Symbolic dynamics for $\beta$-shifts and self-normal numbers

More than 30 years ago R\'enyi [1] introduced the representations of real numbers with an arbitrary base $\beta > 1$ as a generalization of the $p$-adic representations. One of the most studied problems in this field is the link between expansions to base $\beta$ and ergodic properties of th...

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Bibliographic Details
Published in:Ergodic theory and dynamical systems 1997-06, Vol.17 (3), p.675-694
Main Author: SCHMELING, JÖRG
Format: Article
Language:English
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Summary:More than 30 years ago R\'enyi [1] introduced the representations of real numbers with an arbitrary base $\beta > 1$ as a generalization of the $p$-adic representations. One of the most studied problems in this field is the link between expansions to base $\beta$ and ergodic properties of the corresponding $\beta$-shift. In this paper we will follow the bibliography of Blanchard [2] and give an affirmative answer to a question on the size of the set of real numbers $\beta$ having complicated symbolic dynamics of their $\beta$-shifts.
ISSN:0143-3857
1469-4417
DOI:10.1017/S0143385797079182