Geodesic
Invariant densities for random $\beta$-expansions
Journal of the European Mathematical Society, Tome 9 (2007) no. 1, pp. 157-176

Voir la notice de l'article provenant de la source EMS Press

Let β>1 be a non-integer. We consider expansions of the form ∑i=1∞​βidi​​, where the digits (di​)i≥1​ are generated by means of a Borel map Kβ​ defined on {0,1}N×[0,⌊β⌋/(β−1)]. We show existence and uniqueness of an absolutely continuous Kβ​-invariant probability measure w.r.t. mp​⊗λ, where mp​ is the Bernoulli measure on {0,1}N with parameter p (01) and λ is the normalized Lebesgue measure on [0,⌊β⌋/(β−1)]. Furthermore, this measure is of the form mp​⊗μβ,p​, where μβ,p​ is equivalent with λ. We establish the fact that the measure of maximal entropy and mp​⊗λ are mutually singular. In case 1 has a finite greedy expansion with positive coefficients, the measure mp​⊗μβ,p​ is Markov. In the last section we answer a question concerning the number of universal expansions, a notion introduced in [EK].
DOI : 10.4171/jems/76
Classification : 28-XX, 11-XX, 37-XX, 00-XX
Keywords: greedy expansions, lazy expansions, absolutely continuous invariant measures, measures of maximal entropy, Markov chains, universal expansions 
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Karma Dajani; Martijn de Vries. Invariant densities for random $\beta$-expansions. Journal of the European Mathematical Society, Tome 9 (2007) no. 1, pp. 157-176. doi: 10.4171/jems/76

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