AI 中文总结
该研究针对黎曼球上2:2全纯对应模交配族的离散轨迹，构造了双曲平面到其补集的典范解析映射，诱导出镶嵌并提出相关结构猜想，揭示了其与经典曼德博集合的拓扑关联。
AI 中文摘要
模曼德博集合$M_\Gamma$是黎曼球上2:2全纯对应$\mathcal{F}_a$的模交配族的连通轨迹，它同胚于经典曼德博集合$M$。克莱因组合轨迹$\mathcal{K}$（即$\mathcal{F}_a$族的“离散轨迹”）是$a$平面上$M_\Gamma$的捏合邻域，在根点处捏合。受Douady和Hubbard构造著名共形双射$\Phi: \mathbb{C}\setminus M \to \mathbb{C}\setminus \overline{\mathbb{D}}$的启发，我们构造了从$\mathcal{K}\setminus M_\Gamma$到双曲平面$\mathbb{H}$的典范映射$\Psi$，并证明$\Psi$是解析的。该映射$\Psi$通过拉回模群不变的$\mathbb{H}$的镶嵌，诱导出$\mathcal{K}\setminus M_\Gamma$的镶嵌。我们提出了一系列关于$\mathcal{K}$的结构、其边界以及$\Psi(\mathcal{K}) \subset \mathbb{H}$的猜想。
英文摘要
The modular Mandelbrot set $M_Γ$, the connectedness locus of the modular mating family of 2 : 2 holomorphic correspondences $\mathcal{F}_a$ on the Riemann sphere, is homeomorphic to the classical Mandelbrot set $M$. The Klein combination locus $\mathcal{K}$ (the "discreteness locus" of the family $\mathcal{F}_a$) is a pinched neighborhood of $M_Γ$ in the $a$-plane, pinched at the root point. We construct a canonical map $Ψ$ from $\mathcal{K}\setminus M_Γ$ into the hyperbolic plane $\mathbb{H}$, inspired by the construction of Douady and Hubbard for their celebrated conformal bijection $Φ: \mathbb{C}\setminus M \to \mathbb{C}\setminus \overline{\mathbb{D}}$, and we prove that $Ψ$ is analytic. This map $Ψ$ induces a tessellation of $\mathcal{K}\setminus M_Γ$ by pulling back a tessellation of $\mathbb{H}$ invariant under the modular group. We develop a series of conjectures concerning the structure of $\mathcal{K}$, its boundary, and $Ψ(\mathcal{K}) \subset \mathbb{H}$.
Comments42 pages, 15 figures
