Approximation of non-archimedean Lyapunov exponents and applications over global fields
| Autor: | Gabriel Vigny, Yûsuke Okuyama, Thomas Gauthier |
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| Přispěvatelé: | Centre de Mathématiques Laurent Schwartz (CMLS), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), NASA Goddard Space Flight Center (GSFC), Kyoto Institute of Technology, Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352 (LAMFA), Université de Picardie Jules Verne (UPJV)-Centre National de la Recherche Scientifique (CNRS) |
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
General Mathematics [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] Dynamical Systems (math.DS) Lyapunov exponent Absolute value (algebra) 01 natural sciences Minimax approximation algorithm symbols.namesake FOS: Mathematics Number Theory (math.NT) Mathematics - Dynamical Systems 0101 mathematics Algebraically closed field [MATH]Mathematics [math] Function field Projective variety ComputingMilieux_MISCELLANEOUS Mathematics Mathematics - Number Theory Degree (graph theory) Applied Mathematics 010102 general mathematics [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] Moduli space symbols |
| Zdroj: | Transactions of the American Mathematical Society Transactions of the American Mathematical Society, American Mathematical Society, 2020, 373 (12), pp.8963-9011. ⟨10.1090/tran/8232⟩ |
| ISSN: | 0002-9947 |
| DOI: | 10.1090/tran/8232⟩ |
| Popis: | Let $K$ be an algebraically closed field of characteristic 0 that is complete with respect to a non-archimedean absolute value. We establish a locally uniform approximation formula of the Lyapunov exponent of a rational map $f$ of $\mathbb{P}^1$ of degree $d>1$ over $K$, in terms of the multipliers of $n$-periodic points of $f$, with an explicit control in terms of $n$, $f$ and $K$. As an immediate consequence, we obtain an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of rational maps over $K$. Combined with our former archimedean version, this non-archimedean quantitative approximation allows us to show: - a quantified version of Silverman's and Ingram's recent comparison between the critical height and any ample height on the moduli space $\mathcal{M}_d(\bar{\mathbb{Q}})$, - two improvements of McMullen's finiteness of the multiplier maps: reduction to multipliers of cycles of exact given period and an effective bound from below on the period, - a characterization of non-affine isotrivial rational maps defined over the function field $\mathbb{C}(X)$ of a normal projective variety $X$ in terms of the growth of the degree of the multipliers. Some typos fixed |
| Databáze: | OpenAIRE |
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