Periodicity of \(\beta\)-expansions for certain Pisot units. (English. French summary) Zbl 1360.37103
Summary: Given \(\beta >1\), let \(T_\beta\)
\[
\begin{aligned} T_\beta : [0,1[&\to [0,1[ \\ x &\to \beta x-\lfloor\beta_x\rfloor.\end{aligned}
\]
The iteration of this transformation gives rise to the greedy \(\beta\)-expansion. There has been extensive research on the properties of this expansion and its dependence on the parameter \(\beta\).
In [Bull. Lond. Math. Soc. 12, 269–278 (1980; Zbl 0494.10040)], K. Schmidt analyzed the set of periodic points of \(T_\beta\), where \(\beta\) is a Pisot number. In an attempt to generalize some of his results, we study, for certain Pisot units, a different expansion that we call linear expansion \[ x=\sum\limits_{i\geq 0} e_i\beta^{-i}, \] where each \(e_i\) can be superior to \(\lfloor\beta\rfloor\), its properties and the relation with \(\mathrm{Per}(\beta)\).
In [Bull. Lond. Math. Soc. 12, 269–278 (1980; Zbl 0494.10040)], K. Schmidt analyzed the set of periodic points of \(T_\beta\), where \(\beta\) is a Pisot number. In an attempt to generalize some of his results, we study, for certain Pisot units, a different expansion that we call linear expansion \[ x=\sum\limits_{i\geq 0} e_i\beta^{-i}, \] where each \(e_i\) can be superior to \(\lfloor\beta\rfloor\), its properties and the relation with \(\mathrm{Per}(\beta)\).
MSC:
| 37E05 | Dynamical systems involving maps of the interval |
| 37C15 | Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems |
| 11C20 | Matrices, determinants in number theory |
| 11R06 | PV-numbers and generalizations; other special algebraic numbers; Mahler measure |
| 11A63 | Radix representation; digital problems |
Citations:
Zbl 0494.10040References:
| [1] | Adamczewski, B., Frougny, C., Siegel, A., Steiner, W. ,Rational numbers with purely periodic {\(\beta\)}-expansion, Bull. London Math. Soc. 42 (2010), 538-552. · Zbl 1211.11010 |
| [2] | Akiyama, S., Pisot Numbers and greedy algorithm, In Number Theory (Eger,1996), pages 9-21. de Gruyter, Berlin, 1998. · Zbl 0919.11063 |
| [3] | Akiyama, S., Self affine tilling and Pisot numeration system, In Number Theory and its applications (Kyoto, 1997), volume 2 of Dev.Math., pages 7-17. Kluwer Acad.Publ., Dordrecht, 1999. |
| [4] | Akiyama, S., Cubic Pisot units with finite beta expansions, in Algebraic number theory and Diophantine analysis, (Graz, 1988), pages 11-26. de Gruyter, Berlin, 2000 · Zbl 1001.11038 |
| [5] | Akiyama,S., Barat, G., Berth\'{}e, V. and Siegel, A., Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions, Monatsh. Math. 155, (2008) 377-419 · Zbl 1190.11005 |
| [6] | Bertrand, A., D\'{}eveloppement en base de Pisot et r\'{}epartition m\'{}odulo 1, C. R. Acad. Sci. Paris S\'{}er. A-B Math 285(6) (1977), A419-A421. · Zbl 0362.10040 |
| [7] | Bertrand-Mathis, A., D\'{}eveloppement en base {\(\theta\)}, r\'{}epartition modulo um de la suite (x{\(\theta\)}n)n\geq0, langages codes e {\(\theta\)}-shift, Bull.Soc. Math France 114 (1986) 271-323 · Zbl 0628.58024 |
| [8] | Boyd, D. W.,Salem numbers of degree four have periodic expansions, Th\'{}eorie des nombres (Quebec, PQ, 1987), 57-64 de Gruyter, Berlin, 1987. |
| [9] | Boyd, D. W., A characterization of two related classes of Salem numbers, Journal Number Theory 50 (1995), no2, 309-317 · Zbl 0824.11069 |
| [10] | Boyd, D. W., On the beta expansion for Salem numbers of degree 6, Mathematics of Computation, vol. 65(1996), no214, 861–875 . · Zbl 0848.11048 |
| [11] | Boyd, D. W., The beta expansion for Salem numbers, Organic mathematics (Burnaby, BC, 1995), 117–131, CMS Conf. Proc., 20, Amer. Math. Soc., Providence, RI, 1997. · Zbl 1053.11536 |
| [12] | Frougny, C., Solomyak, B., Finite {\(\beta\)}-expansions, Ergodic Theory and Dynamical Systems, 12 (1992), 45-82. |
| [13] | Lind, D., Marcus, B., Symbolic Dynamics and Coding, Cambridge University Press, (1995). · Zbl 1106.37301 |
| [14] | Parry, W., On the {\(\beta\)}-expansions of real numbers, Acta Math. Hungar. 11 (1960), 401-416. · Zbl 0099.28103 |
| [15] | Praggastis, B., Numeration systems and Markov partitions from self-similar tillings, Trans. amer. Math. Soc., 351(8), (1999), 3315–3349. · Zbl 0984.11008 |
| [16] | Renyi, A., Representations for real numbers and their ergodic, Acta Math. Sci. Hung., 8 (1957). |
| [17] | Schmidt, K., On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc. 12 (1980), no. 4, 269-278. · Zbl 0494.10040 |
| [18] | Thurston, W., Groups, tilings and finite state automata, AMS Colloquium lectures, 1989. |
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