Further remarks on rigidity of Hénon maps

Abstract

For a Hénon map H in C 2 , we characterize the polynomial automorphisms of 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 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 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.