Harmonic Limits of Dynamical and Control Systems

In this thesis, we will analyze an approach to describe the rotational behaviour of dynamical systems and control systems, namely the concept of rotational factor maps. The general idea is to find a complex-valued map F on the state space that maps the dynamics onto a rotation around the origin in the complex plane. We will call such a map a rotational factor map. More formally, these rotational factor maps are eigenfunctions of the Koopman operator. This concept of rotational factor maps is closely connected to harmonic limits, which are ergodic sums (for discrete-time systems) or integrals (for systems in continuous time). It turns out that the existence of rotational factor maps is equivalent to the existence of non-zero harmonic limits. So we use harmonic limits to analyse the spectral properties of dynamical systems given by the iteration of a map, by a semi-flow or by a control system.