Advertisement
Published in:
2020 | OriginalPaper | Chapter
8. Semigroups of Holomorphic Functions
Authors : Filippo Bracci, Manuel D. Contreras, Santiago Díaz-Madrigal
Published in: Continuous Semigroups of Holomorphic Self-maps of the Unit Disc
Publisher: Springer International Publishing
Activate our intelligent search to find suitable subject content or patents.
Select sections of text to find matching patents with Artificial Intelligence. powered by
Select sections of text to find additional relevant content using AI-assisted search. powered by
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.