Beta-expansions, natural extensions and multiple tilings associated with Pisot units
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- by Charlene Kalle and Wolfgang Steiner PDF
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Abstract:
From the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit $\beta$ and the greedy $\beta$-transformation. In this paper, we consider different transformations generating expansions in base $\beta$, including cases where the associated subshift is not sofic. Under certain mild conditions, we show that they give multiple tilings. We also give a necessary and sufficient condition for the tiling property, generalizing the weak finiteness property (W) for greedy $\beta$-expansions. Remarkably, the symmetric $\beta$-transformation does not satisfy this condition when $\beta$ is the smallest Pisot number or the Tribonacci number. This means that the Pisot conjecture on tilings cannot be extended to the symmetric $\beta$-transformation.
Closely related to these (multiple) tilings are natural extensions of the transformations, which have many nice properties: they are invariant under the Lebesgue measure; under certain conditions, they provide Markov partitions of the torus; they characterize the numbers with purely periodic expansion, and they allow determining any digit in an expansion without knowing the other digits.
References
- 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. MR 1888763
- 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. MR 1738803
- Shigeki Akiyama, Cubic Pisot units with finite beta expansions, Algebraic number theory and Diophantine analysis (Graz, 1998) de Gruyter, Berlin, 2000, pp. 11–26. MR 1770451
- Shigeki Akiyama, On the boundary of self affine tilings generated by Pisot numbers, J. Math. Soc. Japan 54 (2002), no. 2, 283–308. MR 1883519, DOI 10.2969/jmsj/05420283
- Shigeki Akiyama, Hui Rao, and Wolfgang Steiner, A certain finiteness property of Pisot number systems, J. Number Theory 107 (2004), no. 1, 135–160. MR 2059954, DOI 10.1016/j.jnt.2004.02.001
- Shigeki Akiyama and Klaus Scheicher, Symmetric shift radix systems and finite expansions, Math. Pannon. 18 (2007), no. 1, 101–124. MR 2321960
- Veronica Baker, Marcy Barge, and Jaroslaw Kwapisz, Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to $\beta$-shifts, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 7, 2213–2248. Numération, pavages, substitutions. MR 2290779
- 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). MR 447134
- Valérie Berthé and Anne Siegel, Tilings associated with beta-numeration and substitutions, Integers 5 (2005), no. 3, A2, 46. MR 2191748
- Karma Dajani and Cor Kraaikamp, From greedy to lazy expansions and their driving dynamics, Expo. Math. 20 (2002), no. 4, 315–327. MR 1940010, DOI 10.1016/S0723-0869(02)80010-X
- K. Dajani and C. Kalle. A note on the greedy $\beta$-transformation with arbitrary digits. SMF Sem. et Congres, 19:81–102, 2008.
- Pál Erdös, István Joó, and Vilmos Komornik, Characterization of the unique expansions $1=\sum ^\infty _{i=1}q^{-n_i}$ and related problems, Bull. Soc. Math. France 118 (1990), no. 3, 377–390 (English, with French summary). MR 1078082
- Kenneth Falconer, Techniques in fractal geometry, John Wiley & Sons, Ltd., Chichester, 1997. MR 1449135
- 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. MR 1395048, DOI 10.1017/S0143385700008920
- 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. MR 1440771, DOI 10.1017/S0143385797069691
- 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. MR 1444059, DOI 10.1017/S0143385797069812
- Natalie Priebe Frank and E. Arthur Robinson Jr., Generalized $\beta$-expansions, substitution tilings, and local finiteness, Trans. Amer. Math. Soc. 360 (2008), no. 3, 1163–1177. MR 2357692, DOI 10.1090/S0002-9947-07-04527-8
- Christiane Frougny and Boris Solomyak, Finite beta-expansions, Ergodic Theory Dynam. Systems 12 (1992), no. 4, 713–723. MR 1200339, DOI 10.1017/S0143385700007057
- Christiane Frougny and Wolfgang Steiner, Minimal weight expansions in Pisot bases, J. Math. Cryptol. 2 (2008), no. 4, 365–392. MR 2549463, DOI 10.1515/JMC.2008.017
- Franz Hofbauer, The maximal measure for linear mod one transformations, J. London Math. Soc. (2) 23 (1981), no. 1, 92–112. MR 602242, DOI 10.1112/jlms/s2-23.1.92
- M. Hollander. Linear numeration systems, finite beta-expansions, and discrete spectrum of substitution dynamical systems. Ph.D. thesis, Washington University, 1996.
- Shunji Ito and Hui Rao, Purely periodic $\beta$-expansions with Pisot unit base, Proc. Amer. Math. Soc. 133 (2005), no. 4, 953–964. MR 2117194, DOI 10.1090/S0002-9939-04-07794-9
- Shunji Ito and H. Rao, Atomic surfaces, tilings and coincidence. I. Irreducible case, Israel J. Math. 153 (2006), 129–155. MR 2254640, DOI 10.1007/BF02771781
- Douglas Lind and Brian Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995. MR 1369092, DOI 10.1017/CBO9780511626302
- Yves Meyer, Algebraic numbers and harmonic analysis, North-Holland Mathematical Library, Vol. 2, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1972. MR 0485769
- 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. MR 1460032
- R. Daniel Mauldin and S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988), no. 2, 811–829. MR 961615, DOI 10.1090/S0002-9947-1988-0961615-4
- W. Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416 (English, with Russian summary). MR 142719, DOI 10.1007/BF02020954
- Marco Pedicini, Greedy expansions and sets with deleted digits, Theoret. Comput. Sci. 332 (2005), no. 1-3, 313–336. MR 2122508, DOI 10.1016/j.tcs.2004.11.002
- Brenda Praggastis, Numeration systems and Markov partitions from self-similar tilings, Trans. Amer. Math. Soc. 351 (1999), no. 8, 3315–3349. MR 1615950, DOI 10.1090/S0002-9947-99-02360-0
- G. Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France 110 (1982), no. 2, 147–178 (French, with English summary). MR 667748
- V. A. Rohlin, Exact endomorphisms of a Lebesgue space, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 499–530 (Russian). MR 0143873
- Klaus Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc. 12 (1980), no. 4, 269–278. MR 576976, DOI 10.1112/blms/12.4.269
- 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). MR 2073838
- Boris Solomyak, Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems 17 (1997), no. 3, 695–738. MR 1452190, DOI 10.1017/S0143385797084988
- 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). MR 2721985
- Wolfgang Steiner, Parry expansions of polynomial sequences, Integers 2 (2002), Paper A14, 28. MR 1945950
- W. Thurston. Groups, tilings and finite state automata. AMS Colloquium lectures, 1989.
Additional Information
- Charlene Kalle
- Affiliation: Department of Mathematics, Utrecht University, Postbus 80.000, 3508 TA Utrecht, The Netherlands
- Address at time of publication: Institute of Mathematics, Leiden University, Postbus 9512, 2300RA, Leiden, The Netherlands
- Email: kallecccj@math.leidenuniv.nl
- Wolfgang Steiner
- Affiliation: LIAFA, CNRS, Université Paris Diderot – Paris 7, Case 7014, 75205 Paris Cedex 13, France
- MR Author ID: 326598
- Email: steiner@liafa.jussieu.fr
- Received by editor(s): July 31, 2009
- Received by editor(s) in revised form: January 26, 2010
- Published electronically: January 6, 2012
- Additional Notes: The first author was partly supported by the EU FP6 Marie Curie Research Training Network CODY (MRTN 2006 035651).
The second author was supported by the French Agence Nationale de la Recherche, grant ANR–06–JCJC–0073 “DyCoNum”. - © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 2281-2318
- MSC (2010): Primary 11A63, 11R06, 28A80, 28D05, 37B10, 52C22, 52C23
- DOI: https://doi.org/10.1090/S0002-9947-2012-05362-1
- MathSciNet review: 2888207