Exponential mixing of all orders on K\"ahler manifolds: (quasi-)plurisubharmonic observables

TL;DR

This paper proves that for certain holomorphic automorphisms on compact K"ahler manifolds, the measure of maximal entropy exhibits exponential mixing of all orders for a broad class of observables, leading to statistical limit theorems.

Contribution

It establishes exponential mixing of all orders for d.s.h. observables on K"ahler manifolds, a significant advancement in understanding complex dynamical systems.

Findings

Exponential mixing of all orders for d.s.h. observables.

Validation of the central limit theorem for these observables.

Enhanced understanding of statistical properties of holomorphic automorphisms.

Abstract

Let $f$ be a holomorphic automorphism of a compact K\"ahler manifold with simple action on cohomology and $\mu$ its unique measure of maximal entropy. We prove that $\mu$ is exponentially mixing of all orders for all d.s.h.\ observables, i.e., functions that are locally differences of plurisubharmonic functions. As a consequence, every d.s.h.\ observable satisfies the central limit theorem with respect to $\mu$.