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We provide a David extension result for circle homeomorphisms conjugating twodynamical systems such that parabolic periodic points go to parabolic periodicpoints, but hyperbolic points can go to parabolics as well. We use this result,in particular, to prove the existence of a new class of welding homeomorphisms,to establish an explicit dynamical connection between critically fixedanti-rational maps and kissing reflection groups, to show conformalremovability of the Julia sets of geometrically finite polynomials and of thelimit sets of necklace reflection groups, to produce matings ofanti-polynomials and necklace reflection groups, and to give a new proof of theexistence of Suffridge polynomials (extremal points in certain spaces ofunivalent maps).

David extension of circle homeomorphisms, welding, mating, and removability