内函数的稳定收敛
Stable convergence of inner functions
摘要：
设$\mathscr J$是导数位于Nevanlinna类中的内函数的集合。本文讨论了$\mathscr J$上的一个自然拓扑，其中如果$F_n$的临界结构收敛于$F$的临界结构，则$F_n \to F$是自然拓扑.我们表明，这正是发生时的临界结构的$F_n$均匀集中在Korenblum星。证明使用刘维尔的单位圆盘的全纯自映射和高斯曲率方程的解决方案之间的对应关系。基于Korenblum和Roberts的工作，我们证明了这种拓扑结构也控制着加权Bergman空间的不变子空间的行为，这些子空间是由单个内部函数生成的。
英文摘要：
Let $\mathscr J$ be the set of inner functions whose derivative lies in the Nevanlinna class. In this paper, we discuss a natural topology on $\mathscr J$ where $F_n \to F$ if the critical structures of $F_n$ converge to the critical structure of $F$. We show that this occurs precisely when the critical structures of the $F_n$ are uniformly concentrated on Korenblum stars. The proof uses Liouville's correspondence between holomorphic self-maps of the unit disk and solutions of the Gauss curvature equation. Building on the works of Korenblum and Roberts, we show that this topology also governs the behaviour of invariant subspaces of a weighted Bergman space which are generated by a single inner function.
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