Sofic Measures and Densities of Level Sets

Abstract

Abstract The Bernoulli convolution associated to the real β > 1 and the probability vector (p 0, . . . , pd −1) is a probability measure ηβ,p on ℝ, solution of the self-similarity relation η = ∑ k = 0 d − 1 p k ⋅ η ○ S k − 1 $\eta = \sum\nolimits_{k = 0}^{d - 1} {p_k \cdot \eta \circ S_k^{ - 1} } $ , where S k ( x ) = x + k β $S_k (x) = {{x + k} \over \beta }$ . If β is an integer or a Pisot algebraic number with finite Rényi expansion, ηβ,p is sofic and a Markov chain is naturally associated. If β = b ∈ ℕ and p 0 = ⋯ = p d − 1 = 1 d $p_0 = \cdots = p_{d - 1} = {1 \over d}$ , the study of ηb,p is close to the study of the order of growth of the number of representations in base b with digits in {0, 1, . . . , d − 1}. In the case b = 2 and d = 3 it has also something to do with the metric properties of the continued fractions.