Quasiconformal Surgery and Complex Dynamics

Quasiconformal surgery, Shishikura fundamental lemma, Sullivan general principle, and Sullivan straightening theorem characterizing quasiregular quasirational functions. Polynomial-like functions. Hybrid conjugacy as a quasiconformal mapping of open sets with the filled Julia sets of respective polynomial-like functions that are holomorphic conjugacies of the restrictions to the filled Julia sets, and proof that every polynomial-like function is hybridly conjugated to a polynomial. Applications to: parametrization of the Mandelbrot set hyperbolic components by the attracting periodic orbit multiplier, as in 1981 by Sullivan and, independently, Douady and Hubbard; Sullivan nonwandering domains theorem, of 1985; modifying the multiplier of an attracting periodic orbit; modifying superattracting to attracting periodic orbits; modifying the rotation ring module; sewing a Siegel disk at an invariant Jordan curve as initiated by Ghys in 1984 and justified with a contribution of Herman in 1986; making a Herman ring from a Siegel disk (and analogously for their cycles), as by Shishikura in 1987; optimal upper bounds of the number of attracting or neutral periodic orbits and of the number of Herman ring cycles of rational functions, as by Shishikura in 1987; sewing a continuum in a Julia set, as by McMullen in 1988.