Eleftherios Theodosiadis: Geometry of multi-slit Loewner chains and semigroups of finite shift

Abstract.

This thesis deals with two topics in complex analysis that are related to families of Riemann maps that depend on some parameter function called the driving function of the family. One is Loewner’s theory and the other is the theory of semigroups of holomorphic self-maps of the unit disc. This thesis consists of four works, three of which lie in the intersection of the two theories and the other one refers solely on semigroups.

In Paper I, we deal with the Loewner equation both in the unit disc (the radial case) and in the upper half-plane (the chordal case). The solutions to these equations, which depend on the space and time variables, are called (radial or chordal) Loewner chains. Its main purpose is to present explicitly solutions to certain choices of driving functions and additionally visualize their geometry as time evolves. In particular, we deal with conformal maps with finitely many slits, for both cases. Thus, the aforementioned evolution involves the growth of multiple curves either in the unit disc or in the upper half-plane. Secondly, we discover the semigroup nature of these families, which we utilize in order to connect the radial with the chordal case through a Möbius transform, although in the general theory this is not always possible.

The second paper is a continuation of Paper I, where we extend the study of the chordal Loewner chains of Paper I to chains with infinitely many slits. Again, we study the geometry of the chains as time evolves and we find the same geometric behaviour as in Paper I. However, this study is more complicated and requires a different approach that involves techniques from classical complex analysis and the use of the harmonic measure.

In Paper III we are concentrated in a specific type of semigroups. We call those semigroups of finite shift. In the general theory of semigroups, several authors have studied the rate of convergence of a semigroup to the Denjoy-Wolff point, in terms of the Euclidean distance. In this direction, we also examine the rate of convergence for this case, in terms of the Euclidean distance, the hyperbolic distance and also in terms of the harmonic measure.

In Paper IV, we present some computational examples of Loewner chains. Some of them are related to those appearing in Papers I and II. We work similarly in the sense that we solve the Loewner equation for some certain driving functions. In addition, we have collected some Loewner chains that do not appear in the literature and we recover their driving functions. Our intention is to visualize these elementary examples in an effort to compare the geometry of the chains with their driving functions.