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Pré-Publication, Document De Travail Année : 2011

Dynamical properties of the negative beta transformation

Résumé

We analyse basic dynamical properties of the negative beta transformation, which has been studied recently by Ito and Sadahiro. Contrary to the classical beta transformation, 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 for any $\beta>1$ the $(-\beta)$-transformation is exact, 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 $(-\beta)$-number, which is a $\beta>1$ such that the $(-\beta)$-expansion of $1$ is eventually periodic, is a Perron number. This generalises a well-known property of Parry numbers (which were called $\beta$-numbers by Parry). Nevertheless, the properties $\beta$-number and $(-\beta)$-number are not equivalent.
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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. 2011. ⟨hal-00555127v1⟩
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