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Dynamics of Schwarz reflections: the mating phenomena

Seung-Yeop Lee, Mikhail Lyubich, Nikolai G. Makarov, Sabyasachi Mukherjee

TL;DR

The paper develops Schwarz reflection dynamics for quadrature-domain boundaries, unveiling a rich anti-holomorphic mating theory. It proves that the deltoid’s Schwarz reflection is the unique conformal mating of the ideal triangle group with the quadratic anti-polynomial $f_0(z)=\bar z^2$, and then extends the framework to a circle-and-cardioid family $\mathcal{S}$ to realize further matings with $\bar z^2+c$. By analyzing tiling/non-escaping sets, Fatou components, limit sets, and the parameter space (connectedness locus), the authors establish a robust dynamical and geometric picture, including local connectivity, conformal removability, and a dynamical-uniformization scheme for the tiling set. They also connect the mating picture to laminations and the Minkowski question-mark function via a circle homeomorphism $\mathcal{E}$, enabling a combinatorial bridge to anti-polynomial dynamics. The work thereby broadens mating theory to anti-holomorphic and reflection-group dynamics, providing explicit hyperbolic/Misiurewicz cases and concrete mating examples in the circle-and-cardioid family.

Abstract

We initiate the exploration of a new class of anti-holomorphic dynamical systems generated by Schwarz reflection maps associated with quadrature domains. More precisely, we study Schwarz reflection with respect to the deltoid, and Schwarz reflections with respect to the cardioid and a family of circumscribing circles. We describe the dynamical planes of the maps in question, and show that in many cases, they arise as unique conformal matings of quadratic anti-holomorphic polynomials and the ideal triangle group.

Dynamics of Schwarz reflections: the mating phenomena

TL;DR

The paper develops Schwarz reflection dynamics for quadrature-domain boundaries, unveiling a rich anti-holomorphic mating theory. It proves that the deltoid’s Schwarz reflection is the unique conformal mating of the ideal triangle group with the quadratic anti-polynomial , and then extends the framework to a circle-and-cardioid family to realize further matings with . By analyzing tiling/non-escaping sets, Fatou components, limit sets, and the parameter space (connectedness locus), the authors establish a robust dynamical and geometric picture, including local connectivity, conformal removability, and a dynamical-uniformization scheme for the tiling set. They also connect the mating picture to laminations and the Minkowski question-mark function via a circle homeomorphism , enabling a combinatorial bridge to anti-polynomial dynamics. The work thereby broadens mating theory to anti-holomorphic and reflection-group dynamics, providing explicit hyperbolic/Misiurewicz cases and concrete mating examples in the circle-and-cardioid family.

Abstract

We initiate the exploration of a new class of anti-holomorphic dynamical systems generated by Schwarz reflection maps associated with quadrature domains. More precisely, we study Schwarz reflection with respect to the deltoid, and Schwarz reflections with respect to the cardioid and a family of circumscribing circles. We describe the dynamical planes of the maps in question, and show that in many cases, they arise as unique conformal matings of quadratic anti-holomorphic polynomials and the ideal triangle group.

Paper Structure

This paper contains 62 sections, 74 theorems, 72 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Ideal triangle group
  3. The reflection map $\rho$, and symbolic dynamics
  4. The conjugacy $\mathcal{E}$
  5. Quadrature domains and Schwarz reflections
  6. Dynamics of deltoid reflection
  7. Schwarz reflection with respect to the deltoid
  8. Deltoid
  9. Schwarz reflection
  10. Covering properties of $\sigma$
  11. Tiling set
  12. Non-escaping set, basin of infinity
  13. Singular points
  14. Main results
  15. Proof of Theorem \ref{['deltoid_dynamical_partition']}
  16. ...and 47 more sections

Key Result

Theorem 1.1

1) The dynamical plane of the Schwarz reflection $\sigma$ of the deltoid can be partitioned as where $T^\infty$ is the tiling set, $A(\infty)$ is the basin of infinity, and $\Gamma$ is their common boundary (which we call the limit set). Moreover, $\Gamma$ is a conformally removable Jordan curve. 2) $\sigma$ is the unique conformal mating of the reflection map $\rho: \overline{\mathbb D}\setm

Figures (11)

  • Figure 1: The rational map $\varphi$ semi-conjugates the reflection map $1/\overline{z}$ of $\mathbb{D}$ to the Schwarz reflection map $\sigma$ of $\Omega$ .
  • Figure 2: Left: The $\mathcal{G}$-tessellation of $\mathbb{D}$ and the formation of a few initial tiles are shown. Right: The dual tree to the $\mathcal{G}$-tessellation of $\mathbb{D}$.
  • Figure 3: Left: Each connected component of $X_1=\varphi^{-1}(T^0)\cap\overline{\mathbb{D}}$ is mapped univalently onto $T^0$ by $\varphi$. Moreover, $\varphi$ is a two-to-one branched covering from the connected set $X_2$ onto $\Omega$, and the only ramification point is the origin. Right: Schwarz dynamics of the deltoid with the tiles of various ranks shaded.
  • Figure 4: The image of the geodesic ray $\pmb{\gamma}^{\mathbf{z}_0}$ under $\rho^{\circ N}$ shadows the actual geodesic ray $\pmb{\gamma}^{\mathbf{z}_N'}$.
  • Figure 5: Left: The Farey tree. Right: The dyadic tree.
  • ...and 6 more figures

Theorems & Definitions (157)

  • Theorem 1.1: Dynamics of deltoid reflection
  • Theorem 1.2: Connectivity of the non-escaping set
  • Theorem 1.3: Fatou components and critical orbits
  • Theorem 1.4: Limit sets of geometrically finite maps
  • Definition 2.1: Tiles
  • Definition 2.2: Rays
  • Remark 2.3
  • Definition 3.1: Schwarz functions and quadrature domains
  • Definition 3.2: Quadrature functions
  • Theorem 3.3: Characterization of quadrature domains
  • ...and 147 more