Quasiconformal mappings and Complex Dynamics
Albert Clop (UAB), Núria Fagella (UB) and Xavier Tolsa, ICREA (UAB)
Tue-Thu 12h-14h, from Februrary 17, 2015.
Room T1, Facultat de Matemàtiques, UB
- Some elements of complex analysis. Normal families and Montel’s theorem. Piccard’s theorem. Riemann surfaces and the uniformization theorem. Conformal geometry: the hyperbolic metric, Schwarz-Pick lemma, Koebe’s distortion theorem.
- Quasiconformal mappings. Equivalent definitions. Modulus of an annulus, ACL maps, Sobolev maps. Relationship between quasisymmetric and quasiconformal maps. Hölder regularity: Mori’s theorem. The Beurling transform: some basic Calderón-Zygmund theory. The measurable Riemann mapping theorem and the Stoilow factorization. Quasiconformal extensions of Douady-Erle and Beurling-Ahlfors. Holomorphic motions and the λ-lemma.
- Complex dynamics. Iteration of maps: Generalities. Conjugacies and equivalences. Circle homeomorphisms and rotation number. Holomorphic dynamics: the dynamical plane. Fatou and Julia sets and their properties. Local theory. Classification of Fatou components. Quasiconformal surgery. Applications: No wandering domains for rational maps; polynomial-like mappings. Families of holomorphic dynamics: parameter spaces. Parametrization of hyperbolic components using quasiconformal surgery. The quadratic family: the Mandelbrot set.
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- J.B. Conway. Functions of one complex variable (I and II). Springer-Verlag, 1978 and 1995.
- J. Milnor. Dynamics in one complex variable. Annals of Mathematical Studies, Princeton University Press, 2006.