Topological Dynamics via Structured Koopman Subsystems

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Dokumentart: PhDThesis
Date: 2021-12-15
Language: German
Faculty: 7 Mathematisch-Naturwissenschaftliche Fakultät
Department: Mathematik
Advisor: Nagel, Rainer (Prof. Dr.)
Day of Oral Examination: 2021-09-24
DDC Classifikation: 510 - Mathematics
Keywords: Topologische Dynamik , Funktionalanalysis , Subsystem , Ljapunov-Funktion , Ergodentheorie
Other Keywords: verallgemeinerte rekurrente Menge
topologische Ergodizität
ergodic theory
topological dynamics
functional analysis
Lyapunov function
Koopman operator
generalized recurrent set
Lyapunov algebra
fixed space
topological ergodicity
quotient system
Conley decomposition
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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.

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