# What can Koopmanism do for attractors in dynamical systems?

## TL;DR

This paper explores how Koopman operator theory can be used to analyze the long-term behavior and various types of attractors in dynamical systems through linear operator semigroups.

## Contribution

It provides a characterization of different attractor concepts and their minimal attractors using Koopman semigroup techniques, enabling linear analysis of nonlinear systems.

## Key findings

- Characterization of asymptotically stable attractors
- Comparison of Milnor attractors and centers of attraction
- Minimal attractor descriptions for each attractor type

## Abstract

We characterize the longterm behavior of a semiflow on a compact space $K$ by asymptotic properties of the corresponding Koopman semigroup. In particular, we compare different concepts of attractors, such as asymptotically stable attractors, Milnor attractors and centers of attraction. Furthermore, we give a characterization for the minimal attractor for each mentioned property. The main aspect is that we only need techniques and results for linear operator semigroups, since the Koopman semigroup permits a global linearization for a possibly non-linear semiflow.