A class of adding machines and Julia sets

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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

Abstract
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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 and 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.
[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.
[3]	D. A. Caprio, A class of adding machine and Julia sets, preprint, arXiv:1508.05062.
[4]	D. A. Caprio and A. Messaoudi, Julia Sets for a class of endomorphisms on $\mathbb{C}^2$, work in progress.
[5]	C. Frougny, Systṁes de numération linéaires et automates finis, Ph.D thesis, Université Paris 7, Rapport LITP 89-69, 1989.
[6]	C. Frougny, Fibonacci representations and finite automata, IEEE Trans. Inform. Theory, 37 (1991), 393-399. doi: 10.1109/18.75263.
[7]	P. J. Grabner, P. Liardet and R. F. Tichy, Odometers and systems of numeration, Acta Arithmetica, 70 (1995), 103-123.
[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.
[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.
[10]	G. F. Lawler, Introduction to Stochastic Processes, $1^{nd}$ edition, Chapman and Hall, 1995.
[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.
[12]	A. Messaoudi and D. Smania, Eigenvalues of Fibonacci stochastic adding machine, Stochastics and Dynamics, 10 (2010), 291-313. doi: 10.1142/S0219493710002966.
[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.
[14]	A. Messaoudi and G. Valle, Spectra of stochastic adding machines based on Cantor Systems of Numeration, preprint, arXiv:1307.6876.
[15]	J. Milnor, Dynamics in one complex variable, $3^{nd}$ edition, Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006.
[16]	S. Morosawa, Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge Studies in Advanced Mathematics, vol. 66, Cambridge University Press, Cambridge, 2000.
[17]	J. Y. Ouvrard, Probabilités, Cassini, vol. 2, Paris, 2009.
[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.

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.
[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.
[3]	D. A. Caprio, A class of adding machine and Julia sets, preprint, arXiv:1508.05062.
[4]	D. A. Caprio and A. Messaoudi, Julia Sets for a class of endomorphisms on $\mathbb{C}^2$, work in progress.
[5]	C. Frougny, Systṁes de numération linéaires et automates finis, Ph.D thesis, Université Paris 7, Rapport LITP 89-69, 1989.
[6]	C. Frougny, Fibonacci representations and finite automata, IEEE Trans. Inform. Theory, 37 (1991), 393-399. doi: 10.1109/18.75263.
[7]	P. J. Grabner, P. Liardet and R. F. Tichy, Odometers and systems of numeration, Acta Arithmetica, 70 (1995), 103-123.
[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.
[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.
[10]	G. F. Lawler, Introduction to Stochastic Processes, $1^{nd}$ edition, Chapman and Hall, 1995.
[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.
[12]	A. Messaoudi and D. Smania, Eigenvalues of Fibonacci stochastic adding machine, Stochastics and Dynamics, 10 (2010), 291-313. doi: 10.1142/S0219493710002966.
[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.
[14]	A. Messaoudi and G. Valle, Spectra of stochastic adding machines based on Cantor Systems of Numeration, preprint, arXiv:1307.6876.
[15]	J. Milnor, Dynamics in one complex variable, $3^{nd}$ edition, Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006.
[16]	S. Morosawa, Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge Studies in Advanced Mathematics, vol. 66, Cambridge University Press, Cambridge, 2000.
[17]	J. Y. Ouvrard, Probabilités, Cassini, vol. 2, Paris, 2009.
[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.

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