Comptes Rendus
no. 11
Functional analysis/Dynamical systems
Generalized adding machines and Julia sets
[Odomètre stochastique et ensembles de Julia]

Présenté par : Étienne Ghys

Ali Messaoudi1 ; Glauco Valle2
1 Universidade Estadual Paulista – Departamento de Matemática do Instituto de Biociências Letras e Ciências Exatas, Rua Cristóvão Colombo, 2265, Jardim Nazareth, 15054-000 São José do Rio Preto, SP, Brazil
2 Universidade Federal do Rio de Janeiro – Departamento de Métodos Estatísticos do Instituto de Matemática, Caixa Postal 68530, 21945-970, Rio de Janeiro, Brazil
Comptes Rendus. Mathématique, Volume 354 (2016) no. 11, pp. 1096-1100.

Nous définissons l'odomètre stochastique associé à un système de numération de Cantor. Nous calculons les parties du spectre de l'opérateur de transition associé à cet odométre dans différents espaces de Banach classiques. Nous montrons que le spectre est lié aux ensembles de Julia fibrés.

We define stochastic adding machines based on Cantor Systems of numeration. Our aim here is to compute the parts of spectra of the transition operators associated with these stochastic adding machines in different classical Banach spaces. We show that these spectra are connected to fibered Julia sets.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2016.09.009
Ali Messaoudi 1 ; Glauco Valle 2

1 Universidade Estadual Paulista – Departamento de Matemática do Instituto de Biociências Letras e Ciências Exatas, Rua Cristóvão Colombo, 2265, Jardim Nazareth, 15054-000 São José do Rio Preto, SP, Brazil
2 Universidade Federal do Rio de Janeiro – Departamento de Métodos Estatísticos do Instituto de Matemática, Caixa Postal 68530, 21945-970, Rio de Janeiro, Brazil 
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     title = {Generalized adding machines and {Julia} sets},
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Ali Messaoudi; Glauco Valle. Generalized adding machines and Julia sets. Comptes Rendus. Mathématique, Volume 354 (2016) no. 11, pp. 1096-1100. doi : 10.1016/j.crma.2016.09.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.09.009/

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[4] G.H. Hardy; E.M. Wright An Introduction to the Theory of Numbers, Oxford University Press, Oxford, UK, 1954

[5] P. Killeen; T. Taylor A stochastic adding machine and complex dynamics, Nonlinearity, Volume 13 (2000) no. 6, pp. 1889-1903

[6] P. Kirschenhofer; R.F. Tichy On the distribution of digits in Cantor representations of integers, J. Number Theory, Volume 18 (1984), pp. 121-134

[7] A. Messaoudi; O. Sester; G. Valle Stochastic adding machines and fibered Julia sets, Stoch. Dyn., Volume 13 (2013) no. 3, p. 26 pages

[8] A. Messaoudi; G. Valle Spectra of stochastic adding machines based on Cantor systems of numeration, 2013 | arXiv

[9] O. Sester Hyperbolicité des polynômes fibrés, Bull. Soc. Math. Fr., Volume 127 (1999), pp. 393-428

[10] K. Yosida Functional Analysis, Springer, 1980

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