Relation between H\'{e}non maps with biholomorphic escaping sets

TL;DR

This paper establishes that Hénon maps with biholomorphically equivalent escaping sets are related through affine automorphisms, revealing a structural connection between their dynamical properties.

Contribution

It proves a classification result linking Hénon maps with biholomorphic escaping sets via affine conjugation, advancing understanding of their complex dynamics.

Findings

Hénon maps with biholomorphic escaping sets are affine conjugates.

The structure of escaping sets determines the conjugacy class of Hénon maps.

Biholomorphic equivalence of escaping sets implies affine automorphism relation.

Abstract

Let $H$ and $F$ be two H\'{e}non maps with biholomorphically equivalent escaping sets, then there exist affine automorphisms $A_1$ and $A_2$ in $\mathbb{C}^2$ such that \[ F=A_1\circ H \circ A_2 \] in $\mathbb{C}^2$.