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Dynamical properties of the negative beta-transformation

Published online by Cambridge University Press:  14 October 2011

LINGMIN LIAO  and
WOLFGANG STEINER
LINGMIN LIAO
Affiliation:
LAMA, CNRS UMR 8050, Université Paris-Est—Créteil—Val-de-Marne, UFR Sciences et Technologie, 61 Avenue du Général de Gaulle, 94010 Créteil Cedex, France (email: lingmin.liao@u-pec.fr)
WOLFGANG STEINER
Affiliation:
LIAFA, CNRS UMR 7089, Université Paris Diderot – Paris 7, Case 7014, 75205 Paris Cedex 13, France (email: steiner@liafa.jussieu.fr)
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Abstract

We analyse dynamical properties of the negative beta-transformation, which has been studied recently by Ito and Sadahiro. Contrary to the classical beta-transformation, the density of the absolutely continuous invariant measure of the negative beta-transformation may be zero on certain intervals. By investigating this property in detail, we prove that the (−β)-transformation is exact for all β>1, confirming a conjecture of Góra, and intrinsic, which completes a study of Faller. We also show that the limit behaviour of the (−β)-expansion of 1 when β tends to 1 is related to the Thue–Morse sequence. A consequence of the exactness is that every Yrrap number, which is a β>1 such that the (−β) -expansion of 1 is eventually periodic, is a Perron number. This extends a well-known property of Parry numbers. However, the set of Parry numbers is different from the set of Yrrap numbers.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 5 , October 2012 , pp. 1673 - 1690
Copyright
Copyright © Cambridge University Press 2011

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