The structure of mean equicontinuous group actions

We study mean equicontinuous actions of locally compact $\sigma$-compact amenable groups on compact metric spaces. In this setting, we establish the equivalence of mean equicontinuity and topo-isomorphy to the maximal equicontinuous factor and provide a characterization of mean equicontinuity of an action via properties of its product. This characterization enables us to show the equivalence of mean equicontinuity and the weaker notion of Besicovitch-mean equicontinuity in fairly high generality, including actions of abelian groups as well as minimal actions of general groups. In the minimal case, we further conclude that mean equicontinuity is equivalent to discrete spectrum with continuous eigenfunctions. Applications of our results yield a new class of non-abelian mean equicontinuous examples as well as a characterization of those extensions of mean equicontinuous actions which are still mean equicontinuous.


Introduction
Isometric actions on compact metric spaces constitute fundamental objects of study in the field of dynamical systems. In fact, despite possessing structurally simple dynamics, they relate to deep problems of general mathematical interest. Already rigid rotations on the circle have close connections to continued fraction expansions (see, for example, [Ser85]), the rich theory of discrepancy of sequences (see, e.g., [DT97] and references therein), or the Three Distance Problem and its versatile generalizations (see, for instance, [AB98]), to name but a few. With their dynamical simplicity on the one hand and the relevance of such problems on the other hand, it is natural to take actions by isometries as a point of departure in the endeavor to understand topological dynamical systems in general.
Actually, a substantial part of the abstract theory of topological dynamics can be understood as dealing with the following issue: given a general action which is not isometric, how close is this action to an isometric one? An essential tool in answering this question is the so-called maximal equicontinuous factor (or, topologically equivalent, the maximal isometric factor) of a given action. Now, with this canonical factor at hand, we may restate the above question in the following way: what is the regularity of the corresponding factor map?
Of course, various regularity features can be (and have been) considered. On the topological side, it is natural to investigate the existence of points where this factor map is one-to-one and this leads to the notion of almost automorphic actions [Vee65]. Once an invariant measure µ is given, one can also ask for injectivity of the factor map for almost all points with respect to µ and in many contexts this is referred to as regularity of the system.
With a more measure-theoretical flavor, we may study factor maps that establish a measure isomorphy with respect to all invariant measures and their push-forward on the maximal equicontinuous factor. This is the starting point of the current article and we present a comprehensive treatment of actions which allow for such factor maps. Our first main result gives a characterization of these actions in terms of a weakening of isometry known as mean equicontinuity (Theorem 1.1). Our subsequent results then unfold the notion of mean equicontinuity in terms of product systems (Theorem 1.2) and provide a spectral characterization of mean equicontinuity (Theorem 1.4) for minimal actions. A priori, the concept of mean equicontinuity comes in two variants, one known as Weyl-mean equicontinuity and the other as Besicovitch-mean equicontinuity. Along the way, we derive sufficient conditions for these two notions to agree (Theorem 1.3).
The concepts of Weyl-and Besicovitch-mean equicontinuity were introduced in [LTY15] for integer actions. In fact, in this case the notion of Besicovitch-mean equicontinuity is immediately seen to be equivalent to the concept of mean Lyapunov-stability which was already introduced in 1951 by Fomin [Fom51] in the context of Z-actions with discrete spectrum. Later, a first systematic treatment was carried out by Auslander [Aus59].
Our results tie in with various recent streaks of investigations: for Z-actions, there is the fundamental work of Downarowicz and Glasner on mean equicontinuity [DG16], providing a detailed study in the minimal case. Our results generalize these results from the group of integers to general locally compact σ-compact amenable groups. In the main structural characterization given in Theorem 1.1, we can also completely remove the minimality condition. Further, in our treatment of the relation between Weyl-and Besicovitch-mean equicontinuity, we can remove the minimality condition in many cases as well and thereby generalize [QZ18] which treats the case of general (that is, not necessarily minimal) Z-actions.
Concerning abelian groups, mean equicontinuity and its relation to the spectral theory of dynamical systems (in particular, to discrete spectrum) has been studied by various groups [GR17,Len16,GRM18]. Indeed, these works feature weaker versions of mean equicontinuity in order to characterize discrete spectrum. So, the restriction of our spectral result to the abelian case, given in Corollary 1.6, can be seen as a natural complement to these works. More specifically, our spectral characterization shows -in the minimal case-that mean equicontinuity is equivalent to unique ergodicity and discrete spectrum together with the continuity of eigenfunctions (see also [DG16] for the case of Z-actions).
Discrete spectrum is particularly relevant in the context of aperiodic order. This field has attracted substantial attention in the last decades due to the discovery of substances -later called quasicrystals-featuring this type of order (see the recent survey collection [KLS15] and the monograph [BG13] for background and further details). A basic quantity in the study of aperiodic order is the diffraction measure of an aperiodic configuration and a key task is to understand when the diffraction measure is a pure point measure. Due to a collective effort over the last twenty years, this turns out to be equivalent to discrete spectrum of an associated dynamical systems, see for instance [BL17] for a recent survey.
There is no axiomatic framework for aperiodic order (yet). However, typical systems studied in the context of aperiodic order have further regularity properties such as minimality and unique ergodicity. As discussed below (see Remark 6.4), one may argue that our spectral characterization shows that mean equicontinuous systems are the "right" systems to model minimal systems with aperiodic order.
Finally, we want to stress that our spectral characterization is not restricted to the abelian case but also covers the non-abelian case. It hence provides a spectral counterpart to the investigation of the diffraction measure for certain systems recently carried out by Björklund, Hartnick and Pogorzelski in [BHP17,BHP18].

Basic notation and definitions
(F n ) n∈N , called a (left) Følner sequence, of non-empty compact sets in G such that lim n→∞ m(gF n F n ) m(F n ) = 0 for all g ∈ G, where denotes the symmetric difference and m is a (left) Haar measure of G (we may synonymously write |F | for the Haar measure m(F ) of a measurable set F ⊆ G).
Since G acts by homeomorphisms, for each g ∈ G the map g : X x → gx is Borel bimeasurable. We call a Borel probability measure µ on X invariant under G (or G-invariant) if µ(A) = µ(gA) for every Borel measurable subset A ⊆ X and g ∈ G. We say a G-invariant measure µ is ergodic if all Borel sets A with µ(A gA) = 0 (g ∈ G) verify µ(A) = 0 or µ(A) = 1. It is well known that the amenability of G ensures the existence of a G-invariant measure for (X, G). Further, the set of invariant measures is convex and an invariant measure is ergodic if and only if it is an extremal point of the set of invariant measures. In particular, if (X, G) has a unique invariant measure, this measure is necessarily ergodic and (X, G) is referred to as uniquely ergodic. Finally, we call a closed invariant set A ⊆ X uniquely ergodic if (A, G) is uniquely ergodic. For further information of measure-theoretic properties of dynamical systems, see also [EW11].

Main results
Given a dynamical system (X, G) and a Følner sequence F = (F n ) n∈N , we call (X, G) Besicovitch-F-mean equicontinuous or just F-mean equicontinuous if for all ε > 0 there exists δ ε > 0 such that for all x, y ∈ X with d(x, y) < δ ε . The dependence on the Følner sequence immediately motivates the next definition which will also be the integral notion in this article. We say (X, G) is Weyl-mean equicontinuous or just mean equicontinuous if for all ε > 0 there is δ ε > 0 such that for all x, y ∈ X with d(x, y) < δ ε we have D(x, y) := sup{D F (x, y) | F is a Følner sequence} < ε.
Before we can proceed, a few comments are in order. First, note that D F and D are pseudometrics. Moreover, as is not hard to see, D is G-invariant, that is, D(gx, gy) = D(x, y) for all x, y ∈ X and g ∈ G (for the convenience of the reader, we provide a proof of this fact, see Proposition 3.12). Indeed, if G is abelian, it is immediately seen that (1) already defines a G-invariant pseudometric (simply for algebraic reasons) which simplifies many proofs for abelian G. In the non-abelian situation, this does not hold anymore in general. Yet, it turns out that under fairly general assumptions on (X, G) it actually is true if (X, G) is mean equicontinuous (see Theorem 1.3). It is an interesting observation that in this case, however, the reason behind the invariance of D F is not so much algebraic but ergodic in nature (see Section 5).
In the following main structural result, we will see that mean equicontinuity of a system (X, G) is intimately linked to a regularity property of the topological factor map π : X → T onto its maximal equicontinuous factor (T, G). For the definition of this regularity property, we need to introduce the following notion. Two probability spaces (X, B X , µ) and (Y, B Y , ν) are called isomorphic (mod 0) if there are measurable sets M ⊆ X and N ⊆ Y with µ(M ) = ν(N ) = 1 and a bi-measurable bijection h : M → N which is measure preserving, that is, In this case, we call h an isomorphism (mod 0) with respect to µ and ν. We also refer to an everywhere defined measurable map h : X → Y as an isomorphism (mod 0) with respect to µ and ν if h(x) = h (x) with x ∈ M for some h and M as above.
Suppose now that (X, G) is a topological extension of (Y, G) via a factor map h : X → Y and let µ be a G-invariant measure on X. We say (X, G) is a topo-isomorphic extension of (Y, G) with respect to µ if h is also an isomorphism with respect to µ and h(µ) where h(µ) denotes the push-forward of µ. In this case, we call h a topo-isomorphy with respect to µ. In case that no measure is specified, (X, G) is called a topo-isomorphic extension of (Y, G) and h a topo-isomorphy if h : X → Y is a topo-isomorphy with respect to every Ginvariant measure µ on X. Observe that the push-forward of an invariant measure µ under a topo-isomorphy is ergodic if and only if µ is ergodic.
Theorem 1.1 (Mean equicontinuity and topo-isomorphy). The topological dynamical system (X, G) is (Weyl-) mean equicontinuous if and only if it is a topo-isomorphic extension of its maximal equicontinuous factor (T, G).
Let us point out that the proof of this theorem also shows that the maximal equicontinuous factor of a mean equicontinuous system is in a natural sense the quotient of X by the pseudometric D (see the corresponding discussion in Section 3.3).
The concept of topo-isomorphy is at the interface of topological and measure-theoretical aspects of dynamical systems. This kind of "hybrid" notion was also recently studied by Downarowicz and Glasner in [DG16] where a similar statement to the above is proven for minimal dynamical systems with G = Z. We would like to mention that the direction from topo-isomorphy to mean equicontinuity (Theorem 3.7) is proven in a completely different way than in [DG16], while the proof that mean equicontinuity implies topo-isomorphy is close to the ones of [LTY15, Theorem 3.8] and [DG16, Proposition 2.5], see Section 3.
It is very worth noting that Theorem 1.1 is by far not only of abstract importance but actually offers a direct way to establish the mean equicontinuity of many well-known minimal group actions. To emphasize this, we briefly present a (non-exhaustive) list of minimal group actions where mean equicontinuity can always be derived by using the structural characterization provided in Theorem 1.1. Starting with Z-actions, two very common example classes which are well-known to be mean equicontinuous are Sturmian subshifts and regular Toeplitz subshifts, see for instance [Fog02,Ků03,Dow05] for further information and references. Non-symbolic examples can be found in the class of so-called Auslander systems (see [Aus88] and [HJ97]).
Before we go beyond Z-actions, we want to stress that minimal mean equicontinuous systems are always uniquely ergodic, see Corollary 1.5 (iii). To present the reader a nonminimal and moreover, intrinsically non-uniquely ergodic system, we provide a symbolic Z-action which has infinitely many ergodic measures in Example 5.11.
Concerning actions by more general groups, Theorem 1.1 also constitutes a basis for providing a novel and straightforward construction method for a class of non-abelian mini-mal mean equicontinuous systems (outlined in Subsection 7.1). Moreover, we can continue the list of examples from above: higher-dimensional subshifts, i.e., Z n -actions, which are mean equicontinuous can for instance be obtained from regular Toeplitz arrays, see [Cor06]. Furthermore, the theory of quasicrystals contains many natural examples of mean equicontinuous R n -actions, like the R 2 -actions obtained from Penrose tilings [Rob96] or the chair tiling [Rob99]. For more information concerning tilings and Delone sets in R n , see [BG13].
It is also possible to consider Delone sets (and canonical actions induced by them) in more general groups than R n . Especially, so-called regular model sets immediately yield mean equicontinuous group actions, see [Sch99] for locally compact abelian groups and [BHP18] for locally compact second countable groups. The latter reference, in particular, contains examples of the Heisenberg group acting in a mean equicontinuous way.
Finally, we would like to mention that minimal tame systems are always topo-isomorphic extensions of their maximal equicontinuous factor if the corresponding acting group is amenable (see [Gla18,Corollary 5.4 (2)] as well as the short discussion at end of Subsection 7.3). Systems belonging to this family are for instance Sturmian-like symbolic Z n -actions [GM18]. Moreover, in [FK18] it is shown that for each minimal equicontinuous action by an amenable group there exists a tame (and hence mean equicontinuous) extension which is not equicontinuous.
Our next main result gives a characterization of mean equicontinuity of a system in terms of its product system (see Section 4 for details). To the authors' knowledge, this result does not have a predecessor in any special situation. However, it is, of course, well in line with a plethora of results on characterizing properties of a dynamical system via properties of its product. We need the following notion: a system (X, G) is pointwise uniquely ergodic if the orbit closure Gx of every point x ∈ X is uniquely ergodic. For such systems we denote by µ x the unique ergodic measure supported on the orbit closure of x ∈ X.
Theorem 1.2 (Mean equicontinuity and the product system). The system (X, G) is mean equicontinuous if and only if • the product system (X × X, G) is pointwise uniquely ergodic • and the map (x, y) → µ (x,y) is continuous (with respect to the weak-*topology).
As the metric d is continuous on X × X, the previous theorem together with the standard result on the existence of averages of continuous functions for uniquely ergodic dynamical systems implies for mean equicontinuous systems that the lim sup in (1) is actually a limit and does not depend on the chosen Følner sequence, whence, in particular, it follows D = D F for any left Følner sequence F (see also Section 4 for a related discussion).
Moreover, the previous result allows us to derive the following theorem on the independence of Følner sequences. Theorem 1.3 (Mean equicontinuity and F-mean equicontinuity). Let (X, G) be a dynamical system and assume that • there is an invariant measure µ with full support, i.e. supp(µ) = X, or that • the group G is abelian.
Then (X, G) is mean equicontinuous if and only if (X, G) is F-mean equicontinuous for some left Følner sequence F.
Observe that if (X, G) is minimal, the extra assumption of a measure with full support is evidently fulfilled. It is noteworthy that the extra effort needed to overcome the lack of commutativity in this work is most visible in the proof of the above statement. We would also like to remark that in the recent article [QZ18], a similar statement has independently (and by different means) been proven to hold if G = Z. Under the assumption of a minimal Z-action, it is known due to [DG16].
In the minimal case we can provide another characterization of mean equicontinuity. This is a characterization in terms of spectral theory, or more specifically, in terms of a decomposition of the space L 2 (X, µ). The corresponding proof can be found in Section 6.
Theorem 1.4 (Mean equicontinuity and spectral theory). Assume (X, G) is minimal. Then (X, G) is mean equicontinuous if and only if (X, G) has a unique invariant measure µ and L 2 (X, µ) can be written as an orthogonal sum of finite dimensional, G-invariant subspaces consisting of continuous functions. Now, for minimal systems we may combine all the previous theorems to obtain a slightly simplified list of equivalent characterizations of mean equicontinuity (note that the statements (i)-(iii) are a generalization of Theorem 2.1 in [DG16] which is treating Z-actions).
Corollary 1.5. Let (X, G) be a minimal system. Then the following are equivalent: (iii) (X, G) is uniquely ergodic and topo-isomorphic to its MEF with respect to its unique invariant measure µ.
(v) (X, G) has a unique invariant measure µ and L 2 (X, µ) can be written as an orthogonal sum of finite dimensional, G-invariant subspaces consisting of continuous functions.
We finish this section with a discussion of the spectral characterization of mean equicontinuity when G is abelian. This case is particularly important due to its relevance for the study of aperiodic order. Let G be the dual group of G, i.e., the group of all continuous group homomorphisms from G to the unit circle and let (X, G) be a dynamical system with an invariant probability measure µ. Then f ∈ L 2 (X, µ) with f = 0 is called an eigenfunction to the eigenvalue ξ ∈ G if f (g·) = ξ(g)f (·) for all g ∈ G. Here, the equality is understood in the sense of L 2 functions. If such an f is continuous with for all x ∈ X and g ∈ G it is called a continuous eigenfunction. The dynamical system (X, G) with G-invariant measure µ is said to have discrete spectrum with continuous eigenfunctions if there exists an orthonormal basis for L 2 (X, µ) of continuous eigenfunctions.
Corollary 1.6. Let G be abelian. A minimal system (X, G) is mean equicontinuous if and only if it is uniquely ergodic and has discrete spectrum with continuous eigenfunctions.
For subshifts associated to non-periodic primitive substitutions, a classical result by Host [Hos86] states that all eigenvalues possess a continuous eigenfunction. Hence, the previous corollary implies that these subshifts are mean equicontinuous if and only if they have pure point spectrum (since they are always minimal). This yields, for instance, that the subshifts associated to the Fibonacci and Tribonacci substitution are mean equicontinuous. For more information, see for instance [Fog02] and [Que10]. Further, a generalization of Host's result to primitive tiling substitutions of R n with finite local complexity can be found in [Sol07].

Acknowledgments
This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 750865. Furthermore, MG acknowledges support by the DFG grants JA 1721/2-1 and GR 4899/1-1. Moreover, the authors would also like to thank Dominik Kwietniak for bringing the example depicted in Figure 1 to their attention.

Some basic preliminaries on ergodic theory
In this section, we discuss some definitions and statements of the ergodic theory of general actions by locally compact σ-compact amenable groups. In particular, we will be concerned with averages along Følner sequences where we pay special attention to an exposition which only requires a very fundamental set of tools. In particular, we will only make use of the Mean Ergodic Theorem in the following and avoid the more sophisticated Pointwise Ergodic Theorem by Lindenstrauss [Lin01].
In order to provide an alternative characterization of topo-isomorphic extensions, let us make the following classical measure-theoretic observation whose proof is provided for the convenience of the reader.
Proposition 2.1. Suppose X and Y are compact metric spaces, µ is a Borel probability measure on X and h : X → Y is measurable. Then, the operator is unitary if and only if h is an isomorphism mod 0 with respect to µ and h(µ).
Proof. We only show that unitarity implies isomorphy (mod 0), the "if"-part is obvious. First, we have to fix some notation. For a compact metric space (Z, d) we denote by B(Z) the Borel σ-algebra and byB(Z) the associated measure algebra (see, for example, [Wal82] for the notion of measure algebras). Since U µ : , and coincides with h on M . This proves the statement.
Recall that any locally compact group G admits a left (right) Haar measure (defined uniquely up to a positive multiplicative constant) denoted by m (m r ) which is left (right) invariant, that is, for all ϕ ∈ L 1 (G, m) and g ∈ G we have ϕ(gs)dm(s) = ϕ(s)dm(s) ( ϕ(sg)dm r (s) = ϕ(s)dm r (s)), where L 1 (G, m) is the space of all Haar integrable functions on G. Note that from time to time we will also refer to the left/right Haar measure by using the notation |·| if there is no risk of ambiguity.
In the introduction we have already encountered the notion of a left Følner sequence (F n ) n∈N in G consisting of non-empty compact sets in G such that (3) There are also right Følner sequences which fulfill an analogue condition to (3) where the left Haar measure and the multiplication from the left is replaced by the right Haar measure and multiplication from the right, respectively. From now on, the standard assumption is that we deal with left Haar measures and left Følner sequences if not stated otherwise. Let (X, d) be a compact metric space. By C(X) we denote the set of all complex-valued continuous functions on X equipped with the uniform topology which is induced by the sup norm · ∞ . Given a Borel probability measure µ on X and ϕ ∈ C(X), we set µ(ϕ) = ϕ dµ.
The next theorem is well known for Z-actions and can be proven for the group actions considered in this article by adapting the corresponding arguments from [Wal82] and [Fur81], see also the short discussion regarding Theorem 2.16 in [MR13].
Theorem 2.2. Let (X, G) be a dynamical system. The following statements are equivalent: Further, if one of the above conditions hold, then the convergence in (ii) is uniform in x ∈ X, independent of the left Følner sequence (F n ) n∈N , and we have c = µ(ϕ).
For the sake of completeness, we provide a proof of the next statement.
Proposition 2.3. Let (X, G) be a dynamical system. Suppose for each ϕ ∈ C(X) there is a right Følner sequence (F n ) n∈N and a constant c ∈ R with for all x ∈ X. Then (X, G) has a unique G-invariant measure µ and µ(ϕ) = c.
Proof. As mentioned in the introduction, (X, G) allows for a G-invariant measure µ on X. Now, using Fubini and dominated convergence, we have Since finite Borel measures on compact metric spaces are uniquely determined by integrating continuous functions, we obtain that µ is the only G-invariant measure on X.
Throughout this work, we will encounter Birkhoff averages of continuous functions, i.e., limits of the above kind, at several places. For that reason, we introduce the following notation: given a left Følner sequence F and a continuous function ϕ on X, we set for x ∈ X and n ∈ N. Furthermore, we introduce the following functions on X We simply write A(F, ϕ)(x) for the above limits, provided they coincide (as in the previous statements). If F is a right Følner sequence, we refer to the analogous quantities (where the left Haar measure is replaced by the right Haar measure) by the same symbols. For a dynamical system (X, G) with an ergodic measure µ and a left Følner sequence F in G, we say a point x ∈ X is (µ-)generic with respect to F if for every continuous function ϕ on X the limit A(F, ϕ)(x) exists and equals µ(ϕ). It is worth noting and easy to see that every µ-generic point has a dense orbit in the support of µ. For the purpose of being self-contained, we provide a proof of the next well-known statement. Note that a direct consequence of this statement is the well-known singularity of ergodic measures.
Theorem 2.4. Let (X, G) be a topological dynamical system with an ergodic measure µ. Then every left Følner sequence F = (F n ) n∈N allows for a subsequence F = (F n ) n∈N with respect to which µ-almost every point is generic.
Proof. Let F be a left Følner sequence and (ϕ ) ∈N be a dense sequence in C(X) (which exists due to Stone-Weierstrass). By the Mean Ergodic Theorem, for all x in a full measure set X ϕ1 ⊆ X. By inductively repeating the above argument, we get that for for all ∈ N and x ∈ X C(X) . Note that for every fixed x ∈ X we have that A(F , ϕ)(x), A(F , ϕ)(x), and µ(ϕ) depend continuously on ϕ ∈ C(X). Altogether, we thus have for every ϕ ∈ C(X) and every x ∈ X C(X) that We will need the following auxiliary statement which is immediately linked to the ergodic representation of invariant measures, see for instance [Far62] for more information.

Topo-isomorphic extensions
In the following we establish the equivalence of (Weyl-) mean equicontinuity and topoisomorphy and thus prove our main structural result (Theorem 1.1). To that end, we first gather some basics on topo-isomorphic extensions in Subsection 3.1. Then Theorem 3.7 (in Subsection 3.2) yields one direction of the main theorem. Theorem 3.14 (in Subsection 3.3) yields the other direction.
Theorem 1.1 naturally suggests to also look at the relation between mean equicontinuous systems and their topo-isomorphic extensions. We show that the preservation of the maximal equicontinuous factor is a characteristic of such extensions (see Subsection 3.4).

Basics on topo-isomorphic extensions
In this section we explain the structure of topo-isomorphic extensions over equicontinuous systems. Roughly speaking, such systems are partitioned into uniquely ergodic components and this will be relevant in our considerations hereafter.
Proposition 3.1. Suppose (X, G) is a topo-isomorphic extension of (Y, G) via the factor map h : X → Y . If µ 1 and µ 2 are two distinct ergodic G-invariant measures on X, then the image measures h(µ 1 ) and h(µ 2 ) differ as well.
We will make use of the following classical lemma (see, for example, [Aus88]) which gives that the notions of transitivity and minimality coincide for equicontinuous systems.
Lemma 3.2. If (X, G) is equicontinuous, then for each x ∈ X we have that Gx is minimal.
Regarding the next statements, see also Theorem 14 (Decomposition Theorem) in [Aus59] for the case of Z-actions.
Proposition 3.3. Assume that (X, G) is a topo-isomorphic extension of an equicontinuous system (Y, G) with factor map h.
Proof. (a) Since factor maps preserve transitivity, this follows from Lemma 3.2.
is equicontinuous, minimal subsets are uniquely ergodic (this classical fact also follows from Theorem 5.6 below). Hence, h maps every invariant measure on A to the same invariant measure on h(A). By Proposition 3.1, A is uniquely ergodic.
Conversely, suppose A is uniquely ergodic. As any orbit closure carries an invariant measure, Gx and Gy have a non-empty intersection for Theorem 3.4. Assume that (X, G) is a topo-isomorphic extension of an equicontinuous system (Y, G) with factor map h : X → Y . Then the following statements are true.
is not possible. To that end, assume the contrary.
) is a closed invariant subset of X that contains A µ and A ν . Hence, it is not uniquely ergodic. This contradicts the previous proposition.
As a consequence of the preceding theorem we can decompose topo-isomorphic extensions of equicontinuous systems into uniquely ergodic components. Let us introduce the following notation: whenever (X, G) is a dynamical system and µ an ergodic measure, X µ denotes the set of all x ∈ X whose orbit closure Gx supports µ and no other invariant measure. In other words, X µ comprises the set of all points which are µ-generic with respect to each Følner sequence.
Corollary 3.5. Let (X, G) be a topo-isomorphic extension of an equicontinuous system (Y, G). Then the sets X µ partition X, that is X = µ X µ , where µ runs over all ergodic measures on X. Further, each X µ is the preimage of a minimal subset of Y and any such preimage coincides with an X µ .
Proof. According to the previous theorem, (X, G) is pointwise uniquely ergodic which immediately gives that the sets X µ partition X.
For the second part, Lemma 3.2 yields that it suffices to show that to each minimal set M ⊆ Y there is a unique ergodic measure µ on X such that h −1 (M ) = X µ , with h the factor map from X to Y . Proposition 3.3 (b) yields that there is a unique ergodic measure µ on X with h −1 (M ) ⊆ X µ . Clearly, for any x ∈ X whose orbit closure supports µ, we must have Corollary 3.6. Suppose (X, d) is a compact, connected metric space and (X, G) is mean equicontinuous. Then (X, G) has either a unique ergodic measure (minimal set) or uncountably many ergodic measures (minimal sets).
Proof. Recall that the support of an ergodic measure is always transitive (due to the generic points) and that every minimal set supports an ergodic measure. Due to the pointwise unique ergodicity of mean equicontinuous systems (see Theorem 3.14 and Theorem 3.4 (a)), this implies that there is a one-to-one correspondence between minimal sets and ergodic measures. Hence, it suffices to show the statement for ergodic measures.
Due to Corollary 3.5, we have a bijection between the ergodic measures of (X, G) and the minimal sets of its maximal equicontinuous factor (T, G). By Lemma 3.2, T allows for a partition by minimal sets which are clearly compact and pairwise disjoint. Since T is connected (as the continuous image of X), a classical result by Sierpinski [Sie18] yields that such a partition consists of either one or uncountably many partition elements.

Topo-isomorphy implies mean equicontinuity
Now, we show one direction of our main structural result. To that end, we make use of Proposition 2.1 and rephrase the assertion that topo-isomorphy implies mean equicontinuity as follows.
Theorem 3.7. Let (X, G) be a dynamical system and (T , G) an equicontinuous factor with factor map π such that for every G-invariant measure µ the operator U µ : L 2 (T , π(µ)) → L 2 (X, µ) : f → f • π is unitary. Then (X, G) is mean equicontinuous and (T , G) is the associated MEF.
Before we can turn to the proof of Theorem 3.7, we need two further ingredients. The first ingredient is another characterization of mean equicontinuity which makes use of the continuous functions on X. For that purpose we define the pseudometric D f associated to a function f ∈ C(X) by The following statement is well known, see [DI88, Proposition 1] and [GRM17, Theorem 2.14]. We include a proof for the convenience of the reader. Proof. It is not hard to see that the topology generated on X by D as well as the mean equicontinuity of (X, G) is independent of the particular choice of the metric d (provided d generates the original topology on X). This will be used throughout the proof. y). This implies D f ≤ D. As D is continuous, this implies (ii).
(ii)⇒(i): Choose a sequence (f n ) n∈N of continuous functions on X which separate points and satisfy f n ∞ ≤ 1 for all n ∈ N. Then for any c n > 0 with n c n < ∞ we have that n c n |f n (x) − f n (y)| (4) defines a metric equivalent to d. We can hence assume w.l.o.g. that d is given by (4). Now, clearly D ≤ c n D fn =: D, where D is continuous by (ii) and the summability of (c n ) n∈N .
The last statement has been shown along the proof.
Remark 3.9. The above shows that the topology on X generated by D agrees with the topology generated by the collection of D f 's with f ∈ C(X).
The other ingredient needed for the proof of Theorem 3.7 -and in some sense the main insight of the present section-is the following lemma.
Proof. By (a) of Theorem 3.4 the orbit closure of x i (i = 1, 2) supports a unique ergodic measure µ i . W.l.o.g. we may assume that µ 1 = µ 2 (if µ 1 = µ 2 , the following argument works in an analogous and slightly simplified way). By unitarity of the U µi 's and denseness of continuous functions in L 2 (T , π (µ i )), we can find g i ∈ C(T ) with By Cauchy-Schwarz, we then obtain Set M i = π Gx i . Then M 1 and M 2 are disjoint by Theorem 3.4 (b) since µ 1 = µ 2 . Let S i (i = 1, 2) be continuous functions on T with S i | Mi = 1 and S 1 | M2 = S 2 | M1 = 0. Set g = S 1 g 1 + S 2 g 2 . Now, for any t ∈ G we have Consequently, we obtain We show that all three terms become small for x 1 sufficiently close to x 2 . By unique ergodicity on orbit closures and Theorem 2.2, we obtain The term D g•π can be treated as follows. If π (x 1 ) is close to π (x 2 ), we obtain that tπ (x 1 ) is close to tπ (x 2 ) for all t ∈ G (by equicontinuity). As g is continuous (and hence uniformly continuous) on T , this implies that g •π (tx 1 ) = g(tπ (x 1 )) is close to g •π (tx 2 ) = g(tπ (x 2 )) for all t ∈ G and we are done.
Proof of Theorem 3.7. We first show that (X, G) is mean equicontinuous. By Proposition 3.8, it suffices to show that D f is continuous for any f ∈ C(X). Let such an f be given and consider an arbitrary ε > 0. We have to show that if x 1 , x 2 ∈ X are close, then D f (x 1 , x 2 ) < ε. This, however, is clear from Lemma 3.10 as for x 1 close to x 2 we clearly have π (x 1 ) close to π (x 2 ) due to the continuity of π .
It remains to show that (T , G) is the MEF. As discussed in Subsection 1.1, it suffices to show that inf t∈G d(tx, ty) = 0 whenever π (x) = π (y). Now, π (x) = π (y) implies D f (x, y) = 0 for all continuous f on X (by Lemma 3.10) and hence D(x, y) = 0 due to Proposition 3.8. From this and the definition of D we easily find inf t∈G d(tx, ty) = 0.
Remark 3.11. We defined a topo-isomorphy h to be a topological factor map which is an isomorphism mod 0 with respect to µ and h(µ) for every invariant measure µ. It is natural to ask whether Theorem 3.7 still remains true if we relax the assumptions on h by considering h to be a factor map which is only an isomorphism mod 0 with respect to µ and h(µ) for every ergodic measure µ. In the proof of Lemma 3.10, the topo-isomorphy with respect to every invariant measure was (implicitly) used twice: once, to ensure pointwise unique ergodicity and once, to ensure that the supports of two distinct ergodic measures have disjoint images under h (see also Theorem 3.4). In fact, the latter implies the former and hence implies mean equicontinuity if we additionally assume h to be a topo-isomorphy onto an equicontinuous factor with respect to every ergodic measure. However, it is not true that pointwise unique ergodicity and topo-isomorphy to an equicontinuous factor with respect to ergodic measures only yields that the supports of distinct ergodic measures have distinct images, as can be seen in Figure 1.

Mean equicontinuity implies topo-isomorphy
In this section we establish that mean equicontinuity implies topo-isomorphy of the dynamical system to its MEF. Together with Theorem 3.7 from the previous subsection, this proves our main structural result Theorem 1.1. We first note that D is actually G-invariant.
Proposition 3.12 (Invariance of D). Let (X, G) be a dynamical system. Then D satisfies D(tx, ty) = D(x, y) for all x, y ∈ X and t ∈ G.
Proof. Recall that there exists a unique ∆ : G → (0, ∞) (called modular function) whose defining property is that for all Haar measurable h : G → [0, ∞). A short computation and canceling of modular functions then gives that for all s ∈ G and all Haar measurable bounded g : G → [0, ∞) whenever F is a compact subset of G with positive Haar measure. This shows that where Fs denotes the sequence (F n s) n∈N . Now, (F n s) n∈N is clearly a Følner sequence as well. Hence, the desired statement follows as the definition of D involves all Følner sequences. Let a dynamical system (X, G) be given. For x, y ∈ X write x∼ y if D(x, y) = 0. If (X, G) is mean equicontinuous, then clearly the quotient map β : X → X/∼ is continuous. By the invariance of D due to Proposition 3.12, the action of G on X/∼ given by gβ(x) := β(gx) is well defined and isometric. Hence, (X/∼, G) is an equicontinuous factor of (X, G).
Proof. As discussed in Section 1.1 it suffices to show that inf t∈G d(tx, ty) = 0 whenever β(x) = β(y). This, however, is clear.
Proof. Fix a G-invariant measure µ and let µ = µ z dν(z) be the disintegration of µ over its image measure ν := π(µ) (see, e.g., [Fur81]). We consider the relative product measure µ × ν µ supported in the relative product of X over T Recall that µ × ν µ is invariant under the action of G on X × T X given by g(x, y) := (gx, gy) for each (x, y) ∈ X × T X and g ∈ G, see Proposition 5.14 in [Fur81].
We claim that µ× ν µ is only supported on the diagonal {(x, x) ∈ X×X | x ∈ X} ⊆ X× T X. For a contradiction assume this is not the case. Then there exists an open set A in X × T X which has a positive distance to the diagonal and fulfills (µ × ν µ)(A) > 0. Using Lemma 2.5, this yields that there is an ergodic measureμ on X × T X withμ(A) > 0. According to Theorem 2.4,μ-almost every point isμ-generic with respect to some Følner sequence F. Now, for every such (x, y) ∈ X × T X we have This is in contradiction to the previous proposition because π(x) = π(y) ⇔ D(x, y) = 0 in case that (X, G) is mean equicontinuous. Now, observe that the only measures supported in π −1 (z) whose Cartesian squares are supported in the diagonal of X × T X are delta measures. Thus, µ z is a delta measure for ν-almost every z ∈ T. Finally, the map which assigns to each z the support of µ z is an isomorphism with respect to ν and µ whose inverse coincides with π for µ-a.e. point.

Further properties and first non-minimal examples
Here, we discuss first consequences of the results of the previous subsections. In particular, we show that the preservation of the maximal equicontinuous factor is a characteristic feature of topo-isomorphic extensions of mean equicontinuous systems. Furthermore, we discuss some examples of non-minimal mean equicontinuous systems for G = Z.
Theorem 3.15 (Characterization of mean equicontinuous extensions). Let (X, G) be an extension of a mean equicontinuous system (Y, G). Then, (X, G) is topo-isomorphic to (Y, G) if and only if it is mean equicontinuous and its MEF agrees with that of (Y, G).
Proof. Assume first that (X, G) is a topo-isomorphic extension of (Y, G). By Theorem 3.14, (Y, G) is a topo-isomorphic extension of its MEF (T, G). Clearly, (X, G) is also a topoisomorphic extension of (T, G). The statement now follows from Theorem 3.7.
Consider now the situation that (X, G) is mean equicontinuous and the MEFs of (Y, G) and (X, G) agree. Let h be the factor map from X to Y and π a factor map from Y to T (the MEF of both systems). Note that both π and π • h : X → T are topo-isomorphies, according to Theorem 3.14. This implies that h is a topo-isomorphy, too.
Corollary 3.16. If two equicontinuous dynamical systems (X, G) and (Y, G) are topoisomorphic, then they are in fact topological conjugate.
Proof. By the previous theorem such systems share the same MEF. By equicontinuity, however, they agree with their MEF.
Remark 3.17. The corollary is reminiscent of the rigidity phenomenon which is well known for ergodic abelian equicontinuous group actions, see for instance [FK02].
In order to state another consequence of Theorem 3.15 we need the following observation.
Proposition 3.18. Any topological factor of a mean equicontinuous system is mean equicontinuous as well.
Proof. Let (Y, G) be a factor of a mean equicontinuous system (X, G) with factor map h. By Proposition 3.8, it suffices to show that D f is continuous for any continuous f on Y . As the factor map h is a continuous surjective map between compact spaces, continuity of D f is equivalent to continuity of D f • h = D f •h and the statement follows from a further application of Proposition 3.8.
Given this proposition, Theorem 3.15 has the following immediate consequence (systems fitting into the setting of the following statement can be found in [DD02, Section 5]).
Corollary 3.19. If (X, G) is mean equicontinuous and an extension of (Y, G) with the same MEF, then (X, G) is a topo-isomorphic extension of (Y, G).
Much of the previous work on mean equicontinuity is concerned with minimal Z-actions. Therefore, we would like to close this section with a discussion of two simple kinds of examples of well-known systems which are mean equicontinuous but not minimal. The first example will be still transitive (in fact, as we will see, all but one point have a dense orbit) and the second kind of examples will have no dense orbits but will still be uniquely ergodic. For a non-uniquely ergodic system, see Example 5.11. For a general introduction to substitution systems, see for example [Ků03]. There are two infinite sequences in {0, 1} N which are invariant with respect to the Cantor substitution: the constant sequence (111 . . .) and the sequence ω obtained by applying the substitution successively to the letter 0 and its images, i.e., 0 → 010 → 010111010 → 010111010111111111010111010 → · · · . Now, there is a standard method to obtain a two-sided subshift (Σ ω , σ) from ω, see for instance [Ků03, Proposition 3.71]. That is, Σ ω is a closed subset of {0, 1} Z (equipped with the product topology) which is invariant under the action of the left shift σ : {0, 1} Z → {0, 1} Z . By making use of the concrete structure of ω, it is not difficult to see that all points in Σ ω , except the constant sequence (. . . 111 . . .), have a dense orbit and that the letter 0 occurs with zero density in each sequence of Σ ω . The former implies that Σ ω is uncountable and the latter that (Σ ω , σ) is uniquely ergodic, with the unique invariant measure the delta measure supported on (. . . 111 . . .). Thus (Σ ω , σ) is a non-trivial topo-isomorphic extension of its trivial MEF and hence, mean equicontinuous.
Example 3.21. By a classical result of Denjoy [Den32], there exist examples of C 1 circle diffeomorphisms which have a rigid rotation (S 1 , R α ), with α ∈ R irrational, as a factor but are not conjugate to it. Herman [Her79] showed later that these examples can even be made C 1+ε for any ε < 1. We will refer to these kind of systems as Denjoy examples.
All Denjoy examples have a unique minimal set C ⊂ S 1 and a unique invariant measure µ supported on C. We claim that any Denjoy system (S 1 , f ) is mean equicontinuous because of the following reason. Since the factor map π : S 1 → S 1 , extending (S 1 , R α ) to (S 1 , f ), is monotone, we have that π −1 (θ) for θ ∈ S 1 is either a singleton or an interval. This immediately implies that the set of non-invertible points {θ ∈ S 1 : #π −1 (θ) > 1} is countable. Accordingly, we get that h is invertible on a full measure set with respect to µ (since h(µ) is the Lebesgue measure on S 1 , the unique invariant measure of R α ).
McSwiggen has shown that there are Denjoy homeomorphisms on higher-dimensional tori that share the same properties just mentioned, in particular, that the set of non-invertible points is countable. This means these systems are mean equicontinuous, too. For examples on the two-torus, see [McS93] as well as [NV94,NS96] for more information concerning these systems. For examples defined on general k-tori, k ≥ 2, see [McS95].

Mean equicontinuity via product systems
In Theorem 3.4 we have seen that any mean equicontinuous system is pointwise uniquely ergodic. Here, we show that pointwise unique ergodicity of the product system together with a continuity property is an equivalent characterization of mean equicontinuity.
Proof. Similarly as in the proof of Proposition 3.8 (i)⇒(ii), we see that for each ε > 0 there is δ > 0 such that whenever d(x, y) < δ. This immediately gives the continuity of A(F, ϕ) and A(F, ϕ).
In the following, if (X, G) is pointwise uniquely ergodic, then the map x → µ x from X into the space of all Borel probability measures on X (equipped with the weak-*topology) is defined to send each x ∈ X to the unique G-invariant measure µ x supported on Gx.
Theorem 4.2. For a system (X, G) the following conditions are equivalent: (i) (X, G) is mean equicontinuous.
Proof. First, assume that (X, G) is mean equicontinuous. This easily implies that the product system (X × X, G) is also mean equicontinuous. According to Theorem 3.4 (a), this in turn yields that (X × X, G) is pointwise uniquely ergodic. Now, consider some left Følner sequence F. From the previous proposition we have that for every ϕ ∈ C(X × X) the functions A(F, ϕ)(·) and A(F, ϕ)(·) are continuous on X × X. Hence, using Theorem 2.2, for a sequence of points (x n , y n ) ∈ X × X converging to (x, y) as n → ∞ we have Since ϕ ∈ C(X × X) was arbitrary, µ (xn,yn) converges weakly to µ (x,y) as n → ∞ .
Let us conclude with a few comments on the natural question of why we have to formulate the assumptions of Theorem 4.2 (ii) for the product system. Obviously, a system is automatically pointwise uniquely ergodic if its product system has this property (if additionally (x, y) → µ (x,y) is continuous, then x → µ x is continuous as well). However, the converse is not true. For example, the product of a uniquely ergodic weakly mixing system with itself is ergodic with respect to the product measure. Hence, there are points whose orbit is dense in the full product and hence supports the product measure as well as the diagonal measure.
Furthermore, the next example shows that pointwise unique ergodicity of the product system (and hence, of the original system) and continuous dependence of the map x → µ x does not imply continuity of the map (x, y) → µ (x,y) . Example 4.3. Let C ⊆ S 1 be a Cantor set which does not contain rationals 1 and consider the skew-product F : C × S 1 → C × S 1 : (x, θ) → (x, θ + x). Clearly, the corresponding Z-action is pointwise uniquely ergodic (the unique invariant measure supported on the orbit closure of (x, θ) is given by µ (x,θ) = δ x × m S 1 ) and the map (x, θ) → µ (x,θ) is continuous. Furthermore, the product system is topologically conjugate tô and still pointwise uniquely ergodic. However, the map (x 1 , x 2 , θ 1 , θ 2 ) → µ (x1,x2,θ1,θ2) cannot be continuous as this would imply mean equicontinuity (due to Theorem 4.2) while the MEF of (C × S 1 , F ) is trivial so that the corresponding factor map is not a topo-isomorphy.

Relating Besicovitch-and Weyl-mean equicontinuity
By its very definition, Weyl-mean equicontinuity is a stronger assumption than Besicovitch-F-mean equicontinuity. Quite remarkably, it turns out that Besicovitch-F-mean equicontinuity, i.e., control over one Følner sequence F suffices to conclude Weyl-mean equicontinuity in many situations. A detailed study is given in this section and the presented results yield a proof of Theorem 1.3. By means of this result, we provide a non-trivial non-uniquely ergodic mean equicontinuous systems at the end of this section.
In the following, we will also speak of F-mean equicontinuity with respect to a right Følner sequence F, where the definition is completely analogous to the definition using left Følner sequences given in (1).
Theorem 5.1. Let (X, G) be F-mean equicontinuous for some left Følner sequence F and let there be a G-invariant measure µ with supp(µ) = X. Then (X, G) is mean equicontinuous.
Remark 5.2. A comment on the assumption supp(µ) = X may be in order. As is well known, every dynamical system (X, G) possesses a G-invariant measure µ of maximal support, that is, a measure µ such that supp(µ) contains the support of any other G-invariant measure. This support is clearly unique and coincides with the closure of the union of all supports of ergodic measures. While in general, supp(µ) may not fill the whole space X, we can, of course, restrict attention to the maximal support and then apply the above theorem. By the Poincaré Recurrence Theorem, one may think of this as restricting to the recurrent dynamics of the system (X, G).
Recall that for minimal dynamical systems every invariant measure has full support.
Corollary 5.3. If (X, G) is minimal, then F-mean equicontinuity for some left Følner sequence F implies mean equicontinuity.
Next we will collect some further assertions needed for the proof of Theorem 5.1. The following elementary lemma makes up for the (possible) lack of separability of G. Recall that G is assumed to be σ-compact.
Lemma 5.4. Let (X, G) be a dynamical system. Then there exists a countable subgroup T ≤ G such that T x = Gx for every x ∈ X.
Proof. Since G is σ-compact, there exists an exhausting sequence (K n ) n∈N of compact subsets of G. Given ε > 0, set T n ⊆ K n to be a finite subset such that for each s ∈ K n there is t ∈ T n with sup x∈X d(sx, tx) < ε. Note that T n is well defined due to the continuity of the defining action of (X, G) as well as the compactness of K n and X. Set T ε := n∈N T n . Then T := n∈N T 1/n is countable and verifies T x = Gx for every x ∈ X. Letting T be the group generated by T proves the statement.
Proposition 5.5. Suppose (X, G) is F-mean equicontinuous with respect to some left Følner sequence F. Then the support of each ergodic measure µ is uniquely ergodic.
Proof. By possibly restricting to the support of µ, we may assume without loss of generality that X = supp(µ). By possibly going over to a subsequence of F, we may further assume without loss of generality that there is a full measure set X µ of µ-generic points with respect to F (see Theorem 2.4).
From Proposition 4.1 we know that for each ϕ ∈ C(X) the maps A(F, ϕ)(·) and A(F, ϕ)(·) are continuous. Hence, with T as in Lemma 5.4 and x 0 ∈ t∈T tX µ we have that for all x from the set T x 0 ⊆ X µ . Note that T x 0 is dense because of Lemma 5.4 and the fact that µ-generic points are transitive. By the continuity of A(F, ϕ)(·) and A(F, ϕ)(·), we get that A(F, ϕ)(x), in fact, exists and coincides with µ(ϕ) for all x ∈ X. As ϕ ∈ C(X) was arbitrary, Theorem 2.2 yields the unique ergodicity of (X, G).
Theorem 5.6. Suppose (X, G) is F-mean equicontinuous with respect to some left Følner sequence F. Consider a point x ∈ supp(µ) where µ is an arbitrary G-invariant measure. Then the orbit closure Gx is uniquely ergodic.
Proof. By Theorem 2.2, it suffices to show that A(F, ϕ)(·) exists and is constant on Gx for each ϕ ∈ C Gx . In fact, by Tietze's Extension Theorem, it is enough to consider ϕ ∈ C(X). Observe that by Lemma 2.5 there is a sequence (x n ) n∈N in X with x n → x for n → ∞ such that each x n lies in the support of an ergodic measure. By Proposition 4.1, the functions A(F, ϕ)(·) and A(F, ϕ)(·) and hence, A(F, ϕ)(g ·) and A(F, ϕ)(g ·) are continuous for every g ∈ G, so that where we used the unique ergodicity on ergodic components (Proposition 5.5) in the second equality. This proves equality of A(F, ϕ)(·) and A(F, ϕ)(·) on Gx. Similarly, we see that A(F, ϕ)(·) and A(F, ϕ)(·) are constant on Gx. As both functions are continuous, this shows that A(F, ϕ)(·) exists and is constant on Gx for each ϕ ∈ C(X).
Proof of Theorem 5.1. By Theorem 4.2, it suffices to show that (X × X, G) is pointwise uniquely ergodic and that the map (x, y) → µ (x,y) is continuous. To that end, we first note that with (X, G) the product system (X × X, G) is F-mean equicontinuous as well. Moreover, by the assumptions, the measure µ × µ has full support on X × X which implies that (X × X, G) is pointwise uniquely ergodic, by Theorem 5.6.
It remains to show the continuity of the map (x, y) → µ (x,y) . By pointwise unique ergodicity and Theorem 2.2, we have for any ϕ ∈ C(X × X) that Hence, the continuity follows from Proposition 4.1 applied to (X × X, G).
In Theorem 5.1 we had to assume full support of the measure to deduce mean equicontinuity from F-mean equicontinuity for some left Følner sequence F. This is not needed if we know that a system is F-mean equicontinuous for a right Følner sequence F. Details are discussed next.
Proposition 5.7. Let F = (F n ) n∈N be a right Følner sequence so that (X, G) is F-mean equicontinuous. If (X, G) is transitive, then it has a unique G-invariant measure µ and there is a subsequence F = (F n ) n∈N of F such that for each ϕ ∈ C(X).
Proof. Given ϕ ∈ C(X), we know by Proposition 4.1 that the maps A(F, ϕ)(·) and A(F, ϕ)(·) are continuous. Moreover, as F is a right Følner sequence, A(F, ϕ) and A(F, ϕ) are invariant and hence-due to the transitivity of (X, G)-constant. Now, by the Stone-Weierstrass Theorem, C(X) is separable so that there exists a dense sequence of functions (ϕ n ) n∈N in C(X). Observe that there is a subsequence F 1 of F with A(F 1 , ϕ 1 ) = A(F 1 , ϕ 1 ). Recursively, we obtain a subsequence F n+1 of F n with A(F n+1 , ϕ n+1 ) = A(F n+1 , ϕ n+1 ) for each n ∈ N. By setting F = (F n n ) n∈N , we eventually have a right Følner sequence F with respect to which A(F , ϕ n ) = A(F , ϕ n ) for all n ∈ N. As A(F, ϕ) and A(F, ϕ) depend continuously on ϕ, we have for all ϕ ∈ C(X). By using Proposition 2.3, we obtain the desired statement.
Theorem 5.8. If (X, G) is F-mean equicontinuous for some right Følner sequence F, then (X, G) is mean equicontinuous.
Proof. Given the preceding result, the proof is almost literally the same as the one of Theorem 5.1.
Recall that a Følner sequence F is two-sided if it is a left and right Følner sequence (this implies in particular that G is unimodular). Clearly, if G is abelian, every Følner sequence is two-sided. We immediately obtain the following corollaries.
Corollary 5.9. Suppose G is unimodular and let (X, G) be F-mean equicontinuous for a two-sided Følner sequence F. Then (X, G) is mean equicontinuous.
Corollary 5.10. If G is abelian and (X, G) is F-mean equicontinuous for some Følner sequence F, then (X, G) is mean equicontinuous.
Proof of Theorem 1.3. This theorem is now an immediate consequence of Theorem 5.1 and Corollary 5.10.
Last, we would like to address the question of whether there are non-trivial non-uniquely ergodic mean equicontinuous systems (that is, non-uniquely ergodic mean equicontinuous systems which are neither finite unions of uniquely ergodic systems nor products of such). The following example demonstrates that such non-trivial neither minimal nor uniquely ergodic systems exist.
Example 5.11. Given a sequence x = (x k ) k∈Z ∈ {0, 1} Z and p ∈ N, let us set the p-periodic part of x to be Per(x, p) := {k ∈ Z | x k = x k+np (n ∈ Z)}. We put T to be the closure of in {0, 1} Z (equipped with the product topology). Observe that for every x ∈ T and each n ∈ N, we have that there is exactly one k ∈ [0, 2 n − 1] \ Per(x, 2 n ).
Clearly, T is σ-invariant where σ : {0, 1} Z → {0, 1} Z denotes the left shift. We show that (T , σ) is mean equicontinuous by proving that it is F-mean equicontinuous for F = ([0, 2 n − 1]) n∈N , see Corollary 5.10. To that end, define D n to be the pseudometric given by where we consider d to be the Cantor metric with d(x, y) := 2 − min{|k| | k∈Z and x k =y k } . By definition, lim sup n→∞ D n (x, y) = D F (x, y) for x, y ∈ {0, 1} Z . Now, given x 1 , x 2 ∈ T with d(x 1 , x 2 ) ≤ 2 −2 n , observe that there are at most two elements in [0, 2 n − 1] \ (Per(x 1 , 2 n ) ∪ Per(x 2 , 2 n )) so that for all k ≥ n. Now, given y ∈ T , let (x n ) n∈N be a sequence in T with d(x n , y) ≤ 2 −2 n . Observe that D n (x n , y) ≤ 1/2 n · 2 n =1 2 − ≤ 2 −n as well as d(x n , x k ) ≤ 2 −2 n for k ≥ n. Hence, D k (x n , y) ≤ D k (x n , x k ) + D k (x k , y) ≤ 2 −n+3 + 2 −k for all k ≥ n so that D F (x n , y) ≤ 2 −n+3 . This yields the F-mean equicontinuity of (T , σ). Observe that T contains a dense set of points which are periodic with respect to σ as well as a dense set of infinite (i.e., non-periodic) subshifts (in fact, regular Toeplitz subshifts).

Mean equicontinuity and discrete spectrum
In this section we establish a relation between mean equicontinuity and discrete spectrum. A dynamical system (X, G) together with an invariant measure µ is said to have discrete spectrum if L 2 (X, µ) can be written as an orthogonal sum of finite dimensional, G-invariant subspaces V α , where α runs through some index set, see [Mac64] for further details. As before, we will denote by (T, G) the maximal equicontinuous factor of (X, G) and by π : X → T a corresponding factor map.
We will need the following well-known fact which follows from the general theory of Ellis semigroups of equicontinuous systems (see, for example, [Aus88,): if (T , G) is minimal and equicontinuous, then T is homeomorphic to a homogeneous space, that is, there is a compact group E(T ) and a closed subgroup F ≤ E(T ) (in general not normal) such that T is homeomorphic to the set of left cosets E(T )/F . If G is abelian, then E(T ) is abelian and T is homeomorphic to E(T ).
Theorem 6.1. Suppose (X, G) is minimal. Then, the following assertions are equivalent: (i) The system (X, G) is mean equicontinuous.
(ii) (X, G) is uniquely ergodic and, if µ denotes the unique invariant probability measure, then L 2 (X, µ) can be written as an orthogonal sum of finite dimensional, G-invariant subspaces V α , consisting of continuous functions (α runs through some index set I).
Proof. (i)⇒(ii): Several results of the previous sections imply that every minimal mean equicontinuous system is uniquely ergodic. Let us hence denote by µ the unique G-invariant measure on X. Now, observe that if L 2 (T, π(µ)) can be decomposed as an orthogonal sum of finite dimensional G-invariant subspaces consisting of continuous functions, then this holds true for L 2 (X, µ) as well. This follows from the unitarity of U µ (defined as in (2); see also Theorem 3.7 and Proposition 2.1) and the fact that U µ maps continuous functions to continuous functions (due to the continuity of π). Therefore, it suffices to find a corresponding decomposition of L 2 (T, π(µ)). If T is homeomorphic to the compact group E(T) from above, this decomposition is provided by the classical Peter-Weyl Theorem. In case that T is homeomorphic to a homogeneous space, the decomposition is obtained by a standard extension of the Peter-Weyl Theorem to homogeneous spaces.
(ii)⇒(i): For each α ∈ I we define the pseudometric d α on X via As V α is finite dimensional and consists of continuous functions, each d α is continuous. As V α is G-invariant, each d α is G-invariant. Further, observe that the separability of L 2 (X, µ) implies that I is countable. Thus, we may consider the pseudometric D = α c α ·d α , where (c α ) α∈I is some summable sequence of positive numbers.
We can hence introduce an invariant and closed equivalence relation on X by x ∼ y :⇐⇒ d α (x, y) = 0 for all α ∈ I (⇐⇒ D (x, y) = 0).
Then Y := X/∼ is a compact space which we may consider equipped with the metric D in the obvious way. Further, (Y, G) (where the action of G on Y is defined in the canonical way) is an isometric and hence equicontinuous factor, as D is G-invariant. Let h : X → Y be the factor map and note that V : L 2 (Y, h(µ)) → L 2 (X, µ) with V f = f • h is unitary (as we only identify points which can not be distinguished by elements of the V α ). Now, the application of Theorem 3.7 yields (i).
(a) The assumption of minimality of (X, G) can be slightly weakened to unique ergodicity. In fact, for (ii)⇒(i) we did not need the minimality of (X, G). For (i)⇒(ii) note that (T, G) is still minimal (see Proposition 3.3).
(b) The Peter-Weyl Theorem used in the proof of (i)⇒(ii) actually gives one more feature of the finite dimensional subspaces appearing in (ii). They can be assumed to be irreducible. Here, a G-invariant subspace V of L 2 (X, µ) is called irreducible if it can not be written as an orthogonal sum of two non-trivial G-invariant subspaces.
In the case of abelian G, we obtain a somewhat stronger statement. As this is of interest in various contexts, we include a discussion.
Corollary 6.3. Let G be abelian. Suppose (X, G) is minimal. Then, the following assertions are equivalent: (i) The system (X, G) is mean equicontinuous.
(ii) The system (X, G) is uniquely ergodic and, if µ denotes the unique invariant probability measure, then L 2 (X, µ) has an orthonormal basis of continuous eigenfunctions.
Proof. Clearly, condition (ii) of the present corollary is stronger than condition (ii) of the previous theorem. Thus, it suffices to show (i)⇒(ii). This can be seen as in the proof of the previous theorem after noting that T is homeomorphic to the compact group E(T).
With this in mind, statement (ii) is a direct consequence of the duality theory for compact abelian groups. Alternatively, one may also argue that the irreducible subspaces appearing in Theorem 6.1 (ii) must be one-dimensional in the abelian case.
Remark 6.4. The last three decades have seen tremendous interest in the field of aperiodic order, also known as mathematical quasicrystals (see [BG13,KLS15] for extensive discussions). The common way to model aperiodic order is via dynamical systems over the group R n . In typical examples, these systems will be uniquely ergodic and minimal. In any case, such a system comes with a diffraction measure. As mentioned in the introduction, a key effort is to show that the diffraction measure is a pure point measure. This in turn has been proven to be equivalent to discrete spectrum of the underlying dynamical system. Hence, discrete spectrum is at the core of aperiodic order. In the further analysis of the diffraction measure, continuity of the eigenfunctions turns out to play a role. Indeed, it is exactly under this condition that a convincing positive answer to the so-called Bombieri-Tayler Conjecture can be given [Len09] (see [Rob99] for related earlier results as well.) Given this situation, the class of minimal uniquely ergodic systems with discrete spectrum and continuous eigenfunctions (which is characterized in the preceding corollary) presents itself as a very natural candidate for models of aperiodic order. if and only if for every z 0 ∈ Z there is a clopen neighborhood U of z 0 and an element g ∈ G such that sz = gz for all z ∈ U . We make use of the following structural result. Recall that a group G acts freely on Z if gz = z for some z ∈ Z and g ∈ G implies that g is the identity. Last, we present a straightforward instructive application of Theorem 7.2. Let us point out that all the considerations in the following example directly generalize to higher-dimensional odometers and associated regular Toeplitz configurations, see [Cor06].
Example 7.3. We assume that the reader is familiar with the theory of odometers/adding machines, see for instance [Ků03,Dow05] for further information. We consider the dyadic odometer (2 N , Z). That is, 2 N is the compact group obtained as the inverse limit and n ∈ Z acts on θ ∈ 2 N by θ → θ + n, where we consider n as an element of 2 N . Now, the isometric subgroup [[(2 N , Z)]] I contains, among others, the element s given by Obviously, s(θ + 1) = 1 + sθ. Hence, [[(2 N , Z)]] I is a non-abelian amenable group which acts mean equicontinuously (according to Theorem 7.2) on, in particular, the shift orbit closure of any regular Toeplitz sequence whose MEF is given by (2 N , Z) (for concrete examples, see also [Ků03,Dow05]). Obviously, these orbit closures are Cantor spaces as well.
To obtain examples where the domain is not totally disconnected, we can consider Auslander systems, see [HJ97], which also have odometers as their MEF and are mean equicontinuous.

Irregular extensions
Suppose (X, G) is an extension of (Y, G) via the factor map h : X → Y . We say (X, G) is a regular extension of (Y, G) if for every G-invariant measure µ on X we have that h(µ)({y ∈ h(X) : #h −1 (y) > 1}) = 0; otherwise we say (X, G) is an irregular extension. Given y ∈ Y , we refer to h −1 (y) as its fiber.
Note that a regular extension is automatically a topo-isomorphic extension. Examples of regular extensions of equicontinuous systems are Sturmian subshifts, regular Toeplitz subshifts and the Denjoy systems described in Example 3.21. There are also irregular topo-isomorphic extensions of equicontinuous systems. The Cantor substitution subshift in Example 3.20 is a transitive irregular extension of the trivial system. Minimal examples can be found in [DK15,DG16], where [DK15, Example 5.1] has almost surely (with respect to the unique invariant measure of its MEF) countable fibers but still a residual set of points whose fibers are singletons. In contrast, in the examples constructed in [DG16, Section 3], every fiber is uncountable. Indeed, in this subsection, we will show that almost every fiber of an irregular extension must be at least countable. For the convenience of the reader, we provide a proof of the next statement.
Lemma 7.4. Let (X, G) be an extension of (Y, G) via the factor map h : X → Y and let µ be an ergodic G-invariant measure on Y . Suppose h −1 (y) is finite for µ-almost every y ∈ Y . Then there is n 0 ∈ N such that µ-almost everywhere we have #h −1 = n 0 .
Proof. Observe that h gives rise to an upper semi-continuous and hence Borel measurable map γ from Y to the space of compact subsets of X (endowed with the Hausdorff metric), defined by γ(y) = π −1 (y) for each y ∈ Y . By Lusin's Theorem, there is a compact set K ⊆ Y with µ(K ) > 0 such that γ | K is continuous. Set Y := {y ∈ Y : #h −1 (y) < ∞}. By the assumptions, µ(Y ) = 1. Since µ is an inner regular measure, we may assume w.l.o.g. that K ⊆ Y . Let K ⊆ K be the support of the measure µ| K . Clearly, µ(K) > 0.
Theorem 7.5. Let (X, G) be an irregular extension of (Y, G) via the factor map h : X → Y , that is, h(µ)({y ∈ h(X) : #h −1 (y) > 1}) > 0 for some G-invariant measure µ on X. If h(µ)almost all fibers of h are finite, then h is not a topo-isomorphy.
Proof. We may assume without loss of generality that µ is ergodic, because of Lemma 2.5. For a contradiction, assume that (X, G) is a topo-isomorphic extension of (Y, G) via h.
Observe that ν is G-invariant and ν = µ. Moreover, h(ν) = h(µ) and this implies ν is ergodic since h is assumed to be a topo-isomorphy. However, this yields a contradiction, according to Proposition 3.1.
We immediately obtain the next two statements, using Corollary 3.19 for the second one.
Corollary 7.6. Assume (X, G) has a unique G-invariant measure µ and is an irregular topoisomorphic extension of a mean equicontinuous system (Y, G) via a factor map h : X → Y . Then for h(µ)-almost every y ∈ Y we have that h −1 (y) is infinite.
Corollary 7.7. Suppose that (X, G) is an irregular extension of (Y, G) via the factor map h : X → Y and suppose the MEF of (X, G) and (Y, G) coincide. If (Y, G) is mean equicontinuous and the fibers of h are finite, then (X, G) can not be mean equicontinuous.
An example fitting into the setting of the second corollary is the Thue-Morse subshift which is a 2-1 extension of a regular Toeplitz subshift with the same maximal equicontinuous factor (see, for instance, [BG13] for more information). In particular, we get that the Thue-Morse system is not mean equicontinuous.

Maximally almost periodic groups
In this last section we show that if a group G acts minimally, mean equicontinuously and effectively on a compact metric space (X, d), then it is necessarily maximally almost periodic.
Recall that G acts effectively on X if for each g ∈ G, there is x ∈ X with gx = x. Recall further that a topological group G is maximally almost periodic (MAP) if G admits a continuous and injective homomorphism into a compact Hausdorff group, see for instance [vN34]. Note that a locally compact MAP group is necessarily unimodular [LR68]. We will make use of the following characterization of maximal almost periodicity [Hua79]: a topological group G is MAP if and only if G admits an equicontinuous and effective action on a compact Hausdorff space.
Theorem 7.8. Let (X, G) be a dynamical system and denote by (T, G) its maximal equicontinuous factor. If (X, G) is mean equicontinuous, allows for an invariant measure of full support and G acts effectively on X, then G also acts effectively on T. In particular, this implies that G is maximal almost periodic and unimodular.
Proof. As before, we denote by π a factor map from X to T. Let µ be an invariant measure with full support. Since π is a topo-isomorphy, there are subsets M ⊆ X and N ⊆ T of full µ-and π(µ)-measure, respectively, such that the restriction of π to M is a bijection from M onto π(M ) = N . Now, assume there is g ∈ G with gy = y for all y ∈ T. Observe that such g has to verify gx = x for µ-almost all x ∈ M , since π restricted to M is injective and since -by the invariance of µ-almost every point of M is mapped into M under the action of g. As µ is of full support, every full-measure set is dense in X. Thus, the continuity of g implies gx = x for all x ∈ X. As G acts effectively on X, this gives g = e.
Recall that for a minimal dynamical system all measures have full support.
Corollary 7.9. If G acts minimally, mean equicontinuously and effectively on X, then G is maximal almost periodic and unimodular.
We would like to close with a partial answer to the following question [GM18, Question 8.1]: which discrete countable groups G have effective tame minimal actions? Here, the term tame refers to a certain low dynamical complexity of a dynamical system (see, e.g., [Gla18]). Now, according to [Gla18,Corollary 5.4 (2)], if (X, G) is tame and G amenable, then (X, G) is a topo-isomorphic extension of its MEF and hence mean equicontinuous, due to Theorem 3.7 (for Z-actions, see also [Gla18, Corollary 5.10]). Thus, from Theorem 7.8 we obtain that among the amenable, discrete countable groups exactly the maximally almost periodic ones allow for an effective tame minimal action.