内函数的临界结构
Critical structures of inner functions
摘要：
M.海因斯说，直到后合成与莫比乌斯变换，有限的布拉施克产品是唯一确定的临界点。K. Dyakonov建议将这个结果扩展到无限程度可能很有趣，但是，需要小心，因为内部函数可能具有相同的临界集。在这项工作中，我们尝试参数化内函数的1-生成的不变子空间的加权Bergman空间$A^2_1$。我们的技术是基于刘维尔对应，它提供了一个复分析和非线性椭圆偏微分方程之间的桥梁。
英文摘要：
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 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 $A^2_1$. Our technique is based on the Liouville correspondence which provides a bridge between complex analysis and non-linear elliptic PDE.
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