# Equilibrium measures of meromorphic self-maps on non-Kahler manifolds

## TL;DR

This paper extends the theory of equilibrium measures to dominant meromorphic self-maps on non-Kähler manifolds, demonstrating existence and properties similar to the Kähler case using new analytical tools.

## Contribution

It introduces weakly d.s.h. functions and Sobolev test functions to establish equilibrium measures for non-Kähler manifolds, broadening the scope of complex dynamics.

## Key findings

- Existence of equilibrium measure for dominant meromorphic maps on non-Kähler manifolds.
- Equilibrium measures satisfy properties analogous to the Kähler case.
- Identification of a class of holomorphic self-maps with dominant topological degree on Hopf manifolds.

## Abstract

Let $X$ be a compact complex non-K\"ahler manifold and $f$ a dominant meromorphic self-map of $X$. Examples of such maps are self-maps of Hopf manifolds, Calabi-Eckmann manifolds, non-tori nilmanifolds and their blowups. We prove that if $f$ has a dominant topological degree, then $f$ possesses an equilibrium measure $\mu$ satisfying well-known properties as in the K\"ahler case. The key ingredients are the notion of weakly d.s.h. functions substituting d.s.h. functions in the K\"ahler case and the use of suitable test functions in Sobolev spaces. A large enough class of holomorphic self-maps with dominant topological degree on Hopf manifolds is also given.