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Green currents for holomorphic automorphisms of compact Kähler manifolds. (English) Zbl 1066.32024
Authors’ 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 $${h}(f)$$ of $$f$$ is positive, then for some $$m\geq 1$$, there is a positive closed $$(m,m)$$-current $$T_m$$ which satisfies the relation $f^* T_m=\exp({h}(f)) T_m.$ 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.
Reviewer: Zhuan Ye (DeKalb)

MSC:
 32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables 32Q15 Kähler manifolds 32U40 Currents 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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