, Proc. Sympos. Appl. Math., Amer. Math. Soc. Providence, RI, vol. 60, 2004, p. 81–119. Zbl 1076.37010

[145] D. Roy« Approximation to real numbers by cubic algebraic integers. II », Ann. of Math. 158 (2003), p. 1081–1087. Zbl 1044.11061 | MR 2031862

[146] W. Rudin« Some theorems on Fourier coefficients », Proc. Amer. Math. Soc. 10 (1959), p. 855–859. Zbl 0091.05706 | MR 116184

[147] T. Sadahiro« Multiple points of tilings associated with Pisot numeration systems », Theoret. Comput. Sci. 359 (2006), p. 133–147. Zbl 1220.11009 | MR 2251606

[148] Y. Sano, P. Arnoux & S. Ito« Higher dimensional extensions of substitutions and their dual maps », J. Anal. Math. 83 (2001), p. 183–206. Zbl 0987.11013 | MR 1828491

[149] K. Scheicher & J. M. Thuswaldner« Canonical number systems, counting automata and fractals », Math. Proc. Cambridge Philos. Soc. 133 (2002), p. 163–182. Zbl 1001.68070 | MR 1900260

[150] K. Schmidt« On periodic expansions of Pisot numbers and Salem numbers », Bull. London Math. Soc. 12 (1980), p. 269–278. Zbl 0494.10040 | MR 576976

[151] —, « Algebraic coding of expansive group automorphisms and two-sided beta-shifts », Monatsh. Math. 129 (2000), p. 37–61. Zbl 1010.37005

[152] —, « Algebraic coding of expansive group automorphisms and two-sided beta-shifts », Monatsh. Math. 129 (2000), p. 37–61. Zbl 1010.37005 | MR 1741033

[153] M. Senechal« What is...a quasicrystal? », Notices Amer. Math. Soc. 53 (2006), p. 886–887. Zbl 1137.82001 | MR 2253164

[154] A. Siegel« Représentation des systèmes dynamiques substitutifs non unimodulaires », Ergodic Theory Dynam. Systems 23 (2003), p. 1247–1273. MR 1997975

[155] —, « Pure discrete spectrum dynamical system and periodic tiling associated with a substitution », Ann. Inst. Fourier (Grenoble) 54 (2004), p. 341–381. Numdam | Zbl 1083.37009 | MR 2073838

[156] V. F. Sirvent« Geodesic laminations as geometric realizations of Pisot substitutions », Ergodic Theory Dynam. Systems 20 (2000), p. 1253–1266. Zbl 0963.37013 | MR 1779402

[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. Zbl 1038.37008 | MR 1941235

[158] V. F. Sirvent & Y. Wang« Self-affine tiling via substitution dynamical systems and Rauzy fractals », Pacific J. Math. 206 (2002), p. 465–485. Zbl 1048.37015 | MR 1926787

[159] S. Smale« Differentiable dynamical systems », Bull. Amer. Math. Soc. 73 (1967), p. 747–817. Zbl 0202.55202 | MR 228014

[160] B. De Smit« The fundamental group of the Hawaiian earring is not free », Internat. J. Algebra Comput. 2 (1992), p. 33–37. Zbl 0738.20033 | MR 1167526

[161] B. Solomyak« Dynamics of self-similar tilings », Ergodic Theory Dynam. Systems 17 (1997), p. 695–738. Zbl 0884.58062 | MR 1452190

[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. Zbl 39.0283.01

[165] —, « Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen », Norske Vid. Selsk. Skr. Mat. Nat. Kl. 43 (1912), p. 1–67. Zbl 44.0462.01

[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. Numdam | Zbl 1161.37016 | | MR 2289436

[168] W. A. Veech« Interval exchange transformations », J. Anal. Math. 33 (1978), p. 222–272. Zbl 0455.28006 | MR 516048

[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 266227

[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. Zbl 0971.11007 | MR 1738827