Filled Julia set of some class of H\'enon-like map

TL;DR

This paper analyzes the structure of filled Julia sets for a specific class of Hénon-like maps, showing they are unions of stable and unstable manifolds of fixed and periodic points when c is between -1 and 0.

Contribution

It characterizes the filled Julia sets of a class of Hénon-like maps as unions of manifolds associated with fixed and 3-periodic points for certain parameter ranges.

Findings

Forward filled Julia set is union of stable manifolds of fixed and 3-periodic points.

Backward filled Julia set is union of unstable manifolds of fixed and 3-periodic points.

Results hold for parameter c in (-1, 0).

Abstract

In this work we consider a class of endomorphisms of $\mathbb{R}^2$ defined by $f(x,y)=(xy+c,x)$, where $c\in\mathbb{R}$ is a real number and we prove that when $-1<c<0$, the forward filled Julia set of $f$ is the union of stable manifolds of fixed and $3-$periodic points of $f$. We also prove that the backward filled Julia set of $f$ is the union of unstable manifolds of the saddle fixed and $3-$periodic points of $f$.