Linear Recursive Odometers and Beta-Expansions

Abstract

The aim of this paper is to study the connection between different properties related to β-expansions. In particular, the relation between two conditions, both ensuring purely discrete spectrum of the odometer, is analyzed. The first one is the so-called Hypothesis B for the G-odometers and the second one is denoted by (QM) and it has been introduced in the framework of tilings associated to Pisot β-numerations.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] AKIYAMA, S.: Cubic Pisot units with finite beta expansions, in: Algebraic Number Theory and Diophantine Analysis (Franz Halter-Koch ed. et al.), Proceedings of the international conference, Graz, Austria, August 30-September 5, 1998, de Gruyter, Berlin, 2000, pp. 11–26.

  • [2] AKIYAMA, S.: Positive finiteness of number systems, in: Number Theory. Tradition and Modernization (Zhang, Wenpeng ed. et al.), Papers from the 3rd China-Japan Seminar on Number Theory, Xian, China, February 1216, 2004. Dev. Math. Vol. 15, Springer, New York, NY 2006. pp. 1–10.

  • [3] BARAT, G.—GRABNER, P.: Combinatorial and probabilistic properties of systems of numeration, Ergodic Theory Dynam. Systems 36 (2016), 422–457.

  • [4] BARAT, B.—GRABNER, P. J.—HELLEKALEK, P.: Pierre Liardet (1943–2014) in memoriam, EMS Newsletter, 2015.

  • [5] BARGE, M.: The Pisot conjecture for β-substitutions, arXiv:1505.04408, 2015.

  • [6] BERTRAND-MATHIS, A.: Développement en base θ; répartition modulo un de la suite (n)n≥ 0 ; langages codés et θ-shift, Bull. Soc. Math. France 114 (1986), no. 3, 271–323.

  • [7] FROUGNY, F.—SOLOMYAK, B.: Finite beta-expansions, Ergodic Theory Dynam. Systems 12 (1992), 713–723.

  • [8] GRABNER, P.—LIARDET, P.—TICHY, R.: Odometers and systems of numeration, Acta Arith. 70 (1995), 103–123.

  • [9] HEJDA, T.—STEINER, W.: Beta-expansions of rational numbers in quadratic Pisot bases, arXiv:1411.2419, 2014.

  • [10] HOFER, M.—IACÒ, M. R.—TICHY, R.: Ergodic properties of β-adic Halton sequences, Ergodic Theory Dynam. Systems 35 (2015), no. 3, 895–909.

  • [11] ITO, S.—TAKAHASHI, Y.: Markov subshifts and realization of β-expansions, J. Math. Soc. Japan 26 (1974), 33–55.

    • Crossref
    • Export Citation
  • [12] MINERVINO, M.—STEINER, W.: Tilings for Pisot beta numeration, Indag. Math. (N.S.) 25 (2014), no. 4, 745–773.

  • [13] PARRY, W.: On the β-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416.

    • Crossref
    • Export Citation
  • [14] RÉNYI, A.: Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar 8 (1957), 477–493.

    • Crossref
    • Export Citation
  • [15] SIEGEL, A.—THUSWALDNER, J. M.: Topological properties of Rauzy fractals, Mém. Soc. Math. Fr. (N.S.) 118 (2009), 140 pages.

OPEN ACCESS

Journal + Issues

Search