  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.
