Quantitative equidistribution of periodic points for rational maps

TL;DR

The paper obtains an explicit rate of quantitative equidistribution for periodic points of a complex rational map $f$ toward its equilibrium measure $\mu_f$, demonstrating a rate of $(nd^{-n})^{1/2}$ against $\mathscr{C}^1$ observables and $(\log n/n)^{1/2}$ for Hölder observables. The approach couples Favre–Rivera-Letelier–type equidistribution for quasi-Fekete configurations with a generalized product formula for finitely generated fields, and crucially uses the Hölder regularity of the dynamical Green function $g_f$ and Baker’s estimate for the Hsia kernel. By proving that Galois-invariant finite sets of preperiodic points are quasi-Fekete, the authors transfer arithmetic information (via the generalized product formula) into analytic equidistribution rates, recovering known results over number fields and extending them to finitely generated fields. The results provide explicit, field-uniform convergence rates for periodic points and establish a robust framework connecting arithmetic dynamics with complex potential theory.

Abstract

We show that periodic points of period $n$ of a complex rational map of degree $d$ equidistribute towards the equilibrium measure $μ_f$ of the rational map with a rate of convergence of $(nd^{-n})^{1/2}$ for $\mathscr{C}^1$-observables. This is a consequence of a quantitative equidistribution of Galois invariant finite subsets of preperiodic points à la Favre and Rivera-Letelier. Our proof relies on the Hölder regularity of the quasi-psh Green function of a rational map, an estimate of Baker concerning Hsia kernel, as well as on the product formula and its generalization by Moriwaki for finitely generated fields over $\mathbb{Q}$.

Theorems & Definitions (5)

• proof
• proof
• proof : Proof of the Claim
• proof : Proof of Theorem \ref{['tm:Galois-quantitative']}
• proof : Proof of Theorem \ref{['Main-theorem']}