H\'enon maps with biholomorphic Kato surfaces

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.