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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

P. Berger, Persistence of laminations, Bull. Braz. Math. Soc. (N.S.), 41 (2010), 259-319. doi: 10.1007/s00574-010-0013-0.  Google Scholar

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L. Carleson and T. W. Gamelin, Complex Dynamics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9 .  Google Scholar

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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.  Google Scholar

[6]

E. de Faria and W. de Melo, Mathematical Tools for One-Dimensional Dynamics, Cambridge University Press, Cambridge, 2008. doi: 10.1017/cbo9780511755231.  Google Scholar

[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.  Google Scholar

[8]

V. Guedj, Dynamics of quadratic polynomial mappings of $\mathbbC^2$, Michigan Math. J., 52 (2004), 627-648. doi: 10.1307/mmj/1100623417.  Google Scholar

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V. Guedj, Propriétés ergodiques des applications rationnelles, in Panor. Synthèses, 30 (2010), 97-202.  Google Scholar

[10]

V. Guillemin and A. Pollack, Differential Topology, AMS Chelsea Publishing, Providence, 2010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/cbo9780511809187.  Google Scholar

[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.  Google Scholar

[15]

A. Messaoudi and D. Smania, Eigenvalues of Fibonacci stochastic adding machine, Stoch. Dyn., 10 (2010), 291-313. doi: 10.1142/S0219493710002966.  Google Scholar

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S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge University Press, Cambridge, 2000. doi: d.doi.org/10.1017/s0143385700001036.  Google Scholar

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C. Robinson, An Introduction to Dynamical Systems, Continuous and Discrete, Second edition, American Mathematical Society, Providence, 2012.  Google Scholar

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D. Ruelle, Elements of Differentiable Dynamics and Bifurcation Theory, Academic Press Inc., Boston, 1989.  Google Scholar

show all references

References:
[1]

L. V. Ahlfors, Lectures on Quasiconformal Mappings, Second edition, American Mathematical Society, Providence, 2006. doi: 10.1090/ulect/038.  Google Scholar

[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.  Google Scholar

[3]

P. Berger, Persistence of laminations, Bull. Braz. Math. Soc. (N.S.), 41 (2010), 259-319. doi: 10.1007/s00574-010-0013-0.  Google Scholar

[4]

L. Carleson and T. W. Gamelin, Complex Dynamics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9 .  Google Scholar

[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.  Google Scholar

[6]

E. de Faria and W. de Melo, Mathematical Tools for One-Dimensional Dynamics, Cambridge University Press, Cambridge, 2008. doi: 10.1017/cbo9780511755231.  Google Scholar

[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.  Google Scholar

[8]

V. Guedj, Dynamics of quadratic polynomial mappings of $\mathbbC^2$, Michigan Math. J., 52 (2004), 627-648. doi: 10.1307/mmj/1100623417.  Google Scholar

[9]

V. Guedj, Propriétés ergodiques des applications rationnelles, in Panor. Synthèses, 30 (2010), 97-202.  Google Scholar

[10]

V. Guillemin and A. Pollack, Differential Topology, AMS Chelsea Publishing, Providence, 2010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/cbo9780511809187.  Google Scholar

[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.  Google Scholar

[15]

A. Messaoudi and D. Smania, Eigenvalues of Fibonacci stochastic adding machine, Stoch. Dyn., 10 (2010), 291-313. doi: 10.1142/S0219493710002966.  Google Scholar

[16]

S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge University Press, Cambridge, 2000. doi: d.doi.org/10.1017/s0143385700001036.  Google Scholar

[17]

C. Robinson, An Introduction to Dynamical Systems, Continuous and Discrete, Second edition, American Mathematical Society, Providence, 2012.  Google Scholar

[18]

D. Ruelle, Elements of Differentiable Dynamics and Bifurcation Theory, Academic Press Inc., Boston, 1989.  Google Scholar

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