The dynamics alluded to by the title of the course refers to dynamical systems
that arise from iterating a holomorphic self-map of a complex manifold. In this
course, the manifolds underlying these dynamical systems will be of complex
dimension 1. The foundations of complex dynamics are best introduced in the
setting of compact spaces. Iterative dynamical systems on compact Riemann
surfaces other than the Riemann sphere – viewed here as the one-point
compactification of the complex plane – are relatively simple. We shall study
what this means. Thereafter, the focus will shift to rational functions: these
are the holomorphic self-maps of the Riemann sphere. Along the way, some of the
local theory of fixed points will be presented. In the case of rational maps,
some ergodic-theoretic properties of the orbits under iteration will be
studied. The development of the latter will be self-contained. The properties/
theory coverd will depend on the time available and on the audience’s interest.
Suggested books and references:
J. Milnor, Dynamics in One Complex Variable, Annals of Mathematics Studies no. 160, Princeton University Press, 2006.
A.F. Beardon, Iteration of Rational Functions: Complex Analytic Dynamical Systems, Graduate Texts in Mathematics no. 132, Springer-Verlag, 1991.