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Green currents for holomorphic automorphisms of compact Kähler manifolds


Authors: Tien-Cuong Dinh and Nessim Sibony
Journal: J. Amer. Math. Soc. 18 (2005), 291-312
MSC (2000): Primary 37F10, 32H50, 32Q15, 32U40
Published electronically: December 7, 2004
MathSciNet review: 2137979
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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

\begin{displaymath}f^* T_p=\lambda T_p, \ \lambda>1,\end{displaymath}

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

\begin{displaymath}f^* T_m=\exp({h}(f)) T_m.\end{displaymath}

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.


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Additional Information

Tien-Cuong Dinh
Affiliation: Mathématique - Bât. 425, UMR 8628, Université Paris-Sud, 91405 Orsay, France
Email: TienCuong.Dinh@math.u-psud.fr

Nessim Sibony
Affiliation: Mathématique - Bât. 425, UMR 8628, Université Paris-Sud, 91405 Orsay, France
Email: Nessim.Sibony@math.u-psud.fr

DOI: http://dx.doi.org/10.1090/S0894-0347-04-00474-6
Keywords: Green current, equilibrium measure, mixing.
Received by editor(s): November 20, 2003
Published electronically: December 7, 2004
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.