The main object in this talk will be the mating of piecewise
(antanalytic dynamical systems of the unit disk. While quasiconformal
maps can be used for the mating of two hyperbolic dynamical systems,
they are insufficient for mating a hyperbolic dynamical system with a
parabolic one. Instead, we achieve the mating using the notion of a
David homeomorphism, which is a generalization of a quasiconformal
homeomorphism that allows unbounded quasiconformal dilatation. The main
theorem that we will discuss provides extensions of a general class of
dynamically defined circle homeomorphisms to David homeomorphisms of the
unit disk. An implication of this theory is that limit sets of a certain
class of Kleinian reflection groups (called necklace reflection groups)
are conformally removable. The talk is based on joint work with Misha
Lyubich, Sergei Merenkov, Sabyasachi Mukherjee, and Christina Karafyllia.