Further remarks on rigidity of Henon maps

Abstract

For a Henon 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 Henon 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 Henon maps coincide, then they almost commute.