×

zbMATH — the first resource for mathematics

Beta-expansions, natural extensions and multiple tilings associated with Pisot units. (English) Zbl 1295.11010
The present article deals with generalized \(\beta\)-expansions. In particular, the authors construct natural extensions and show multiple tilings associated with them.
Let \(T: X\to X\) be defined by \(Tx=\beta x-a\) for \(x\in X_a\) and \(a\in A\), where \(A\) is a finite subset of \(\mathbb{R}\). This generalizes several kinds of transformations like the classical greedy \(\beta\)-transformation, linear modulo \(1\) transformations, minimal weight expansions and symmetric \(\beta\)-transformations.
In the first part they provide several properties of these transformation like admissibility, ordering, continuity. They also characterize when the shift space is sofic.
The second part deals with natural extensions. They start by characterizing those points which are eventually and purely periodic. Then they construct the natural extension and consider its domain. Finally they provide natural extensions for the different examples originating from the first part.
In the third part they consider multiple tilings. The authors provide a sufficient criterion for the presence of a multiple tiling of a hyperplane and of the torus. At the end of this part they construct such a multiple tiling.

MSC:
11A67 Other number representations
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
28A80 Fractals
28D05 Measure-preserving transformations
37B10 Symbolic dynamics
52C23 Quasicrystals and aperiodic tilings in discrete geometry
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Pierre Arnoux, Valérie Berthé, Hiromi Ei, and Shunji Ito, Tilings, quasicrystals, discrete planes, generalized substitutions, and multidimensional continued fractions, 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. · Zbl 1017.68147
[2] Shigeki Akiyama, Self affine tiling and Pisot numeration system, Number theory and its applications (Kyoto, 1997) Dev. Math., vol. 2, Kluwer Acad. Publ., Dordrecht, 1999, pp. 7 – 17. · Zbl 0999.11065
[3] Shigeki Akiyama, Cubic Pisot units with finite beta expansions, Algebraic number theory and Diophantine analysis (Graz, 1998) de Gruyter, Berlin, 2000, pp. 11 – 26. · Zbl 1001.11038
[4] Shigeki Akiyama, On the boundary of self affine tilings generated by Pisot numbers, J. Math. Soc. Japan 54 (2002), no. 2, 283 – 308. · Zbl 1032.11033
[5] Shigeki Akiyama, Hui Rao, and Wolfgang Steiner, A certain finiteness property of Pisot number systems, J. Number Theory 107 (2004), no. 1, 135 – 160. · Zbl 1052.11055
[6] Shigeki Akiyama and Klaus Scheicher, Symmetric shift radix systems and finite expansions, Math. Pannon. 18 (2007), no. 1, 101 – 124. · Zbl 1164.11007
[7] Veronica Baker, Marcy Barge, and Jaroslaw Kwapisz, Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to \?-shifts, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 7, 2213 – 2248. Numération, pavages, substitutions. · Zbl 1138.37008
[8] Anne Bertrand, Développements en base de Pisot et répartition modulo 1, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 6, A419 – A421 (French, with English summary). · Zbl 0362.10040
[9] Valérie Berthé and Anne Siegel, Tilings associated with beta-numeration and substitutions, Integers 5 (2005), no. 3, A2, 46. · Zbl 1139.37008
[10] Karma Dajani and Cor Kraaikamp, From greedy to lazy expansions and their driving dynamics, Expo. Math. 20 (2002), no. 4, 315 – 327. · Zbl 1030.11035
[11] K. Dajani and C. Kalle. A note on the greedy \( \beta \)-transformation with arbitrary digits. SMF Sem. et Congres, 19:81-102, 2008. · Zbl 1247.37006
[12] Pál Erdös, István Joó, and Vilmos Komornik, Characterization of the unique expansions 1=\sum ^{\infty }\?\?\(_{1}\)\?^{-\?\?} and related problems, Bull. Soc. Math. France 118 (1990), no. 3, 377 – 390 (English, with French summary). · Zbl 0721.11005
[13] Kenneth Falconer, Techniques in fractal geometry, John Wiley & Sons, Ltd., Chichester, 1997. · Zbl 0869.28003
[14] Leopold Flatto and Jeffrey C. Lagarias, The lap-counting function for linear mod one transformations. I. Explicit formulas and renormalizability, Ergodic Theory Dynam. Systems 16 (1996), no. 3, 451 – 491. · Zbl 0865.58016
[15] Leopold Flatto and Jeffrey C. Lagarias, The lap-counting function for linear mod one transformations. II. The Markov chain for generalized lap numbers, Ergodic Theory Dynam. Systems 17 (1997), no. 1, 123 – 146. · Zbl 0979.37017
[16] Leopold Flatto and Jeffrey C. Lagarias, The lap-counting function for linear mod one transformations. III. The period of a Markov chain, Ergodic Theory Dynam. Systems 17 (1997), no. 2, 369 – 403. · Zbl 0979.37018
[17] Natalie Priebe Frank and E. Arthur Robinson Jr., Generalized \?-expansions, substitution tilings, and local finiteness, Trans. Amer. Math. Soc. 360 (2008), no. 3, 1163 – 1177. · Zbl 1138.37010
[18] Christiane Frougny and Boris Solomyak, Finite beta-expansions, Ergodic Theory Dynam. Systems 12 (1992), no. 4, 713 – 723. · Zbl 0814.68065
[19] Christiane Frougny and Wolfgang Steiner, Minimal weight expansions in Pisot bases, J. Math. Cryptol. 2 (2008), no. 4, 365 – 392. · Zbl 1170.11003
[20] Franz Hofbauer, The maximal measure for linear mod one transformations, J. London Math. Soc. (2) 23 (1981), no. 1, 92 – 112. · Zbl 0431.54025
[21] M. Hollander. Linear numeration systems, finite beta-expansions, and discrete spectrum of substitution dynamical systems. Ph.D. thesis, Washington University, 1996.
[22] Shunji Ito and Hui Rao, Purely periodic \?-expansions with Pisot unit base, Proc. Amer. Math. Soc. 133 (2005), no. 4, 953 – 964. · Zbl 1099.11062
[23] Shunji Ito and H. Rao, Atomic surfaces, tilings and coincidence. I. Irreducible case, Israel J. Math. 153 (2006), 129 – 155. · Zbl 1143.37013
[24] Douglas Lind and Brian Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995. · Zbl 1106.37301
[25] Yves Meyer, Algebraic numbers and harmonic analysis, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1972. North-Holland Mathematical Library, Vol. 2. · Zbl 0267.43001
[26] Robert V. Moody, Meyer sets and their duals, The mathematics of long-range aperiodic order (Waterloo, ON, 1995) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 489, Kluwer Acad. Publ., Dordrecht, 1997, pp. 403 – 441. · Zbl 0880.43008
[27] R. Daniel Mauldin and S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988), no. 2, 811 – 829. · Zbl 0706.28007
[28] W. Parry, On the \?-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401 – 416 (English, with Russian summary). · Zbl 0099.28103
[29] Marco Pedicini, Greedy expansions and sets with deleted digits, Theoret. Comput. Sci. 332 (2005), no. 1-3, 313 – 336. · Zbl 1080.11009
[30] Brenda Praggastis, Numeration systems and Markov partitions from self-similar tilings, Trans. Amer. Math. Soc. 351 (1999), no. 8, 3315 – 3349. · Zbl 0984.11008
[31] G. Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France 110 (1982), no. 2, 147 – 178 (French, with English summary). · Zbl 0522.10032
[32] V. A. Rohlin, Exact endomorphisms of a Lebesgue space, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 499 – 530 (Russian).
[33] Klaus Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc. 12 (1980), no. 4, 269 – 278. · Zbl 0494.10040
[34] Anne Siegel, Pure discrete spectrum dynamical system and periodic tiling associated with a substitution, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 2, 341 – 381 (English, with English and French summaries). · Zbl 1083.37009
[35] Boris Solomyak, Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems 17 (1997), no. 3, 695 – 738. · Zbl 0884.58062
[36] Anne Siegel and Jörg M. Thuswaldner, Topological properties of Rauzy fractals, Mém. Soc. Math. Fr. (N.S.) 118 (2009), 140 (English, with English and French summaries). · Zbl 1229.28021
[37] Wolfgang Steiner, Parry expansions of polynomial sequences, Integers 2 (2002), Paper A14, 28. · Zbl 1107.11307
[38] W. Thurston. Groups, tilings and finite state automata. AMS Colloquium lectures, 1989.
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.