Non-compact topological skew products

Topological skew products are continuous group actions which extend a compact flow (i.e. a compact space with a continuous group action) by a locally compact topological group and commute with the translation on the extension group. Skew products occur naturally for discrete flows (actions of the group of integers), as well as in the context of continuous flows of systems of ordinary differential equations. The main tool in the study of compact flows, the Ellis group, is not available for actions on non-compact spaces. However, skew products can much better studied than general actions non-compact spaces. The starting point is the theory of compact minimal flows (i.e. the orbit of each point is dense in the compact space), which is applied to the compact flow in the base of the skew product. This theory provides structure theorems under certain conditions (distal, point distal, only minimal) on the compact flow. The aim of this project is the development of the ensuing structural theory of topological skew products of distal flows toward more general situations. Applications of this theory may arise in all those areas of science which rely on mathematical modeling with non-compact dynamical systems.

The theory of dynamical systems abstracts the observation of a time evolution in the mathematical notion of a group action. In a group action, the addition in the time coordinate (or their generalisation in the mathematical structure of a group operation) is mapped consistently into the transformation of the system under its time evolution. A key distinction has to be made between compact (I.e. bounded) and non-compact systems. The main tool in topological dynamics, the Ellis group, is available only for compact systems. This research project focused on an important class of non-compact systems, the class of topological skew products.Topological skew products are continuous group actions on the product of a compact system and a locally compact group (this term includes the real line or the Euclidean space). As a key property of a skew product, the group action on the product and the translation in the extension group commute. Based on this property, skew products can be studied through the application of the theory of compact minimal group actions (i.e. the orbit of each point is dense in the compact space). This has been accomplished in a former research project with the application of Furstenberg structure theorem for distal minimal group actions (I.e. two distinct points cannot become arbitrarily close to each other under a common time evolution). The objective of this project was, inter alia, the generalisation of this result for point-distal minimum group actions (here the previously described property is valid only for the "majority" of the all points). This generalisation could not be achieved in full generality, but for an important class of examples of point-distal minimal group actions.