Weak Gibbs property and system of numeration

Abstract

We study the selfsimilarity and the Gibbs properties of several measures defined on the product space $Ω\_r:=\{0,1,...,\break r-1\}^{\mathbb N}$. This space can be identified with the interval $[0,1]$ by means of the numeration in base $r$. The last section is devoted to the Bernoulli convolution in base $\beta =\frac{1+\sqrt{5}}{2}$, called the Erd\H os measure, and its analogue in base $-\beta =-\frac{1+\sqrt{5}}{2}$, that we study by means of a suitable system of numeration.