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Algebraic correspondences and Schwarz reflections: Where rational dynamics meets Kleinian groups

Luna Lomonaco, Sabyasachi Mukherjee

TL;DR

The article surveys the dynamics of algebraic correspondences, emphasizing matings between rational maps and Kleinian groups within both holomorphic and antiholomorphic frameworks. It develops a unified internal/external map paradigm and demonstrates explicit one-parameter families (e.g., $\mathcal{F}_a$, Circle-and-Cardioid, $\Sigma_d^*$) that realize matings with modular, Hecke, and necklace groups, including Schwarz reflection dynamics. It analyzes parameter spaces such as the modular Mandelbrot set $\mathcal{M}_{\Gamma}$ and the parabolic Mandelbrot set $\mathcal{M}_1$, proving a homeomorphism between them and describing a tessellation of the exterior via partial Böttcher maps, thereby linking deformation spaces to moduli of correspondences. The work also connects Julia sets to limit sets through Nielsen-type dynamics, yielding applications to conformal removability and welding, and outlines a general four-step program to extend these matings to broader group/map combinations, higher genus, and transcendental dynamics. Overall, the paper lays a comprehensive foundation for a broad combination program in complex dynamics, with deep structural parallels between rational dynamics and Kleinian groups and significant implications for parameter-space geometry and geometric function theory.

Abstract

We present an overview of the rapidly evolving field of dynamics of algebraic correspondences, with a focus on matings between rational maps and Kleinian groups. These correspondences exhibit rich dynamics, both within the Sullivan dictionary and beyond. We highlight unifying structures in their parameter spaces, showing how moduli spaces of rational maps and Kleinian groups naturally connect. We also outline a range of applications of the techniques developed in this framework and conclude with several promising directions.

Algebraic correspondences and Schwarz reflections: Where rational dynamics meets Kleinian groups

TL;DR

The article surveys the dynamics of algebraic correspondences, emphasizing matings between rational maps and Kleinian groups within both holomorphic and antiholomorphic frameworks. It develops a unified internal/external map paradigm and demonstrates explicit one-parameter families (e.g., , Circle-and-Cardioid, ) that realize matings with modular, Hecke, and necklace groups, including Schwarz reflection dynamics. It analyzes parameter spaces such as the modular Mandelbrot set and the parabolic Mandelbrot set , proving a homeomorphism between them and describing a tessellation of the exterior via partial Böttcher maps, thereby linking deformation spaces to moduli of correspondences. The work also connects Julia sets to limit sets through Nielsen-type dynamics, yielding applications to conformal removability and welding, and outlines a general four-step program to extend these matings to broader group/map combinations, higher genus, and transcendental dynamics. Overall, the paper lays a comprehensive foundation for a broad combination program in complex dynamics, with deep structural parallels between rational dynamics and Kleinian groups and significant implications for parameter-space geometry and geometric function theory.

Abstract

We present an overview of the rapidly evolving field of dynamics of algebraic correspondences, with a focus on matings between rational maps and Kleinian groups. These correspondences exhibit rich dynamics, both within the Sullivan dictionary and beyond. We highlight unifying structures in their parameter spaces, showing how moduli spaces of rational maps and Kleinian groups naturally connect. We also outline a range of applications of the techniques developed in this framework and conclude with several promising directions.

Paper Structure

This paper contains 26 sections, 10 theorems, 7 equations, 8 figures.

Table of Contents

  1. Introduction.
  2. The first family of matings.
  3. The family $\mathcal{F}_a$.
  4. Matings between parabolic rational maps and the modular group.
  5. Dynamics of modular matings.
  6. The modular Mandelbrot set.
  7. Tessellation of the exterior of the modular Mandelbrot set.
  8. A general framework of mating: internal and external maps.
  9. Mating phenomena in the antiholomorphic world: Schwarz reflection dynamics.
  10. Quadrature domains and Schwarz reflection maps.
  11. The deltoid example: mating $\overline{z}^2$ with the ideal triangle reflection group $\pmb{G}_2$.
  12. Special families of Schwarz reflections.
  13. The Circle-and-Cardioid family: mating quadratic anti-polynomials with $\pmb{G}_2$.
  14. Antiholomorphic counterpart of the family $\mathcal{F}_a$.
  15. The family $\Sigma_d^\ast$: mating $\overline{z}^d$ with necklace groups.
  16. ...and 11 more sections

Key Result

Theorem 2.1

For $a = 4$, the correspondence $\mathcal{F}_a: z \rightarrow w$ given by is a mating between $\mathop{\mathrm{PSL}}\nolimits(2,\mathbb{Z})$ and $P_{-2}(z)=z^2-2$. Further, all matings between the modular group and quadratic polynomials belong to the family $\mathcal{F}_a$.

Figures (8)

  • Figure 1: Left: Tessellation of $\mathop{\mathrm{PSL}}\nolimits_2(\mathbb{Z})$. Center and Right: Mandelbrot set, basilica, and a disconnected Julia set.
  • Figure 2: Top: Schematic fundamental domains $\Delta_Q$ for $Cov_0^Q$ and $\Delta_a$ for $J_a$. Bottom: Dynamics of $\mathcal{F}_a$.
  • Figure 3: Above, from the left: the modular Mandelbrot set $\mathcal{M}_{\Gamma}$, and the limit sets for correspondences $\mathcal{F}_a$ where $a$ is the center of the hyperbolic components of $\mathcal{M}_{\Gamma}$ of periods $1, 2, 3$, respectively. Below, from the left: the parabolic Mandelbrot set $\mathcal{M}_1$, and the filled Julia sets $K_A$ for $P_A$ hybrid equivalent to $\mathcal{F}_a$ above.
  • Figure 4: Top left: Dynamical tessellation via partial Böttcher map $\varphi_a$. Top right: Parameter tessellation on $\mathbb{D}(4,3)$. Bottom: Points that approximate $\partial \mathcal{K}$. Pictures are courtesy of Ivan Pedro Suarez Navarro.
  • Figure 5: Top left: The Nielsen map $\pmb{\mathcal{N}}_2$ of the ideal triangle group $\pmb{G}_2$. Top middle: The brown region is the droplet and its complement is the quadrature domain $\mathcal{U}$ of §\ref{['deltoid_subsec']}. The tiling set $T^\infty(\sigma)$, which resembles the tiling of $\mathbb{D}$ by ideal triangles, is the Jordan domain bounded by the green fractal. Top right: A branch of the associated correspondence $\mathfrak{C}$ preserves a yellow component and is conformally conjugate to $\overline{z}^2$. The branches of $\mathfrak{C}$ act as anti-conformal self-maps on the green/blue region, generating a properly discontinuous group action conformally equivalent to $\pmb{G}_2$. Bottom left: Part of the connectedness locus (and exterior tessellation) of the C&C family of §\ref{['circle_and_cardioid_subsec']}. Bottom right: Mating of $\overline{z}^2-1$ and $\pmb{\mathcal{N}}_2$ in the C&C family (droplet in white).
  • ...and 3 more figures

Theorems & Definitions (12)

  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Definition 5.1
  • Theorem 5.2
  • Theorem 5.3
  • Theorem 5.4
  • Theorem 5.5
  • Theorem 5.6
  • ...and 2 more