Quantitative equidistribution of periodic points for rational maps

TL;DR

This paper proves that periodic points of complex rational maps become uniformly distributed according to the equilibrium measure, with a quantifiable rate of convergence depending on the degree and period.

Contribution

It provides a quantitative rate of equidistribution for periodic points of rational maps, extending previous qualitative results using advanced potential theory and number theory tools.

Findings

Periodic points equidistribute towards the equilibrium measure

Convergence rate is proportional to ^{-n/2} for period n

Method combines potential theory, Baker's estimate, and Moriwaki's product formula

Abstract

We show that periodic points of period $n$ of a complex rational map of degree $d$ equidistribute towards the equilibrium measure $\mu_f$ 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 \`a la Favre and Rivera-Letelier. Our proof relies on the H\"older 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 $\mathbb{Q}$.