Quantitative equidistribution of periodic points for rational maps

2025/05/05 by Thomas Gauthier, Gauthier, Thomas, Gabriel Vigny +1 · 1 citation
Mathematics · #Mathematical Dynamics and Fractals #Analytic and geometric function theory

paper · pdf · doi:10.48550/arxiv.2505.02608

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 \mathscrC¹-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 ℚ.

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