Some dynamical properties of pseudo-automorphisms in dimension 3

Truong, Tuyen

doi:10.1090/S0002-9947-2014-06340-X

Some dynamical properties of pseudo-automorphisms in dimension $3$
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by Tuyen Trung Truong PDF

Trans. Amer. Math. Soc. 368 (2016), 727-753

Abstract:

Let $X$ be a compact Kähler manifold of dimension $3$ and let $f:X\rightarrow X$ be a pseudo-automorphism. Under the mild condition that $\lambda _1(f)^2>\lambda _2(f)$, we prove the existence of invariant positive closed $(1,1)$ and $(2,2)$ currents, and we also discuss the (still open) problem of intersection of such currents. We prove a weak equi-distribution result for Green $(1,1)$ currents of meromorphic selfmaps, not necessarily algebraic $1$-stable, of a compact Kähler manifold of arbitrary dimension and discuss how a stronger equidistribution result may be proved for pseudo-automorphisms in dimension $3$. As a byproduct, we show that the intersection of some dynamically related currents is well-defined with respect to our definition here, even though not obviously to be seen so using the usual criteria.

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