Densities of currents and complex dynamics

TL;DR

This paper generalizes the concept of densities of currents to non-Kähler manifolds, introduces exotic periodic points, and analyzes their asymptotic behavior and invariants in complex dynamics.

Contribution

It extends densities of currents to broader settings, introduces exotic periodic points, and studies their asymptotics and invariants in complex dynamics.

Findings

Established asymptotic formulas for periodic points

Proved algebraic entropy is a bi-meromorphic invariant

Extended densities of currents to non-Kähler manifolds

Abstract

We extend the Dinh-Sibony notion of densities of currents to the setting where the ambient manifold is not necessarily K\"ahler and study the intersection of analytic sets from the point of view of densities of currents. As an application, we introduce the notion of exotic periodic points of a meromorphic self-map. We then establish the expected asymptotic for the sum of the number of isolated periodic points and the number of exotic periodic points for holomorphic self-maps with a simple action on the cohomology groups on a compact K\"ahler manifold. We also show that the algebraic entropy of meromorphic self-maps of compact complex surfaces is a finite bi-meromorphic invariant.