Some dynamical properties of pseudo-automorphisms in dimension 3

Abstract

Let $X$ be a compact K\"ahler 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 equidistribution result (which is essentially known in the literature) for Green $(1,1)$ currents of meromorphic selfmaps, not necessarily 1-algebraic stable, of a compact K\"ahler 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 are well-defined with respect to our definition here, even though not obviously to be seen so using the usual criteria.