A rigidity theorem for Hénon maps

Abstract

The purpose of this note is to explore further the rigidity properties of Hénon maps from [5]. For instance, we show that if H and F are Hénon maps with the same Green measure (µH = µF ), or the same filled Julia set (KH = KF ), or the same Green function (GH = GF ), then H 2 and F 2 have to commute. This in turn, gives that H and F have the same non-escaping sets. Further we prove that, either of the association of a Hénon map H to its Green measure µH or to its filled Julia set KH or to its Green function GH is locally injective.

On the automorphism group of certain Short $\mathbb C^2$'s

“…This also proves that L(K − ) = K − and by Theorem 1.1 of [7], we have a 2 = 0. Further, by Theorem 1.1 from [7], there exist linear maps…”

“…Much like Fatou-Bieberbach domains, Short C 2 's come in a variety of shapes and sizes, and possess a number of intriguing properties as can be seen from the examples in [1], [5], [6], [7], [8] and [13]. The purpose of this paper is to study the holomorphic automorphism group of Short C 2 's that arise in (ii) above.…”

Further remarks on rigidity of Hénon maps

“…The goal of the present article is to improve and extend the rigidity result of Hénon maps obtained in [7].…”

“…H are completely invariant under H. In [7], for a Hénon map H, we proved the following rigidity theorem.…”

“…Recently, in [7], an analogue of the above-mentioned rigidity phenomenon has been proved for the Hénon maps in C 2 which, by the classification theorem of Freidland-Milnor ( [12]), is the most important class of automorphisms in C 2 from the point of view of dynamics. The class of Hénon maps consists of the polynomial automorphisms of C 2 of the form…”