Rigidity of Julia sets for Hénon type maps

Tien-Cuong Dinh; Nessim Sibony

doi:10.3934/jmd.2014.8.499

Article Contents

2014, Volume 8, Issue 3&4: 499-548. Doi: 10.3934/jmd.2014.8.499

This issue Previous Article Lectures on dynamics, fractal geometry, and metric number theory Next Article Center Lyapunov exponents in partially hyperbolic dynamics

Rigidity of Julia sets for Hénon type maps

Tien-Cuong Dinh^1, and
Nessim Sibony^2,

1.
Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076
2.
Université Paris-Sud, Mathématique - Bâtiment 425, 91405 Orsay

Received: December 31, 2012

Revised: July 31, 2013

Published: March 2015

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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.

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Mathematics Subject Classification: Primary: 37-02, 37F10; Secondary: 32H30, 32H50, 32U90.

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