Linear Recursive Odometers and Beta-Expansions

Maria Rita Iacò; Wolfgang Steiner; Robert F. Tichy

doi:10.1515/udt-2016-0010

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Linear Recursive Odometers and Beta-Expansions

Uniform distribution theory

Volume 11 (2016): Issue 1 (June 2016)

By: Maria Rita Iacò, Wolfgang Steiner and Robert F. Tichy

Open Access

|Jan 2017

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.

References

[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.
Search in Google Scholar Back to article
[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. 1–10.10.1007/0-387-30829-6_1
Search in Google Scholar Back to article
[3] BARAT, G.—GRABNER, P.: Combinatorial and probabilistic properties of systems of numeration, Ergodic Theory Dynam. Systems 36 (2016), 422–457.10.1017/etds.2014.69
Search in Google Scholar Back to article
[4] BARAT, B.—GRABNER, P. J.—HELLEKALEK, P.: Pierre Liardet (1943–2014) in memoriam, EMS Newsletter, 2015.
Search in Google Scholar Back to article
[5] BARGE, M.: The Pisot conjecture for β-substitutions, arXiv:1505.04408, 2015.
Search in Google Scholar Back to article
[6] BERTRAND-MATHIS, A.: Développement en base θ; répartition modulo un de la suite (xθⁿ)_n≥₀; langages codés et θ-shift, Bull. Soc. Math. France 114 (1986), no. 3, 271–323.
Search in Google Scholar Back to article
[7] FROUGNY, F.—SOLOMYAK, B.: Finite beta-expansions, Ergodic Theory Dynam. Systems 12 (1992), 713–723.10.1017/S0143385700007057
Search in Google Scholar Back to article
[8] GRABNER, P.—LIARDET, P.—TICHY, R.: Odometers and systems of numeration, Acta Arith. 70 (1995), 103–123.10.4064/aa-70-2-103-123
Search in Google Scholar Back to article
[9] HEJDA, T.—STEINER, W.: Beta-expansions of rational numbers in quadratic Pisot bases, arXiv:1411.2419, 2014.
Search in Google Scholar Back to article
[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.
Search in Google Scholar Back to article
[11] ITO, S.—TAKAHASHI, Y.: Markov subshifts and realization of β-expansions, J. Math. Soc. Japan 26 (1974), 33–55.10.2969/jmsj/02610033
Open DOI Search in Google Scholar Back to article
[12] MINERVINO, M.—STEINER, W.: Tilings for Pisot beta numeration, Indag. Math. (N.S.) 25 (2014), no. 4, 745–773.
Search in Google Scholar Back to article
[13] PARRY, W.: On the β-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416.10.1007/BF02020954
Open DOI Search in Google Scholar Back to article
[14] RÉNYI, A.: Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar 8 (1957), 477–493.10.1007/BF02020331
Open DOI Search in Google Scholar Back to article
[15] SIEGEL, A.—THUSWALDNER, J. M.: Topological properties of Rauzy fractals, Mém. Soc. Math. Fr. (N.S.) 118 (2009), 140 pages.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.1515/udt-2016-0010 | Journal eISSN: 2309-5377 | Journal ISSN: 1336-913X

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Language: English

Page range: 175 - 186

Submitted on: Apr 14, 2016

Accepted on: Apr 26, 2016

Published on: Jan 13, 2017

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 2 issues per year

Keywords:

beta-expansions,

odometer,

purely discrete spectrum,

finiteness property

Related subjects:

Mathematics,

Algebra and number theory,

Discrete mathematics,

Probability and statistics,

Numerical and computational mathematics

© 2017 Maria Rita Iacò, Wolfgang Steiner, Robert F. Tichy, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 11 (2016): Issue 1 (June 2016)