Hyperbolic Geometry and Iteration Theory

Abstract

In this chapter we introduce some basic tools necessary for our study. We start recalling the concept of Riemann surfaces, focus mainly on the geometry of the unit disc, the complex plane and the Riemann sphere. Next, from Schwarz’s Lemma, we define the hyperbolic metric and hyperbolic distance of the unit disc, and extend, à la Kobayashi, the concept of hyperbolic distance to Riemann surfaces. We turn then our attention to the analytical and dynamical properties of holomorphic self-maps of the unit disc. We introduce the notion of horocycles and Stolz’s regions, and we prove the Lindelöf Theorem, which allows one to infer the existence of non-tangential limits provided the limit along some curve exists. Then we prove Julia’s Lemma and the Julia-Wolff-Carathéodory Theorem, which can be seen as boundary version of the Schwarz Lemma. With those tools at hand, we consider iteration of holomorphic self-maps of the unit disc, and prove the Denjoy-Wolff Theorem, which says that, except trivial cases, the orbits of a holomorphic self-map of the unit disc converge to a same point on the closed unit disc. Finally, we discuss boundary fixed points (and, more generally, boundary contact points) of holomorphic self-maps of the unit disc when no continuous extension to the boundary is assumed.