Comptes Rendus
no. 1
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

Eric Olivier1
1 The Chinese University of Hong Kong, Hong Kong, China
Comptes Rendus. Mathématique, Volume 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
Eric Olivier 1

1 The Chinese University of Hong Kong, Hong Kong, China 
@article{CRMATH_2003__336_1_63_0,
     author = {Eric Olivier},
     title = {On the {Gibbs} properties of the {Erd\"os} measure},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {63--68},
     publisher = {Elsevier},
     volume = {336},
     number = {1},
     year = {2003},
     doi = {10.1016/S1631-073X(02)00002-X},
     language = {en},
}
TY  - JOUR
AU  - Eric Olivier
TI  - On the Gibbs properties of the Erdös measure
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 63
EP  - 68
VL  - 336
IS  - 1
PB  - Elsevier
DO  - 10.1016/S1631-073X(02)00002-X
LA  - en
ID  - CRMATH_2003__336_1_63_0
ER  - 
%0 Journal Article
%A Eric Olivier
%T On the Gibbs properties of the Erdös measure
%J Comptes Rendus. Mathématique
%D 2003
%P 63-68
%V 336
%N 1
%I Elsevier
%R 10.1016/S1631-073X(02)00002-X
%G en
%F CRMATH_2003__336_1_63_0
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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)00002-X/

[1] J. Alexander; J. York Fat baker's transformations, Ergodic Theory Dynamical Systems, Volume 4 (1984), pp. 1-23

[2] P. Erdös On a family of Bernoulli convolutions, Amer. J. Math., Volume 61 (1939), pp. 974-976

[3] D.-J. Feng, E. Olivier, Multifractal analysis of the weak Gibbs measures and phase transition – application to some Bernoulli convolutions, Ergodic Theory Dynamical Systems, 2001, submitted

[4] B. Jessen; A. Wintner Distribution functions and the Riemann zeta function, Trans. Amer. Math. Soc., Volume 38 (1935), pp. 48-88

[5] K.-S. Lau; S.-M. Ngai Lq-spectrum of the Bernoulli convolution associated with the Golden Ratio, Stud. Math., Volume 131 (1998), pp. 17-29

[6] F. Ledrappier; A. Porzio A dimension formula for Bernoulli convolution, J. Statist. Phys., Volume 76 (1994), pp. 225-251

[7] E. Olivier, N. Sidorov, A. Thomas, On the Gibbs properties of Bernoulli convolutions related to β-numeration, Preprint CUHK, 2002

[8] Y. Peres; W. Schlag; B. Solomyak Sixty years of Bernoulli convolutions, Progr. Probab., 46, Birkhäuser, 2000, pp. 40-65

[9] N. Sidorov; A. Vershik Ergodic properties of the Erdös measure, the entropy of the Golden Shift and related problems, Monatsh. Math., Volume 126 (1998), pp. 215-261

[10] M. Yuri Zeta functions for certain non-hyperbolic systems and topological Markov approximations, Ergodic Theory Dynamical Systems, Volume 17 (1997), pp. 997-1000

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

A property of the spectra of non-Pisot numbers

Toufik Zaïmi

C. R. Math (2010)