Rigidity of Julia sets for Henon type maps

We prove that the Julia set of a Henon type automorphism on C^2 is very rigid: it supports a unique positive ddc-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 Kaehler 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 Henon type maps on C^k and of automorphisms of compact Kaehler manifolds.