Sofic Measures and Densities of Level Sets

The Bernoulli convolution associated to the real β > 1 and the integer d ≥ β is a probablilty measure ηβ,d on R, solution of the self-similarity relation η = d−1 ∑ k=0 pk ·η◦Sk where (p0, . . . , pd−1) is a probabililty vector and Sk(x) = x+k β . If β is an integer or a Pisot algebraic number, the study of this measure is close to the study of the order of growth of the number of representations in base β with digits in {0, 1, . . . , d− 1}. In the case β = 2 and d = 3 it is also related to the continued fractions. 0. Introduction The different sections of this paper are relatively independent. A sofic probability measure on a space {0, 1, . . . , b − 1}N, i.e. the image of a Markov probability measure by a shift-commuting continuous map, is representable by products of matrices as explained in Theorem 5. Now the measures defined by Bernoulli convolution [12], i.e. ηβ,d := ∞ A n=1 ( d−1 ∑