Complex Dynamics and Hyperbolic Geometry

Math 275 - MWF 12:00-1:00 pm - 216 Science Center
Harvard University - Spring 2000

Instructor: Curtis T McMullen (

  • 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

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