Topological Dynamics via Structured Koopman Subsystems

Abstract

This thesis deals with the interplay of quotient systems of a topological dynamical system and subsystems of its corresponding Koopman system. It begins with a historical „prelude“ (in German) where biographical aspects of the involved mathematicians are highlighted. In Chapter 1 topological dynamical systems and their corresponding Koopman systems are introduced and the correspondence of quotient systems and subsystems is explained. Chapter 2 is devoted to the simplest subsystem of a Koopman system, the fixed space. A dynamical description of the corresponding quotient system of the dynamical system is derived via a hierarchy of transfinite orbits. In particular, this leads to the characterization of a one-dimensional fixed space. In Chapter 3 the Lyapunov algebra is defined which is generated by so-called Lyapunov functions. Its properties, special cases and its connection to the generalized recurrent set are discussed. Also algebras which are generated by a single Lyapunov function are considered and extended Lyapunov functions are introduced. Finally, decompositions of the state space obtained by the Lyapunov algebra are studied.