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The natural extension of the random beta-transformation

Part of: Elementary number theory Random dynamical systems Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  04 November 2022

YOUNÈS TIERCE Open the ORCID record for YOUNÈS TIERCE [Opens in a new window]
YOUNÈS TIERCE*
Affiliation:
Univ. Rouen Normandie, CNRS, LMRS, UMR 6085, F-76000 Rouen, France
*
e-mail: [email protected]
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Abstract

We construct a geometrico-symbolic version of the natural extension of the random $\beta $-transformation introduced by Dajani and Kraaikamp [Random $\beta $-expansions. Ergod. Th. & Dynam. Sys. 23(2) (2003) 461–479]. This construction provides a new proof of the existence of a unique absolutely continuous invariant probability measure for the random $\beta $-transformation, and an expression for its density. We then prove that this natural extension is a Bernoulli automorphism, generalizing to the random case the result of Smorodinsky [$\beta $-automorphisms are Bernoulli shifts. Acta Math. Acad. Sci. Hungar. 24 (1973), 273–278] about the greedy transformation.

Keywords

Ergodic theorybeta expansionsBernoulli processes

MSC classification

Primary: 37H99: None of the above, but in this section
Secondary: 11A63: Radix representation; digital problems 60B99: None of the above, but in this section
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 11 , November 2023 , pp. 3861 - 3896
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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