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.

中文翻译：

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非阿基米德李雅普诺夫指数的近似及其在全球领域的应用

令 $K$ 是特征为 0 的代数闭域，它相对于非阿基米德绝对值是完备的。我们建立了一个 Lyapunov 指数的局部一致近似公式，即在 $n$- 的乘数方面，$\mathbb{P}^1$ 的度数 $d>1$ 的有理映射 $f$ 超过 $K$ $f$ 的周期点，在 $n$、$f$ 和 $K$ 方面有明确的控制。作为一个直接的结果，我们获得了对 $K$ 上的一维有理映射族中极点附近的 Lyapunov 指数爆炸的估计。结合我们以前的阿基米德版本，这种非阿基米德定量近似使我们能够展示： - 西尔弗曼和英格拉姆最近比较临界高度和模空间上任何充足高度的量化版本 $\mathcal{M}_d(\bar {\mathbb{Q}})$,