Julia sets for Fibonacci endomorphisms of $\mathbb{C}^2$ and $\mathbb{R}^2$

TL;DR

This paper investigates the complex and real dynamics of Fibonacci endomorphisms in two dimensions, focusing on their Julia sets and how they relate to perturbations of classical Anosov torus maps.

Contribution

It introduces a detailed analysis of Julia sets for a family of Fibonacci endomorphisms in both complex and real settings, extending understanding of their dynamical properties.

Findings

Characterization of Julia sets for the family $f_c$

Relationship between Fibonacci endomorphisms and classical Anosov maps

Insights into the stability and bifurcations of these dynamical systems

Abstract

We study the dynamics of the family $f_c(x, y)= (xy+c, x)$ of endomorphisms of $\mathbb{R}^2$ and $\mathbb{C}^2$, where $c$ is a real or complex parameter. Such maps can be seen as perturbations of the map $f_0(x,y)=(xy,x)$, which is a complexification of the Anosov torus map $(u,v) \mapsto (u+v,u)$.