A conjugacy between piecewise quasiconformal expansive circle coverings has a quasiconformal or David extension to the disk under Markov partition and endpoint-type conditions, with applications to Blaschke matings and parabolic basins.
Shishikura, Bifurcation of parabolic fixed points , The Mandelbrot set, theme and variations, London Mathematical Society Lecture Note Series, vol
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Piecewise quasiconformal dynamical systems of the unit circle
A conjugacy between piecewise quasiconformal expansive circle coverings has a quasiconformal or David extension to the disk under Markov partition and endpoint-type conditions, with applications to Blaschke matings and parabolic basins.