反全纯对应与配对 I：实现定理
Antiholomorphic correspondences and mating I: realization theorems
AI总结：
本文结合反全纯动力学的四个分支，引入了一般反全纯对应类，并建立了 Riemann 球面上双次数 d:d 对应作为映射与群配对的实现定理。
摘要：
在本文中，我们将反全纯动力学的四个不同分支结合在一起：全局反有理映射、反射群、求积域中的 Schwarz 反射以及反全纯对应。我们建立了 Riemann 球面上（对于 $d\geq 2$）双次数 $d$:$d$ 对应作为映射与群的配对的第一个一般性实现定理。为了实现这一点，我们引入并研究了一般反全纯对应类的动力学；即具有反全纯局部分支的多值映射。这种对应与单复变量中的一类单值反全纯映射密切相关；即单连通求积域的 Schwarz 反射映射。利用这种联系，我们证明了所有具有连通 Julia 集（任意次数）的抛物反全纯有理映射与 Hecke 群的反全纯类似物的配对可以被实现为这样的对应。当抛物映射被替换为具有连通 Julia 集的临界非回归反全纯多项式时，我们得出了相同的结论。
英文摘要：
In this paper, we bring together four different branches of antiholomorphic dynamics: of global anti-rational maps, reflection groups, Schwarz reflections in quadrature domains, and antiholomorphic correspondences. We establish the first general realization theorems for bi-degree $d$:$d$ correspondences on the Riemann sphere (for $d\geq 2$) as matings of maps and groups. To achieve this, we introduce and study the dynamics of a general class of antiholomorphic correspondences; i.e., multi-valued maps with antiholomorphic local branches. Such correspondences are closely related to a class of single-valued antiholomorphic maps in one complex variable; namely, Schwarz reflection maps of simply connected quadrature domains. Using this connection, we prove that matings of all parabolic antiholomorphic rational maps with connected Julia sets (of arbitrary degree) and antiholomorphic analogues of Hecke groups can be realized as such correspondences. We also draw the same conclusion when parabolic maps are replaced with critically non-recurrent antiholomorphic polynomials with connected Julia sets.
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