Comptes Rendus
Probability Theory/Dynamical Systems
On the Gibbs properties of the Erdös measure
Comptes Rendus. Mathématique, Volume 336 (2003) no. 1, pp. 63-68.

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.

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.

Published online:
DOI: 10.1016/S1631-073X(02)00002-X

Eric Olivier 1

1 The Chinese University of Hong Kong, Hong Kong, China
     author = {Eric Olivier},
     title = {On the {Gibbs} properties of the {Erd\"os} measure},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {63--68},
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     volume = {336},
     number = {1},
     year = {2003},
     doi = {10.1016/S1631-073X(02)00002-X},
     language = {en},
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Eric Olivier. On the Gibbs properties of the Erdös measure. Comptes Rendus. Mathématique, Volume 336 (2003) no. 1, pp. 63-68. doi : 10.1016/S1631-073X(02)00002-X.

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