Measures of maximal entropy for random $\beta$-expansions

Abstract

Let β > 1 be a non-integer. We consider β-expansions of the form ∞ i=1 d i /β i , where the digits (d i ) i≥1 are generated by means of a Borel map K β defined on {0, 1} N ×[0, β /(β − 1)]. We show that K β has a unique mixing measure ν β of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure ν β the digits (d i ) i≥1 form a uniform Bernoulli process. In case 1 has a finite greedy expansion with positive coefficients, the measure of maximal entropy is Markov. We also discuss the uniqueness of β-expansions.

Expansions of Real Numbers in Non-Integer Bases

“…Properties L1 and L2 are well known and have appeared in several articles and with quite short proofs, e.g. [6] and [1], where in the latter paper simple and short dynamical proofs are given. We give the above proofs for two reasons; first, they are elementary and second, to make this exposition self-contained.…”

“…Let q ∈ (1,2]. By an expansion with respect to q, or q-expansion, of a positive real number x we mean a sequence (e i ) i≥1 ⊆ {0, 1} satisfying…”

“…In 1991, Erdös, Horváth, and Joo [3] showed that for almost all q ∈ (1, 2], there are uncountably many different q-expansions, and surprisingly, there exist as well uncountably many exceptional q ∈ (0, 1) for which there is only one q-expansion. In 1998, Komornik and Loreti [5] determined the smallest base q ∈ (1,2] for which the q-expansion of 1 is unique. In 1999, Komornik and Loreti [6] gave a sufficient condition for which the number 1 has exactly two different q-expansions as well as using this information to construct the smallest base q for which the number 1 has exactly two different q-expansions.…”

“…Let q ∈ (1,2]. By a q-expansion of 1, we mean a sequence (e i ) i≥1 of integers in {0, 1} satisfying the equality 1 =…”

Beta-Shifts, Their Languages, and Computability

“…A wellknown class of symbolic dynamical systems is that of the β-shifts introduced by Rényi [26], developed by Parry in the seminal paper [25], and studied intensively thereafter, see for example [15,28,37,21,5,10,29,30,8,7,1].…”