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

A class of adding machines and Julia sets

Danilo Antonio Caprio 1,

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 $\mathbb{C}^2$.
Keywords: fibered Julia sets., stochastic adding machines, Markov chains, Julia sets, spectrum of transition operators.
Mathematics Subject Classification: Primary: 37A30, 37F50; Secondary: 60J10, 47A1.
Citation: Danilo Antonio Caprio. A class of adding machines and Julia sets. Discrete & Continuous Dynamical Systems, 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), 2449-2471. 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), 47-68. 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, Université Paris 7, Rapport LITP 89-69, 1989. Google Scholar

[6]

C. Frougny, Fibonacci representations and finite automata, IEEE Trans. Inform. Theory, 37 (1991), 393-399. 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), 103-123.  Google Scholar

[8]

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

[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), 430-459. Google Scholar

[10]

G. F. Lawler, Introduction to Stochastic Processes, $1^{nd}$ edition, Chapman and Hall, 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), 1250021, 26 pages. doi: 10.1142/S0219493712500219.  Google Scholar

[12]

A. Messaoudi and D. Smania, Eigenvalues of Fibonacci stochastic adding machine, Stochastics and Dynamics, 10 (2010), 291-313. 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), 5247-5269. 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, Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006.  Google Scholar

[16]

S. Morosawa, Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge Studies in Advanced Mathematics, vol. 66, Cambridge University Press, Cambridge, 2000.  Google Scholar

[17]

J. Y. Ouvrard, Probabilités, Cassini, vol. 2, Paris, 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) 179-182.  Google Scholar

show all references

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), 2449-2471. 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), 47-68. 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, Université Paris 7, Rapport LITP 89-69, 1989. Google Scholar

[6]

C. Frougny, Fibonacci representations and finite automata, IEEE Trans. Inform. Theory, 37 (1991), 393-399. 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), 103-123.  Google Scholar

[8]

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

[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), 430-459. Google Scholar

[10]

G. F. Lawler, Introduction to Stochastic Processes, $1^{nd}$ edition, Chapman and Hall, 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), 1250021, 26 pages. doi: 10.1142/S0219493712500219.  Google Scholar

[12]

A. Messaoudi and D. Smania, Eigenvalues of Fibonacci stochastic adding machine, Stochastics and Dynamics, 10 (2010), 291-313. 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), 5247-5269. 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, Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006.  Google Scholar

[16]

S. Morosawa, Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge Studies in Advanced Mathematics, vol. 66, Cambridge University Press, Cambridge, 2000.  Google Scholar

[17]

J. Y. Ouvrard, Probabilités, Cassini, vol. 2, Paris, 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) 179-182.  Google Scholar

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