On the Gibbs properties of the Erdös measure

Abstract

We consider the infinite convolved Bernoulli measures (Bernoulli convolutions) related to β-numeration. A Markovian matrix decomposition of these measures is obtained when β is a Pisot number whose associated β-shift is of finite type. We study the special case of the Erdös measure (i.e., when β is the golden ratio) that we prove to be weak Gibbs, insuring the multifractal formalism to hold. To cite this article: E. Olivier, C. R. Acad. Sci. Paris, Ser. I 336 (2003).