Invariant densities for random $\beta$-expansions

Abstract

Let β > 1 be a non-integer. We consider expansions of the formare generated by means of a Borel map K β defined on {0, 1} N × [0, ⌊β⌋/(β − 1)]. We show existence and uniqueness of an absolutely continuous K β -invariant probability measure w.r.t. mp ⊗ λ, where mp is the Bernoulli measure on {0, 1} N with parameter p (0 < p < 1) and λ is the normalized Lebesgue measure on [0, ⌊β⌋/(β − 1)]. Furthermore, this measure is of the form mp ⊗µ β,p , where µ β,p is equivalent with λ. We establish the fact that the measure of maximal entropy and mp⊗λ are mutually singular. In case the number 1 has a finite greedy expansion with positive coefficients, the measure mp ⊗ µ β,p is Markov. In the last section we answer a question concerning the number of universal expansions, a notion introduced in [EK].then the lazy transformation L β : J β → J β is defined by1991 Mathematics Subject Classification. Primary:28D05, Secondary:11K16, 28D20, 37A35, 37A45.

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