Probability Theory/Dynamical Systems
On the Gibbs properties of the Erdös measure
[Sur les propriétés de Gibbs de la mesure d'Erdös]

Présenté par : Jean-Pierre Kahane

Olivier, Eric1
1 The Chinese University of Hong Kong, Hong Kong, China
Comptes Rendus. Mathématique, Tome 336 (2003) no. 1, pp. 63-68.

Nous considérons les mesures obtenues comme une convolution d'une infinité de mesures de Bernoulli (convolutions de Bernoulli) liées à la β-numération. Une décomposition matricielle markovienne de ces mesures est établie, quand β est un nombre de Pisot dont le β-shift associé est de type fini. Nous concluons en démontrant que la mesure d'Erdös (i.e., quand β est le nombre d'or) est faiblement de Gibbs, assurant ainsi que le formalisme multifractal est valide.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)00002-X
Olivier, Eric 1

1 The Chinese University of Hong Kong, Hong Kong, China 
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Olivier, Eric. On the Gibbs properties of the Erdös measure. Comptes Rendus. Mathématique, Tome 336 (2003) no. 1, pp. 63-68. doi : 10.1016/S1631-073X(02)00002-X. http://archive.numdam.org/articles/10.1016/S1631-073X(02)00002-X/

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