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Article Dans Une Revue Ergodic Theory and Dynamical Systems Année : 2012

Dynamical properties of the negative beta transformation

Résumé

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 $(-\beta)$-transformation is exact for all $\beta>1$, confirming a conjecture of Góra, and intrinsic, which completes a study of Faller. We also show that the limit behaviour of the $(-\beta)$-expansion of $1$ when $\beta$ tends to $1$ is related to the Thue-Morse sequence. A consequence of the exactness is that every Yrrap number, which is a $\beta>1$ such that the $(-\beta)$-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.
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Dates et versions

hal-00555127 , version 1 (12-01-2011)
hal-00555127 , version 2 (25-01-2011)
hal-00555127 , version 3 (28-06-2011)

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Lingmin Liao, Wolfgang Steiner. Dynamical properties of the negative beta transformation. Ergodic Theory and Dynamical Systems, 2012, 32 (5), pp.1673-1690. ⟨10.1017/S0143385711000514⟩. ⟨hal-00555127v3⟩
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