Description and goals:
This course is a rapid introduction to the dynamics of holomorphic self-maps
of Riemann surfaces, emphasizing the case of rational maps of the sphere.
It provides an opportunity to learn some of the main ideas in the field and
to see how tools from geometric complex analysis and quasiconformal theory
are utilized in concrete dynamical applications.
The course will consist of 14 two-hour lectures. Accordingly,
we start with the rudiments of the theory and take a path that gets us to some
interesting global results as quickly as possible. Toward the end, we take glimpses
into more advanced topics.
Here is an outline of possible topics:
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Uniformization of Riemann surfaces, 2D hyperbolic geometry,
Schwarz lemma, normal families, classical Koebe distortion
theorems, modulus and extremal length.
-
Basic properties of Julia and Fatou sets, classification of dynamics
on hyperbolic surfaces.
-
Quasiconformal maps in the plane, Beltrami differentials, conformal
structures, the measurable Riemann mapping theorem
-
Deformation of rational maps, Sullivan's no wandering domain theorem,
basic quasiconformal surgery.
-
Holomorphic motions and lambda-lemma, applications in stability
theory of holomorphic dynamical systems.
-
Irrational rotations, combinatorial and arithmetical aspects, critical circle maps,
Siegel disks and Herman rings.
General prerequisite: A good knowledge of graduate-level
complex analysis and curiosity to learn about a beautiful piece of
modern mathematics. For the most part the course will not require any
background in dynamical systems, but some knowledge of it will be
helpful.
Lecturer: Saeed Zakeri
Graduate Center Office: 4302
Email: saeed DOT zakeri AT qc DOT cuny DOT edu
Class meetings: Fridays 11:00 - 1:00 in room 5417
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