Ergodic properties of random holomorphic endomorphisms of $\Bbb{P}^k$

TL;DR

This paper investigates the ergodic behavior of random holomorphic endomorphisms on complex projective space, constructing invariant currents and proving a central limit theorem for certain observables.

Contribution

It introduces new invariant currents with convergence properties and establishes conditions for Hölder continuity of their quasi-potentials, along with a CLT for observables.

Findings

Constructed positive closed currents with invariance and convergence properties

Provided a sufficient condition for Hölder continuity of quasi-potentials

Proved a central limit theorem for d.s.h and Hölder observables

Abstract

We study ergodic properties of compositions of holomorphic endomorphisms of the complex projective space chosen independently at random according to some probability distribution. Along the way, we construct positive closed currents which have good invariance and convergence properties. We provide a sufficient condition for these currents to have H\"{o}lder continuous quasi-potentials. We also prove central limit theorem for d.s.h and H\"{o}lder continuous observables.