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The Arakelov-Zhang pairing and Julia sets
Proceedings of the American Mathematical Society ( IF 1 ) Pub Date : 2021-06-22 , DOI: 10.1090/proc/15518
Andrew Bridy , Matt Larson

Abstract:The Arakelov-Zhang pairing $\langle \psi ,\phi \rangle$ is a measure of the “dynamical distance” between two rational maps $\psi$ and $\phi$ defined over a number field $K$. It is defined in terms of local integrals on Berkovich space at each completion of $K$. We obtain a simple expression for the important case of the pairing with a power map, written in terms of integrals over Julia sets. Under certain disjointness conditions on Julia sets, our expression simplifies to a single canonical height term; in general, this term is a lower bound. As applications of our method, we give bounds on the difference between the canonical height $h_\phi$ and the standard Weil height $h$, and we prove a rigidity statement about polynomials that satisfy a strong form of good reduction.


中文翻译:

Arakelov-Zhang 配对和 Julia 集

摘要:Arakelov-Zhang 配对 $\langle \psi ,\phi \rangle$ 是对在数域 $K$ 上定义的两个有理映射 $\psi$ 和 $\phi$ 之间“动态距离”的度量。它是根据每次完成 $K$ 时 Berkovich 空间上的局部积分定义的。我们获得了一个简单的表达式,用于与功率图配对的重要情况,用 Julia 集上的积分表示。在 Julia 集上的某些不相交条件下,我们的表达式简化为单个规范高度项;一般来说,这个术语是一个下界。作为我们方法的应用,我们给出了规范高度 $h_\phi$ 和标准 Weil 高度 $h$ 之间的差异的界限,并且我们证明了关于满足强归约形式的多项式的刚性陈述。
更新日期:2021-07-27
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