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Dynamics for \(\beta \)-shifts and Diophantine approximation. (English) Zbl 1140.11035

The authors investigate the \(\beta \)-expansion of an algebraic number in an algebraic base \(\beta \). Using tools from Diophantine approximation, they prove several results that may suggest a strong difference between the asymptotic behaviour of eventually periodic expansions and that of non-eventually periodic expansions. Among interesting results proved by the authors, we give the following one.
Let \(\beta\) be a Pisot or a Salem number. Let \(\alpha\) be an algebraic number in \([0,1)\). Then the \(\beta\)-expansion of \(\alpha\) can be generated by a finite automaton if and only if it is eventually periodic.

MSC:

11J81 Transcendence (general theory)
11J13 Simultaneous homogeneous approximation, linear forms
37B10 Symbolic dynamics
68R15 Combinatorics on words
11B85 Automata sequences
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References:

[1] Verger-Gaugry, Ann. Inst. Fourier (Grenoble) 56 pp 2565– (2006) · Zbl 1177.11013
[2] DOI: 10.1017/S0143385797079182 · Zbl 0908.58017
[3] DOI: 10.1007/BF02020954 · Zbl 0099.28103
[4] DOI: 10.1007/BF01534862
[5] Bertrand-Mathis, Bull. Soc. Math. France 114 pp 271– (1986)
[6] DOI: 10.1007/BF02588048 · Zbl 1195.11093
[7] Borel, C. R. Acad. Sci. Paris 230 pp 591– (1950)
[8] DOI: 10.1007/BF02020331 · Zbl 0079.08901
[9] DOI: 10.1016/0304-3975(89)90038-8 · Zbl 0682.68081
[10] Gazeau, J. Th?or. Nombres Bordeaux 16 pp 125– (2004) · Zbl 1075.11007
[11] Adamczewski, Ann. of Math. (2) 165 pp 547– (2007) · Zbl 1195.11094
[12] DOI: 10.1016/j.tcs.2004.03.035 · Zbl 1068.68112
[13] DOI: 10.1007/BF01454845 · JFM 55.0115.01
[14] DOI: 10.2307/2695302 · Zbl 0997.11052
[15] Ito, J. Math. Soc. Japan 27 pp 20– (1975) · Zbl 0292.10040
[16] DOI: 10.1023/A:1015594913393 · Zbl 1010.11038
[17] Evertse, Compos. Math. 101 pp 225– (1996)
[18] Adamczewski, C. R. Acad. Sci. Paris 339 pp 11– (2004) · Zbl 1119.11019
[19] DOI: 10.1016/j.aam.2006.04.001 · Zbl 1117.68059
[20] Berth?, Integers 5 pp none– (2005)
[21] Bertrand-Mathis, Ergod. Th. and Dynam. Sys. 8 pp 35– (1988) · Zbl 0657.28014
[22] Lagarias, Experiment. Math. 10 pp 355– (2001) · Zbl 1015.11036
[23] DOI: 10.1090/S0025-5718-96-00700-4 · Zbl 0848.11048
[24] DOI: 10.1112/blms/12.4.269 · Zbl 0494.10040
[25] Bailey, Experiment. Math. 10 pp 175– (2001) · Zbl 1047.11073
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