Holomorphic Dynamics and Hyperbolic Geometry

Rational maps are self-maps of the Riemann sphere of the form z → p(z)/q(z) where p(z) and q(z) are polynomi-als. Kleinian groups are discrete subgroups of PSL(2,C), acting as isometries of 3-dimensional hyperbolic space and as conformal automorphisms of its boundary, the Riemann sphere. Both theories experienced remarkable advances in the last two decades of the 20th century and are very active areas of continuing research. The aim of the course is to introduce some of the main techniques and results in the two areas, emphasising the strong connections and parallels between them. Topics to be covered in 5 two-hour lectures (Added April 2013. The lecture notes that follow are divided into 8 chapters. They comprise the material that was covered during the 5 week lecture course in February-March 2013. This was essentially the list of topics below, with some covered in more depth than others.) 1. Dynamics of rational maps: The Riemann sphere and rational maps (basic essentials from complex analysis); conformal automorphisms of the sphere, plane and disc; Schwarz’s Lemma; the Poincare ́ metric on the upper half-plane and unit disc; conjugacies, fixed points and periodic orbits (basic essentials from dynamical systems); spherical metric; equicontinuity; Fatou and Julia sets (definition). 2. Fatou and Julia sets: Normal families and Montel’s Theorem; characterisations and proporties of Fato