We prove a Carathéodory-type extension of branched quasisymmetric (BQS) homeomorphisms between two domains in proper, locally path-connected metric spaces as homeomorphisms between their prime end closures. We also give a Carathéodory-type extension of geometric quasiconformal mappings between two such domains provided the two domains are both Ahlfors Q-regular and support a Q-Poincare inequality when equipped with their respective Mazurkiewicz metrics. We also provide examples to demonstrate the strengths and weaknesses of prime end closures in this context.

中文翻译：

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关于素数终止边界的Carathéodory型扩展定理

我们证明了在适当的，局部路径连接的度量空间中的两个域之间的分支拟对称（BQS）同胚的Carathéodory型扩展作为它们的主要末端封闭之间的同胚。我们还给出了两个这样的域之间的几何拟保形映射的Carathéodory型扩展，条件是两个域都是Ahlfors Q - regular且当配备各自的Mazurkiewicz度量时支持Q- Poincare不等式。我们还提供了一些示例来说明这种情况下主要端盖的优点和缺点。