Quasiconformal mappings and Complex Dynamics

Lecturers
Albert Clop (UAB), Núria Fagella (UB) and Xavier Tolsa, ICREA (UAB)
Schedule
Tue-Thu 12h-14h, from Februrary 17, 2015.
Location
Room T1, Facultat de Matemàtiques, UB
Contents
  1. 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.
  2. 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.
  3. 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.
Bibliography
  1.  L.V. Ahlfors. Lectures on Quasiconformal Mappings. American Mathematical Soc., 1966.
  2. K. Astala, T. Iwaniec and G. Martin. Elliptic partial differential equations and quasiconformal mappings in the plane. Princeton University Press, 2009.
  3.  A.F. Beardon. Iteration of rational functions: Complex analytic dynamical systems. Springer, 1991.
  4. B. Branner and N. Fagella. Quasiconformal surgery in holomorphic dynamics. Cambridge University Press, 2014.
  5. L. Carleson and T.W. Gamelin. Complex dynamics. Springer, 1995.
  6. J.B. Conway. Functions of one complex variable (I and II). Springer-Verlag, 1978 and 1995.
  7. J. Milnor. Dynamics in one complex variable. Annals of Mathematical Studies, Princeton University Press, 2006.