Journal of Modern Dynamics (JMD)

Rigidity of Julia sets for Hénon type maps

Pages: 499 - 548, Issue 3/4, September/December 2014      doi:10.3934/jmd.2014.8.499

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Tien-Cuong Dinh - Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076, Singapore (email)
Nessim Sibony - Université Paris-Sud, Mathématique - Bâtiment 425, 91405 Orsay, France (email)

Abstract: We prove that the Julia set of a Hénon type automorphism on $\mathbb{C}^2$ is very rigid: it supports a unique positive $dd^c$-closed current of mass 1. A similar property holds for the cohomology class of the Green current associated with an automorphism of positive entropy on a compact Kähler surface. Relations between this phenomenon, several quantitative equidistribution properties and the theory of value distribution will be discussed. We also survey some rigidity properties of Hénon type maps on $\mathbb{C}^k$ and of automorphisms of compact Kähler manifolds.

Keywords:  Hénon map, holomorphic automorphism, Julia set, Green current, Nevanlinna theory, rigidity.
Mathematics Subject Classification:  Primary: 37-02, 37F10; Secondary: 32H30, 32H50, 32U90.

Received: January 2013;      Revised: August 2013;      Available Online: April 2015.