非阿基米德动力学中的内在约化与内在深度
The intrinsic reductions and the intrinsic depths in non-archimedean dynamics
AI总结：
本文为非阿基米德动力学中Rumely模理论刻画及退化极限定理提供更简单证明，引入内在约化与方向性内在深度概念，并建立两个关键定理。
摘要：
在这篇短文中，我们旨在给出Rumely关于结果函数$\operatorname{ordRes}_\phi$在Berkovich双曲空间上的II型极小轨迹的模理论刻画的更具概念性和更简单的证明，其中$\phi$是定义在配备非平凡非阿基米德绝对值的代数闭完备域上的$\mathbb{P}^1$上的有理函数；同时，我们也旨在给出一个退化极限定理的更简单且更自然的证明，该定理在DeMarco--Faber之后以改进的形式出现，涉及与复有理函数的亚纯族相关的$\mathbb{P}^1(\mathbb{C})$上唯一最大熵测度族。我们在Berkovich射影直线上的每一点引入非阿基米德有理函数$\phi$的内在约化及其方向性内在深度，这些概念基于Berkovich射影直线的树结构和解析结构定义，适用于上述目标。随后，我们在非阿基米德动力学中建立两个定理，这两个定理在上述目标中均发挥关键作用。
英文摘要：
In this short paper, we aim at giving a more conceptual and simpler proof of Rumely's moduli theoretic characterization of type II minimal locus of the resultant function $\operatorname{ordRes}_ϕ$ on the Berkovich hyperbolic space for a rational function $ϕ$ on $\mathbb{P}^1$ defined over an algebraically closed and complete field that is equipped with a non-trivial and non-archimedean absolute value, and also aim at giving a much simpler and more natural proof of a degenerating limit theorem, in an improved form after DeMarco--Faber, for the family of the unique maximal entropy measures on $\mathbb{P}^1(\mathbb{C})$ associated to a meromorphic family of complex rational functions. We introduce the intrinsic reduction of a non-archimedean rational function $ϕ$ at each point in the Berkovich projective line and its directionwise intrinsic depths, which are suitable notions for the above aims and defined in terms of the tree and analytic structures of the Berkovich projective line. Then we establish two theorems in non-archimedean dynamics, both of which play key roles in the above aims.
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