Characterization, long-term behavior, and approximation of Koopman operators / vorgelegt von Julian Hölz. Wuppertal, April 2026
Inhalt
- Introduction
- Preliminaries and Notation
- 1 Topological Dynamical Systems
- 1.1 The Koopman and the Perron–Frobenius Operators
- 1.1.1 The Topology on the State Space
- 1.1.2 The Koopman Operator
- 1.1.3 The Perron–Frobenius Operator
- 1.2 Duality of the Perron–Frobenius and Koopman Operators
- 1.2.1 Norming Dual Pairs and Weak Topologies
- 1.2.2 The Koopman and Perron–Frobenius Operators as Adjoints
- 1.2.3 Weakly Closed Ideals and Weakly Continuous Functionals
- 1.3 Characterizing Topological Dynamics with Operators
- 1.4 The Koopman Operator on the Bounded Measurable Functions
- 1.5 Notes
- 2 Measure-Preserving Dynamical Systems
- 2.1 Measure-Preserving Systems and their Koopman Operators
- 2.2 Mean Ergodicity, Ergodicity and Mixing
- 2.3 From Topological to Measure-Preserving Dynamics
- 2.4 Notes
- 3 Entropies of Dynamical Systems
- 3.1 Measure-Theoretic Entropy
- 3.2 Approximation Entropy
- 3.2.1 Definition and Basic Properties
- 3.2.2 Lower Bounds for the Approximation Entropy
- 3.2.3 A Spectral-Theoretic Perspective on Approximation Entropy
- 3.3 Interplay between Measure-Theoretic and Approximation Entropy
- 3.4 Topological Entropy
- 3.5 Notes
- 4 Data-Driven Approximation of Koopman Operators
- 4.1 The Extended Dynamic Mode Decomposition
- 4.1.1 How to Collect Data
- 4.1.2 A Description of the Algorithm
- 4.1.3 A Discussion on Ergodic Convergence Rates
- 4.1.4 The Structure of the Optimal Approximation
- 4.2 Lower Dimension Bound by Measure-Theoretic Entropy
- 4.3 Lower Dimension Bound by Approximation Entropy
- 4.4 Notes
- 5 Entropy via Banach Lattices
- 5.1 Abstract Entropy on Banach Lattices
- 5.2 Measure-Theoretic Entropy
- 5.3 Topological Entropy
- 5.4 Notes
- 6 Uniform Ergodicity for Koopman Operators
- 6.1 Uniform Ergodicity of Lattice Homomorphisms
- 6.2 Uniform Ergodicity for Topological Dynamics
- 6.3 Uniform Ergodicity of Measure-Preserving Dynamics
- 6.4 Quasi-Compactness and Compactness
- 6.5 Notes
- Bibliography
- Index