Towards a Koopman theory for dynamical systems on completely regular spaces

TL;DR

This paper extends Koopman theory to continuous semiflows on completely regular spaces, providing a framework to analyze PDE-derived dynamics through operator semigroups and their properties.

Contribution

It introduces Koopman semigroups for semiflows on completely regular spaces, characterizes their generators, and demonstrates applications to attractors in dynamical systems.

Findings

Koopman semigroups are well-defined on these spaces

Infinitesimal generators can be characterized algebraically and lattice-theoretically

Application to attractors illustrates the approach's usefulness

Abstract

The Koopman linearization of measure-preserving systems or topological dynamical systems on compact spaces has proven to be extremely useful. In this article we look at dynamics given by continuous semiflows on completely regular spaces which arise naturally from solutions of PDEs. We introduce Koopman semigroups for these semiflows on spaces of bounded continuous functions. As a first step we study their continuity properties as well as their infinitesimal generators. We then characterize them algebraically (via derivations) and lattice theoretically (via Kato's equality). Finally, we demonstrate-using the example of attractors-how this Koopman approach can be used to examine properties of dynamical systems.