Further remarks on rigidity of Hénon maps

Abstract

For a Hénon map H in C2, we characterize the polynomial automorphisms of C2 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 ShortC2 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 ShortC2'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.