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Selberg and Brolin on value distribution of complex dynamics

Yûsuke Okuyama

TL;DR

This work connects Selberg-type covering theory with quantitative value distribution in complex dynamics on $P^1$, providing explicit rates for equidistribution of both iterated preimages and periodic points of rational maps. By refining previous proximity arguments and employing Selberg's theorem, it proves $m(f^n,a)=O(n)$ leading to $O(nd^{-n})$-type convergence for preimages and $m(f^n,Id)=O(\eta^n)$ yielding $O((\eta/d)^n)$-rates for periodic points under an exponentwise hypothesis H. The results hold unconditionally for unicritical polynomials and extend to Hölder test functions, offering a robust quantitative bridge between Nevanlinna theory and complex dynamics. These explicit rates enhance our understanding of how rapidly dynamical distributions converge to the equilibrium measure $\mu_f$ and have potential extensions to higher-dimensional projective spaces.

Abstract

The Brolin-Lyubich-Freire--Lopes--Mañé equidistribution theorem for iterated preimages of a given non-exceptional value and Lyubich's periodic point version of it are foundational in the study of dynamics of rational functions of degree more than one on the complex projective line, and Drasin and the author studied a quantification of the former in a formalism of Nevanlinna theory or more specifically with the aid of Selberg's theorem. In this paper, we point out that the argument in that previous study have already yielded a better quantification of the Brolin-Lyubich-Freire--Lopes--Mañé equidistribution theorem, and also point out that a similar argument also yields a quantification of Lyubich's theorem under an exponentwise version of the so called hypothesis H.

Selberg and Brolin on value distribution of complex dynamics

TL;DR

This work connects Selberg-type covering theory with quantitative value distribution in complex dynamics on , providing explicit rates for equidistribution of both iterated preimages and periodic points of rational maps. By refining previous proximity arguments and employing Selberg's theorem, it proves leading to -type convergence for preimages and yielding -rates for periodic points under an exponentwise hypothesis H. The results hold unconditionally for unicritical polynomials and extend to Hölder test functions, offering a robust quantitative bridge between Nevanlinna theory and complex dynamics. These explicit rates enhance our understanding of how rapidly dynamical distributions converge to the equilibrium measure and have potential extensions to higher-dimensional projective spaces.

Abstract

The Brolin-Lyubich-Freire--Lopes--Mañé equidistribution theorem for iterated preimages of a given non-exceptional value and Lyubich's periodic point version of it are foundational in the study of dynamics of rational functions of degree more than one on the complex projective line, and Drasin and the author studied a quantification of the former in a formalism of Nevanlinna theory or more specifically with the aid of Selberg's theorem. In this paper, we point out that the argument in that previous study have already yielded a better quantification of the Brolin-Lyubich-Freire--Lopes--Mañé equidistribution theorem, and also point out that a similar argument also yields a quantification of Lyubich's theorem under an exponentwise version of the so called hypothesis H.
Paper Structure (5 sections, 6 theorems, 47 equations)

This paper contains 5 sections, 6 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Quantitative equidistribution of iterated preimages
  3. Quantitative equidistribution of periodic points
  4. Proof of Theorem \ref{['sec:intro']}
  5. Proof of Theorem \ref{['th:periodic']}

Key Result

Theorem 1.1

Let $V$ be a bounded and at most finitely connected domain in $\mathbb{C}$ whose boundary components are piecewise real analytic Jordan closed curves in $\mathbb{C}$, so that for every $y\in V$, the (positive real valued) Green function $G_V(\cdot,y)$ on $V$ with pole $y$ exists and extends continuo

Theorems & Definitions (15)

  • Theorem 1.1: Selberg Selberg44, for a modern formulation, see Weitsman72
  • Theorem 1
  • Remark 1.2
  • Theorem 2
  • Corollary 1
  • Remark 2.1
  • Remark 2.2
  • proof : Proof of Theorem \ref{['th:proximity']}
  • Definition 3.1
  • Remark 3.2
  • ...and 5 more