Super-potentials for currents on compact Kaehler manifolds and dynamics of automorphisms

TL;DR

This paper introduces super-potentials for positive closed currents on compact Kähler manifolds and applies this framework to analyze the dynamics of holomorphic automorphisms, including properties of Green currents and equilibrium measures.

Contribution

It develops a calculus for currents using super-potentials and applies it to establish regularity, entropy, ergodicity, and hyperbolicity results in complex dynamics.

Findings

Regularity and uniqueness of Green currents

Entropy and ergodicity of equilibrium measures

Hyperbolicity properties in automorphism dynamics

Abstract

We introduce a notion of super-potential (canonical function) associated to positive closed (p,p)-currents on compact Kaehler manifolds and we develop a calculus on such currents. One of the key points in our study is the use of deformations in the space of currents. As an application, we obtain several results on the dynamics of holomorphic automorphisms: regularity and uniqueness of the Green currents. We also get the regularity, the entropy, the ergodicity and the hyperbolicity of the equilibrium measures.