ANTIHOLOMORPHIC CORRESPONDENCES AND MATING I: REALIZATION THEOREMS

In this paper, we bring together four different branches of antiholomor- phic dynamics: of global anti-rational maps, reflection groups, Schwarz reflections in quadrature domains, and antiholomorphic correspondences. We establish the first generalrealizationtheoremsforbi-degree d:d correspondences on the Riemann sphere (for d ≥ 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.