A new proof of a theorem of Hubbard-Oberste-Vorth

TL;DR

This paper presents a novel proof of a theorem related to Hénon maps, showing the Julia set as a fixed point of a contraction and offering new characterizations useful for future research.

Contribution

It introduces a new proof method for the Hubbard-Oberste-Vorth theorem and provides alternative characterizations of Julia sets for perturbed hyperbolic polynomials.

Findings

Julia set $J^{+}$ is the fixed point of a contracting operator

New characterizations of Julia sets $J$ and $J^{+}$

Applicable to perturbations of hyperbolic polynomials

Abstract

We give a new proof of a theorem of Hubbard-Oberste-Vorth [HOV2] for H\'enon maps that are perturbations of a hyperbolic polynomial and recover the Julia set $J^{+}$ inside a polydisk as the image of the fixed point of a contracting operator. We also give different characterizations of the Julia sets $J$ and $J^{+}$ which prove useful for later applications.