On the Fibonacci complex dynamical systems

El Houcein El  Abdalaoui; Sylvain  Bonnot; Ali  Messaoudi; Olivier  Sester

doi:10.3934/dcds.2016.36.2449

Article Contents

2016, Volume 36, Issue 5: 2449-2471. Doi: 10.3934/dcds.2016.36.2449

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On the Fibonacci complex dynamical systems

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

Received: April 30, 2014

Revised: August 31, 2015

Published: May 2017

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Abstract

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.

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

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