Antiholomorphic correspondences and mating I: realization theorems

Abstract

In this paper, we bring together four different branches of antiholomorphic dynamics: of global anti-rational maps, reflection groups, Schwarz reflections in quadrature domains, and antiholomorphic correspondences. We establish the first general realization theorems for bi-degree $d$:$d$ correspondences on the Riemann sphere (for $d\ge 2$) as matings of maps and groups. To achieve this, we introduce and study the dynamics of a general class of antiholomorphic correspondences; i.e., multi-valued maps with antiholomorphic local branches. Such correspondences are closely related to a class of single-valued antiholomorphic maps in one complex variable; namely, Schwarz reflection maps of simply connected quadrature domains. Using this connection, we prove that matings of all parabolic antiholomorphic rational maps with connected Julia sets (of arbitrary degree) and antiholomorphic analogues of Hecke groups can be realized as such correspondences. We also draw the same conclusion when parabolic maps are replaced with critically non-recurrent antiholomorphic polynomials with connected Julia sets.

Final version, 52 pages, 11 figures

References in corpus (7)

• Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle
• Topology of quadrature domains
• Dynamics of Schwarz reflections: the mating phenomena
• On Dynamical Gaskets Generated by Rational Maps, Kleinian Groups, and Schwarz Reflections
• Univalent Polynomials and Hubbard Trees
• Schwarz reflections and anti-holomorphic correspondences
• Antiholomorphic correspondences and mating I: realization theorems

Cited by in corpus (2)

• Antiholomorphic correspondences and mating I: realization theorems
• Mating parabolic rational maps with Hecke groups