Some dynamical properties of pseudo-automorphisms in dimension 3

TL;DR

This paper investigates the dynamical properties of pseudo-automorphisms on compact Kähler threefolds, establishing invariant currents, discussing intersection problems, and exploring equidistribution phenomena under certain spectral conditions.

Contribution

It proves the existence of invariant positive closed currents for pseudo-automorphisms in dimension 3 under specific spectral conditions and discusses intersection and equidistribution issues.

Findings

Existence of invariant positive closed currents under certain spectral conditions.

Weak equidistribution results for Green currents of meromorphic maps.

Intersection of dynamically related currents is well-defined in this setting.

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.