Equidistribution problems of complex dynamics in higher dimension

TL;DR

This paper surveys two decades of research on equidistribution in complex dynamics, focusing on the behavior of orbits, subvarieties, and currents, and discusses recent techniques, concepts, and open problems.

Contribution

It provides a comprehensive overview of the development of equidistribution theory in higher-dimensional complex dynamics, highlighting new tools and unresolved questions.

Findings

Summary of key results in equidistribution of orbits and currents

Introduction of new techniques for analyzing positive closed currents

Identification of open problems in the field

Abstract

Equidistribution of the orbits of points, subvarieties or of periodic points in complex dynamics is a fundamental problem. It is often related to strong ergodic properties of the dynamical system and to a deep understanding of analytic cycles, or more generally positive closed currents, of arbitrary dimension and degree. The later topic includes the study of the potentials and super-potentials of positive closed currents, their intersection with or without dimension excess. In this paper, we will survey some results and tools developed during the last two decades. Related concepts, new techniques and open problems will be presented.