Sofic measures and densities of level sets
Résumé
The Bernoulli convolution associated to the real $\beta>1$ and the integer $d\ge\beta$ is a probablilty measure $\eta_{\beta,d}$ on $\mathbb R$, solution of the self-similarity relation $\displaystyle\eta=\sum_{k=0}^{d-1}p_k\cdot\eta\circ S_k$ where $(p_0,\dots,p_{d-1})$ is a probabililty vector and $S_k(x)=\frac{x+k}\beta$. If $\beta$ 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 $\beta$ with digits in $\{0,1,\dots,d-1\}$. In the case $\beta=2$ and $d=3$ it is also related to the continued fractions.
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