# American Institute of Mathematical Sciences

November  2016, 36(11): 5951-5970. doi: 10.3934/dcds.2016061

## A class of adding machines and Julia sets

 1 Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, RJ, Brazil

Received  August 2015 Revised  May 2016 Published  August 2016

In this work we define a stochastic adding machine associated to the Fibonacci base and to a probabilities sequence $\overline{p}=(p_i)_{i\geq 1}$. We obtain a Markov chain whose states are the set of nonnegative integers. We study probabilistic properties of this chain, such as transience and recurrence. We also prove that the spectrum associated to this Markov chain is connected to the fibered Julia sets for a class of endomorphisms in $\mathbb{C}^2$.
Citation: Danilo Antonio Caprio. A class of adding machines and Julia sets. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5951-5970. doi: 10.3934/dcds.2016061
##### References:
 [1] H. El Abdalaoui, S. Bonnot, A. Messaoudi and O. Sester, On the Fibonacci complex dynamical systems,, Discrete and Continuous Dynamical Systems - A, 36 (2016), 2449. doi: 10.3934/dcds.2016.36.2449. Google Scholar [2] H. El Abdalaoui and A. Messaoudi, On the spectrum of stochastic perturbations of the shift and Julia Sets,, Fundamenta Mathematicae, 218 (2012), 47. doi: 10.4064/fm218-1-3. Google Scholar [3] D. A. Caprio, A class of adding machine and Julia sets, preprint,, , (). Google Scholar [4] D. A. Caprio and A. Messaoudi, Julia Sets for a class of endomorphisms on $\mathbbC^2$,, work in progress., (). Google Scholar [5] C. Frougny, Systṁes de numération linéaires et automates finis,, Ph.D thesis, (1989), 89. Google Scholar [6] C. Frougny, Fibonacci representations and finite automata,, IEEE Trans. Inform. Theory, 37 (1991), 393. doi: 10.1109/18.75263. Google Scholar [7] P. J. Grabner, P. Liardet and R. F. Tichy, Odometers and systems of numeration,, Acta Arithmetica, 70 (1995), 103. Google Scholar [8] P. R. Killeen and T. J. Taylor, A stochastic adding machine and complex dynamics,, Nonlinearity, 13 (2000), 1889. doi: 10.1088/0951-7715/13/6/302. Google Scholar [9] P. R. Killeen and T. J. Taylor, How the propagation of error through stochastic counters affects time discrimination and other psychophysical judgements,, Psychological Review, 107 (2000), 430. Google Scholar [10] G. F. Lawler, Introduction to Stochastic Processes,, $1^{nd}$ edition, (1995). Google Scholar [11] A. Messaoudi, O. Sester and G. Valle, Spectrum of stochastic adding machines and fibered Julia sets,, Stochastics and Dynamics, 13 (2013). doi: 10.1142/S0219493712500219. Google Scholar [12] A. Messaoudi and D. Smania, Eigenvalues of Fibonacci stochastic adding machine,, Stochastics and Dynamics, 10 (2010), 291. doi: 10.1142/S0219493710002966. Google Scholar [13] A. Messaoudi and R. M. A. Uceda, Stochastic adding machine and 2-dimensional Julia sets,, Discrete and Continuous Dynamical Systems - A, 34 (2014), 5247. doi: 10.3934/dcds.2014.34.5247. Google Scholar [14] A. Messaoudi and G. Valle, Spectra of stochastic adding machines based on Cantor Systems of Numeration, preprint,, , (). Google Scholar [15] J. Milnor, Dynamics in one complex variable,, $3^{nd}$ edition, (2006). Google Scholar [16] S. Morosawa, Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics,, Cambridge Studies in Advanced Mathematics, (2000). Google Scholar [17] J. Y. Ouvrard, Probabilités,, Cassini, (2009). Google Scholar [18] E. Zeckendorff, Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas,, Bull. Soc. Royale Sci. Liège, 42 (1972), 179. Google Scholar

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##### References:
 [1] H. El Abdalaoui, S. Bonnot, A. Messaoudi and O. Sester, On the Fibonacci complex dynamical systems,, Discrete and Continuous Dynamical Systems - A, 36 (2016), 2449. doi: 10.3934/dcds.2016.36.2449. Google Scholar [2] H. El Abdalaoui and A. Messaoudi, On the spectrum of stochastic perturbations of the shift and Julia Sets,, Fundamenta Mathematicae, 218 (2012), 47. doi: 10.4064/fm218-1-3. Google Scholar [3] D. A. Caprio, A class of adding machine and Julia sets, preprint,, , (). Google Scholar [4] D. A. Caprio and A. Messaoudi, Julia Sets for a class of endomorphisms on $\mathbbC^2$,, work in progress., (). Google Scholar [5] C. Frougny, Systṁes de numération linéaires et automates finis,, Ph.D thesis, (1989), 89. Google Scholar [6] C. Frougny, Fibonacci representations and finite automata,, IEEE Trans. Inform. Theory, 37 (1991), 393. doi: 10.1109/18.75263. Google Scholar [7] P. J. Grabner, P. Liardet and R. F. Tichy, Odometers and systems of numeration,, Acta Arithmetica, 70 (1995), 103. Google Scholar [8] P. R. Killeen and T. J. Taylor, A stochastic adding machine and complex dynamics,, Nonlinearity, 13 (2000), 1889. doi: 10.1088/0951-7715/13/6/302. Google Scholar [9] P. R. Killeen and T. J. Taylor, How the propagation of error through stochastic counters affects time discrimination and other psychophysical judgements,, Psychological Review, 107 (2000), 430. Google Scholar [10] G. F. Lawler, Introduction to Stochastic Processes,, $1^{nd}$ edition, (1995). Google Scholar [11] A. Messaoudi, O. Sester and G. Valle, Spectrum of stochastic adding machines and fibered Julia sets,, Stochastics and Dynamics, 13 (2013). doi: 10.1142/S0219493712500219. Google Scholar [12] A. Messaoudi and D. Smania, Eigenvalues of Fibonacci stochastic adding machine,, Stochastics and Dynamics, 10 (2010), 291. doi: 10.1142/S0219493710002966. Google Scholar [13] A. Messaoudi and R. M. A. Uceda, Stochastic adding machine and 2-dimensional Julia sets,, Discrete and Continuous Dynamical Systems - A, 34 (2014), 5247. doi: 10.3934/dcds.2014.34.5247. Google Scholar [14] A. Messaoudi and G. Valle, Spectra of stochastic adding machines based on Cantor Systems of Numeration, preprint,, , (). Google Scholar [15] J. Milnor, Dynamics in one complex variable,, $3^{nd}$ edition, (2006). Google Scholar [16] S. Morosawa, Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics,, Cambridge Studies in Advanced Mathematics, (2000). Google Scholar [17] J. Y. Ouvrard, Probabilités,, Cassini, (2009). Google Scholar [18] E. Zeckendorff, Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas,, Bull. Soc. Royale Sci. Liège, 42 (1972), 179. Google Scholar
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