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Towards a Koopman theory for dynamical systems on completely regular spaces.
Bálint Farkas1, Henrik Kreidler1
1Fakultät für Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaußstraße 20, 42119 Wuppertal, Germany.
Summary
Koopman semigroups are introduced for continuous semiflows on regular spaces, extending Koopman linearization to partial differential equations. This framework aids in analyzing dynamical systems, including attractors.
Area of Science:
- Dynamical Systems Theory
- Functional Analysis
- Partial Differential Equations (PDEs)
Background:
- Koopman linearization is valuable for measure-preserving and topological dynamical systems.
- Continuous semiflows arise naturally from solutions of PDEs on completely regular spaces.
Purpose of the Study:
- Introduce Koopman semigroups for continuous semiflows on completely regular spaces.
- Analyze the properties of these Koopman semigroups.
- Demonstrate the application of the Koopman approach to study dynamical systems.
Main Methods:
- Study continuity properties and infinitesimal generators of Koopman semigroups.
- Algebraic characterization using derivations.
- Lattice-theoretic characterization via Kato's equality.
Main Results:
- Koopman semigroups are defined and their continuity properties investigated.
- Infinitesimal generators are analyzed.
- Algebraic and lattice-theoretic characterizations are established.
Conclusions:
- The Koopman semigroup approach provides a powerful tool for analyzing dynamical systems arising from PDEs.
- This method is applicable to understanding properties like attractors.
- Extends the utility of Koopman linearization to a broader class of systems.
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