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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. The concept of an attractor of a dynamical system has been of great interest in the last 50 years. After the term first occurred in 1964 in [ABS64, p.55], cf. [Mil85, p.177], many variations and modifications have been defined, each of them yielding different examples. The survey article “On the Concept of Attractor” from 1985 by J. Milnor, cf. [Mil85], treats the history of its many definitions and even adds an additional concept. He justifies this by describing some of the stability properties in the previous definitions as “awkward”, [Mil85, p.178]. In his opinion prior definitions seem to be too restrictive omitting some interesting examples, [Mil85, p.178]. The scope of this article is to establish a systematic hierarchy of the attractors mentioned in [Mil85] and, additionally, treat so-called minimal centers of attraction. We will do so by “translating” these concepts into operator theoretic terms by linearizing the non-linear dynamical system globally. We call this process “Koopmanism”. Indeed, this idea first appeared around 1930 in the papers [Koo31] by B. O. Koopman and [Neu32] by J. von Neumann and provided the precise mathematical terms to treat the so-called ergodic hypothesis from L. Boltzmann formulated in [Bol85]. It is based on the distinction between the state space K of a (physical) system and the associated observable space O being a (vector) space of real or complex valued functions on K. If the non-linear semiflow φt : K → K , t ∈ R+ , describes the dynamics on the state space, the maps f 7→ T (t)f := f ◦ φt , f ∈ O , 1 2 become linear operators and, if O remains invariant, (T (t))t≥0 is a one-parameter semigroup of linear operators on O. This idea led to the proof of the classical ergodic theorems of J. von Neumann [vNeu32] and G. D. Birkhoff [Bir31] and even gave rise to ergodic theory as a mathematical discipline. The recent state of the art of this operator theoretic approach, called “Koopmanism”, to ergodic theory is presented in the monograph [EFHN15]. In this paper we show how this can be used to classify and discuss attractors in topological dynamical systems. For this purpose we consider a pair (K, (φt)t≥0) consisting of a compact topological Hausdorff space K and a continuous semiflow (φt)t≥0 on K, cf. [Nag+86, Def.3.1]. On the Banach space C(K) of all real-valued continuous functions on K we obtain the corresponding C0-semigroup (T (t))t≥0, called Koopman semigroup, given by the operators T (t)f := f ◦ φt for f ∈ C(K) , t ≥ 0 . An attractor is a closed and (φt)t≥0-invariant subset ∅ 6= M ⊆ K possessing a certain asymptotic property. See [Mil85] for a survey of various concepts. In our perspective, every such subset of K correlates with the closed and (T (t))t≥0-invariant ideal IM := {f ∈ C(K) | f |M ≡ 0} in the Banach algebra C(K). Essential to this matter is that all closed ideals in C(K) are of the form IM where M is a closed subset of K (cf. [EFHN15, Theorem 4.8]). M ⊆ K closed subset ↔ IM ⊆ C(K) closed ideal M (φt)t≥0-invariant ↔ IM (T (t))t≥0-invariant. Given an attractor M , our idea is to restrict the Koopman semigroup to the corresponding closed ideal IM and characterize the long-term behavior of (φt)t≥0 around M by asymptotic properties of (T (t))t≥0 restricted to IM . So our Leitmotiv can be explained as

What can Koopmanism do for attractors in dynamical systems?