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Equidistribution speed of iterated preimages for rational maps on the Riemann sphere

Mai Hao, Zhuchao Ji

TL;DR

This work provides a near-optimal quantitative equidistribution speed for iterated preimages under rational maps on the Riemann sphere ${\mathbb P}^1$. It delivers a sharp $O(n d^{-n})$ rate for points not in super-attracting cycles in general, and strengthens to $O(d^{-n})$ for geometrically finite maps, including all hyperbolic maps, by combining potential-theoretic truncations with diameter estimates of inverse branches. The authors compare geometric, diameter-based methods with Nevanlinna-theoretic approaches and derive explicit constants; independent results by Okuyama support the $O(nd^{-n})$ rate. The results advance quantitative understanding of equidistribution towards the maximal entropy measure $\mu_f$ and have potential implications for the effective dynamics of rational maps on ${\mathbb P}^1$ and related moduli spaces.

Abstract

The exponential equidistribution speed of iterated preimages for holomorphic endomorphisms on $\mathbb{P}^k$ was established by Drasin-Okuyama for $k=1$, and by Dinh-Sibony for arbitrary $k$. In this paper, we obtain a near-optimal equidistribution speed with order $O(nd^{-n})$ in dimension one for points that are not super-attracting periodic. For geometrically finite rational maps (including all hyperbolic rational maps), we prove that the equidistribution speed order is $O(d^{-n})$ for points that are not super-attracting, attracting, or parabolic periodic.

Equidistribution speed of iterated preimages for rational maps on the Riemann sphere

TL;DR

This work provides a near-optimal quantitative equidistribution speed for iterated preimages under rational maps on the Riemann sphere . It delivers a sharp rate for points not in super-attracting cycles in general, and strengthens to for geometrically finite maps, including all hyperbolic maps, by combining potential-theoretic truncations with diameter estimates of inverse branches. The authors compare geometric, diameter-based methods with Nevanlinna-theoretic approaches and derive explicit constants; independent results by Okuyama support the rate. The results advance quantitative understanding of equidistribution towards the maximal entropy measure and have potential implications for the effective dynamics of rational maps on and related moduli spaces.

Abstract

The exponential equidistribution speed of iterated preimages for holomorphic endomorphisms on was established by Drasin-Okuyama for , and by Dinh-Sibony for arbitrary . In this paper, we obtain a near-optimal equidistribution speed with order in dimension one for points that are not super-attracting periodic. For geometrically finite rational maps (including all hyperbolic rational maps), we prove that the equidistribution speed order is for points that are not super-attracting, attracting, or parabolic periodic.
Paper Structure (5 sections, 12 theorems, 42 equations)

This paper contains 5 sections, 12 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Denker--Przytycki--Urbański's lemma for Fatou critical points
  3. Diameter of the pull-back of a ball
  4. Proof of Theorem \ref{['thm:main']}
  5. Proof of Theorem \ref{['thm:gf']}

Key Result

Theorem 1.1

Let $f$ be a rational map on ${\mathbb P}^1$ of degree $d\geq 2$. Let $a\in {\mathbb P}^1$ which is not a super-attracting periodic point. Then for any $1<\lambda<d$, there exists a constant $C=C(f,a,\lambda)>0$ such that for any ${\mathcal{C}}^2$ observable $\phi:{\mathbb P}^1\to {\mathbb R}$,

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • ...and 8 more