Topological Dynamics via Structured Koopman Subsystems

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URI: http://hdl.handle.net/10900/121768
http://nbn-resolving.de/urn:nbn:de:bsz:21-dspace-1217682
http://dx.doi.org/10.15496/publikation-63134
Dokumentart: PhDThesis
Date: 2021-12-15
Language: German
English
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
Lyapunovalgebra
Fixraum
Zerlegung
topologische Ergodizität
Quotientensystem
Koopmanoperator
ergodic theory
topological dynamics
functional analysis
subsystem
Lyapunov function
Koopman operator
generalized recurrent set
Lyapunov algebra
fixed space
decomposition
topological ergodicity
quotient system
Conley decomposition
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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.

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