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Title: Approximation of non-archimedean Lyapunov exponents and applications over global fields

Abstract: 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 mutiplier 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.
 Subjects: Dynamical Systems (math.DS); Number Theory (math.NT) Cite as: arXiv:1803.06859 [math.DS] (or arXiv:1803.06859v1 [math.DS] for this version)

Submission history

From: Thomas Gauthier [view email]
[v1] Mon, 19 Mar 2018 10:29:07 GMT (62kb)