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A family of non-sofic beta expansions

Published online by Cambridge University Press:  04 August 2014

SHIGEKI AKIYAMA
SHIGEKI AKIYAMA*
Affiliation:
Institute of Mathematics, University of Tsukuba, Tennodai 1-1-1, Tsukuba, Ibaraki, 305-8571, Japan email akiyama@math.tsukuba.ac.jp
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Abstract

Let ${\it\beta}_{n}>1$ be a root of $x^{n}-x-1$ for $n=4,5,\ldots$ ; we will prove that ${\it\beta}_{n}$ is not a Parry number, i.e., the associated beta transformation does not correspond to a sofic symbolic system. A generalization is shown in the last section.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 2 , April 2016 , pp. 343 - 354
Copyright
© Cambridge University Press, 2014 

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References

Bertin, M.-J., Decomps-Guilloux, A., Grandet-Hugot, M., Pathiaux-Delefosse, M. and Schreiber, J.-P.. Pisot and Salem Numbers. Birkhäuser, Basel, 1992.CrossRefGoogle Scholar
Bertrand, A.. Développements en base de Pisot et répartition modulo 1. C. R. Acad. Sci. Paris Sér. A–B 285(6) (1977), A419–A421.Google Scholar
Boyd, D. W.. Perron units which are not Mahler measures. Ergod. Th. & Dynam. Sys. 6(4) (1986), 485–488.CrossRefGoogle Scholar
Boyd, D. W.. Salem numbers of degree four have periodic expansions. Number Theory. Walter de Gruyter, Berlin, 1989, pp. 57–64.Google Scholar
Boyd, D. W.. On the beta expansion for Salem numbers of degree 6. Math. Comp. 65 (1996), 861–875.CrossRefGoogle Scholar
Filaseta, M., Ford, K. and Konyagin, S.. On an irreducibility theorem of A. Schinzel associated with coverings of the integers. Illinois J. Math. 44(3) (2000), 633–643.CrossRefGoogle Scholar
Hurwitz, A. and Courant, R.. Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen. Interscience Publishers, Inc., New York, 1944.Google Scholar
Ito, Sh. and Takahashi, Y.. Markov subshifts and realization of 𝛽-expansions. J. Math. Soc. Japan 26(1) (1974), 33–55.CrossRefGoogle Scholar
Lang, S.. Algebra (Graduate Texts in Mathematics, 211), 3rd edn. Springer, New York, 2002.CrossRefGoogle Scholar
Ljunggren, W.. On the irreducibility of certain trinomials and quadrinomials. Math. Scand. 8 (1960), 65–70.CrossRefGoogle Scholar
Mills, W. H.. The factorization of certain quadrinomials. Math. Scand. 57(1) (1985), 44–50.CrossRefGoogle Scholar
Parry, W.. On the 𝛽-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416.CrossRefGoogle Scholar
Schinzel, A.. On the reducibility of polynomials and in particular of trinomials. Acta Arith. 11 (1965), 1–34.CrossRefGoogle Scholar
Schinzel, A.. Reducibility of polynomials and covering systems of congruences. Acta Arith. 13 (1967/1968), 91–101.CrossRefGoogle Scholar
Schinzel, A.. Polynomials with Special Regard to Reducibility (Encyclopedia of Mathematics and its Applications, 77). Cambridge University Press, Cambridge, 2000, with an appendix by Umberto Zannier.CrossRefGoogle Scholar
Schmidt, K.. On periodic expansions of Pisot numbers and Salem numbers. Bull. Lond. Math. Soc. 12 (1980), 269–278.CrossRefGoogle Scholar
Selmer, E. S.. On the irreducibility of certain trinomials. Math. Scand. 4 (1956), 287–302.CrossRefGoogle Scholar
Smyth, C. J.. On the product of the conjugates outside the unit circle of an algebraic integer. Bull. Lond. Math. Soc. 3 (1971), 169–175.CrossRefGoogle Scholar
Solomyak, B.. Conjugates of beta-numbers and the zero-free domain for a class of analytic functions. Proc. Lond. Math. Soc. 68 (1994), 477–498.CrossRefGoogle Scholar
Whittaker, E. T. and Watson, G. N.. A Course in Modern Analysis. Cambridge University Press, Cambridge, 1927.Google Scholar