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$\log$-Hölder regularity of currents and equidistribution towards Green currents

Marco Vergamini

TL;DR

The paper develops a quantitative theory of equidistribution for holomorphic dynamics on both automorphisms of compact Kähler manifolds and endomorphisms of projective spaces by introducing and exploiting log-Hölder regularity of currents via super-potentials. It defines and analyzes log-Hölder-continuous super-potentials, proves Skoda-type estimates, and shows that push-forwards and pull-backs preserve this regularity, enabling exponential-type convergence toward Green currents when tested against log-Hölder observables with bounded $dd^c$-mass. For automorphisms with simple cohomological action, the authors obtain explicit rates $ ig|ig( (f^*)^n/d_p^n(S) - rT_+ ig)(\,\\Phi) ig| \\lesssim \\|\\Phi\\|_q \, (\\delta/d_p)^{n q/(q+1)} $, with analogous results for endomorphisms of $\\mathbb{P}^k$ and a broad class of test forms. As applications, they prove exponential mixing of all orders for the equilibrium measure $\\mu=T_+\\wedge T_-$ and derive statistical consequences, underscoring the practical impact of log-Hölder regularity in holomorphic dynamics.

Abstract

Let $f$ be an endomorphism of a projective space or an automorphism of a compact Kähler manifold. We prove that the pull-backs of currents under the iterates of $f$ converge exponentially fast to the Green currents when tested at $\log$-Hölder-continuous observables whose $\mathrm{dd^c}$'s have bounded mass.

$\log$-Hölder regularity of currents and equidistribution towards Green currents

TL;DR

The paper develops a quantitative theory of equidistribution for holomorphic dynamics on both automorphisms of compact Kähler manifolds and endomorphisms of projective spaces by introducing and exploiting log-Hölder regularity of currents via super-potentials. It defines and analyzes log-Hölder-continuous super-potentials, proves Skoda-type estimates, and shows that push-forwards and pull-backs preserve this regularity, enabling exponential-type convergence toward Green currents when tested against log-Hölder observables with bounded -mass. For automorphisms with simple cohomological action, the authors obtain explicit rates , with analogous results for endomorphisms of and a broad class of test forms. As applications, they prove exponential mixing of all orders for the equilibrium measure and derive statistical consequences, underscoring the practical impact of log-Hölder regularity in holomorphic dynamics.

Abstract

Let be an endomorphism of a projective space or an automorphism of a compact Kähler manifold. We prove that the pull-backs of currents under the iterates of converge exponentially fast to the Green currents when tested at -Hölder-continuous observables whose 's have bounded mass.

Paper Structure

This paper contains 13 sections, 109 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on currents
  3. The space of currents
  4. Plurisubharmonic and d.s.h. functions
  5. Superpotentials of currents
  6. Currents with log-Hölder-continuous super-potentials
  7. Regular super-potentials
  8. The operators $\mathcal{L}_\theta$
  9. Skoda-type estimates
  10. Automorphisms of compact Kähler manifolds
  11. Equidistribution results
  12. An application: exponential mixing of all orders
  13. Endomorphisms of $\mathbb{P}^k$

Theorems & Definitions (23)

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