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The natural extension of the random beta-transformation
Part of:
Elementary number theory
Probability theory on algebraic and topological structures
Random dynamical systems
Published online by Cambridge University Press: 04 November 2022
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.
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- Original Article
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- © The Author(s), 2022. Published by Cambridge University Press
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