The book faces the interplay among dynamical properties of semigroups, analytical properties of infinitesimal generators and geometrical properties of Koenigs functions. The book includes precise descriptions of the behavior of trajectories, backward orbits, petals and boundary behavior in general, aiming to give a rather complete picture of all interesting phenomena that occur. In order to fulfill this task, we choose to introduce a new point of view, which is mainly based on the intrinsic dynamical aspects of semigroups in relation with the hyperbolic distance and a deep use of Carathéodory prime ends topology and Gromov hyperbolicity theory. This work is intended both as a reference source for researchers interested in the subject, and as an introductory book for beginners with a (undergraduate) background in real and complex analysis. For this purpose, the book is self-contained and all non-standard (and, mostly, all standard) results are proved in details.

Autorentext

Prof. Filippo Bracci obtained his PhD in Mathematics at the University of Padova in 2001. He is full professor at the University of Rome Tor Vergata since 2007. He was the principal investigator of the ERC project "HEVO". His research interests include complex analysis, several complex variables and holomorphic dynamics.

Prof. Manuel D. Contreras obtained his Ph.D. in Mathematics at the Universidad de Granada in 1993. He is full professor at Departamento de Matemática Aplicada II of the Universidad de Sevilla. His main research interests are complex analysis, geometric function theory, holomorphic dynamics, spaces of analytic functions and operators acting on them.

Prof. Santiago Díaz-Madrigal obtained his Ph.D. in Mathematics at the Universidad de Sevilla in 1990. He is a full professor at Universidad de Sevilla in the department of Matemática Aplicada II since 1998. His current research interests include complex analysis, dynamical systems and probability and, especially, those areas where these topics interact with each other.

Inhalt

Part I: Preliminaries.- 1 Hyperbolic geometry and interation.- 2. Holomorphic functions with non-negative real part.- 3. Univalent functions.- 4. Carathéodory's prime ends theory.- 5. Hyperbolic geometry in simply connected domains.- 6. Quasi-geodesics and localization.- 7. Harmonic measures and Bloch functions.- Part II: Semigroups.- 8 Semigroups of holomorphic functions.- 9 Models and Koenigs functions.- 10 Innitesimal generators.- 11 Extension to the boundary.- 12 Boundary xed points and innitesimal generators.- 13 Fixed points, backward invariant sets and petals.- 14 Contact points.- 15 Poles of the innitesimal generators.- 16 Rate of convergence at the Denjoy-Wolffpoint.- 17 Slopes of orbits at the Denjoy-Wolffpoint.- 18 Topological invariants