Equilibrium measures of meromorphic self-maps on non-Kähler 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.