  We provide a David extension result for circle homeomorphisms conjugating two
dynamical systems such that parabolic periodic points go to parabolic periodic
points, 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 fixed
anti-rational maps and kissing reflection groups, to show conformal
removability of the Julia sets of geometrically finite polynomials and of the
limit sets of necklace reflection groups, to produce matings of
anti-polynomials and necklace reflection groups, and to give a new proof of the
existence of Suffridge polynomials (extremal points in certain spaces of
univalent maps).
