A celebrated theorem of M. Heins says that up to post-composition with a Möbius transformation, a finite Blaschke product is uniquely determined by its critical points. K. Dyakonov suggested that it may interesting to extend this result to infinite degree, however, one needs to be careful since different inner functions may have identical critical sets. In this work, we try parametrizing inner functions by 1-generated invariant subspaces of the weighted Bergman space $A_1^2$. Our technique is based on the Liouville correspondence which provides a bridge between complex analysis and non-linear elliptic PDE.

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M. Heins 的一个著名定理说，在使用莫比乌斯变换进行后合成之前，有限的 Blaschke 积由其临界点唯一地确定。K. Dyakonov 建议将这一结果扩展到无限程度可能很有趣，但是，由于不同的内部函数可能具有相同的临界集，因此需要小心。在这项工作中，我们尝试通过加权 Bergman 空间的 1-生成不变子空间参数化内部函数${\mathrm{一种}}_1^2$. 我们的技术基于 Liouville 对应，它在复分析和非线性椭圆偏微分方程之间架起了一座桥梁。