Semigroups of Holomorphic Functions

Abstract

In this chapter we introduce the primary subject of our study: continuous one-parameter semigroups of holomorphic self-maps of the unit disc. We establish their main basic properties and extend to this context the Denjoy-Wolff Theorem. Then we characterize groups of automorphisms and more generally of linear fractional self-maps of the unit disc. We also briefly consider continuous semigroups of holomorphic self-maps of $\mathbb C$ and $\mathbb C_\infty$, proving that they reduce to groups of Möbius transformations, and we explain why a non-trivial theory of continuous semigroups of holomorphic maps only makes sense for self-maps of the unit disc.