Functional analysis/Dynamical systems
Generalized adding machines and Julia sets
Comptes Rendus. Mathématique, Volume 354 (2016) no. 11, pp. 1096-1100.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2016.09.009
Messaoudi, Ali 1; Valle, Glauco 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
@article{CRMATH_2016__354_11_1096_0,
     author = {Messaoudi, Ali and Valle, Glauco},
     title = {Generalized adding machines and {Julia} sets},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1096--1100},
     publisher = {Elsevier},
     volume = {354},
     number = {11},
     year = {2016},
     doi = {10.1016/j.crma.2016.09.009},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2016.09.009/}
}
TY  - JOUR
AU  - Messaoudi, Ali
AU  - Valle, Glauco
TI  - Generalized adding machines and Julia sets
JO  - Comptes Rendus. Mathématique
PY  - 2016
SP  - 1096
EP  - 1100
VL  - 354
IS  - 11
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2016.09.009/
DO  - 10.1016/j.crma.2016.09.009
LA  - en
ID  - CRMATH_2016__354_11_1096_0
ER  - 
%0 Journal Article
%A Messaoudi, Ali
%A Valle, Glauco
%T Generalized adding machines and Julia sets
%J Comptes Rendus. Mathématique
%D 2016
%P 1096-1100
%V 354
%N 11
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2016.09.009/
%R 10.1016/j.crma.2016.09.009
%G en
%F CRMATH_2016__354_11_1096_0
Messaoudi, Ali; Valle, Glauco. 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. http://www.numdam.org/articles/10.1016/j.crma.2016.09.009/

[1] Abdalaoui, E.H.; Messaoudi, A. On the spectrum of stochastic perturbations of the shift and Julia sets, Fundam. Math., Volume 218 (2012) no. 1, pp. 47-68

[2] Bayart, F.; Matheron, E. Dynamics of Linear Operators, Camb. Tracts Math., vol. 179, Cambridge University Press, Cambridge, UK, 2009

[3] Dunford, N.; Schwartz, J.T. Linear Operators. Part I. General Theory, Wiley, New York, 1988

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

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

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

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

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

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

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

Cited by Sources: