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We will discuss the interplay between three branches of Conformal
Dynamics: iteration of (anti-)rational maps, actions of Kleinian groups,
and dynamics generated by Schwarz reflections in quadrature domains. We
will show examples of Schwarz reflections obtained by matings between
anti-quadratic maps and the triangle modular group, and examples of Julia
realizations for Apollonian-like gaskets. Some of these examples can be
turned into others by means of a David surgery.