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December  2014, 34(12): 5247-5269. doi: 10.3934/dcds.2014.34.5247

Stochastic adding machine and $2$-dimensional Julia sets

Ali Messaoudi 1, and  Rafael Asmat Uceda 2,

1. 

UNESP - Departamento de Matemática do, Instituto de Biociências, Letras e Ciências, Exatas de São José do Rio Preto, Brazil
2. 

Université de Picardie Jules Verne, Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées, CNRS - UMR 7352, France

Received  June 2013 Revised  March 2014 Published  June 2014

In this work we define a stochastic adding machine associated to a quadratic base $(F_n)_{n \geq 0}$ formed by recurrent sequences of order 2. We obtain a Markov chain with states in $\mathbb{Z}^+$ and we prove that the spectrum of the transition operator associated to this Markov chain is connected to the filled Julia sets for a class of endomorphisms in $\mathbb{C}^2$ of which we study topological properties.
Keywords: recurrent sequence, Julia sets, stochastic adding machines, spectrum of transition operator., Markov chains.
Mathematics Subject Classification: Primary: 37A30, 37F50; Secondary: 47A1.
Citation: Ali Messaoudi, Rafael Asmat Uceda. Stochastic adding machine and $2$-dimensional Julia sets. Discrete & Continuous Dynamical Systems, 2014, 34 (12) : 5247-5269. doi: 10.3934/dcds.2014.34.5247
References:
[1]

E. H. El Abdalaoui and A. Messaoudi, On the spectrum of stochastic perturbations of the shift map and Julia sets, Fundamenta Mathematicae, 218 (2012), 47-68. doi: 10.4064/fm218-1-3.  Google Scholar

[2]

El Abdalaoui, S. Bonnot, A. Messaoudi and O. Sester, On the Fibonacci Complex Dynamical Systems, arXiv:1304.4864, 26p, 2013. Google Scholar

[3]

E. Bedford and J. Smillie, John Polynomial diffeomorphisms of $\mathbbC^2$ VI. Connectivity of J, Ann. of Math. (2), 148 (1998), 695-735. doi: 10.2307/121006.  Google Scholar

[4]

R. L. Devaney, An Introduction to Chaothic Dynamical Systems, ABP, 2003.  Google Scholar

[5]

S. Eilenberg, Automata, Languages, and Machines, Volume 1, Academic Press, New York and London, 1974.  Google Scholar

[6]

J. E. Fornaess and N. Sibony, Fatou and Julia sets for entire mappings in $\mathbbC^k$, Math. Ann., 311 (1998), 27-40. doi: 10.1007/s002080050174.  Google Scholar

[7]

A. S. Fraenkel, Systems of numeration, Amer. Math. Monthly, 92 (1985), 105-114. doi: 10.2307/2322638.  Google Scholar

[8]

C. Frougny, Representation of numbers and finite automata, Math. Systems Theory, 25 (1992), 37-60. doi: 10.1007/BF01368783.  Google Scholar

[9]

P. R. Killeen and T. J. Taylor, A stochastic adding machine and complex dynamics, Nonlinearity, 13 (2000), 1889-1903. doi: 10.1088/0951-7715/13/6/302.  Google Scholar

[10]

S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge University Press, 2000.  Google Scholar

[11]

J. Milnor, Dynamics in One Complex Variable, Princeton University Press, 2006.  Google Scholar

[12]

A. Messaoudi and D. Smania, Eigenvalues of Fibonacci stochastic adding machine, Stoch. Dyn., 10 (2010), 291-313. doi: 10.1142/S0219493710002966.  Google Scholar

[13]

A. Messaoudi, O. Sester and G. Valle, Spectrum of stochastic adding machine and fibred Julia sets, Stochastic and Dynamics, 13 (2013), 1250021, 26pp. doi: 10.1142/S0219493712500219.  Google Scholar

[14]

R. A. Uceda, Máquina de Somar, Conjuntos de Julia e Fractais de Rauzy, PhD Thesis, 2011. Google Scholar

[15]

K. Yoshida, Functional Analysis, Springer Verlag, 1980.  Google Scholar

show all references

References:
[1]

E. H. El Abdalaoui and A. Messaoudi, On the spectrum of stochastic perturbations of the shift map and Julia sets, Fundamenta Mathematicae, 218 (2012), 47-68. doi: 10.4064/fm218-1-3.  Google Scholar

[2]

El Abdalaoui, S. Bonnot, A. Messaoudi and O. Sester, On the Fibonacci Complex Dynamical Systems, arXiv:1304.4864, 26p, 2013. Google Scholar

[3]

E. Bedford and J. Smillie, John Polynomial diffeomorphisms of $\mathbbC^2$ VI. Connectivity of J, Ann. of Math. (2), 148 (1998), 695-735. doi: 10.2307/121006.  Google Scholar

[4]

R. L. Devaney, An Introduction to Chaothic Dynamical Systems, ABP, 2003.  Google Scholar

[5]

S. Eilenberg, Automata, Languages, and Machines, Volume 1, Academic Press, New York and London, 1974.  Google Scholar

[6]

J. E. Fornaess and N. Sibony, Fatou and Julia sets for entire mappings in $\mathbbC^k$, Math. Ann., 311 (1998), 27-40. doi: 10.1007/s002080050174.  Google Scholar

[7]

A. S. Fraenkel, Systems of numeration, Amer. Math. Monthly, 92 (1985), 105-114. doi: 10.2307/2322638.  Google Scholar

[8]

C. Frougny, Representation of numbers and finite automata, Math. Systems Theory, 25 (1992), 37-60. doi: 10.1007/BF01368783.  Google Scholar

[9]

P. R. Killeen and T. J. Taylor, A stochastic adding machine and complex dynamics, Nonlinearity, 13 (2000), 1889-1903. doi: 10.1088/0951-7715/13/6/302.  Google Scholar

[10]

S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge University Press, 2000.  Google Scholar

[11]

J. Milnor, Dynamics in One Complex Variable, Princeton University Press, 2006.  Google Scholar

[12]

A. Messaoudi and D. Smania, Eigenvalues of Fibonacci stochastic adding machine, Stoch. Dyn., 10 (2010), 291-313. doi: 10.1142/S0219493710002966.  Google Scholar

[13]

A. Messaoudi, O. Sester and G. Valle, Spectrum of stochastic adding machine and fibred Julia sets, Stochastic and Dynamics, 13 (2013), 1250021, 26pp. doi: 10.1142/S0219493712500219.  Google Scholar

[14]

R. A. Uceda, Máquina de Somar, Conjuntos de Julia e Fractais de Rauzy, PhD Thesis, 2011. Google Scholar

[15]

K. Yoshida, Functional Analysis, Springer Verlag, 1980.  Google Scholar

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