Local regularity of super-potentials and equidistribution of positive closed currents of $\mathbb{P}^k$

TL;DR

This paper studies the local regularity of super-potentials and demonstrates the equidistribution of certain positive closed currents in complex projective space, extending understanding of dynamical systems and complex geometry.

Contribution

It introduces new notions of local regularity of super-potentials and proves their role in equidistribution of positive closed currents under specific holomorphic dynamics.

Findings

Super-potentials with boundedness near invariant sets lead to equidistribution.

Results apply to holomorphic endomorphisms and polynomial automorphisms of complex spaces.

Provides new tools for studying complex dynamical systems and currents.

Abstract

In this paper, we introduce notions about local regularity of the super-potential and prove equidistribution of positive closed $(p, p)$-currents of $\mathbb{P}^k$ whose super-potentials are bounded near an invariant analytic subset, for holomorphic endomorphisms of $\mathbb{P}^k$ and for regular polynomial automorphisms of $\mathbb{C}^k$.