Speeds of Convergence and Backward Orbits of One-Parameter Semigroups of Holomorphic Functions

• Applicant: Dr. Maria Kourou
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 545293643

The main area of research in the proposed project is the theory of one-parameter semigroups of holomorphic functions. Such semigroups form a significant chapter in Geometric Function Theory, and more generally, in Complex Analysis, due to the scientific interest they received in the past few years leading to an extension of their theory in the unit disk as well as domains of higher dimension. We focus our study on the asymptotic behavior of a semigroup and its rate of convergence. When it comes to semigroups in the unit disk, the focal point in our research is the backward-dynamical setting. The first proposed problem concerns the asymptotic behavior of speeds of convergence along backward orbits of a semigroup and aims at information regarding their monotonicity. Furthermore, we explore the phenomenal behavior of backward speeds along non-regular backward orbits, targeting to their deeper understanding. Our following objectives are related to orbits of compact sets in the unit disk under semigroups of holomorphic functions. We provide a configuration of the distance between motions of compacta in terms of potential theory and in particular, extremal length. For this problem, we work on both forward and backward dynamics of a semigroup and investigate all relative positions of compacta. Let us note at this point, that asymptotically in both cases, the compact sets shrink to a certain boundary point. Meanwhile, we aim to provide examples on their distance asymptotically. With the aid of the monotonicity of backward speeds, we extend our research to backward orbits of compact sets. We study the phenomenon of a compact set moving backwards inside the unit disk and how its speed is connected with its initial position. We aim to characterize its speed in terms of the intrinsic characteristics of the semigroup. For this reason, we use several potential theoretic quantities such as harmonic measure, Green's function, and extremal length. Results are expected to lead to a classification of the backward-dynamical setting of the semigroup. Moving up to higher dimensions, we intend to explore the dynamics of semigroups in the unit ball. We investigate motions of compacta under the influence of such a semigroup. Geometric and potential theoretic tools are utilized in order to observe the changes they undergo. More specifically, the volume and the energy of these compacta are studied along with the asymptotic behavior of the aforementioned quantities. In addition, results on the speeds of compacta in the complex plane are adapted in the case of the unit ball. Last but not least, the research findings concerning semigroups of the unit ball are generalized for semigroups of convex domains in n-dimensional complex spaces. Research questions posed in the spectrum of the research proposal compose up-to-date research problems and answers to those form a substantial contribution in the extension of existing knowledge on semigroups of holomorphic functions.

• DFG Programme: Research Grants
• International Connection: Greece, Italy, Spain
• Cooperation Partners: Professor Dr. Leandro Arosio; Professor Dr. Filippo Bracci; Professor Dr. Manuel Domingo Contreras Marquez; Professor Dr. Santiago Diaz-Madrigal; Professor Dr. Pavel Gumenyuk; Dr. Konstantinos Zarvalis
• Co-Investigator: Professor Dr. Oliver Roth