Quasiconformal Surgery and Complex Dynamics

Luis T. Magalhães
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Luis T. Magalhães: University of Lisbon, Instituto Superior Técnico

Chapter Chapter 6 in Quasiconformal Mappings in the Plane and Complex Dynamics, 2025, pp 149-210 from Springer

Abstract: Abstract 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.

Date: 2025
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DOI: 10.1007/978-3-031-80115-0_6

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