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arXiv · 2505.02608

Quantitative equidistribution of periodic points for rational maps

Thomas Gauthier·Gabriel Vigny
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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 $\mathbb{Q}$.

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Thomas Gauthier, Gabriel Vigny. 2025-05-05. Quantitative equidistribution of periodic points for rational maps. https://arxiv.org/abs/2505.02608

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