Let $\mathcal{J}$ be the set of inner functions whose derivative lies in the Nevanlinna class. In this paper, we discuss a natural topology on $\mathcal{J}$ 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.

中文翻译：

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内部功能稳定收敛

让 $\mathcal{Ĵ}$是内部函数集，其派生属于Nevanlinna类。在本文中，我们讨论了一个自然拓扑$\mathcal{Ĵ}$ 哪里 $F_{ñ}\to F$ 如果关键结构 $F_{ñ}$ 汇聚到 $F$。我们证明，这恰好发生在$F_{ñ}$统一集中在科伦布拉姆星上。该证明使用单位圆盘全纯自映射与高斯曲率方程解之间的Liouville对应关系。在Korenblum和Roberts的著作的基础上，我们证明了这种拓扑结构还控制着由单个内部函数生成的加权Bergman空间不变子空间的行为。