Piecewise quasiconformal dynamical systems of the unit circle

We study piecewise quasiconformal covering maps of the unit circle. We provide sufficient conditions so that a conjugacy between two such dynamical systems has a quasiconformal or David extension to the unit disk. Our main result generalizes the main result of Lyubich et al [David extension of circle homeomorphisms, welding, mating, and removability. Mem. Amer. Math. Soc. 313 (2025), 1588], which deals with piecewise analytic maps. As applications, we provide a classification of piecewise quasiconformal maps of the circle up to quasisymmetric conjugacy, we prove a general conformal mating theorem for Blaschke products and we study the quasiconformal geometry of parabolic basins.