# Compact operator semigroups applied to dynamical systems

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## Abstract

In this paper we develop a systematic theory of compact operator semigroups on locally convex vector spaces. In particular we prove new and generalized versions of the mean ergodic theorem and apply them to different notions of mean ergodicity appearing in topological dynamics.

## Keywords

Mean ergodicity Amenable semigroups Tame dynamical systems## 1 Introduction

In [29] and [30] Köhler introduced the *enveloping operator semigroup* of a power bounded operator \(T \in \mathscr {L}(X)\) on a Banach space *X* as the closure of the semigroup \(\mathcal {S}{:}{=}\{(T')^n\mid n \in \mathbb {N}_0\}\) with respect to the operator topology of pointwise convergence induced by the weak* topology on \(X'\). If the operator *T* is the Koopman operator of a topological dynamical system \((K;\varphi )\), i.e., \(Tf = f \circ \varphi \) for each \(f \in \mathrm {C}(K)\), the relation between this enveloping operator semigroup and the classical *Ellis semigroup* is an interesting issue (see [15, 30]).

In addition, A. Romanov used the semigroup given by the convex closure of \(\mathcal {S}\) to examine mean ergodicity of operators with respect to the weak* topology (see [33]) and—in the setting of topological dynamics—with respect to the topology of pointwise convergence (see [34]).

In this paper we generalize the concepts from above and give a new and systematic approach to compact operator semigroups on locally convex spaces. We then discuss its applications to dynamical systems.

It turns out that these semigroups are right topological semigroups (see [3, 4] for an introduction to their theory) with respect to the topology of pointwise convergence. We use this as our starting point and introduce the new concept of abstract Köhler semigroups with their basic properties (see Propositions 2.4 and 2.5). In the subsequent section we discuss examples appearing in topological dynamics: the *Köhler semigroup*, the *Ellis semigroup* and the *Jacobs semigroup* (see Proposition 3.8). We give a new operator theoretic characterization of the Ellis semigroup (see Definition 3.4) and prove that a metric topological dynamical system is tame if and only its convex Köhler semigroup is a Fréchet–Urysohn space (see Proposition 3.11).

Our main results are contained in the last two sections. In Sect. 4 we return to the abstract viewpoint and generalize a mean ergodic theorem established by Romanov [33] for a single operator on a dual Banach space with the weak* topology to left amenable operator semigroups acting on locally convex spaces (see Theorem 4.3). The result is then applied to extend a mean ergodic theorem of Schreiber [38] to right amenable operator semigroups on barrelled locally convex spaces (see Theorem 4.7).

In the final section we study unique ergodicity, norm mean ergodicity and weak* mean ergodicity for right amenable semitopological semigroups acting on a compact space. This approach as well as the result that transitive weak* mean ergodic systems are already uniquely ergodic (see Theorem 5.6) seem to be new.

At the end of this section we discuss mean ergodicity of tame metric systems. We give a new proof for unique ergodicity of minimal tame metric dynamical systems generalizing results of Glasner [15], Huang [23] as well as Kerr and Li [26] to the case of amenable semigroup actions (see Corollary 5.4). Finally we extend results of Romanov [34] and characterize weak* mean ergodicity for tame amenable semigroup actions (see Theorem 5.10).

In this paper all vector spaces are complex. Moreover, all topological vector spaces and compact spaces are assumed to be Hausdorff. Given topological vector spaces *X* and *Y* we denote the continuous linear mappings from *X* to *Y* by \(\mathscr {L}(X,Y)\) and set \(\mathscr {L}(X) {:}{=}\mathscr {L}(X,X)\).

We recall that a special class of locally convex spaces arises from dual pairs (see [40], Chapter IV). More generally, take two vector spaces *X* and *Y* and a bilinear mapping \(\langle \,\cdot \, ,\, \cdot \, \rangle :X\times Y\longrightarrow \mathbb {C}\) that separates *X*, i.e., for each \(0 \ne x \in X\) there is \(y \in Y\) with \(\langle x,y\rangle \ne 0\), and call the pair (*X*, *Y*) a *left-separating pair*. The seminorms \(\rho _{y}\) given by \(\rho _y(x){:}{=}|\langle x,y\rangle |\) for \(x \in X\) and \(y \in Y\) define a (Hausdorff) locally convex topology \(\sigma (X,Y)\) on *X*.

The canonical duality between a Banach space *X* and its dual \(X'\) clearly defines left-separating pairs \((X,X')\) and \((X',X)\). Other examples are discussed in Sect. 3.

## 2 Abstract Köhler semigroups

We start with a general definition of Köhler semigroups. Examples and applications are given in the sections below.

### Definition 2.1

Let *X* be a locally convex space, \(\mathcal {S} \subset \mathscr {L}(X)\) a subsemigroup and equip \(X^X\) with the product topology. The *(abstract) Köhler semigroup* of \(\mathcal {S}\) is the closure \(\mathcal {K}(\mathcal {S}){:}{=}\overline{\mathcal {S}} \subset X^X\).

If *X* carries the weak topology induced by a left-separating pair (*X*, *Y*) we denote the Köhler semigroup associated to a semigroup \(\mathcal {S} \subset \mathscr {L}(X)\) by \(\mathcal {K}(\mathcal {S};X,Y)\).

*S*equipped with a topology is called

*right topological*if the mapping

*left topological*if

*semitopological*if it is both, left topological and right topological (see Section 1.3 of [4]).

*topological center*\(\Lambda (S)\) of a right topological semigroup

*S*is

### Lemma 2.2

*X*the following assertions hold.

- (i)
\(\mathcal {K}(\mathcal {S})\) is a right topological semigroup.

- (ii)
The topological center \(\Lambda (\mathcal {K}(\mathcal {S}))\) contains \(\mathcal {S}\).

- (iii)
\(\mathcal {K}(\mathcal {S})\) is compact if and only if \(\mathcal {S}x\) is relatively compact for each \(x \in X\).

- (iv)
If \(\mathcal {K}(\mathcal {S}) \subset \mathscr {L}(X)\), then \(\mathcal {K}(\mathcal {S})\) is semitopological.

- (v)
If \(\mathcal {S}\) is abelian, then \(\mathcal {K}(\mathcal {S})\) is semitopological if and only if it is abelian.

For operator semigroups on certain classes of locally convex spaces one can say more and we recall some definitions from this theory. For an introduction we refer to [25, 40].

### Definition 2.3

*X*is called

- (i)
*barrelled*if every radial, convex, circled and closed set is a zero neighborhood. - (ii)
*Montel*if it is barrelled and every bounded subset is relatively compact. - (iii)
*quasi-complete*if every closed bounded subset is complete (with respect to the uniformity defined by the zero neighborhoods of*X*).

Banach spaces and even Fréchet spaces are barrelled. Important examples of Montel spaces are the space of all holomorphic functions \(\mathrm {H}(\Omega )\) on an open connected subset \(\Omega \subset \mathbb {C}\) equipped with the topology of compact convergence and the space of smooth functions \(\mathrm {C}^\infty (\Omega )\) on an open set \(\Omega \subset \mathbb {R}^n\) equipped with the topology of compact convergence and, in all derivatives. We refer to Section 11.5 of [25] for more about these spaces.

Besides Banach and Fréchet spaces the dual space of a Banach space equipped with the weak* topology is quasi-complete.

### Proposition 2.4

*X*.

- (i)The Köhler semigroup \(\mathcal {K}(\mathcal {S})\) consists of continuous mappings on
*X*if one of the following conditions is fulfilled.- (a)
For each pointwise bounded subset \(M \subset \mathscr {L}(X)\) the pointwise closure \(\overline{M} \subset X^X\) is contained in \(\mathscr {L}(X)\).

- (b)
The space

*X*is barrelled. - (c)
*X*carries the weak topology \(\sigma (X,X')\) associated to a barrelled topology on*X*. - (d)
The semigroup \(\mathcal {S}\subset \mathscr {L}(X)\) is equicontinuous.

- (a)
- (ii)
If

*X*is Montel, then \(\mathcal {K}(\mathcal {S})\) is a compact semitopological subsemigroup of \(\mathscr {L}(X)\).

### Proof

Clearly, condition (a) implies \(\mathcal {K}(\mathcal {S}) \subset \mathscr {L}(X)\). We show that the spaces in (b) and (c) satisfy (a). For a barrelled space *X* this is a consequence of the Banach–Steinhaus Theorem (see Theorem III.4.6 in [40]).

*M*of \(\sigma (X,X')\)-continuous operators and take \(T_{\alpha } \in M\) for \(\alpha \in A\) and \(T \in X^X\) with \(\lim _{\alpha } \langle T_{\alpha }x,x'\rangle = \langle Tx,x'\rangle \) for all \(x \in X\) and \(x' \in X'\). By passing to the (algebraic) adjoint operator \(T^* :X' \longrightarrow X^*\), we obtain

For given \(x' \in X'\) the net \((T_{\alpha }'x')_{\alpha \in A}\) is contained in the bounded set \(M'x' = \{S'x'\mid S \in M\} \subset X'\) which is (since *X* is barrelled) relatively \(\sigma (X',X)\)-compact (see the corollary to IV.1.6 in [40]). Thus we find a subnet of \((T_{\alpha }x')_{\alpha \in A}\) converging in the \(\sigma (X',X)\)-topology to an element \(y' \in X'\) which yields \(T^*x' = y' \in X'\). We obtain \(T^*X' \subset X'\) which shows that *T* is \(\sigma (X,X')\)-continuous (IV.2.1 in [40]) and so (ii) is proved.

In the situation of (d) we obtain \(\mathcal {K}(\mathcal {S}) \subset \mathscr {L}(X)\) by Theorem III.4.3 of [40].

From the definition of a Montel space and part (i) we immediately obtain (ii). \(\square \)

At the end of this section we prove some relations between different Köhler semigroups. We recall that a continuous mapping between two right topological semigroups is a *homomorphism of right topological semigroups* if it is multiplicative. A surjective homomorphism is called *epimorphism*.

### Proposition 2.5

- (i)Assume that \(X_2\) is a subspace of \(X_1\) and \(\mathcal {S}_2 = \mathcal {S}_1|_{X_2}\). If \(\mathcal {K}(\mathcal {S}_1)\) and \(\mathcal {K}(\mathcal {S}_2)\) are compact, thenis an epimorphism of right topological semigroups.$$\begin{aligned} \mathcal {K}(\mathcal {S}_1) \longrightarrow \mathcal {K}(\mathcal {S}_2), \quad S \mapsto S|_{X_2} \end{aligned}$$
- (ii)Assume that \(\Phi \in \mathscr {L}(X_1,X_2)\) is a surjective map and \(\Psi :\mathcal {S}_1 \longrightarrow \mathcal {S}_2\) is an epimorphism of semigroups with \(\Phi (Sx) = \Psi (S) \Phi (x)\) for all \(S \in \mathcal {S}_1\) and \(x \in X_1\). If \(\mathcal {K}(\mathcal {S}_1)\) is compact, thenwith \(\hat{\Psi }(S)(\Phi (x)){:}{=}\Phi (Sx)\) for \(S \in \mathcal {S}_1\) and \(x \in X\) is an epimorphism of right topological semigroups with \(\hat{\Psi }|_{\mathcal {S}_1} = \Psi \).$$\begin{aligned} \hat{\Psi }:\mathcal {K}(\mathcal {S}_1) \longrightarrow \mathcal {K}(\mathcal {S}_2), \quad S \mapsto \hat{\Psi }(S) \end{aligned}$$

### Proof

## 3 Compact operator semigroups in topological dynamics

We now apply the concepts and results from Sect. 2 in topological dynamics.

### Definition 3.1

*topological dynamical system*is a pair (

*K*;

*S*) of a compact space

*K*and a semitopological semigroup

*S*acting on

*K*such that the mapping

Occasionally we identify an element \(s \in S\) with the continuous mapping \(K\longrightarrow K, \, x \mapsto sx\).

### Definition 3.2

*K*;

*S*) be a topological dynamical system.

- (i)
To each \(s \in S\) we associate the

*Koopman operator*\(T_{s} \in \mathscr {L}(\mathrm {C}(K))\) by setting \(T_{s}f(x) {:}{=}f(sx)\) for \(s \in S\) and \(x \in K\). - (ii)The
*Koopman semigroup*\(\mathcal {T}_{{S}}\) associated to (*K*;*S*) is$$\begin{aligned} \mathcal {T}_{S}{:}{=}\{T_s\mid s \in S\} \subset \mathscr {L}(\mathrm {C}(K)). \end{aligned}$$

For an introduction to the theory of dynamical systems we refer to [6, 11, 17]. For an operator theoretic approach to this topic see [9].

*K*. We also identify the Banach lattice of discrete measures on

*K*with

*S*-invariant probability measure \(\mu \in \mathrm {C}(K)'\) we identify the Banach lattice of measures absolutely continuous with respect to \(\mu \) with \(\mathrm {L}^1(K,\mu )\). We then obtain operator semigroups associated to the left-separating pairs \((\mathrm {C}(K)',\mathrm {C}(K))\), \((\ell ^1(K),\mathrm {C}(K))\) and \((\mathrm {L}^1(K,\mu ),\mathrm {C}(K))\).

### Lemma 3.3

*K*;

*S*) and the corresponding Koopman semigroup \(\mathcal {T}_S\). The following assertions are true.

- (i)
The semigroup \(\mathcal {T}_{S}'\) has relatively \(\sigma (\mathrm {C}(K)',\mathrm {C}(K))\)-compact convex orbits \(\mathrm {co}\, \mathcal {T}_{S}'\mu \) for all \(\mu \in \mathrm {C}(K)'\).

- (ii)
The space \(\ell ^1(K)\) is \(\mathcal {T}_{S}'\)-invariant and the semigroup \(\mathcal {T}_{S}'|_{\ell ^1(K)}\) has relatively \(\sigma (\ell ^1(K),\mathrm {C}(K))\)-compact orbits \(\mathcal {T}_{S}'(a_x)_{x \in K}\) for all \((a_x)_{x \in K} \in \ell ^1(K)\).

- (iii)
For each

*S*-invariant probability measure \(\mu \in \mathrm {C}(K)'\) the space \(\mathrm {L}^1(K,\mu )\) is \(\mathcal {T}_{S}'\)-invariant and the semigroup \(\mathcal {T}_{S}'|_{\mathrm {L}^1(K,\mu )}\) has relatively \(\sigma (\mathrm {L}^1(K,\mu ),\mathrm {C}(K))\)-compact convex orbits \(\mathrm {co}\,\mathcal {T}_{S}'h\) for all \(h \in \mathrm {L}^1(K,\mu )\).

### Proof

The first assertion holds since all operators in \(\mathcal {T}_{\mathcal {S}}\) are contractions and the unit ball of \(\mathrm {C}(K)'\) is weak* compact by the Banach–Alaoglu theorem (see Theorem III.4.3 of [40]).

Finally consider an *S*-invariant probability measure \(\mu \in \mathrm {C}(K)'\). If a measure \(\nu \in \mathrm {C}(K)'\) is absolutely continuous with respect to \(\mu \), then so is \(T_s'\nu \) for each \(s \in S\). To see this, take a Borel measurable \(\mu \)-null set *A*. Since \(\mu \) is invariant, we obtain \(\mu (s^{-1}(A)) = 0\) and by absolute continuity of \(\nu \) we conclude \(T_s'\nu (A) = \nu (s^{-1}(A)) = 0\).

We now introduce various enveloping semigroups.

### Definition 3.4

*K*;

*S*) we associate the following semigroups.

- (i)
The

*Köhler semigroup*\(\mathcal {K}(K;S){:}{=}\mathcal {K}(\mathcal {T}_{S}';\mathrm {C}(K)',\mathrm {C}(K))\). - (ii)
The

*convex Köhler semigroup*\(\mathcal {K}_{\mathrm {c}}(K;S){:}{=}\mathcal {K}(\mathrm {co} \, \mathcal {T}_{S}';\mathrm {C}(K)',\mathrm {C}(K))\). - (iii)
The

*Ellis semigroup*\(\mathcal {E}(K;S){:}{=}\mathcal {K}(\mathcal {T}_{S}';\ell ^1(K),\mathrm {C}(K))\). - (iv)
For an

*S*-invariant probability measure \(\mu \in \mathrm {C}(K)'\) the*Jacobs semigroup*\(\mathcal {J}(K,\mu ;S){:}{=}\mathcal {K}(\mathcal {T}_{S}';\mathrm {L}^1(K,\mu ),\mathrm {C}(K))\). - (v)
For an

*S*-invariant probability measure \(\mu \in \mathrm {C}(K)'\) the*convex Jacobs semigroup*\(\mathcal {J}_{\mathrm {c}}(K,\mu ;S){:}{=}\mathcal {K}(\mathrm {co}\,\mathcal {T}_{S}';\mathrm {L}^1(K,\mu ),\mathrm {C}(K))\).

### Remark 3.5

### Remark 3.6

*K*;

*S*) given as the closure \(\overline{S}\subset K^K\) (see [10]). If we identify

*K*with its homeomorphic copy \(\{\delta _x\mid x \in K\} \subset \mathrm {C}(K)'\) (where \(\mathrm {C}(K)'\) is equipped with the weak* topology), one readily checks that the mapping

### Remark 3.7

Proposition 2.5 shows that taking subsystems and factors of a topological dynamical system (*K*; *S*) induces epimorphisms of the corresponding Köhler and Ellis semigroups.

The following result is an immediate consequence of Proposition 2.5 (i) and shows the relations between these semigroups.

### Proposition 3.8

*K*;

*S*) be a topological dynamical system and \(\mu \in \mathrm {C}(K)'\) an

*S*-invariant probability measure. We then have the following diagram of epimorphisms of right topological semigroups where the arrows are given by the canonical restriction maps.

It is natural to ask when these epimorphisms are isomorphisms. If the topological dynamical system (*K*; *S*) is *weakly almost periodic*, i.e., if \(\mathcal {T}_Sf\) is relatively weakly compact for each \(f \in \mathrm {C}(K)\), we obtain the following.

### Proposition 3.9

If (*K*; *S*) is a weakly almost periodic system, then the canonical epimorphism \(\mathcal {K}(K;S) \longrightarrow \mathcal {E}(K;S)\) is an isomorphism.

If moreover \(\mu \in \mathrm {C}(K)'\) is a strictly positive invariant probability measure, then the canonical epimorphisms \(\mathcal {K}(K;S) \longrightarrow \mathcal {J}(\mu )\) and \(\mathcal {K}_\mathrm {c}(K;S) \longrightarrow \mathcal {J}_\mathrm {c}(\mu )\) are also isomorphisms.

### Proof

*K*;

*S*) is weakly almost periodic, \(\mathcal {K}(\mathcal {T}_S;\mathrm {C}(K),\mathrm {C}(K)')\) is a compact semitopological semigroup and one readily checks that

If \(\mu \in \mathrm {C}(K)'\) is a strictly positive invariant probability measure, then \(\mathrm {L}^1(K,\mu )\) is also weak* dense in \(\mathrm {C}(K)'\), and thus \(\mathcal {K}(K;S) \longrightarrow \mathcal {J}(\mu )\) is injective.

Since, by Krein’s theorem (see Theorem 11.4 in [40] or Theorem C.11 in [9]), the sets \(\mathrm {co}\,\mathcal {T}_Sf\) are also relatively weakly compact, a similar argument shows that the remaining map is an isomorphism. \(\square \)

The question when the epimorphism \(\mathcal {K}(K;S) \longrightarrow \mathcal {E}(K;S)\) is an isomorphism was first posed by Pym [32] and then answered by Köhler [30] for metrizable topological dynamical systems satisfying the following condition which is weaker than weak almost periodicity.

### Definition 3.10

A metric topological dynamical system (*K*; *S*) is *tame* if for each \(f \in \mathrm {C}(K)\) the orbit \(\mathcal {T}_Sf\) is relatively sequentially compact with respect to the product topology of \(\mathbb {C}^K\).

Such systems have been studied in detail by Glasner and Megrelishvili [14, 15, 16, 18, 19, 20, 21] and later by Huang in [23], Kerr and Li [26] as well as by Romanov [34].

*X*is a

*Fréchet–Urysohn space*(see page 53 of [12]) if each subset \(A \subset X\) satisfies

*K*is called

*Rosenthal compact*if it can be continuously embedded into the space of Baire 1 functions \(\mathrm {B}_1(X)\) on a Polish space

*X*. By results of Bourgain et al. [2] every Rosenthal compact space is a Fréchet–Urysohn space. Moreover, closed subspaces and countable products of Rosenthal compact spaces are Rosenthal compact (see Section c-17 of [22] for more properties of such spaces). This leads to the following characterizations of tameness.

### Proposition 3.11

*K*;

*S*) the following are equivalent.

- (i)
The system (

*K*;*S*) is tame. - (ii)
\(\mathcal {K}_{\mathrm {c}}(K;S)\) is a Rosenthal compact space.

- (iii)
\(\mathcal {K}(K;S)\) is a Rosenthal compact space.

- (iv)
\(\mathcal {E}(K;S)\) is a Fréchet–Urysohn space.

### Proof

As above we identify *K* with its homeomorphic copy \(\{\delta _x \mid x \in K\}\) in \(\mathrm {C}(K)'\). In particular, \((R'f|_K)(x) = \langle f, R \delta _x\rangle \) for \(R \in \mathcal {K}_{\mathrm {c}}(K;S)\), \(f \in \mathrm {C}(K)\) and \(x \in K\).

*K*is metric). We will show that

Since closed subspaces of Rosenthal compact spaces are Rosenthal compact, we immediately obtain that (ii) implies (iii). Moreover, continuous images of compact Fréchet–Urysohn spaces are again Fréchet–Urysohn, so (iii) implies (iv) by Proposition 3.8.

*S*converging to \(\psi \) whereby

### Remark 3.12

The equivalence of assertions (i) and (iv) is well-known. Our proof is based on the arguments of Glasner and Megrelishvili (see Theorem 3.2 in [18]), but extends the result to the convex Köhler semigroup showing thereby that the classes D2 and D3 of dynamical systems in [34] are actually the same.

The next result is known (see Theorem 1.5 of [14] for the case of a group action), but for the sake of completeness we give a short proof.

### Proposition 3.13

*K*;

*S*) be a tame metric topological dynamical system and for \(\psi \in \mathrm {E}(K;S)\) define the

*Koopman operator*\(T_\psi \in \mathscr {L}(\mathrm {C}(K), \mathrm {B}_1(K))\) by \(T_{\psi }f{:}{=}f \circ \psi \) for \(\psi \in \mathrm {E}(K;S)\) and \(f \in \mathrm {C}(K)\). Then

### Proof

A map from a sequential space to a Hausdorff space is continuous if and only if it is sequentially continuous (see Proposition 1.6.15 in [12]). It is therefore a direct consequence of Lebesgue’s Theorem that *J* is continuous. Multiplicativity is trivial and, since \(J(S) = \mathcal {T}_S'\), *J* is surjective. \(\square \)

### Remark 3.14

Even for a tame system (*K*; *S*) and a strictly positive invariant probability measure \(\mu \in \mathrm {C}(K)'\) the epimorphism \(\mathcal {K}(K;S) \longrightarrow \mathcal {J}(K,\mu ;S)\) is not injective in general. In fact, if *S* is abelian, then so is \(\mathcal {J}(K,\mu ;S)\) (see Lemma 2.2 (iv)), but \(\mathcal {K}(K;S)\) generally not (see [15], Example 4.5).

In Sect. 5 we use the semigroups introduced above to study qualitative properties (e.g., mean ergodicity) of dynamical systems.

## 4 Mean ergodic semigroups

Inspired by the approach of R. Nagel to mean ergodic semigroups (see [31] and the supplement of Chapter 8 of [9]) we use techniques developed by Romanov [33] as well as Schreiber [37, 38] to discuss mean ergodicity of operator semigroups on locally convex spaces (see also [8, 36], Section 2.1.2 in [13, 27]; see [1] for mean ergodicity of one-parameter semigroups).

### Definition 4.1

*X*be a locally convex space and \(\mathcal {S} \subset \mathscr {L}(X)\) be an operator semigroup.

- (i)A net \((T_{\alpha })_{\alpha \in A} \subset \overline{\mathrm {co}\, \mathcal {S}}^{X^X}\) is called
- (a)a
*left ergodic net for*\(\mathcal {S}\) iffor each \(x \in X\) and \(T \in \mathcal {S}\).$$\begin{aligned} \lim _{\alpha } (\mathrm {Id}-T)T_{\alpha }x = 0 \end{aligned}$$ - (b)a
*right ergodic net for*\(\mathcal {S}\) iffor each \(x \in X\) and \(T \in \mathcal {S}\).$$\begin{aligned} \lim _{\alpha } T_{\alpha }(\mathrm {Id}-T)x = 0 \end{aligned}$$ - (c)
a

*two-sided ergodic net for*\(\mathcal {S}\) if it is left and right ergodic for \(\mathcal {S}\).

- (a)
- (ii)The semigroup \(\mathcal {S}\) is called
- (a)
*left mean ergodic*if each left ergodic net for \(\mathcal {S}\) is pointwise convergent. - (b)
*right mean ergodic*if each right ergodic net for \(\mathcal {S}\) is pointwise convergent.

- (a)

We recall some examples (see [38], Examples 1.2).

### Example 4.2

- (i)Consider \(\mathcal {S} = \{T^n\mid n \in \mathbb {N}_0\}\) for an operator \(T \in \mathscr {L}(X)\) with bounded orbits \(\mathcal {S}x\) for \(x \in X\) on a locally convex space
*X*. The sequence \((A_N)_{N \in \mathbb {N}}\) of*Cesàro means*defined byfor \(x \in X\) and \(N \in \mathbb {N}\) is a two-sided ergodic sequence for \(\mathcal {S}\).$$\begin{aligned} A_Nx {:}{=}\frac{1}{N}\sum _{n=0}^{N-1}T^nx \end{aligned}$$ - (ii)For a pointwise bounded strongly continuous semigroup \((T(t))_{t \ge 0}\) (i.e., \(t\mapsto T(t)x\) is continuous for each \(x \in X\)) on a quasi-complete space
*X*(see Definition 2.3 (iii)) we setfor \(x \in X\) and \(s \in (0, \infty )\) where the integral is understood in the sense of Bourbaki (see Proposition III.3.7 in [5]). Then \((A_s)_{s > 0}\) is a two-sided ergodic net for the semigroup \(\{T(t)\mid t \ge 0\}\).$$\begin{aligned} A_sx {:}{=}\frac{1}{s}\int _0^sT(t)x\,\mathrm {d}t \end{aligned}$$ - (iii)The net of
*Abel means*\((S_r)_{r \in (1,\infty )}\) for an operator \(T \in \mathscr {L}(X)\) with bounded orbits on a quasi-complete, barrelled locally convex space*X*defined byfor \(x \in X\) and \(r \in (1,\infty )\) is a two-sided ergodic net for \(\mathcal {S} = \{T^n\mid n \in \mathbb {N}_0\}\).$$\begin{aligned} S_rx{:}{=}(r-1)\sum _{n=0}^{\infty }\frac{1}{r^{n+1}}T^nx \end{aligned}$$ - (iv)Example (ii) can be generalized as follows. Consider a locally compact group
*G*with left Haar measure \(\mu \) and let \(S \subset G\) be a subsemigroup. Assume further that there is a \(\mu \)*-Følner net*\((F_{\alpha })_{\alpha \in A}\) in*S*, i.e., \(F_{\alpha } \subset S\) is a compact set with positive finite measure for each \(\alpha \in A\) andfor each \(s \in S\). Given a pointwise bounded representation$$\begin{aligned} \lim _{\alpha } \frac{\mu (F_{\alpha } \triangle sF_{\alpha })}{\mu (F_{\alpha })} = 0 \end{aligned}$$of$$\begin{aligned} S \longrightarrow \mathscr {L}(X), \quad s \mapsto T(s) \end{aligned}$$*S*on a quasi-complete locally convex space*X*we can definefor each \(x \in X\) and each \(\alpha \in A\) and thereby obtain a left ergodic net for \(\{T(s)\mid s \in S\}\).$$\begin{aligned} \mathcal {F}_{\alpha }x{:}{=}\frac{1}{\mu (F_{\alpha })} \int _{F_{\alpha }} T(s)x\, \mathrm {d}\mu . \end{aligned}$$

Recall that a semitopological semigroup *S* is *left amenable* if the space \(\mathrm {C}_{\mathrm {b}}(S)\) of bounded continuous functions on *S* has a *left invariant mean*, i.e., a positive element \(m \in \mathrm {C}_{\mathrm {b}}(S)'\) with \(m(\mathbb {1}) = 1\) and \(m(L_Sf) = m(f)\) for each \(f \in \mathrm {C}_{\mathrm {b}}(S)\) and \(s \in S\), where \(L_sf(t) {:}{=}f(st)\) for every \(t \in S\) (see Section 2.3 of [4]). *Right amenability and (two-sided) amenability* are defined analogously.

We now assume that the semigroup \(\mathcal {S} \subset \mathscr {L}(X)\) endowed with the topology of pointwise convergence is left amenable which is always the case if \(\mathcal {S}\) is abelian. In this situation we can characterize the convergence of all left ergodic nets through an algebraic property of the Köhler semigroup \(\mathcal {K}(\mathrm {co}\, \mathcal {S})\).

### Theorem 4.3

*X*and a left amenable semigroup \(\mathcal {S}\subset \mathscr {L}(X)\) with relatively compact convex orbits \(\mathrm {co} \,\mathcal {S}x\) for all \(x \in X\) the following are equivalent.

- (i)
The semigroup \(\mathcal {S}\) is left mean ergodic.

- (ii)
The semigroup \(\mathcal {K}(\mathrm {co}\,\mathcal {S})\) has a zero

*Q*, i.e., \(QS=SQ =Q\) for every \(S \in \mathcal {K}(\mathrm {co}\,\mathcal {S})\).

*kernel*of \(\mathcal {K}(\mathrm {co} \, \mathcal {S})\), i.e., the intersection of all ideals (see Notation 1.2.3 in [4]).

### Lemma 4.4

*X*and a left amenable semigroup \(\mathcal {S}\subset \mathscr {L}(X)\) with relatively compact convex orbits \(\mathrm {co} \,\mathcal {S}x\) for \(x \in X\). Then the kernel of \(\mathcal {K}(\mathrm {co}\,\mathcal {S})\) is given by

### Proof

*I*is a right zero. Moreover,

*I*is an ideal and we thus have \(\ker (\mathcal {K}(\mathrm {co}\, \mathcal {S})) \subset I\). On the other hand, each right zero

*Q*of \(\mathcal {K}(\mathrm {co}\, \mathcal {S})\) is a minimal idempotent and thus satisfies

### Lemma 4.5

*X*and a left amenable semigroup \(\mathcal {S}\subset \mathscr {L}(X)\) with relatively compact convex orbits \(\mathrm {co} \,\mathcal {S}x\) for \(x \in X\). For a net \((T_{\alpha })_{\alpha \in A} \subset \mathcal {K}(\mathrm {co} \, \mathcal {S})\) the following are equivalent.

- (i)
The net \((T_{\alpha })_{\alpha \in A}\) is left ergodic.

- (ii)
All accumulation points of \(\{T_{\alpha }\mid \alpha \in A\}\) for the operator topology of pointwise convergence are contained in \(\ker (\mathcal {K}(\mathrm {co}\, \mathcal {S}))\).

### Proof

*Q*of \((T_{\alpha })_{\alpha \in A}\), the operator \((\mathrm {Id}-T)Q\) is an accumulation point of \(((\mathrm {Id}-T)T_{\alpha })_{\alpha \in A}\). The assumption yields \((\mathrm {Id}-T)Q= 0\) and therefore \(Q = TQ\). Since \(T \in \mathcal {S}\) was arbitrary, we obtain \(Q \in \ker (\mathcal {K}(\mathrm {co}\, \mathcal {S}))\) by Lemma 4.4.

Assume now that there is \(T \in \mathcal {S}\) such that \(((\mathrm {Id}-T)T_{\alpha })_{\alpha \in A}\) does not converge to zero. We then find a zero neighborhood *U* and a subnet \((T_{\beta })_{\beta \in B}\) of \((T_{\alpha })_{\alpha \in A}\) with \((\mathrm {Id}-T)T_{\beta } \notin U\) for all \(\beta \in B\). By compactness of \(\mathcal {K}(\mathrm {co}\, \mathcal {S})\) we find an accumulation point *Q* of this subnet satisfying \((\mathrm {Id}-T)Q \ne 0\). Hence \(TQ \ne Q\) and thus \((T_{\alpha })_{\alpha \in A}\) has an accumulation point which is not contained in \(\ker (\mathcal {K}(\mathrm {co}\, \mathcal {S}))\). \(\square \)

### Proof

(of Theorem 4.3). By Lemma 4.4 the second assertion is equivalent to \(\ker (\mathcal {K}(\mathrm {co}\,\mathcal {S}))\) being the singleton \(\{Q\}\) and thus Lemma 4.5 shows the implication “(ii) \(\Rightarrow \) (i)”.

*P*generating the topology on

*X*. We define a partial order on the set

*A*into a directed set. For each triple \(\alpha = (Y,M,k) \in A\) we find \(T_{1,\alpha },T_{2,\alpha } \in \mathrm {co} \, \mathcal {S}\) with

The following definitions are natural.

### Definition 4.6

*X*be a barrelled space. A semigroup \(\mathcal {S} \subset \mathscr {L}(X)\) is called

- (i)
*weak* mean ergodic*if \(\mathcal {S}'\) is left mean ergodic with respect to the \(\sigma (X',X)\)-topology. - (ii)
*weakly mean ergodic*if \(\mathcal {S}\) is right mean ergodic with respect to the \(\sigma (X,X')\)-topology. - (iii)
*strongly mean ergodic*if \(\mathcal {S}\) is right mean ergodic with respect to the given topology on*X*.

*X*we use the notation

### Theorem 4.7

- (i)
There is a two-sided ergodic net \((T_{\alpha })_{\alpha \in A}\) for \(\mathcal {S}\) with respect to the weak topology such that \((T_{\alpha }x)_{\alpha \in A}\) converges weakly for each \(x \in X\).

- (ii)
The semigroup \(\mathcal {S}\) is weakly mean ergodic.

- (iii)
The semigroup \(\mathcal {S}\) is strongly mean ergodic.

- (iv)
The semigroup \(\mathcal {K}(\mathrm {co}\,\mathcal {S};X,X')\) has a zero

*P*. - (v)
The semigroup \(\mathcal {K}(\mathrm {co}\,\mathcal {S}';X',X)\) has a zero

*Q*which is weak* continuous. - (vi)
The fixed space \({\text {fix}}(\mathcal {S})\) separates \({\text {fix}}(\mathcal {S}')\).

- (vii)
\(X = {\text {fix}}(\mathcal {S}) \oplus \overline{\mathrm {lin}} {\text {rg}}(\mathrm {Id}- \mathcal {S})\).

We start with the following lemma.

### Lemma 4.8

- (i)
There exist right ergodic nets for \(\mathcal {S}\) with respect to the topology \(\tau \).

- (ii)
Let

*D*be the set of all \(x \in X\) for which \(\lim _\alpha T_\alpha x\) exists for every net \((T_\alpha )_{\alpha \in A}\) which is right ergodic with respect to \(\tau \). Then*D*is closed. - (iii)
Let \(D_0\) be the set of all \(x \in X\) for which \(\lim _\alpha T_\alpha x=0\) for every net \((T_\alpha )_{\alpha \in A}\) which is right ergodic with respect to \(\tau \). Then \(D_0\) is closed.

- (iv)
\({\text {fix}}(\mathcal {S}) \cap \overline{\mathrm {lin}} {\text {rg}}(\mathrm {Id}- \mathcal {S}) = \{0\}\) and \({\text {fix}}(\mathcal {S}) \oplus \overline{\mathrm {lin}} {\text {rg}}(\mathrm {Id}- \mathcal {S}) \subset D\).

### Proof

We first observe that there are right ergodic nets for \(\mathcal {S}\) with respect to the \(\sigma (X,X')\)-topology. In fact, by Lemma 4.5 we find a left ergodic net \((T_\alpha ')_{\alpha \in A}\) for \(\mathcal {S}' \subset \mathscr {L}(X')\) with respect to the \(\sigma (X',X)\)-topology such that \(T_{\alpha }' \in \mathrm {co}\,\mathcal {S}'\) for all \(\alpha \in A\). The net \((T_{\alpha })_{\alpha \in A}\) of pre-adjoints is then right ergodic for \(\mathcal {S}\) with respect to the \(\sigma (X,X')\)-topology.

The proof of Theorem 1.4 in [38] (which still works in the case of locally convex spaces) now shows that there are actually even right ergodic nets for \(\mathcal {S}\) with respect to the topology \(\tau \).

Now if \((T_\alpha )_{\alpha \in A}\) is a right ergodic net for \(\mathcal {S}\) with respect to the topology \(\tau \), then the set \(\{T_\alpha \mid \alpha \in A\} \subset \mathscr {L}(X)\) is equicontinuous since \((X,\tau )\) is barrelled (see Theorem III.4.2 of [40]). Therefore \(\lim _\alpha T_\alpha x\) exists for each \(x \in \overline{D}\) by Theorem III.4.5 of [40] and *D* is closed. Similarly we see that \(D_0\) is closed.

Since \(\overline{\mathrm {lin}} {\text {rg}}(\mathrm {Id}- \mathcal {S}) \subset D_0\) by (ii), we have \({\text {fix}}(\mathcal {S}) \cap \overline{\mathrm {lin}} {\text {rg}}(\mathrm {Id}- \mathcal {S}) = \{0\}\). Moreover, we obtain \({\text {fix}}(\mathcal {S}) \oplus \overline{\mathrm {lin}} {\text {rg}}(\mathrm {Id}- \mathcal {S}) \subset D\). \(\square \)

### Proof

*P*is continuous with respect to the weak topology by Proposition 2.4. Moreover, we obtain

Now assume that (v) is satisfied. Let \(Q= P'\in \mathcal {K}(\mathrm {co}\,\mathcal {S};X';X)\) be the weak* continuous zero and take \(0 \ne x' \in {\text {fix}}(\mathcal {S}')\). We find \(x \in X\) with \(\langle x,x'\rangle \ne 0\) and \(y{:}{=}Px \in {\text {fix}}(\mathcal {S})\) then satisfies \(\langle y,x'\rangle = \langle x,Qx'\rangle = \langle x,x'\rangle \ne 0\). Hence we have (vi).

Suppose that (vi) holds. Take \(x' \in X'\) vanishing on \({\text {fix}}(\mathcal {S}) \oplus \overline{\mathrm {lin}} {\text {rg}}(\mathrm {Id}- \mathcal {S})\). In particular \(\langle x-Sx,x'\rangle = 0\) for all \(x \in X\) and \(S \in \mathcal {S}\) and hence \(x' \in {\text {fix}}(\mathcal {S}')\) and \(x'=0\) since \({\text {fix}}(\mathcal {S})\) separates \({\text {fix}}(\mathcal {S}')\) and \(x'\) vanishes on \({\text {fix}}(\mathcal {S})\). Thus \({\text {fix}}(\mathcal {S}) \oplus \overline{\mathrm {lin}} {\text {rg}}(\mathrm {Id}- \mathcal {S})\) is dense in *X* by the Hahn–Banach Theorem, and, by Lemma 4.8 (ii) \(D=X\). Thus (vi) implies (iii).

Theorem 4.3 shows that (v) implies (ii) and therefore the equivalence of assertions (i)–(vi). The statement about the limit also follows from Theorem 4.3.

### Corollary 4.9

Every amenable operator semigroup \(\mathcal {S} \subset \mathscr {L}(X)\) on a barrelled locally convex space *X* with relatively weakly compact convex orbits is strongly mean ergodic.

### Proof

*P*is a zero in \(\mathcal {K}(\mathrm {co}\,\mathcal {S};X,X')\). \(\square \)

### Remark 4.10

If *X* is a reflexive barrelled space, then every bounded amenable semigroup is strongly mean ergodic by Corollary 4.9 (see Theorem IV.5.6 in [40]).

We present an example where Theorem 4.7 is applicable.

### Example 4.11

## 5 Mean ergodicity in topological dynamics

*K*;

*S*) the mapping

*S*and equip \(\mathcal {T}_S\) with the strong operator topology (see Theorem 4.17 in [9]). In particular, if

*S*is left amenable, then \(\mathcal {T}_S\) is right amenable.

A topological dynamical system (*K*; *S*) is said to be *weak** (resp. *norm*) *mean ergodic* if the Koopman semigroup \(\mathcal {T}_S \subset \mathscr {L}(\mathrm {C}(K))\) is weak* (resp. strongly) mean ergodic. Moreover, the system (*K*; *S*) is *uniquely ergodic* if there is a unique *S*-invariant probability measure \(\mu \in \mathrm {C}(K)'\). Using the convex Köhler semigroup (see Definition 3.4) we obtain the following characterization of unique ergodicity.

### Proposition 5.1

*K*;

*S*) be a topological dynamical system with

*S*left amenable. The following assertions are equivalent.

- (i)
There is a net \((T_{\alpha })_{\alpha \in A} \subset \mathrm {co}\,\mathcal {T}_S\) for \(\mathcal {T}_{S}\) which is right ergodic with respect to the weak topology such that for each \(f \in \mathrm {C}(K)\) the net \((T_\alpha f)_{\alpha \in A}\) converges weakly to a constant function.

- (ii)
The system (

*K*;*S*) is uniquely ergodic. - (iii)
The semigroup \(\mathcal {K}_{\mathrm {c}}(K;S)\) has a zero which is a rank one operator.

*K*for each \(f \in \mathrm {C}(K)\) and for each net \((T_{\alpha })_{\alpha \in A}\) which is right ergodic for \(\mathcal {T}_S\) with respect to the norm topology.

### Proof

Suppose that (ii) is valid and let \(Q_1,Q_2 \in \ker (\mathcal {K}_{\mathrm {c}}(K;S))\). For each probability meausure \(\mu \in \mathrm {C}(K)'\) the measures \(Q_1\mu , Q_2\mu \in \mathrm {C}(K)'\) are invariant probability measures and thus \(Q_1\mu = Q_2\mu \). This implies \(Q_1 = Q_2\) and therefore \(\mathcal {K}_{\mathrm {c}}(K;S)\) has a zero *Q*. If \(\mu _1,\mu _2 \in \mathrm {C}(K)'\) are two probability measures, we also obtain \(Q\mu _1 = Q\mu _2\). As a result *Q* has rank one.

*Q*is rank one, we obtain \(Q\nu = Q\delta _x = \mu \) for each probability measure \(\nu \in \mathrm {C}(K)'\). Now consider the operator \(P \in \mathscr {L}(\mathrm {C}(K))\) given by

*Q*is weak* continuous. Thus (i) and the remaining assertion follow from Theorem 4.7. \(\square \)

### Remark 5.2

*K*;

*S*) with

*S*left amenable.

- (i)
(

*K*;*S*) is weak* mean ergodic if and only if \(\mathcal {K}_{\mathrm {c}}(K;S)\) has a zero (see Theorem 4.3). - (ii)
(

*K*;*S*) is norm mean ergodic if and only if \(\mathcal {K}_{\mathrm {c}}(K;S)\) has a weak* continuous zero (see Theorem 4.7). - (iii)
(

*K*;*S*) is uniquely ergodic if and only if \(\mathcal {K}_{\mathrm {c}}(K;S)\) has a zero which is a rank one operator (see Proposition 5.1).

Recall that a topological dynamical system (*K*; *S*) is *minimal* if *K* has no non-trivial closed *S*-invariant subsets. The following consequence of Proposition 5.1 is a variation of [28], Proposition 3.2, for two-sided ergodic sequences (see also the remark below Corollary 3.3 in [33] and the paper by Iwanik [24]).

### Corollary 5.3

Consider a minimal topological dynamical system (*K*; *S*) with *S* left amenable. If there is an operator sequence \((T_n)_{n \in \mathbb {N}} \subset \mathrm {co}\,\mathcal {T}_S\) which is two-sided ergodic for \(\mathcal {T}_S\) with respect to the \(\sigma (\mathrm {C}(K),\ell ^1(K))\)-topology such that \((T_nf)_{n \in \mathbb {N}}\) converges pointwise for each \(f \in \mathrm {C}(K)\), then (*K*; *S*) is uniquely ergodic.

### Proof

Take a sequence \((T_n)_{n \in \mathbb {N}}\) as above and set \(Pf(x){:}{=}\lim _{n \rightarrow \infty } T_nf(x)\) for \(x \in K\) and \(f \in \mathrm {C}(K)\). Then *P* maps \(\mathrm {C}(K)\) to the space of Baire 1 functions \(\mathrm {B}_1(K)\).

For \(f\in \mathrm {C}(K)\) and \(x_1,x_2 \in K\) the pre-images \(M_i{:}{=}(Pf)^{-1}(Pf(x_i))\) are non-empty, *S*-invariant \(\mathrm {G}_{\delta }\) sets for \(i=1,2\) and—by minimality of (*K*; *S*)—dense. In Baire spaces the intersection of two dense \(\mathrm {G}_{\delta }\) sets is dense and in particular non-empty. We conclude \(Pf(x_1) = Pf(x_2)\) and therefore *Pf* is constant. By Lebesgue’s Theorem and Proposition 5.1 (*K*; *S*) is uniquely ergodic. \(\square \)

### Corollary 5.4

Every minimal tame metric topological dynamical system (*K*; *S*) with *S* amenable is uniquely ergodic.

### Proof

By Proposition 1.3 of [38] there exists a two-sided ergodic net \((T_{\alpha })_{\alpha \in A}\) with respect to the weak topology \(\sigma (\mathrm {C}(K),\mathrm {C}(K)')\). By passing to a subnet we may assume that the limit \(Q\mu := \lim _{\alpha } T_{\alpha }'\mu \) exists in weak* topology for each \(\mu \in \mathrm {C}(K)'\). Then \(Q \in \mathrm {ker}\,\mathcal {K}_{\mathrm {c}}(K;S)\). Since (*K*; *S*) is tame, \(\mathcal {K}_{\mathrm {c}}(K;S)\) is a Rosenthal compact space and we find a left ergodic sequence \((S_n')_{n \in \mathbb {N}} \subset \mathrm {co}\, \mathcal {T}_S'\) converging to *Q*. Since \(QT_s'=T_s'Q = Q\) for each \(s \in S\), the sequence \((S_n')_{n \in \mathbb {N}}\) is also two-sided ergodic. In particular \((S_n)_{n \in \mathbb {N}}\) is a two-sided ergodic sequence for \(\mathcal {T}_S\) with respect to the \(\sigma (\mathrm {C}(K),\ell ^1(K))\)-topology such that \((S_nf)_{n \in \mathbb {N}}\) converges pointwise for each \(f \in \mathrm {C}(K)\) and hence (*K*; *S*) is uniquely ergodic by Corollary 5.3. \(\square \)

### Remark 5.5

Corollary 5.4 has been proved for abelian group actions by Glasner [16, see Theorem 5.1], Kerr and Li [26, see Theorem 7.19] as well as Huang [23, see Theorem 4.8]. Their proofs are based on a representation type result for minimal tame systems while our proof uses the topological properties of the semigroup \(\mathcal {K}_{\mathrm {c}}(K;S)\).

The following theorem is the main result of this section. We recall that a topological dynamical system (*K*; *S*) is *topologically transitive* if there is \(x \in K\) such that \(\overline{Sx} =K\).

### Theorem 5.6

*K*;

*S*) with

*S*left amenable and the following assertions.

- (i)
(

*K*;*S*) is weak* mean ergodic. - (ii)
(

*K*;*S*) is norm mean ergodic. - (iii)
(

*K*;*S*) is uniquely ergodic.

*K*;

*S*) is topologically transitive, all these assertions are equivalent.

### Remark 5.7

Simple examples show that the three notions of weak* mean ergodicity, norm mean ergodicity and unique ergodicity are truly distinct.

For the proof of Theorem 5.6 we need two lemmas. We write \(\mathrm {P}(K)\) for the probability measures on a compact space *K* and remind the reader that we identify *K* with the space of Dirac measures \(\{\delta _x\mid x \in K\} \subset \mathrm {P}(K)\).

### Lemma 5.8

*K*;

*S*) with

*S*left amenable and a point \(x \in K\). Then the following identities hold.

- (i)
\(\mathcal {K}(K;S)(x) = \overline{Sx} \subset K.\)

- (ii)
\(\mathcal {K}_{\mathrm {c}}(K;S)(x) = \mathrm {P}(\overline{Sx}) \subset \mathrm {P}(K).\)

### Proof

### Lemma 5.9

Consider a weak* mean ergodic topological dynamical system (*K*; *S*) with *S* left amenable. Then for each \(x \in K\) the orbit system \((\overline{Sx};S)\) is uniquely ergodic.

### Proof

Since subystems of weak* mean ergodic systems are again weak* mean ergodic, we may assume—by passing to an orbit system—that (*K*; *S*) is transitive. Take \(x \in K\) with \(K = \overline{Sx}\) and let \(Q \in \mathcal {K}_{\mathrm {c}}(K;S)\) be the zero element. The measure \(\mu {:}{=}Qx \in \mathrm {C}(K)'\) is *S*-invariant.

### Proof

(of Theorem 5.6) It is obvious that (iii) \(\Rightarrow \) (ii) \(\Rightarrow \) (i). If (*K*; *S*) is transitive, then Lemma 5.9 proves the equivalence. \(\square \)

To conclude this section we characterize weak* and “pointwise” mean ergodicity for tame metric systems. Our theorem extends Theorem 4.5 of [34] where a similar result was shown for \(\mathbb {N}_0\)-actions with metrizable Ellis semigroup (called *ordinary systems*).

### Theorem 5.10

*K*;

*S*) with

*S*amenable. Then the following assertions are equivalent.

- (i)
For each operator net \((T_{\alpha })_{\alpha \in A}\) which is right ergodic for \(\mathcal {T}_S\) with respect to the \(\sigma (\mathrm {C}(K),\ell ^1(K))\)-topology, each \(f \in \mathrm {C}(K)\) and each \(x \in K\) the limit \(\lim _{\alpha } (T_\alpha f(x))_{\alpha \in A}\) exists.

- (ii)
The system (

*K*;*S*) is weak* mean ergodic. - (iii)
For each \(x \in X\) the system \((\overline{Sx};S)\) contains a unique minimal set.

Once again we need two lemmas. The first one is a generalization of Lemma 2.3 of [33] to our setting.

### Lemma 5.11

Let (*K*; *S*) be a topological dynamical system and consider \(\psi \in \mathrm {ker}\,\mathrm {E}(K;S)\). Then \(\psi (K)\) is contained in the union of minimal sets.

### Proof

By Theorem 1.2.12 in [4] we find a minimal left ideal *I* of \(\mathrm {E}(K;S)\) containing \(\psi \). However, the set *I*(*x*) is minimal by Proposition 1.6.12 in [4]. \(\square \)

We need a more general version of Corollary 5.4.

### Lemma 5.12

Let (*K*; *S*) be a tame metric topological dynamical system with *S* amenable containing a unique minimal subset. Then (*K*; *S*) is uniquely ergodic.

### Proof

Denote the unique minimal subset by *M* and consider an invariant probability measure \(\mu \in \mathrm {C}(K)'\). Since the support of \(\mu \) is closed and invariant, it contains *M*.

*M*we obtain

Before proceeding to the proof of Theorem 5.10, we observe that pointwise and weak ergodic nets are the same for tame systems.

### Lemma 5.13

Consider a tame metric topological dynamical (*K*; *S*) with *S* amenable. Each operator net \((T_{\alpha })_{\alpha \in A}\) which is right ergodic for \(\mathcal {T}_S\) with respect to the \(\sigma (\mathrm {C}(K),\ell ^1(K))\)-topology is right ergodic for \(\mathcal {T}_S\) with respect to the \(\sigma (\mathrm {C}(K),\mathrm {C}(K)')\)-topology.

### Proof

Take an operator net \((T_{\alpha })_{\alpha \in A}\) which is right ergodic with respect to the \(\sigma (\mathrm {C}(K),\ell ^1(K))\)-topology, \(f \in \mathrm {C}(K)\) and \(s \in S\). We then have \(\lim _{\alpha } (T_{\alpha }(f-T_sf))(x) = 0\) for all \(x \in K\).

*K*;

*S*) is tame, the set \(\overline{\mathrm {co}\,\mathcal {T}_S (\mathrm {Id}-T_s)f}\) is compact in \(\mathrm {B_1}(K)\) and it contains the net \((T_{\alpha }(\mathrm {Id}-T_s)f)_{\alpha \in A}\). The main theorem of [35] therefore implies

### Proof

(of Corollary 5.10) If (*K*; *S*) is weak* mean ergodic, then each orbit is uniquely ergodic by Lemma 5.9 and—since every minimal set supports an invariant probability measure—contains only one minimal set. This proves the implication “(ii) \(\Rightarrow \) (iii)”.

By Lemma 5.12 assertion (iii) implies that each orbit is uniquely ergodic. Thus, for each right \(\sigma (\mathrm {C}(K),\mathrm {C}(K)')\)-ergodic net \((T_{\alpha })_{\alpha \in A}\) for \(\mathcal {T}_S\) we obtain that \((T_{\alpha }f)_{\alpha \in A}\) converges weakly and thus pointwise on each orbit. But then \((T_{\alpha }f)_{\alpha \in A}\) converges pointwise on *K*. Combined with Lemma 5.13 this implies (i).

Finally suppose that (i) holds and take two elements \(Q_1,Q_2 \in \ker (\mathcal {K}_{\mathrm {c}}(K;S))\). Since \(\mathcal {K}_{\mathrm {c}}(K;S)\) is a Fréchet–Urysohn space, we find right ergodic sequences \((T_{i,n}')_{n \in \mathbb {N}}\subset \mathrm {co}\, \mathcal {T}_S'\) for \(\mathcal {T}_S'\) converging to \(Q_i\) for \(i=1,2\). Consider the sequences \((T_{i,n})_{n \in \mathbb {N}}\) consisting of the pre-adjoints and observe that for each \(f \in \mathrm {C}(K)\) we have \(\lim _{n \rightarrow {\infty }} T_{i,n}f = Q_i'f\) with respect to the \(\sigma (\mathrm {C}(K)'',\mathrm {C}(K)')\)-topology for \(i=1,2\). By assumption the sequence obtained by alternating the members of \((T_{1,n}f)_{n \in \mathbb {N}}\) and \((T_{2,n}f)_{n \in \mathbb {N}}\) converges pointwise and thus, by Lebegue’s Theorem, in the \(\sigma (\mathrm {C}(K)'',\mathrm {C}(K)')\)-topology. This yields \(Q_1 = Q_2\). \(\square \)

The next example shows that even for tame systems the weak* convergence of a single ergodic sequence does not ensure weak* mean ergodicity.

### Example 5.14

Consider the space \(\{0,1\}^{\mathbb {N}}\) with the product topology (which is compact and metrizable) and endow it with the shift \(\varphi \) given by

*K*;

*S*) with \(S{:}{=}\{(\varphi |_K)^n\mid n \in \mathbb {N}_0\}\). It is easy to see that

*K*is countable and thus \(\mathrm {E}(K;S) \subset K^K\) has cardinality at most \(\mathfrak {c}\). Therefore the system is tame by Theorem 1.2 of [14].

*K*;

*S*) is not weak* mean ergodic by Theorem 5.10.

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