What can Koopmanism do for attractors in dynamical systems?
Abstract.
We characterize the longterm behavior of a semiflow on a compact space 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 nonlinear semiflow.
in: \addbibresourcekoopman.bib
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 [ausl, p.55], cf. [milnor, 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. [milnor], 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”, [milnor, p.178]. In his opinion prior definitions seem to be too restrictive omitting some interesting examples, [milnor, p.178].
The scope of this article is to establish a systematic hierarchy of the attractors mentioned in [milnor] and, additionally, treat socalled minimal centers of attraction. We will do so by “translating” these concepts into operator theoretic terms by linearizing the nonlinear dynamical system globally. We call this process “Koopmanism”. Indeed, this idea first appeared around 1930 in the papers [koopman] by B. O. Koopman and [neumann] by J. von Neumann and provided the precise mathematical terms to treat the socalled ergodic hypothesis from L. Boltzmann formulated in [boltzmann]. It is based on the distinction between the state space of a (physical) system and the associated observable space being a (vector) space of real or complex valued functions on . If the nonlinear semiflow
describes the dynamics on the state space, the maps
become linear operators and, if remains invariant, is a oneparameter semigroup of linear operators on .
This idea led to the proof of the classical ergodic theorems of J. von Neumann [neumannproof] and G. D. Birkhoff [birkhoffproof] 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 [erg]. 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 consisting of a compact topological Hausdorff space and a continuous semiflow on , cf. [lecture, Def.3.1]. On the Banach space of all realvalued continuous functions on we obtain the corresponding semigroup , called Koopman semigroup, given by the operators
An attractor is a closed and invariant subset possessing a certain asymptotic property. See [milnor] for a survey of various concepts. In our perspective, every such subset of correlates with the closed and invariant ideal in the Banach algebra . Essential to this matter is that all closed ideals in are of the form where is a closed subset of (cf. [erg, Theorem 4.8]).
closed subset  closed ideal  
invariant  invariant. 
Given an attractor , our idea is to restrict the Koopman semigroup to the corresponding closed ideal and characterize the longterm behavior of around by asymptotic properties of restricted to . So our Leitmotiv can be explained as
“”  “” 
The idea to characterize attractivity properties of invariant sets of a flow by stability properties of the Koopman operators restricted to functions vanishing on the attractor is due to A. Mauroy and I. Mezić, see [mez, II.Prop.1]. Their stability corresponds to what we call weak stability later in this article.
In Section 1 we overview the stability theory for strongly continuous operator semigroups with more details on almost weak stability. In the following sections we then apply this theory to attractors in dynamical systems. In Section 2 we characterize absorbing sets and in Section 3 treat wellknown attractivity and stability properties of dynamical systems by “translating” them into stability properties of the restricted Koopman semigroup. We then, in Section 4, prove the existence of minimal attractors and characterize these for each possible asymptotic behavior.
Now we recall some basic facts and fix the notation. Let be a compact Hausdorff space. A family of selfmappings on is called semiflow if and for all . We call a continuous semiflow if
is continuous with respect to the product topology. Thus, a dynamical system is a pair consisting of a compact Hausdorff space and a continuous semiflow . We also call the state space and the elements states. The induced Koopman semigroup on , as defined above, is strongly continuous if and only if is continuous, cf. [lecture, BII, Lem. 3.2].
We recall that is a Calgebra and a Banach lattice for the usual pointwise operations. Given a function and we use the notation
and, analoguously, and .
The sets , , form a basis for the topology on since is completely regular, [erg, Appendix A.2] and [erg, Proof of Lem. 4.12]. This is equivalent to the fact that the zero sets , , form a basis of the closed subsets of or that the topology on coincides with the initial topology induced by . Combining these facts, given a closed subset and an open neighborhood of there exists with and such that , i.e. the sets of the form , , and form a basis for the system of neighborhoods of which we will denote by .
As already mentioned above, for every closed ideal there exists a closed subset such that
see [erg, Thm. 4.8]. The closed set is said to be invariant if for all . It is invariant if and only if the corresponding ideal is invariant, [erg, Lem. 4.18]. We remark that for closed, by , where is the space of all realvalued continuous functions on that vanish at infinity. We identify the dual spaces and of and with the regular Borel measures on and respectively. For a subset we either write or for its complement in .
1. Stability of semigroups
In this section we recall various stability properties of semigroups on a Banach space . The concept of almost weak stability is treated in more detail.
Definition 1.1.
Let be a strongly continuous semigroup on a Banach space . Then is said to be

nilpotent if there exists such that

uniformly exponentially stable if there exists such that

uniformly stable if

strongly stable if

weakly stable if

almost weakly stable if for all pairs there exists a subset with density ^{1}^{1}1The density of a subset is
In the above definition the following chain of implications holds
All implications are strict except b)c) which can be found in [engnag, Chapter V, Section 1]. For examples we refer to [tanja, Chapter III] and [engnag, Chapter V, Section 1].
To later study stability of Koopman semigroups on spaces we need an additional definition.
Definition 1.2.
Let be a dynamical system, the corresponding Koopman semigroup on . Then is said to be almost everywhere pointwise stable if there exist a quasi invariant^{2}^{2}2A measure on is called quasi invariant with respect to if for all if and only if . regular Borel measure on such that for every
We also recall the definition of the growth bound of a strongly continuous semigroup on a Banach space which is defined as
1.1. Almost Weak Stability
For a complete treatment of almost weak stability for semigroups on Banach spaces with relatively weakly compact orbits we refer to [tanja, Chapter III, Section 5] and, for a timediscrete variant to [erg, Chapter 9]. The tools and ideas used in this subsection are based on [erg, Chapter 9].
Proposition 1.3.
Let be semigroup of contractions on some Banach space . Then the following are equivalent

is almost weakly stable,

for all , ,

for all .
Proof.
The equivalence a)b) follows from the so called Koopmanvon Neumann Lemma, see for example [tanja, Chapter III, Lemma 5.2]. The implication b)c) in the time discrete analogue is due to Jones and Lin, cf. [joneslin]. We adapt the proof given in [erg, Prop. 9.17]. Every operator as its adjoint is a contraction and the dual unit ball is compact with respect to the weak* topology. Due to these facts we can define the Koopman system
where
for , , . Fix and define by . By b)
pointwise in and by Lebesgue’s theorem of dominated convergence also weakly and thus in the norm of , cf. [tanja, Chapter I, Theorem 2.25] or [erg, Proposition 8.18]. ∎
Remark 1.4.
Let be semigroup of contractions on some Banach space . The subset
is a closed, invariant subspace of .
Proof.
Let be a convergent sequence in with limit and take . Then there exists such that for all . Now consider . By creftypecap 1.3 c) there exists such that
for all . This implies
∎
Remark 1.5.
Let be semigroup of contractions on some Banach space . In analogy to the previous remark, we define
Both are clearly closed subspaces of .
Remark 1.6.
Let be a dynamical system, the corresponding Koopman semigroup on and a quasi invariant regular Borel measure on . We define
Which is a closed subspace of .
Proposition 1.7.
Let be a semigroup of contractions on a Banach space . If for all there exists a sequence in with as such that for all
then is almost weakly stable.
Proof.
Take and , such that
for all . As in the proof of creftypecap 1.3 we consider the induced Koopman system and the function
If vanishes on , then
for all . We observe that
The functions converge to pointwise in by assumption and by Lebesgue’s Theorem the integral goes to as well. This implies by the theorem of HahnBanach. Thus,
for all . Since was arbitrary, is almost weakly stable. ∎
Remark 1.8.
For semigroups of contractions with relatively weakly compact orbits the assertions in creftypecap 1.7 are equivalent, see [tanja, Chapter III, Section 5]. Here we are able to prove one implication without assuming relatively weakly compact orbits. It still remains open if the opposite implication also holds true in this case.
From now on is a topological dynamical system with compact state space if not otherwise stated and denotes the induced Koopman semigroup on .
2. Absorbing sets
The following section is dedicated to absorbing sets, these are compact invariant subsets of the state space that eventually contain every initial state. We follow the definition in [chueshov, Def. 2.1.1] and differ between two types of such sets.
Definition 2.1.
A closed invariant set is called

absorbing if there exists such that

pointwise absorbing if for all there exists such that
This gives rise to the notion of dissipative systems, cf. [chueshov, Def. 2.1.1].
Definition 2.2.
A dynamical system is called (point) dissipative if it contains a (point) absorbing set.
Proposition 2.3.
Let be a closed invariant set and the restricted Koopman semigroup, i.e. for . Then all the assertions in (I) and all the assertions in (II) are equivalent.


is nilpotent.

is uniformly stable.

.

is absorbing.



For all Dirac measures there exists such that

is pointwise absorbing.

Proof.
We begin with the proof of (I). Clearly, a)b). For the implication b)d) assume not to be absorbing and fix , thus there is with . Since is completely regular there exists with and . Therefore,
Since was arbitrary for all which contradicts b).
The implication d)a) can be seen as follows. Let be such that , thus for every . This implies .
Additionally, clearly a) implies c).
Proof of (II): These equivalences are quite clear since a) implies that for all there exists such that
∎
Next we give a condition under which the two concepts coincide. We recall that a compact space is a Baire space, thus for a sequence of closed subsets , , with
there exists such that has nonempty interior.
Remark 2.4.
Let be closed and invariant. If is pointwise absorbing the sets form a closed cover of .
Proposition 2.5.
Let be closed and invariant and as defined in creftypecap 2.4. The set is absorbing if and only if it is pointwise absorbing and for some .
Proof.
Clearly, if is absorbing it is pointwise absorbing and there exists such that in above construction. For the other implication consider the following. By assumption for every there exists such that
By continuity is open for every and
Since is compact there exist finitely many for some such that
This implies for that
for some and therefore
Define , then
by invariance of . ∎
3. Asymptotics of Dynamical Systems
In this section we consider asymptotic properties of semiflows around closed invariant sets and give operator theoretic characterization of each such property. We also discuss correlations between them.
Definition 3.1.
A closed invariant set is called

uniformly attractive if for all there exists such that

(pointwise) attractive if for all and there exists such that

likely limit set (or Milnor attractor) if there exists a quasi invariant Borel measure on such that for all and almost every there exists with

center of attraction if for all
for all , where denotes the Lebesgue measure on ,

stable in the sense of Lyapunov if for all there exists , such that
The concepts a), b) and e) in creftypecap 3.1 have been established by A. M. Lyapunov in his dissertation ([lya]) in 1892 and have since been broadly applied and investigated for dynamics on locally compact metric spaces. See [bhatia, Chapt. II] or [denker, Chapt. 3, Sect. 2]. The property d) in creftypecap 3.1 appears in G. D. Birkhoff’s monograph “Dynamical Systems” [birkhoff, Chapt. VII] as “central motion” and has been further investigated by H. Hilmy, see for example [hilmy], K. Sigmund, in [sigmund] and by H. Kreidler in [hekr, Sect. 4] to name a few. Definition c) for semiflows on smooth compact manifolds is due to J. Milnor and can be found in [milnor, Section 2].
Remark 3.2.
If is a dynamical system with metric then there exists one null set satisfying the assumptions in creftypecap 3.1 c) that does not depend on since there is a countable neighborhood basis and the countable union of null sets is again a null set.
Remark 3.3.
For the concepts defined in creftypecap 3.1 the following implications hold.
{xy}\xymatrixa) \ar@=>[r] \ar@=>[rd]&b)\ar@=>[r]\ar@=>[rd]&c) & and & a) \ar@<=>[r] &b)+e)
&e) &d)&&
The opposite implications do not hold true in general which can be seen in the following examples. The equivalence of creftypecap 3.1 a) b) + e) will be proven in below creftypecap 3.8 and creftypecap 3.9.
Example 3.4.

Consider the onepoint compactification of and the semiflow defined by
Then is attractive but not uniformly attractive.

Take the onepoint compactification of and the semiflow on , with
Consider the standard Gaussian measure on which is a regular Borel measure on that is quasiinvariant with respect to since it is equivalent to the Lebesgue measure . In particular, . Then is a likely limit set for since and for all but it is neither attractive nor a center of attraction since for all .

In [hilmy, p.287], H. Hilmy gave a concrete example for a center of attraction that is not attractive. We give a simplified version of this example. Take the following differential equation
given in polar coordinates on . The solutions of above differential equation exist for all times and all initial values in and form a semiflow thereon.
The dependence of and is given by
The orbit of an initial state with radius forms a spiral towards the unit circle. On the unit circle the radius is constant and the rate of change of is given by the differential equation
Thus, und are fixed points, because . Therefore, the orbits of states on the unit circle converge to either or . The set is a center of attraction for and it is even minimal with this property. It is easy to see that the minimal attractive subset in this example is .
The next proposition characterizes all above mentioned attractivity properties by means of the corresponding Koopman semigroup.
Proposition 3.5.
Let be a dynamical system, a closed invariant set and the restricted Koopman semigroup, i.e. for .

The following are equivalent.

is strongly stable.

is uniformly attractive.


The following are equivalent

is weakly stable.

is attractive.


The following are equivalent.

is almost everywhere pointwise stable.

is a likely limit set.


The following are equivalent.

is almost weakly stable.

is a center of attraction.

Proof.
Proof of (I): First we show a)b). Take . Since is completely regular there is and such that . By assertion a) there is such that for all . This implies
Therefore, for all . Also b)a) because for every and there is a such that for all . This implies for all and and therefore
for all .
Proof of (II): To prove a)b) take and . Then there exist and such that and since is weakly stable there exists such that
which implies for all . For the opposite implication let , and by b) there exists such that for all and thus
for all Dirac measures and by Lebesgue’s theorem of dominated convergence
for all .
Proof of III: We first prove a) implies b). First take a neighborhood , then there exist and with . By assumption there exists a quasi invariant Borel measure and null set depending on such that for every there is such that
for all . Clearly, this implies for all . The other implication follows similarly.
Proof of (IV): First recall that by . Let be a center of attraction and open, i.e.,
for all and consequently for , which is a compact subset of , the following equivalence holds.
Now take with compact support and denote . Then and thus
Note that the mapping
is continuous and hence measurable, thus for fixed
by the FubiniTonelli Theorem. And thus by Lebesgue’s Theorem of dominated convergence
for all . This implies and since is closed and the continuous functions with compact support are dense in it follows that
On the other hand since is closed invariant ideal there exists closed and invariant such that . Take compact and then there exists a continuous extension and the measurable extension as the characteristic function on . Since the following equation is true for all
hence
which implies
and thus
The open neighborhoods of are exactly the complements of compact sets and since was arbitrary the last equation holds for all open neighborhoods of . Thus is a center of attraction. ∎
To conclude this section we show that the concepts of uniform attractivity and pointwise attractivity coincide if and only if is stable in the sense of Lyapunov. To do so we first characterize stability in the sense of Lyapunov further.
Proposition 3.6.
Let be closed and invariant. Then the following are equivalent.

The set is stable in the sense of Lyapunov.

Every contains an invariant . If is closed, can be chosen closed as well.
Proof.
For the implication a)b) take closed and , such that implies for all . Consider . Then . Therefore, is still a closed neighborhood of which is invariant. The implication b)a) is trivial.
Remark 3.7.
Furthermore, a closed invariant set is Lyapunov stable if and only if
We prove that the assertion implies b) in creftypecap 3.6. Let be an open neighborhood of and assume there is no invariant neighborhood of with . Then for all invariant neighborhoods there exists with . This defines a net which has a convergent subnet, since is compact. Let be the limit of said convergent subnet. On the other hand there exists a convergent subnet with limit in since is compact and the sets are neighborhoods of . This implies which contradicts the fact that . This implies b). ∎
Proposition 3.8.
Let be a dynamical system and a closed invariant subset which is uniformly attractive, then is stable in the sense of Lyapunov.
Proof.
Let be uniformly attractive and assume is not stable in the sense of Lyapunov then there exists , and such that contains no invariant neighborhood. This implies there exists a net and such that
Thus, has a convergent subnet, converging to .
Since there exists such that for all since is uniformly attractive by assumption, the net is bounded and thus there exists a convergent subnet with