Postcritically Finite Rational Functions and Combinatorial Equivalence

Branched coverings of a Riemann surface and correspondence to rational functions when the surface is the Riemann sphere (possibly not respecting the dynamics, as by Thom in 1965; respecting the dynamics via a quasiconformal conjugacy if and only if the dilatation is uniformly bounded, as by Sullivan in 1978 and Tukia in 1980). Proof that for postcritically finite rational functions, alternatively the Julia set is the whole Riemann sphere (ergodicity) or it has area 0. Combinatorial equivalence of marked branched coverings, as by Thurston in 1985. Basic notions of orbifolds. Teichmüller space of the Riemann sphere marked by a subset. Thurston theorem of topological characterization of postcritically finite rational maps not Lattès maps of 1982 proved by Douady and Hubbard in 1993, considering the Thurston obstructions of the spectral radius of the transition matrix of stable multicurves to be less than 1, with quadratic differentials in the Riemann sphere and estimates of lengths of geodesic Jordan curves in the Poincaré metric of hyperbolic Riemann surfaces, proving a slightly more general result of Buff, Cui and Tan of 2011. Levy cycles as Thurston obstructions, found in 1985, and topological polynomials. Marginal fulfillment of Thurston obstruction of functions not Lattès mappings implying the existence of rotation domains, as by McMullen in 1994, relying on meromorphic quadratic differentials with almost all trajectories closed. Application to optimal bounds of the number of zeros of harmonic polynomials of complex variable, as by Geyer in in 2008, using a generalization of the argument principle to harmonic functions, by Suffridge and Thompson in 2000, with application to gravitational lensing.