Quasiconformal mappings and Complex Dynamics
Quasiconformal mappings and Complex Dynamics
Date
Tue-Thu 12h-14h
from Februrary 17, 2015.
Registration
–
Location
Room T1
Facultat de Matemàtiques, UB
This course will explore the interplay between quasiconformal geometry and holomorphic dynamics. After a first introduction to the two subjects we shall discuss how quasiconformal techniques play a crucial role in the understanding of holomorphic dynamical systems.
Date
Tue-Thu 12h-14h
from Februrary 17, 2015.
Registration
–
Location
Room T1
Facultat de Matemàtiques, UB
Lecturers
Albert Clop (UAB)
Núria Fagella (UB)
Xavier Tolsa, ICREA (UAB)
Contents
- 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.
Bibliography
- L.V. Ahlfors. Lectures on Quasiconformal Mappings. American Mathematical Soc., 1966.
- K. Astala, T. Iwaniec and G. Martin. Elliptic partial differential equations and quasiconformal mappings in the plane. Princeton University Press, 2009.
- A.F. Beardon. Iteration of rational functions: Complex analytic dynamical systems. Springer, 1991.
- B. Branner and N. Fagella. Quasiconformal surgery in holomorphic dynamics. Cambridge University Press, 2014.
- L. Carleson and T.W. Gamelin. Complex dynamics. Springer, 1995.
- 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.