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Purely periodic \(\beta \)-expansions in the Pisot non-unit case. (English) Zbl 1197.11139
Summary: It is well known that real numbers with a purely periodic decimal expansion are rationals having, when reduced, a denominator coprime with 10. The aim of this paper is to extend this result to beta-expansions with a Pisot base beta which is not necessarily a unit. We characterize real numbers having a purely periodic expansion in such a base. This characterization is given in terms of an explicit set, called a generalized Rauzy fractal, which is shown to be a graph-directed self-affine compact subset of non-zero measure which belongs to the direct product of Euclidean and \(p\)-adic spaces.

MSC:
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
37B10 Symbolic dynamics
11A63 Radix representation; digital problems
11J70 Continued fractions and generalizations
68R15 Combinatorics on words
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References:
[1] Akiyama, S., Pisot numbers and greedy algorithm, (), 9-21 · Zbl 0919.11063
[2] Akiyama, S., Self affine tiling and Pisot numeration system, (), 7-17 · Zbl 0999.11065
[3] Akiyama, S., Cubic Pisot units with finite beta expansions, (), 11-26 · Zbl 1001.11038
[4] Akiyama, S.; Sadahiro, T., A self-similar tiling generated by the minimal Pisot number, Acta math. inform. univ. ostraviensis, 6, 9-26, (1998) · Zbl 1024.11066
[5] Arnoux, P.; Ito, S., Pisot substitutions and Rauzy fractals, Bull. belg. math. soc. Simon stevin, 8, 2, 181-207, (2001) · Zbl 1007.37001
[6] Baker, V.; Barge, M.; Kwapisz, J., Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to beta-shifts, Ann. inst. Fourier (Grenoble), 56, 2213-2248, (2006) · Zbl 1138.37008
[7] Barat, G.; Berthé, V.; Liardet, P.; Thuswaldner, J., Dynamical directions in numeration, Ann. inst. Fourier (Grenoble), 56, 1987-2092, (2006) · Zbl 1138.37005
[8] Barge, M.; Kwapisz, J., Elements of the theory of unimodular Pisot substitutions with an application to β-shifts, (), 89-99 · Zbl 1116.37011
[9] Barge, M.; Kwapisz, J., Geometric theory of unimodular Pisot substitution, Amer. J. math., 128, 5, 1219-1282, (2006) · Zbl 1152.37011
[10] Bassino, F., Beta-expansions for cubic Pisot numbers, (), 141-152 · Zbl 1152.11342
[11] Berthé, V.; Siegel, A., Tilings associated with beta-numeration and substitutions, Integers, 5, 3, (2005), A2, 46 pp. (electronic) · Zbl 1139.37008
[12] Bertrand, A., Développements en base de Pisot et répartition modulo 1, C. R. acad. sci. Paris Sér. A-B, 285, 6, A419-A421, (1977)
[13] Bertrand, A., Codage des endomorphisms de Pisot du tore \([0, 1 [^r\) et mesures simultanément invariantes pour deux homomorphismes du tore, Math. Z., 231, 369-381, (1999) · Zbl 1044.11072
[14] Bertrand-Mathis, A., Développement en base θ répartition modulo un de la suite (xθn)n⩾0 langages codés et θ-shift, Bull. soc. math. France, 114, 3, 271-323, (1986) · Zbl 0628.58024
[15] Bertrand-Mathis, A., Comment écrire LES nombres entiers dans une base qui n’est pas entière, Acta math. hungar., 54, 3-4, 237-241, (1989) · Zbl 0695.10005
[16] Canterini, V.; Siegel, A., Automate des préfixes-suffixes associé à une substitution primitive, J. théor. nombres Bordeaux, 13, 2, 353-369, (2001) · Zbl 1071.37011
[17] Canterini, V.; Siegel, A., Geometric representation of substitutions of Pisot type, Trans. amer. math. soc., 353, 12, 5121-5144, (2001) · Zbl 1142.37302
[18] Ei, H.; Ito, S., Tilings from some non-irreducible, Pisot substitutions, Discrete math. theor. comput. sci., 7, 81-122, (2005) · Zbl 1153.37323
[19] Fabre, S., Substitutions et β-systèmes de numération, Theoret. comput. sci., 137, 2, 219-236, (1995) · Zbl 0872.11017
[20] Frougny, C., Number representation and finite automata, (), 207-228 · Zbl 0976.11003
[21] Frougny, C., Chapter 7: numeration systems, ()
[22] Holton, C.; Zamboni, L.Q., Geometric realizations of substitutions, Bull. soc. math. France, 126, 2, 149-179, (1998) · Zbl 0931.11004
[23] Ito, S.; Kimura, M., On Rauzy fractal, Japan J. indust. appl. math., 8, 3, 461-486, (1991) · Zbl 0734.28010
[24] Ito, S.; Ohtsuki, M., Modified jacobi – perron algorithm and generating Markov partitions for special hyperbolic toral automorphisms, Tokyo J. math., 16, 2, 441-472, (1993) · Zbl 0805.11056
[25] Ito, S.; Rao, H., Purely periodic β-expansion with Pisot base, Proc. amer. math. soc., 133, 953-964, (2005) · Zbl 1099.11062
[26] Ito, S.; Rao, H., Atomic surfaces, tilings and coincidences I. irreducible case, Israel J. math., 153, 129-156, (2006) · Zbl 1143.37013
[27] Ito, S.; Sano, Y., On periodic β-expansions of Pisot numbers and Rauzy fractals, Osaka J. math., 38, 2, 349-368, (2001) · Zbl 0991.11040
[28] Kenyon, R.; Vershik, A., Arithmetic construction of sofic partitions of hyperbolic toral automorphisms, Ergodic theory dynam. systems, 18, 2, 357-372, (1998) · Zbl 0915.58077
[29] Lindenstrauss, E.; Schmidt, K., Symbolic representations of nonexpansive group automorphisms, Israel J. math., 149, 227-266, (2005) · Zbl 1087.37010
[30] Mauldin, R.D.; Williams, S.C., Hausdorff dimension in graph directed constructions, Trans. amer. math. soc., 309, 2, 811-829, (1988) · Zbl 0706.28007
[31] Messaoudi, A., Propriétés arithmétiques et dynamiques du fractal de Rauzy, J. théor. nombres Bordeaux, 10, 1, 135-162, (1998) · Zbl 0918.11048
[32] Messaoudi, A., Frontière du fractal de Rauzy et système de numération complexe, Acta arith., 95, 3, 195-224, (2000) · Zbl 0968.28005
[33] Praggastis, B., Numeration systems and Markov partitions from self-similar tilings, Trans. amer. math. soc., 351, 8, 3315-3349, (1999) · Zbl 0984.11008
[34] Pytheas Fogg, N., Substitutions in dynamics, arithmetics and combinatorics, () · Zbl 1144.11020
[35] Qu, Y.-H.; Rao, H.; Yang, Y.-M., Periods of β-expansions and linear recurrent sequences, Acta arith., 121, 27-37, (2005) · Zbl 1155.11337
[36] Rauzy, G., Nombres algébriques et substitutions, Bull. soc. math. France, 110, 2, 147-178, (1982) · Zbl 0522.10032
[37] Sano, Y., On purely periodic beta-expansions of Pisot numbers, Nagoya math. J., 166, 183-207, (2002) · Zbl 1029.11040
[38] Sano, Y.; Arnoux, P.; Ito, S., Higher dimensional extensions of substitutions and their dual maps, J. anal. math., 83, 183-206, (2001) · Zbl 0987.11013
[39] Schmidt, K., On periodic expansions of Pisot numbers and salem numbers, Bull. London math. soc., 12, 4, 269-278, (1980) · Zbl 0494.10040
[40] Schmidt, K., Algebraic coding of expansive group automorphisms and two-sided beta-shifts, Monatsh. math., 129, 1, 37-61, (2000) · Zbl 1010.37005
[41] Sidorov, N., Bijective and general arithmetic codings for Pisot toral automorphisms, J. dyn. control syst., 7, 4, 447-472, (2001) · Zbl 1134.37313
[42] Sidorov, N., An arithmetic group associated with a Pisot unit, and its symbolic-dynamical representation, Acta arith., 101, 3, 199-213, (2002) · Zbl 0988.11051
[43] Sidorov, N., Arithmetic dynamics, (), 145-189 · Zbl 1051.37007
[44] Sidorov, N.; Vershik, A., Bijective arithmetic codings of hyperbolic automorphisms of the 2-torus, and binary quadratic forms, J. dyn. control syst., 4, 3, 365-399, (1998) · Zbl 0949.37023
[45] A. Siegel, Représentation géométrique, combinatoire et arithmétique des substitutions de type Pisot, PhD thesis, Université de la Méditerranée, 2000
[46] Siegel, A., Représentation des systèmes dynamiques substitutifs non unimodulaires, Ergodic theory dynam. systems, 23, 4, 1247-1273, (2003) · Zbl 1052.37009
[47] Sirvent, V.F.; Wang, Y., Self-affine tiling via substitution dynamical systems and Rauzy fractals, Pacific J. math., 206, 2, 465-485, (2002) · Zbl 1048.37015
[48] Thurston, W.P., Groups, tilings and finite state automata, (), Lecture notes distributed in conjunction with the Colloquium Series · Zbl 0409.58001
[49] Vershik, A.M., Uniform algebraic approximation of shift and multiplication operators, Dokl. akad. nauk SSSR, Soviet math. dokl., 24, 1981, 97-100, (2003), English translation: · Zbl 0484.47005
[50] Vershik, A.M., Arithmetic isomorphism of hyperbolic automorphisms of a torus and of sofic shifts, Funktsional. anal. i prilozhen., 26, 3, 22-27, (1992) · Zbl 0810.58031
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