Article Contents
Article Contents

# Stochastic adding machine and $2$-dimensional Julia sets

• 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.
Mathematics Subject Classification: Primary: 37A30, 37F50; Secondary: 47A10.

 Citation:

•  [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. [2] El Abdalaoui, S. Bonnot, A. Messaoudi and O. Sester, On the Fibonacci Complex Dynamical Systems, arXiv:1304.4864, 26p, 2013. [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. [4] R. L. Devaney, An Introduction to Chaothic Dynamical Systems, ABP, 2003. [5] S. Eilenberg, Automata, Languages, and Machines, Volume 1, Academic Press, New York and London, 1974. [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. [7] A. S. Fraenkel, Systems of numeration, Amer. Math. Monthly, 92 (1985), 105-114.doi: 10.2307/2322638. [8] C. Frougny, Representation of numbers and finite automata, Math. Systems Theory, 25 (1992), 37-60.doi: 10.1007/BF01368783. [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. [10] S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge University Press, 2000. [11] J. Milnor, Dynamics in One Complex Variable, Princeton University Press, 2006. [12] A. Messaoudi and D. Smania, Eigenvalues of Fibonacci stochastic adding machine, Stoch. Dyn., 10 (2010), 291-313.doi: 10.1142/S0219493710002966. [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. [14] R. A. Uceda, Máquina de Somar, Conjuntos de Julia e Fractais de Rauzy, PhD Thesis, 2011. [15] K. Yoshida, Functional Analysis, Springer Verlag, 1980.