标题： 施瓦茨反射、B-反演及代数对应的广义动力学理论 显示英文标题

标题： A general dynamical theory of Schwarz reflections, B-involutions, and algebraic correspondences

摘要： 本文研究了（反）多项式与Fuchsian群、反射群作为Schwarz反射、B-反演或（反）全纯对应之间的配对，以及它们的参数空间。我们证明了典型（反）多项式（如周期排斥或几何有限的（反）多项式）与来自相应群的圆映射之间配对的存在性。这些配对自然地表现为退化的（反）多项式-like映射，我们还表明此类配对对应的参数空间切片与多项式映射的参数空间有很强的相似性。此外，我们为这些配对提供了代数描述，并构造了代数对应，将典型（反）多项式和亏格为零的orbifold组合在一个共同的动力学平面上，为Fatou关于群和映射统一理论的构想提供了新的具体证据。 显示英文摘要

摘要： In this paper, we study matings of (anti-)polynomials and Fuchsian, reflection groups as Schwarz reflections, B-involutions or as (anti-)holomorphic correspondences, as well as their parameter spaces. We prove the existence of matings of generic (anti-)polynomials, such as periodically repelling, or geometrically finite (anti-)polynomials, with circle maps arising from the corresponding groups. These matings emerge naturally as degenerate (anti-)polynomial-like maps, and we show that the corresponding parameter space slices for such matings bear strong resemblance with parameter spaces of polynomial maps. Furthermore, we provide algebraic descriptions for these matings, and construct algebraic correspondences that combine generic (anti-)polynomials and genus zero orbifolds in a common dynamical plane, providing a new concrete evidence to Fatou's vision of a unified theory of groups and maps.