On the Fibonacci complex dynamical systems

TL;DR

This paper introduces a new family of complex dynamical systems inspired by Fibonacci sequences, extending classical concepts like Julia sets and Green functions, and analyzes their properties, especially for small parameter values.

Contribution

It develops a comprehensive dynamical framework for Fibonacci-like complex functions, extending classical complex dynamics concepts and analyzing new behaviors for small parameters.

Findings

Extended Julia sets and Green functions for this family.

Analyzed properties of the systems for small parameter c.

Extended classical results to this new family.

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 $c\in\C$ 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.