  Let f be a holomorphic automorphism of a compact Kahler manifold (X,\omega)
of dimension k>1. We study the convex cones of positive closed (p,p)-currents
T_p, which satisfy a functional relation $f^*(T_p)=\lambda T_p, \lambda>1,$ and
some regularity condition (PB). Under appropriate assumptions on dynamical
degrees we introduce closed finite dimensional cones, not reduced to zero, of
such currents. In particular, when the topological entropy h(f) of f is
positive, then for some m, there is a positive closed (m,m)-current T_m which
satisfies the relation $f^* T_m=\exp(h(f)) T_m$. Moreover,every quasi-p.s.h.
function is integrable with respect to the trace measure of T_m. When the
dynamical degrees of f are all distinct, we construct an invariant measure
$\mu$ as an intersection of closed currents. We show that this measure is
mixing and gives no mass to pluripolar sets and to sets of small Hausdorff
dimension.
