# Exponential mixing for automorphisms on compact Kaehler manifolds

## TL;DR

This paper proves that holomorphic automorphisms with positive entropy on compact Kähler manifolds exhibit exponential mixing in their equilibrium measures, extending results to higher dimensions under certain conditions.

## Contribution

It establishes exponential mixing for automorphisms on compact Kähler manifolds, generalizing previous results to higher dimensions with specific dynamical degree conditions.

## Key findings

- Equilibrium measure is exponentially mixing for automorphisms on compact Kähler surfaces.
- The result extends to higher-dimensional Kähler manifolds under natural dynamical degree conditions.
- Uses recent developments in pluripotential theory to achieve the proof.

## Abstract

Let f be a holomorphic automorphism of positive entropy on a compact Kaehler surface. We show that the equilibrium measure of f is exponentially mixing. The proof uses some recent development on the pluripotential theory. The result also holds for automorphisms on compact Kaehler manifolds of higher dimension under a natural condition on their dynamical degrees.