Quantitative equidistribution of periodic points for rational maps
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915273686122496 |
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| author | Gauthier, Thomas Vigny, Gabriel |
| author_facet | Gauthier, Thomas Vigny, Gabriel |
| contents | 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 $\mathbb{Q}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02608 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative equidistribution of periodic points for rational maps Gauthier, Thomas Vigny, Gabriel Dynamical Systems Complex Variables Number Theory 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 $\mathbb{Q}$. |
| title | Quantitative equidistribution of periodic points for rational maps |
| topic | Dynamical Systems Complex Variables Number Theory |
| url | https://arxiv.org/abs/2505.02608 |