Equidistribution speed for endomorphisms of projective spaces

TL;DR

This paper estimates how quickly the preimages of a generic point under a non-invertible holomorphic endomorphism of complex projective space become uniformly distributed according to the equilibrium measure.

Contribution

It provides quantitative estimates on the convergence speed of preimages to the equilibrium measure for endomorphisms of projective spaces.

Findings

Convergence speed is quantified for equidistribution of preimages.

Results apply to Zariski generic points in projective space.

Provides bounds on the rate of convergence as n increases.

Abstract

Let f be a non-invertible holomorphic endomorphism of the complex projective space P^k, f^n its iterate of order n and \mu the equilibrium measure of f. We estimate the speed of convergence in the following known result. If a is a Zariski generic point in P^k, the probability measures, equidistributed on the preimages of a under f^n, converge to \mu as n goes to infinity.