AI 中文总结
本文给出双曲 Schwarz 反射的共形配对描述，将其分解为双曲多项式与环状 Schwarz 反射因子，并证明环状反射收敛于 Nielsen 映射，连接双曲与非双曲情形。
AI 中文摘要
我们为具有连通且完全填充 Julia 集的双曲 Schwarz 反射类 $\Sigma_d$ 给出了共形配对描述；这平行于近期在若干非双曲情形中的进展。我们证明 $\Sigma_d$ 的逃逸动力学可以独立地由我们引入的“环状 Schwarz 反射”类来刻画。我们的主要结果是：每个 $\sigma\in\Sigma_d$ 可唯一分解为一个双曲多项式因子和一个环状 Schwarz 反射因子；反之，任意这样的两个因子可以配对以确定唯一这样的 Schwarz 反射。我们还证明环状 Schwarz 反射在适当条件下收敛到 Nielsen 映射，从而连接了双曲与非双曲配对描述。
英文摘要
We give a conformal mating description for the class $Σ_d$ of hyperbolic Schwarz reflections with connected and full filled Julia set; this parallels the recent development in several non-hyperbolic settings. We show that the escaping dynamics of $Σ_d$ can be characterized independently by the class of \emph{annular Schwarz reflections} we introduce. Our main result is that each $σ\inΣ_d$ decomposes into a unique hyperbolic polynomial factor and a unique annular Schwarz reflection factor and, conversely, any two such factors may be mated to determine a unique such Schwarz reflection. We also show that annular Schwarz reflections converge to Nielsen maps in suitable settings, connecting the hyperbolic and non-hyperbolic mating descriptions.
