H\'enon maps with biholomorphic Kato surfaces

TL;DR

This paper establishes a precise correspondence between conjugacy of generalized Hénon maps near infinity and the biholomorphic equivalence of their associated Kato surfaces, providing a complete classification in this context.

Contribution

It proves that Kato surfaces associated to generalized Hénon maps are biholomorphic if and only if the maps are conjugate, and characterizes conjugacy near infinity as affine conjugacy.

Findings

Kato surfaces are biholomorphic iff the maps are conjugate.

Conjugacy near infinity is equivalent to affine conjugacy.

Complete description of Kato surfaces for generalized Hénon maps.

Abstract

Let $F$ and $G$ be generalized H\'enon maps. We show that the Kato surfaces associated to the germs of $F$ and $G$ near infinity are biholomorphic if and only if $F$ and $G$ are conjugate in $\text{Aut}(\mathbb{C}^2)$. This answers a question raised by Favre in X\'enelkis de H\'enon's survey. Moreover, we describe all the Kato surfaces associated to the germ of a generalized H\'enon map near infinity. We also show that two generalized H\'enon maps are conjugate near infinity, if and only if they are affinely conjugate.