Syllabus
Complex Dynamics and Hyperbolic Geometry
Math 275 - MWF 12:00-1:00 pm - 216 Science Center
Harvard University - Spring 2000
Instructor:
Curtis T McMullen
(ctm@math.harvard.edu)
Texts
- Benedetti and Petronio. Lectures on Hyperbolic Geometry. Springer-Verlag, 1992.
- Bedford, Keane and Series. Ergodic Theory, Dynamics and Hyperbolic Surfaces. Oxford University Press, 1991.
- Carleson and Gamelin. Complex Dynamics. Springer-Verlag, 1993.
- Milnor. Lectures on Complex Dynamics. Vieweg, 1999. Distributed by the AMS.
- Otal. Le théorème d'hyperbolisation pour les variétés fibrées de dimension trois , Astérisque volume 235 (1996). Distributed by AMS.
- Thurston. Geometry and Topology of 3-Manifolds. Mimeographed notes, Princeton, 1979.
Prerequisites.
Intended for advanced graduate students.
Acquaintance with complex analysis, hyperbolic
geometry, Lie groups and dynamical systems
will be useful.
Topics.
We will discuss
hyperbolic 3-manifolds and iterated rational maps
in relation to topology, analysis, Teichmüller theory and
ergodic theory.
Topics may include:
-
Hyperbolic manifolds
- Ergodic theory on groups
- Mixing of the geodesic flow
- Quasiconformal maps
- Mostow rigidity
- Ahlfors' finiteness theorem
- Bers' area theorem
- Bounds on cusps
- No invariant line field theorem (Sullivan)
- Thick-thin decomposition
- Geometrically tame ends (Bonahon, Thurston)
- The Ahlfors measure zero conjecture
-
Rational maps
- Classification of stable regions
- No-wandering-domains theorem (Sullivan)
- Holomorphic motions and stability
- Invariant line fields and the hyperbolicity conjecture
- Bounds on indifferent cycles (Epstein)
- Local connectivity and measure of the Julia set (Branner, Hubbard and Yoccoz)
Additional References
- Beardon. The Geometry of Discrete Groups. Springer-Verlag, 1983.
- Gardiner. Teichmüller Theory and Quadratic Differentials. Wiley Interscience, 1987.
- Imayoshi and Taniguchi. Introduction to Teichmüller Spaces. Springer-Verlag, 1992.
-
McMullen.
Complex Dynamics and Renormalization .
Annals of Math Studies 135, Princeton University Press, 1994.
- McMullen. Renormalization and 3-Manifolds which Fiber over the Circle. Annals of Math Studies 142, Princeton University Press, 1996.
- Ratcliffe. Foundations of Hyperbolic Manifolds. Springer-Verlag, 1994.
- Sullivan. On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions. In: Kra and Maskit, editors, Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference. Annals of Math. Studies 97, Princeton, 1981.