Quantitative equidistribution of periodic points for rational maps

Abstract

We show that periodic points of period $n$ of a complex rational map of degree $d$ equidistribute towards the equilibrium measure $\hat{I}{¼}_f$ of the rational map with a rate of convergence of $(n{d}^{-n}{)}^{1/2}$ for ${\mathcal{C}}^1$-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 $\mathbb{Q}$.