Article Contents
Article Contents

On the Fibonacci complex dynamical systems

• 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.
Mathematics Subject Classification: Primary: 37F45, 37F99; Secondary: 37D05, 37D20.

 Citation:

•  [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.