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Approximation of non-archimedean Lyapunov exponents and applications over global fields
Transactions of the American Mathematical Society ( IF 1.363 ) Pub Date : 2020-10-05 , DOI: 10.1090/tran/8232
Thomas Gauthier; Yûsuke Okuyama; Gabriel Vigny

Abstract:Let $ K$ be an algebraically closed field of characteristic 0 that is complete with respect to a non-trivial and non-archimedean absolute value. We establish an approximation of the Lyapunov exponent of a rational map $ f$ of $ \mathbb{P}^1$ of degree $ d>1$ defined over $ K$ in terms of the multipliers of periodic points of $ f$ having the formally exact period $ n$, with an explicit error estimate in terms of $ f,n$, and $ d$. As an immediate consequence, we obtain an estimate on the blow-up of the Lyapunov exponent function near a pole in one-dimensional parameter families of rational maps over $ K$. Combined with an improvement of our former archimedean counterpart, this non-archimedean quantitative approximation of Lyapunov exponents allows us to establish [label=-]
  • a quantification of Silverman's and Ingram's recent comparison between the critical height and any ample height on the dynamical moduli space $ \mathcal {M}_d(\overline {\mathbb{Q}})$ except for the flexible Lattès locus,
  • an improvement of McMullen's finiteness of the multiplier maps in two aspects: reduction to multipliers of cycles having a given formally exact period, and an explicit computation on the magnitude of the formally exact period of cycles, and
  • a characterization of non-affine isotrivial rational maps defined over a function field $ \mathbb{C}(X)$ of a complex normal projective variety $ X$ in terms of the growth of the degree of the multipliers of cycles.

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更新日期:2020-11-21
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