Green currents for holomorphic automorphisms of compact Kähler manifolds

Abstract:

Let $f$ be a holomorphic automorphism of a compact Kähler manifold $(X,\omega )$ of dimension $k\geq 2$. 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, PC). 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 $\mathrm {h}(f)$ of $f$ is positive, then for some $m\geq 1$, there is a positive closed $(m,m)$-current $T_m$ which satisfies the relation \[ f^* T_m=\exp (\mathrm {h}(f)) T_m.\] 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.