We introduce the f -crucial function $${\text {Crucial}}_f$$ Crucial f associated to a rational function $$f\in K(z)$$ f ∈ K ( z ) of degree $$>1$$ > 1 over an algebraically closed field K of possibly positive characteristic that is complete with respect to a non-trivial and non-archimedean absolute value, and give a global and explicit expression of Rumely’s (resultant) function $${\text {ordRes}}_f$$ ordRes f in terms of the hyperbolic metric $$\rho $$ ρ on the Berkovich upper half space $$\mathsf {H}^1$$ H 1 in the Berkovich projective line $${\mathsf {P}}^1={\mathsf {P}}^1(K)$$ P 1 = P 1 ( K ) . We also obtain geometric formulas for Rumely’s weight function $$w_f$$ w f and crucial measure $$\nu _f$$ ν f on $${\mathsf {P}}^1$$ P 1 associated to f , as well as improvements of Rumely’s principal results. As an application to dynamics, we obtain a quantitative equidistribution of the sequence $$(\nu _{f^n})_n$$ ( ν f n ) n of $$f^n$$ f n -crucial measures towards the f -equilibrium (or canonical) measure $$\mu _f$$ μ f on $${\mathsf {P}}^1$$ P 1 .

中文翻译：

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Rumely 权重函数的几何公式​​和非阿基米德动力学中的关键测度

我们引入 f -crucial 函数 $${\text {Crucial}}_f$$ Crucial f 与度数 $$>1$ 的有理函数 $$f\in K(z)$$ f ∈ K ( z ) 相关联$ > 1 在可能正特征的代数闭域 K 上，该域对于非平凡和非阿基米德绝对值是完整的，并给出 Rumely 的（结果）函数 $${\text {ordRes }}_f$$ ordRes f 在 Berkovich 上半空间 $$\mathsf {H}^1$$ H 1 上的双曲度量 $$\rho $$ ρ 在 Berkovich 射影线 $${\mathsf { P}}^1={\mathsf {P}}^1(K)$$ P 1 = P 1 ( K ) 。我们还获得了 Rumely 权重函数 $$w_f$$ wf 的几何公式​​和与 f 相关联的 $${\mathsf {P}}^1$$ P 1 上的关键度量 $$\nu _f$$ ν f ，以及Rumely 主要结果的改进。作为动力学的应用，