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Exponential equidistribution of periodic points for endomorphisms of $\mathbb P^k$

Henry de Thélin, Tien-Cuong Dinh, Lucas Kaufmann

TL;DR

This work proves that for a holomorphic endomorphism $f$ of $\mathbb{P}^k$ with degree $d\ge2$, the repelling periodic points of period $n$ equidistribute toward the equilibrium measure $\mu$ on the small Julia set $J_k$ at an explicit exponential rate. The authors develop a robust inverse-branch framework built on a Manhattan-like cell decomposition and establish a key Monge–Ampère mass bound near analytic sets to control postcritical neighborhoods. They then construct abundant repelling cycles with large multipliers and deduce that the periodic points within $J_k$ become $\mu$-weighted in the limit, while quantifying the negligible contribution from non-repelling or exterior periodic points. Overall, the paper extends known one-dimensional equidistribution results to higher dimensions and provides explicit exponential convergence rates and structural insights into repellers.

Abstract

Let $f$ be a holomorphic endomorphism of $\mathbb P^k$ of algebraic degree $d\geq 2$. We show that the periodic points of $f$ of period $n$ equidistribute towards the equilibrium measure of $f$ exponentially fast as $n$ tends to infinity. This quantifies a theorem of Lyubich for $k=1$ and of Briend-Duval for $k\geq 2$. A byproduct of our proof is the existence of a large number of periodic cycles in the small Julia set with large multipliers.

Exponential equidistribution of periodic points for endomorphisms of $\mathbb P^k$

TL;DR

This work proves that for a holomorphic endomorphism of with degree , the repelling periodic points of period equidistribute toward the equilibrium measure on the small Julia set at an explicit exponential rate. The authors develop a robust inverse-branch framework built on a Manhattan-like cell decomposition and establish a key Monge–Ampère mass bound near analytic sets to control postcritical neighborhoods. They then construct abundant repelling cycles with large multipliers and deduce that the periodic points within become -weighted in the limit, while quantifying the negligible contribution from non-repelling or exterior periodic points. Overall, the paper extends known one-dimensional equidistribution results to higher dimensions and provides explicit exponential convergence rates and structural insights into repellers.

Abstract

Let be a holomorphic endomorphism of of algebraic degree . We show that the periodic points of of period equidistribute towards the equilibrium measure of exponentially fast as tends to infinity. This quantifies a theorem of Lyubich for and of Briend-Duval for . A byproduct of our proof is the existence of a large number of periodic cycles in the small Julia set with large multipliers.
Paper Structure (6 sections, 19 theorems, 57 equations)

This paper contains 6 sections, 19 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Mass of Monge-Ampère measures near analytic sets
  4. Local coordinates, Manhattans and inverse branches
  5. Construction of repelling periodic points
  6. Equidistribution of repelling periodic points

Key Result

Theorem 1.1

Let $f$ be a holomorphic endomorphism of $\mathbb{P}^k$ of algebraic degree $d\geq 2$, $\mu$ be its equilibrium measure and $J_k:={\rm supp}(\mu)$ be its small Julia set. Then, as $n$ tends to infinity, the repelling periodic points of period $n$ of $f$ on $J_k$ are equidistributed with respect to $ for any $\mathscr{C}^\alpha$ test function $\phi$ on $\mathbb{P}^k$, where $\delta_a$ denotes the D

Theorems & Definitions (43)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 33 more