On the Fibonacci complex dynamical systems

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May 2016, 36(5): 2449-2471. doi: 10.3934/dcds.2016.36.2449

On the Fibonacci complex dynamical systems

El Houcein El Abdalaoui ^1, , Sylvain Bonnot ^2, , Ali Messaoudi ^3, and Olivier Sester ^4,

1.	Normandy university,Department of Mathematics, University of Rouen, LMRS, UMR 60 85, Avenue de l'Universite, BP.12, 76801, Saint Etienne du Rouvray, France
2.	Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, São Paulo, SP, Brazil
3.	Departamento de Matemática, Universidade Estadual Paulista, Rua Cristovão Colombo, 2265, CEP 15054-0000, São José do Rio Preto-SP, Brazil
4.	Université Paris-Est, UMR 8050, CNRS, UPEMLV, F-77454,, Marne-la-Vallée, France

Received May 2014 Revised September 2015 Published October 2015

Abstract
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We consider in this paper a sequence of complex analytic functions constructed by the following procedure $f_n(z)=f_{n-1}(z)f_{n-2}(z)+c$, where $c\in\mathbb{C}$ is a parameter. Our aim is to give a thorough dynamical study of this family, in particular we are able to extend the familiar notions of Julia sets and Green function and to analyze their properties. As a consequence, we extend some well-known results. Finally we study in detail the case where $c$ is small.

Keywords: holomorphic motion, endomorphisms of $\mathbb{C}^2$., Julia sets, quasi-circle, complex dynamics.

Mathematics Subject Classification: Primary: 37F45, 37F99; Secondary: 37D05, 37D2.

Citation: El Houcein El Abdalaoui, Sylvain Bonnot, Ali Messaoudi, Olivier Sester. On the Fibonacci complex dynamical systems. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2449-2471. doi: 10.3934/dcds.2016.36.2449

References:

[1]	L. V. Ahlfors, Lectures on Quasiconformal Mappings, Second edition, American Mathematical Society, Providence, 2006. doi: 10.1090/ulect/038.
[2]	E. Bedford and J. Smillie, Polynomial diffeomorphisms of $C^2$: Currents, equilibrium measure and hyperbolicity, Invent. Math., 103 (1991), 69-99. doi: 10.1007/BF01239509.
[3]	P. Berger, Persistence of laminations, Bull. Braz. Math. Soc. (N.S.), 41 (2010), 259-319. doi: 10.1007/s00574-010-0013-0.
[4]	L. Carleson and T. W. Gamelin, Complex Dynamics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9 .
[5]	E. 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.
[6]	E. de Faria and W. de Melo, Mathematical Tools for One-Dimensional Dynamics, Cambridge University Press, Cambridge, 2008. doi: 10.1017/cbo9780511755231.
[7]	J. E. Fornæss and N. Sibony, Complex Hénon mappings in $C^2$ and Fatou-Bieberbach domains, Duke Math. J., 65 (1992), 345-380. doi: 10.1215/S0012-7094-92-06515-X.
[8]	V. Guedj, Dynamics of quadratic polynomial mappings of $\mathbbC^2$, Michigan Math. J., 52 (2004), 627-648. doi: 10.1307/mmj/1100623417.
[9]	V. Guedj, Propriétés ergodiques des applications rationnelles, in Panor. Synthèses, 30 (2010), 97-202.
[10]	V. Guillemin and A. Pollack, Differential Topology, AMS Chelsea Publishing, Providence, 2010.
[11]	S. L. Hruska and R. Roeder, Topology of Fatou components for endomorphisms of $\mathbb{CP}^k$: Linking with the Green's current, Fund. Math., 210 (2010), 73-98. doi: 10.4064/fm210-1-4.
[12]	J. H. Hubbard and R. W. Oberste-Vorth, Hénon mappings in the complex domain. II. Projective and inductive limits of polynomials, in NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 464 (1995), 89-132.
[13]	A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/cbo9780511809187.
[14]	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.
[15]	A. Messaoudi and D. Smania, Eigenvalues of Fibonacci stochastic adding machine, Stoch. Dyn., 10 (2010), 291-313. doi: 10.1142/S0219493710002966.
[16]	S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge University Press, Cambridge, 2000. doi: d.doi.org/10.1017/s0143385700001036.
[17]	C. Robinson, An Introduction to Dynamical Systems, Continuous and Discrete, Second edition, American Mathematical Society, Providence, 2012.
[18]	D. Ruelle, Elements of Differentiable Dynamics and Bifurcation Theory, Academic Press Inc., Boston, 1989.

show all references

References:

[1]	L. V. Ahlfors, Lectures on Quasiconformal Mappings, Second edition, American Mathematical Society, Providence, 2006. doi: 10.1090/ulect/038.
[2]	E. Bedford and J. Smillie, Polynomial diffeomorphisms of $C^2$: Currents, equilibrium measure and hyperbolicity, Invent. Math., 103 (1991), 69-99. doi: 10.1007/BF01239509.
[3]	P. Berger, Persistence of laminations, Bull. Braz. Math. Soc. (N.S.), 41 (2010), 259-319. doi: 10.1007/s00574-010-0013-0.
[4]	L. Carleson and T. W. Gamelin, Complex Dynamics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9 .
[5]	E. 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.
[6]	E. de Faria and W. de Melo, Mathematical Tools for One-Dimensional Dynamics, Cambridge University Press, Cambridge, 2008. doi: 10.1017/cbo9780511755231.
[7]	J. E. Fornæss and N. Sibony, Complex Hénon mappings in $C^2$ and Fatou-Bieberbach domains, Duke Math. J., 65 (1992), 345-380. doi: 10.1215/S0012-7094-92-06515-X.
[8]	V. Guedj, Dynamics of quadratic polynomial mappings of $\mathbbC^2$, Michigan Math. J., 52 (2004), 627-648. doi: 10.1307/mmj/1100623417.
[9]	V. Guedj, Propriétés ergodiques des applications rationnelles, in Panor. Synthèses, 30 (2010), 97-202.
[10]	V. Guillemin and A. Pollack, Differential Topology, AMS Chelsea Publishing, Providence, 2010.
[11]	S. L. Hruska and R. Roeder, Topology of Fatou components for endomorphisms of $\mathbb{CP}^k$: Linking with the Green's current, Fund. Math., 210 (2010), 73-98. doi: 10.4064/fm210-1-4.
[12]	J. H. Hubbard and R. W. Oberste-Vorth, Hénon mappings in the complex domain. II. Projective and inductive limits of polynomials, in NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 464 (1995), 89-132.
[13]	A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/cbo9780511809187.
[14]	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.
[15]	A. Messaoudi and D. Smania, Eigenvalues of Fibonacci stochastic adding machine, Stoch. Dyn., 10 (2010), 291-313. doi: 10.1142/S0219493710002966.
[16]	S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge University Press, Cambridge, 2000. doi: d.doi.org/10.1017/s0143385700001036.
[17]	C. Robinson, An Introduction to Dynamical Systems, Continuous and Discrete, Second edition, American Mathematical Society, Providence, 2012.
[18]	D. Ruelle, Elements of Differentiable Dynamics and Bifurcation Theory, Academic Press Inc., Boston, 1989.

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