Mirrors of conformal dynamics: Interplay between anti-rational maps, reflection groups, Schwarz reflections, and correspondences

TL;DR

This survey explores the deep connections between anti-rational maps, reflection groups, Schwarz reflections, and algebraic correspondences, highlighting their interactions and applications in conformal dynamics.

Contribution

It provides a comprehensive overview of the interplay between four key areas in conformal dynamics, including new examples, relations, and technical tools like David surgery.

Findings

Examples of Schwarz reflections and algebraic correspondences from matings.

Relations between parameter spaces of different dynamical systems.

Applications of the theory to analytic problems.

Abstract

The goal of this survey is to present intimate interactions between four branches of conformal dynamics: iterations of anti-rational maps, actions of Kleinian reflection groups, dynamics generated by Schwarz reflections in quadrature domains, and algebraic correspondences. We start with several examples of Schwarz reflections as well as algebraic correspondences obtained by matings between anti-rational maps and reflection groups, and examples of Julia set realizations for limit sets of reflection groups (including classical Apollonian-like gaskets). We follow up these examples with dynamical relations between explicit Schwarz reflection parameter spaces and parameter spaces of anti-rational maps and of reflection groups. These are complemented by a number of general results and illustrations of important technical tools, such as David surgery and straightening techniques. We also collect several analytic applications of the above theory.