Abstract
We construct a Lebesgue measure preserving natural extension of a skew product system related to the random β-transformation K β . This allows us to give a formula for the density of the absolutely continuous invariant probability measure of K β , answering a question of Dajani and de Vries, and also to evaluate some estimates on the typical branching rate of the set of β-expansions of a real number.
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Kempton, T. On the invariant density of the random β-transformation. Acta Math Hung 142, 403–419 (2014). https://doi.org/10.1007/s10474-013-0377-x
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DOI: https://doi.org/10.1007/s10474-013-0377-x