Exponential mixing property for automorphisms of compact K\"ahler manifolds

TL;DR

This paper proves that for certain holomorphic automorphisms of compact Kähler manifolds with a unique maximal dynamic degree, the equilibrium measure exhibits exponential mixing for all d.s.h. test functions, indicating strong statistical properties.

Contribution

It establishes exponential mixing of the equilibrium measure under specific spectral conditions on the automorphism's action.

Findings

The equilibrium measure is exponentially mixing for all d.s.h. test functions.

Unique maximal dynamic degree with a single eigenvalue of maximal modulus is crucial.

Provides a rigorous proof of exponential decay of correlations in this setting.

Abstract

Let $f$ be a holomorphic automorphism of a compact K\"ahler manifold. Assume moreover that $f$ admits a unique maximal dynamic degree $d_p$ with only one eigenvalue of maximal modulus. Let $\mu$ be its equilibrium measure. In this paper, we prove that $\mu$ is exponentially mixing for all d.s.h.\ test functions.