Further remarks on rigidity of Hénon maps

Abstract

For a Hénon map $H$ in ${\mathbb{C}}^2$, we characterize the polynomial automorphisms of ${\mathbb{C}}^2$ which keep any fixed level set of the Green function of $H$ completely invariant. The interior of any non-zero sublevel set of the Green function of a Hénon map turns out to be a Short ${\mathbb{C}}^2$ and as a consequence of our characterization, it follows that there exists no polynomial automorphism apart from possibly the affine automorphisms which acts as an automorphism on any of these Short ${\mathbb{C}}^2$'s. Further, we prove that if any two level sets of the Green functions of a pair of Hénon maps coincide, then they almost commute.