On automorphisms of super-level sets of Green's functions of Hénon maps

Abstract

The aim of this article is two-fold. First, we obtain a normal form for automorphisms of the escaping sets of Hénon maps in a neighborhood of the attracting fixed point at infinity. Building on this local description, we characterize the global automorphisms of ${\mathbb{C}}^2$ that preserve the escaping sets. The analytic structure of the escaping sets, as established by Hubbard--Oberste-Vorth (\cite{HOV}), plays a crucial role in the proof of these results. In a similar spirit, we investigate the analytic structure of the super-level sets of the Green's functions associated with Hénon maps and give a description of the automorphisms of these domains.