Semi-parabolic tools for hyperbolic H\'enon maps and continuity of Julia sets in $\mathbb{C}^{2}$

TL;DR

This paper establishes continuity and stability results for Julia sets of complex Hénon maps near semi-parabolic fixed points, showing hyperbolicity and connectedness of Julia sets for small parameter perturbations.

Contribution

It introduces new techniques to analyze the continuity and stability of Julia sets in complex Hénon maps near semi-parabolic fixed points, extending one-dimensional convergence concepts to two dimensions.

Findings

Julia sets depend continuously on parameters as t approaches 0

Hénon maps are hyperbolic with connected Julia sets for small nonzero |t|

The family of Hénon maps is stable on Julia sets for nonnegative t

Abstract

We prove some new continuity results for the Julia sets $J$ and $J^{+}$ of the complex H\'enon map $H_{c,a}(x,y)=(x^{2}+c+ay, ax)$, where $a$ and $c$ are complex parameters. We look at the parameter space of dissipative H\'enon maps which have a fixed point with one eigenvalue $(1+t)\lambda$, where $\lambda$ is a root of unity and $t$ is real and small in absolute value. These maps have a semi-parabolic fixed point when $t$ is $0$, and we use the techniques that we have developed in [RT] for the semi-parabolic case to describe nearby perturbations. We show that for small nonzero $|t|$, the H\'enon map is hyperbolic and has connected Julia set. We prove that the Julia sets $J$ and $J^{+}$ depend continuously on the parameters as $t\rightarrow 0$, which is a two-dimensional analogue of radial convergence from one-dimensional dynamics. Moreover, we prove that this family of H\'enon maps is stable on $J$ and $J^{+}$ when $t$ is nonnegative.