On equicontinuous factors of flows on locally path-connected compact spaces

TL;DR

This paper investigates the structure of equicontinuous factors of flows on locally path-connected compact spaces with finite first Betti number, showing they are isomorphic to flows on low-dimensional compact abelian Lie groups.

Contribution

It provides a new proof that the quotient map to the maximal equicontinuous factor is monotone, using a novel characterization of local connectedness via Banach lattices.

Findings

Every equicontinuous factor is isomorphic to a flow on a compact abelian Lie group of dimension less than b_1(K)

A new proof of the monotonicity of the quotient map onto the maximal equicontinuous factor

Characterization of local connectedness in terms of the Banach lattice C(K)

Abstract

We consider a locally path-connected compact metric space $K$ with finite first Betti number $b_1(K)$ and a flow $(K, G)$ on $K$ such that $G$ is abelian and all $G$-invariant functions $f\in\mathrm{C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than $b_1(K)$. For this purpose, we use and provide a new proof for [HJop, Theorem 2.12] which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces $K$ and $L$ that we obtain by characterizing the local connectedness of $K$ in terms of the Banach lattice $\mathrm{C}(K)$.

Equations

$$T_{\varphi }:\text{uC}(L)$$

$$f$$

$$\mathrm{C}:\mathbf{CTop}\to {\mathbf{C}}_{\mathbf{com},1}^{\ast }$$

$$T_p:\text{uC}(L)$$

$$f$$

$$\Psi \text{defeq}{T}_p^{-1}\circ T_q:\text{uC}(M)\to \text{uC}(L)$$

$${\mathcal{A}}_{\mathrm{triv}}\text{defeq}\left\{f\in \text{uC}(K)\text{mmid}\forall s\in S:T_{s}f=f\right\}.$$

$${\mathcal{A}}_{\mathrm{eq}}\text{defeq}\left\{f\in \text{uC}(K)\text{mmid}\left\{T_{s}f\text{mmid}s\in S\right\}\text{is equicontinuous}\right\}$$

$$\displaystyle\mathcal{A}\defeq\overline{\lin}\left\{f\in\uC(K)\mid\forall s\in S\exists\lambda_{s}\in\T\colon T_{s}f=\lambda_{s}f$$

$${\sim }_G\text{defeq}\left\{(x,y)\in M\times M\text{mmid}\overline{Gx}=\overline{Gy}\right\}.$$

$${\mathcal{A}}_L\text{defeq}\{f\in \text{uC}(K)\mid \forall l\in L\forall x,y\in p^{-1}(l):f(x)=f(y)\}$$

$${\sim }_{\mathcal{A}}\text{defeq}\{(x,y)\in K\times K\mid \forall f\in \mathcal{A}:f(x)=f(y)\}.$$

$$\mathrm{irr}(f)\text{defeq}\{g\in \text{uC}(K)\mid g\text{ is an irreducible part of }f\}$$

$$\mathrm{irr}(\mathcal{F})\text{defeq}\bigcup _{f\in \mathcal{F}}\mathrm{irr}(f).$$

$$\mathrm{irr}(f)=\left\{{\text{1}}_{O}f\mid O\text{ is an open connected component of }[f\ne 0]\right\}.$$

$$\left\{g\in \mathrm{irr}(f)\mid \Vert g{\Vert }_{\infty }>ϵ\right\}$$

$$f=\sum _{g\in \mathrm{irr}(f)}g.$$

$${\text{uC}}_{\text{ub}}([f\ne 0])\to \text{uC}(K),g\mapsto gf$$

$$p^{-1}(l)=\bigcap _{U\in \mathcal{U}(l)}{p}^{-1}(U)=\bigcap _{\begin{array}{c}U\in \mathcal{U}(l)\\ \text{closed, connected}\end{array}}{p}^{-1}(U)$$

$${\mathcal{A}}_{\mathrm{eq}}=\left\{f\in \text{uC}(K)\text{mmid}\{T_{s}f\mid s\in S\}\text{ is equicontinuous}\right\}$$

$$\{T_{s}g\mid s\in S\}\subset \bigcup _{s\in S}\mathrm{irr}(T_s(f))=\mathrm{irr}\left(\{T_s(f)\mid s\in S\}\right).$$

$$|g(x)-g(y)|=|f(x)-f(y)|<\frac{ϵ}{2}.$$

$$|g(x)-g(y)|\le |f(x)|+|f(y)|=|f(x)-f(z)|+|f(y)-f(z)|<ϵ.$$

$$\{g_n\mid n\in \mathbb{N}\}\cup \{h_n\mid n\in \mathbb{N}\}$$

$$[g_n\ne 0]\subset [f_n\ne 0]\subset [f_{n+1}\ne 0]=[g_{n+1}\ne 0]\cup [f_{n+1}-g_{n+1}\ne 0].$$

$$[g_n\ne 0]=\left([g_n\ne 0]\cap [g_{n+1}\ne 0]\right)\cup \left([g_n\ne 0]\cap [f_{n+1}-g_{n+1}\ne 0]\right)$$

$${\left(f-\frac{2}{n}{\text{1}}_K\right)}_{+}\le g_n\le \underset{k=1}{\overset{n}{\bigvee }}{g}_k\le f.$$

$$\mathrm{Rc}(p)\text{defeq}\left\{(x,y)\in K\times K\text{mmid}\begin{array}{c}\text{x and y are in the same }\\ \text{connected component of }{p}^{-1}(p(x))\end{array}\right\}$$

$$k=\text{rank}\left({\pi }_1\left({\text{T}}^k\right)\right)=\text{rank}\left(\eta \left(H_1(K_0)\right)\right)\le \text{rank}\left(H_1(K_0)\right)={\text{ub}}_1(K_0)=\frac{{\text{ub}}_1(K)}{{\text{ub}}_0(K)}.$$

$${\mathcal{A}}_{\mathrm{eq}}=\left\{f\in \text{uC}(K)\text{mmid}\left\{T_{\varphi }^{n}f\text{mmid}n\in \mathbb{N}\right\}\text{ is equicontinuous}\right\}$$

$${\text{ub}}_1\left(H/\Gamma \right)=\text{rank}\left(H_1(H/\Gamma )\right)=\text{rank}\left({\pi }_1(H/\Gamma {)}_{\mathrm{ab}}\right)=\text{rank}\left({\Gamma }_{\mathrm{ab}}\right)=\text{rank}\left(\Gamma /[\Gamma ,\Gamma ]\right),$$

Full text

The study of topological dynamical systems via their maximal equicontinuous factors plays an important role for, e.g., tiling dynamical systems (see [MathOfAperiodicOrder, Chapter 5]), Toeplitz flows (see [Downarowicz2005]), or the Furstenberg structure theorem for minimal distal flows. One reason is that, for group actions, the maximal equicontinuous factor coincides with the Kronecker factor. The latter is highly structured, is minimal if and only if it is isomorphic to a minimal rotation on a homogeneous space of some compact group, and also captures spectral-theoretic information. In light of this, it is important to understand how the specific structure and properties of a system can be used to determine its maximal equicontinuous factor. For example, it is known that if $(M,G)$ is a distal minimal flow on a compact manifold $M$, then its maximal equicontinuous factor is a flow on a homogeneous space of some compact Lie group, see [Rees1977, Theorem 1.2] or [Ihrig1984, Theorem 1.2]. If, additionally, the acting group $G$ is abelian, the maximal equicontinuous factor is in fact isomorphic to a flow on a compact abelian Lie group. For non-distal systems, however, few results in this spirit seem to exist. Notably, Hauser and Jäger recently proved the following.

{theorem*}

[[HauserJaeger2017, Theorem 3.1]] Suppose that $f$ is a homeomorphism of the two-torus. If the maximal equicontinuous factor of $(\T^{2},f)$ is minimal, then it must be one of the following three:

1. [(i)]
2. (1)

an irrational translation on the two-torus, 3. (2)

an irrational rotation on the circle, 4. (3)

the identity on a singleton.

Thus, the geometric properties of the two-torus imply that the maximal equicontinuous factor of a flow on it must have a relatively simple structure if it is minimal: It is a rotation on a compact abelian Lie group of dimension less than two. As it turns out, this is representative of the following general phenomenon which is the main result of this article. Recall that every compact abelian Lie group is isomorphic to the product $F\times\T^{m}$ of a finite abelian group $F$ and a torus $\T^{m}$.

{theorem*}

Let $(K,G)$ be a flow such that $K$ is locally path-connected with finite first Betti number $\ub_{1}(K)$, $G$ is abelian, and such that $K$ is metrizable or $G$ is separable. If all $G$-invariant functions $f\in\uC(K)$ are constant, then every equicontinuous factor of $(K,G)$ is isomorphic to a minimal flow on some compact abelian Lie group of dimension less than $\nicefrac{\ub_{1}(K)}{\ub_{0}(K)}$. More precisely, for every equicontinuous factor $(L,G)$ of $(K,G)$, there are a finite abelian group $F$ of order $|F|\leq\ub_{0}(K)$ and an $m\leq\nicefrac{\ub_{1}(K)}{\ub_{0}(K)}$ such that $(L,G)$ is isomorphic to a minimal action of $G$ on $F\times\T^{m}$ via rotations.

This provides a bound on the complexity of the maximal equicontinuous factor in terms of topological invariants of the underlying space $K$ and applies in particular to minimal systems on compact manifolds. As a corollary, we obtain in \crefcor:liegroupquotient and \crefcor:toruscase that the above-cited [HauserJaeger2017, Theorem 3.1] holds analogously for tori of arbitrary dimension and, in an appropriate version, more generally for quotients $H/\Gamma$ of connected, simply connected Lie groups $H$ by discrete, cocompact subgroups $\Gamma\subset H$. Examples for such spaces are in particular given by nilmanifolds, see [HostKra2018, Chapter 10].

Another result of [HauserJaeger2017] and a key element in their proof of [HauserJaeger2017, Theorem 3.1] is that for a large class of flows the factor map onto the maximal equicontinuous factor is monotone, meaning that preimages of points are connected. Similar results were, it seems, first obtained in [McMahonWu1976] where the authors proved that for each extension $p\colon(K,G)\to(L,G)$ of minimal flows that decomposes into a tower of equicontinuous extensions, the quotient map $K\to K/\uS(p)$ is monotone, where $\uS(p)$ denotes the relativized equicontinuous structure relation. Hence, for a distal minimal flow its Furstenberg tower consists entirely of monotone quotient maps, see [Greschonig2014, Proposition 2.2], and in particular the map onto the maximal equicontinuous factor is monotone. Without such structural assumptions, the monotonicity of the maximal equicontinuous factor does not hold in general (take, e.g., the extension of the shift $\tau\colon\Z\to\Z,x\mapsto x+1$ to the one-point compactification of $\Z$). If, however, the underlying space is locally connected, it is shown in [HauserJaeger2017, Theorem 2.12] that the quotient map onto the maximal equicontinuous factor is indeed monotone which is notable since monotone quotient maps relate the geometry of a space to that of a quotient: Every monotone quotient map between suitable spaces induces a surjective homomorphism on the level of fundamental groups, see [Calcut2012, Theorem 1.1] (stated below as \crefthm:calcut). Since this idea will be crucial for the proof of \crefmthm, we also provide an alternative proof for the monotonicity of the maximal equicontinuous factor under the assumption of local connectedness. This is based on two results that are of interest by themselves: We characterize the local connectedness of a compact space $K$ in terms of the Banach lattice $\uC(K)$ and then use this to give a new characterization for the monotonicity of a quotient map between locally connected compact spaces. (For background information on Banach lattices, see [EFHN2015, Section 7.1].) The above-mentioned monotonicity result then is a simple application of these characterizations. We prove these results in \crefsection:monotonicity after collecting some preliminaries in \crefsection:prelim. The main result is proved in \crefsection:mef.

Notation and Terminology. By a topological dynamical system $(K,S,\Phi)$ we mean a continuous action $\Phi\colon S\times K\to K$ of a topological semigroup $S$ on a compact space $K$. (We always assume compact spaces to be Hausdorff.) We usually drop $\Phi$ from the notation and write $sx$ instead of $\Phi(s,x)$ for $s\in S$ and $x\in K$. We simply call $(K,S)$ a flow if $S$ is a group. If we refer to a pair $(K,\phi)$ of a compact space $K$ and a continuous map $\phi\colon K\to K$ as a topological dynamical system, we regard the $\N$ or $\Z$-action on $K$ given by the powers of $\phi$, depending on whether $\phi$ is explicitly specified to be invertible or not. By an extension $p\colon(K,S)\to(L,S)$ of topological dynamical systems we mean a continuous, surjective, $S$-equivariant function $p\colon K\to L$. Given such an extension, we also call $p$ a factor map and $(L,S)$ a factor of $(K,S)$. A system $(K,S)$ is called equicontinuous if the family $\{\Phi(s,\cdot)\mid s\in S\}\subset\uC(K,K)$ is equicontinuous.

If $\phi\colon K\to L$ is a continuous function between compact spaces, we denote by $T_{\phi}$ its Koopman operator

[TABLE]

We assume the reader to be familiar with Koopman operators and the theory of commutative $\uC^{*}$-algebras and refer to [EFHN2015, Chapter 4] for background information. If $H$ is an abelian group, then $\rank(H)$ denotes its torsion-free rank, i.e., the dimension of the $\Q$-vector space $\Q\otimes H$. For a topological space $X$ and $i\in\N_{0}$, we denote by $\ub_{i}(X)$ the $i$-th Betti number of $X$. Note that for a compact, locally path-connected space $X$, $\ub_{0}(X)$ is the (finite) number of connected components of $X$.

Acknowledgements. Part of this work was done during a stay at Kiel University and the author is very grateful to M. Haase for his kind hospitality during this time, as well as for bringing the book [Bronstein1979] and thereby [Rees1977] and [Ihrig1984] to the author’s attention.

1. Factors and invariant subalgebras

Consider the categories $\mathbf{CTop}$ of compact topological spaces and $\mathbf{C^{*}_{com,1}}$ of commutative unital $\mathrm{C}^{*}$-algebras as well as the contravariant functor

[TABLE]

given by $K\mapsto\uC(K)$ and $\phi\mapsto T_{\phi}$ for compact spaces $K$ and $L$ and continuous functions $\phi\colon K\to L$. It is a consequence of the classical Gelfand representation theorem (see [EFHN2015, Theorem 4.23]) that $\uC$ is an antiequivalence of categories. This allows to use operator-theoretic concepts to understand topological dynamical systems (cf. \crefthm:mefequi) but also, conversely, to use geometric tools to obtain results about operators (cf. \crefcor_spectrum). A particular consequence of this antiequivalence that we will use throughout the article is the relationship between factors of a topological dynamical system $(K,S)$ and $S$-invariant unital $\uC^{*}$-subalgebras of $\uC(K)$: Suppose $p\colon(K,S)\to(L,S)$ is a factor map of topological dynamical systems. Then the Koopman operator

[TABLE]

is an $S$-equivariant $\uC^{*}$-embedding and its image $\mathcal{A}_{L}\defeq T_{p}(\uC(L))\subset\uC(K)$ is an $S$-invariant unital $\uC^{*}$-subalgebra of $\uC(K)$, i.e., $T_{s}(\mathcal{A}_{L})\subset\mathcal{A}_{L}$ for each $s\in S$. Moreover, if $q\colon(K,S)\to(M,S)$ is another factor map such that $T_{q}(\uC(M))=T_{p}(\uC(L))$, then

[TABLE]

defines an $S$-equivariant $\uC^{*}$-isomorphism and so there is a unique $S$-equivariant homeomorphism $\eta\colon L\to M$ such that $\Psi=T_{\eta}$, making the following diagram commutative:

[TABLE]

This shows that a factor of $(K,S)$ is, up to isomorphy, uniquely determined by its corresponding $\uC^{*}$-subalgebra of $\uC(K)$. Conversely, every $S$-invariant unital $\uC^{*}$-subalgebra of $\uC(K)$ canonically corresponds to a factor of $(K,S)$ via the Gelfand representation theorem and one thereby obtains, again up to isomorphy, a one-to-one correspondence between factors of $(K,S)$ and $S$-invariant unital $\uC^{*}$-subalgebras of $\uC(K)$. For the convenience of the reader, \crefrem:algebra-quotient below explains a more elementary approach to the this correspondence.

Given that the $\uC^{*}$-subalgebras of many factors such as the maximal trivial factor, the maximal equicontinuous factor, the maximal tame factor, the Kronecker factor, or the Abramov factor admit relatively simple descriptions, it is convenient to study factors and factor maps via their corresponding subalgebras. We do so in \crefcor_algebrachar to give a simple criterion characterizing the monotonicity of a factor via its corresponding $\uC^{*}$-subalgebra. We will then see that this criterion is readily verified for the $\uC^{*}$-subalgebra of the maximal equicontinuous factor.

Example 1.1**.**

Let $(K,S)$ be a topological dynamical system.

1. (1)

The factor consisting of a single point corresponds to the subalgebra $\C\1_{K}\subset\uC(K)$ of constant functions. 2. (2)

Let $p\colon(K,S)\to(L,S)$ be a factor map onto a trivial factor, i.e., one on which $S$ acts trivially. Then $T_{p}(\uC(L))$ is a subalgebra of

[TABLE]

This \enquotefixed algebra corresponds to the maximal trivial factor $(K_{\mathrm{triv}},S)$ of $(K,S)$ through which every factor map onto another trivial factor factorizes. 3. (3)

Similarly, the subalgebra

[TABLE]

corresponds to the maximal equicontinuous factor $(K_{\mathrm{eq}},S)$ of $(K,S)$ since a topological dynamical system $(M,S)$ is equicontinuous if and only if for each $g\in\uC(M)$ the orbit $\{T_{s}g\mid s\in S\}$ is equicontinuous. By the Arzelà-Ascoli theorem, this is equivalent to the orbits $\{T_{s}g\mid s\in S\}$ being relatively compact for each $g\in\uC(M)$. If $S$ acts via homeomorphisms, this means that the maximal equicontinuous factor $(K_{\mathrm{eq}},S)$ coincides with the Kronecker factor which corresponds, for abelian $S$, to the $\uC^{*}$-subalgebra

[TABLE]

spanned by the eigenfunctions of the action of $S$, see [EFHN2015, Corollary 16.32].

Remark 1.2**.**

Let $(M,G)$ be an equicontinuous flow. Then each orbit closure is a minimal subset of $M$ (see [Auslander1988, Lemma 2.3]) and so $M$ decomposes into minimal subsets. Moreover, the orbit closure relation

[TABLE]

is a closed equivalence relation (see [Auslander1988, Exercise 2.6]) and a moment’s though reveals that hence, $M/\sim_{G}$ together with the trivial $G$-action is the maximal trivial factor of $(M,G)$. In particular, $(M,G)$ is minimal if and only if the maximal trivial factor of $(M,G)$ is a point. We note for the proof of \crefmthm that as a consequence, given an arbitrary system $(K,G)$, its maximal equicontinuous factor is minimal if and only if every $G$-invariant function $f\in\uC(K)$ is constant.

Remark 1.3**.**

Given a compact space $K$, one can describe the relationship between compact quotients $L$ of $K$ and unital $\uC^{*}$-subalgebras $\mathcal{A}$ of $\uC(K)$ without using the Gelfand representation theory: If $p\colon K\to L$ is a continuous surjective map onto a compact space $L$, define the unital $\uC^{*}$-subalgebra

[TABLE]

of functions constant on fibers of $p$ and note that $\mathcal{A}_{L}=T_{p}(\uC(L))$. Conversely, if $\mathcal{A}\subset\uC(K)$ is a unital $\uC^{*}$-subalgebra, define the closed equivalence relation

[TABLE]

and set $L_{\mathcal{A}}\defeq K/{\sim_{\mathcal{A}}}$. Then the assignments $L\mapsto\mathcal{A}_{L}$ and $\mathcal{A}\mapsto L_{\mathcal{A}}$ are, up to isomorphy of the compact spaces, mutually inverse. Analogously, one obtains the above-explained correspondence between factors and invariant subalgebras if one considers topological dynamical systems $(K,S)$ on $K$.

2. Local connectedness and monotonicity of factors

As noted in the introduction, one cannot expect the factor map onto the maximal equicontinuous factor of a flow to be monotone in general, i.e., its preimages of points are not generally connected. Therefore, we focus on quotient maps $p\colon K\to L$ between locally connected compact spaces and characterize their monotonicity in terms of the subalgebra $\mathcal{A}_{L}\subset\uC(K)$ of functions constant on fibers of $p$. We then apply this to the maximal equicontinuous factor. Recall the following elementary results on locally connected spaces.

Lemma 2.1**.**

Let $X$ and $Y$ be topological spaces.

1. (a)

$X$* is locally connected if and only if for every open set $O\subset X$ each connected component of $O$ is open in $X$.* 2. (b)

$X$* is locally connected if and only if for every basis $\mathcal{B}$ for the topology of $X$ and every $O\in\mathcal{B}$ each connected component of $O$ is open in $X$.* 3. (c)

If $X$ is compact, $X$ is locally connected if and only if it is uniformly locally connected, i.e., if each entourage $U\in\mathcal{U}_{X}$ contains an entourage $V\in\mathcal{U}_{X}$ such that $V[x]$ is connected for each $x\in X$. 4. (d)

If $X$ is locally connected and $f\colon X\to Y$ is a surjective quotient map, then $Y$ is locally connected. 5. (e)

If $X$ is compact and locally connected, $X$ has only finitely many connected components.

Proof 2.2**.**

For 3a, 3d, and 3e see Theorem 27.9, Corollary 27.11, and Theorem 27.12 of [Willard2004] and for 3c see [James1999, Proposition 9.39]. \Creflem:locallyconnected_b follows from the definition of local connectedness and \creflem:locallyconnected_a.

Since in a compact space $K$ the sets of the form $[f\neq 0]$ for continuous, complex-valued functions $f\in\uC(K)$ constitute a base of the topology, \creflem:locallyconnected3a and 3b provide a natural way to characterize the local connectedness of $K$ purely in terms of the Banach lattice $\uC(K)$. For this purpose, we call $f,g\in\uC(K)$ orthogonal and write $f\perp g$ if they are orthogonal in the Banach lattice $\uC(K)$. (That is, $f\perp g$ if and only if $|f|\wedge|g|=0$ which is equivalent to $fg=0$.) A decomposition $f=g+h$ is called orthogonal if $g$ and $h$ are orthogonal. A function $f\in\uC(K)$ is called reducible if $f=0$ or if there is an orthogonal decomposition $f=g+h$ with nonzero $g,h\in\uC(K)$ and $f$ is called irreducible otherwise. If $f=g+h$ is an orthogonal decomposition and $g$ is irreducible, then $g$ is called an irreducible part of $f$. For a function $f\in\uC(K)$, define

[TABLE]

and for a subset $\mathcal{F}\subset\uC(K)$, set

[TABLE]

A decomposition $f=\sum_{g\in\mathcal{F}}f$ for some at most countable set $\mathcal{F}\subset\uC(K)$ is called irreducible if all $g\in\mathcal{F}$ are irreducible and pairwise orthogonal and the sum converges uniformly to $f$.

Proposition 2.3**.**

Let $K$ be a compact space.

1. (a)

If $f\in\uC(K)$ and $M\subset[f\neq 0]$, then $\1_{M}f\in\uC(K)$ if and only if $M$ is clopen in $[f\neq 0]$. 2. (b)

If $f\in\uC(K)$, then $f$ is irreducible if and only if $[f\neq 0]$ is connected. 3. (c)

For each $f\in\uC(K)$

[TABLE] 4. (d)

For each $f\in\uC(K)$ and $\epsilon>0$ the set

[TABLE]

is finite. In particular, $\operatorname{irr}(f)$ is countable. 5. (e)

$K$* is locally connected if and only if each $f\in\uC(K)$ admits a unique irreducible decomposition. In that case, the irreducible decomposition is given by*

[TABLE]

Proof 2.4**.**

For a fixed $f\in\uC(K)$, the multiplication operator

[TABLE]

is well-defined and if $M\subset[f\neq 0]$ is clopen, then $\1_{M}|_{[f\neq 0]}\in\uC_{\ub}([f\neq 0])$. Therefore, $f\1_{M}\in\uC(K)$. Conversely, if $M\subset[f\neq 0]$ is such that $\1_{M}f\in\uC(K)$, the restriction $\1_{M}f|_{[f\neq 0]}$ is continuous and so dividing by $f|_{[f\neq 0]}$ yields the continuity of $\1_{M}|_{[f\neq 0]}$. This proves 3a which in turn yields 3b.

If $O$ is an open connected component of $[f\neq 0]$, then $\1_{O}f\in\operatorname{irr}(f)$ by 3a and 3b. Conversely, take $g\in\operatorname{irr}(f)$. Then $g=\1_{[g\neq 0]}f$ and so by 3a, $[g\neq 0]$ is a clopen subset of $[f\neq 0]$ and hence a union of connected components of $[f\neq 0]$. However, since $g$ is irreducible, $[g\neq 0]$ is connected and so it is an open connected component of $[f\neq 0]$, proving 3c. Moreover, 3c yields that $\operatorname{irr}(f)$ is a bounded and equicontinuous set in $\uC(K)$ and so by the Arzelà-Ascoli theorem, $\operatorname{irr}(f)$ is relatively compact. Since by 3c, for every two $g,h\in\operatorname{irr}(f)$ with $g\neq h$ one has $\|g-h\|=\max\{\|g\|,\|h\|\}$, 3d follows from the relative compactness of $\operatorname{irr}(f)$.

Now suppose $K$ to be locally connected and take $f\in\uC(K)$. Since $K$ is locally connected, each connected component of $[f\neq 0]$ is open and so by 3c and 3d, the sum $\sum_{g\in\operatorname{irr}(f)}g$ converges uniformly to $f$. Hence, $f$ admits an irreducible decomposition which is readily verified to be unique. Conversely, assume that each $f\in\uC(K)$ admits an irreducible decomposition and let $x\in K$. To show that $K$ is locally connected at $x$, let $U\subset K$ be an open neighborhood of $x$. Since $K$ is completely regular, there exists an $f\in\uC(K)$ with $x\in[f\neq 0]\subset U$. Moreover, since $f$ admits a unique irreducible decomposition, there is a unique $g\in\operatorname{irr}(f)$ such that $g(x)=f(x)\neq 0$. In particular, $x\in[g\neq 0]\subset[f\neq 0]\subset U$ and since $g$ is irreducible, $[g\neq 0]$ is connected, showing that $K$ is locally connected.

After these preparatory notes on local connectedness, we now turn towards the notion of monotonicity and its characterizations. We restrict to compact spaces although many of the arguments are easily adapted to completely regular spaces.

Definition 2.5**.**

Let $X$ and $Y$ be topological spaces and $p\colon X\to Y$ a map. Then $p$ is called monotone if for each $y\in Y$ the preimage $p^{-1}(y)$ is a connected subset of $X$.

If $p\colon K\to L$ is a continuous surjective map, then as noted in \crefsection:prelim and in particular \crefrem:algebra-quotient, $L$ is, up to isomorphy, uniquely determined by the subalgebra $\mathcal{A}_{L}=T_{p}(\uC(L))\subset\uC(K)$ of functions constant on the fibers of $p$. If $K$ is locally connected, this allows to use \creflc_char to characterize the monotonicity of a quotient map $p\colon K\to L$ in terms of the subalgebra $\mathcal{A}_{L}$ and the Koopman operator $T_{p}$.

Proposition 2.6**.**

Let $K$ and $L$ be compact spaces, $K$ locally connected, and $p\colon K\to L$ continuous and surjective. Then the following assertions are equivalent.

1. (a)

$p$* is monotone.* 2. (b)

For every connected set $C\subset L$ the preimage $p^{-1}(C)$ is connected. 3. (c)

For every open, connected set $U\subset L$ the preimage $p^{-1}(U)$ is connected. 4. (d)

For every irreducible $f\in\uC(L)$ the set $p^{-1}([f\neq 0])$ is connected. 5. (e)

$T_{p}$* preserves irreducibility of functions.* 6. (f)

The subalgebra $\mathcal{A}_{L}=T_{p}(\uC(L))\subset\uC(K)$ satisfies $\operatorname{irr}(\mathcal{A}_{L})\subset\mathcal{A}_{L}$.

Proof 2.7**.**

For the implication 3a $\implies$ 3b, suppose $C\subset L$ to be connected and that $p^{-1}(C)=U\cup V$ for disjoint, open sets $U,V\subset p^{-1}(C)$. Then $U$ and $V$ are saturated, i.e., $p^{-1}(p(U))=U$ and $p^{-1}(p(V))=V$, since each fiber of $p$ over $C$ is connected. Hence, the open (!) sets $p(U)$ and $p(V)$ form a cover of $C$ by disoint, open sets. Since $C$ is connected, $U=\emptyset$ or $V=\emptyset$ and so $p^{-1}(C)$ is connected.

The implication 3b $\implies$ 3c is trivial. For the implication 3c $\implies$ 3a, note that for $l\in L$

[TABLE]

as $L$ is locally connected by \creflem:locallyconnected. Since, in a compact space, the intersection of a decreasing family of closed, connected subsets is again connected, $p$ is monotone.

The equivalence of 3d and 3e follows since $[T_{p}(f)\neq 0]=[f\circ p\neq 0]=p^{-1}([f\neq 0])$. Moreover, 3e and 3f are seen to be equivalent using the existence of irreducible decompositions for functions in $\uC(L)$. Finally, the implication 3c $\implies$ 3d is trivial and the converse implication follows analogously to 3c $\implies$ 3a because, $L$ being locally connected and completely regular, the sets of the form $[f\neq 0]$ for irreducible $f\in\uC(L)$ form a basis of the topology of $L$ which allows to copy the argument.

Definition 2.8**.**

Let $K$ be a compact space. A unital $\uC^{*}$-subalgebra $\mathcal{A}\subset\uC(K)$ is called monotone if the canonical quotient map $K\to K/\sim_{\mathcal{A}}$ is monotone.

Corollary 2.9**.**

Let $K$ be a locally connected compact space and $\mathcal{A}\subset\uC(K)$ a unital $\uC^{*}$-subalgebra. Then $\mathcal{A}$ is monotone if and only if it contains the irreducible parts of all its functions.

As mentioned in \crefsection:prelim, many abstractly defined factors in topological dynamics, including the maximal equicontinuous factor, naturally have corresponding $\uC^{*}$-subalgebras that admit simple descriptions. Hence, \crefprop:monotonechar and \crefcor_algebrachar provide a useful way of verifying the monotonicity of factors. For the maximal equicontinuous factor, this yields the following.

Theorem 2.10**.**

Let $(K,S)$ be a topological dynamical system such that $K$ is locally connected and the semigroup $S$ acts on $K$ via monotone maps. Then the factor map onto the maximal equicontinuous factor of $(K,S)$ is monotone.

Proof 2.11**.**

By \crefexample:factor and \crefcor_algebrachar, it suffices to show that the subalgebra

[TABLE]

satisfies $\operatorname{irr}(\mathcal{A}_{\mathrm{eq}})\subset\mathcal{A}_{\mathrm{eq}}$. So let $f\in\mathcal{A}_{\mathrm{eq}}$ and $g\in\operatorname{irr}(f)$. Since the connectedness of a set is preserved by taking preimages under monotone maps, $T_{s}g$ is irreducible for each $s\in S$ and so $T_{s}g\in\operatorname{irr}(T_{s}(f))$. Therefore,

[TABLE]

The latter set is equicontinuous by \creflem:irrequi below and so it follows that the orbit $\{T_{s}g\mid s\in S\}$ of $g$ is equicontinuous as well. Hence, $g\in\mathcal{A}_{\mathrm{eq}}$.

Lemma 2.12**.**

Let $K$ be a locally connected compact space and $\mathcal{F}\subset\uC(K)$. Then $\mathcal{F}$ is equicontinuous if and only if $\operatorname{irr}(\mathcal{F})$ is.

Proof 2.13**.**

Suppose $\mathcal{F}$ to be equicontinuous and take $\epsilon>0$. Then there exists an entourage $V\in\mathcal{U}_{K}$ such that for each $f\in\mathcal{F}$ and $(x,y)\in V$ one has $|f(x)-f(y)|<\frac{\epsilon}{2}$. By \creflem:locallyconnected, we may assume that $V[x]$ is connected for each $x\in K$. Let $g\in\operatorname{irr}(\mathcal{F})$, i.e., $g\in\operatorname{irr}(f_{0})$ for some $f_{0}\in\mathcal{F}$. We claim that $|g(x)-g(y)|<\epsilon$ for all $(x,y)\in V$ which would show that $\operatorname{irr}(\mathcal{F})$ is equicontinuous.

So let $(x,y)\in V$. If $g(x)=g(y)=0$, it holds trivially that $|g(x)-g(y)|<\epsilon$, so assume without loss of generality that $g(x)\neq 0$. If $V[x]$ lies in $[f\neq 0]$, then it lies in the connected component of $[f\neq 0]$ containing $x$ and so $y\in V[x]\subset[g\neq 0]$. Therefore,

[TABLE]

If $V[x]\not\subseteq[f\neq 0]$, there is a $z\in V[x]$ with $f(z)=0$ and so

[TABLE]

Therefore, $\operatorname{irr}(\mathcal{F})$ is equicontinuous. The converse implication follows similarly.

Of course, the most common examples of semigroups acting via monotone maps are given by group actions, so that we obtain a new proof for [HauserJaeger2017, Theorem 2.12].

Corollary 2.14**.**

Let $(K,G)$ be a flow on a locally connected compact space $K$. Then the factor map onto the maximal equicontinuous factor is monotone.

It is known (see [Blokh2005, Theorem 3.16]) that if $(\T^{2},\phi)$ is a minimal topological dynamical system on the two-torus, then $\phi$ is necessarily monotone, and there do exist non-invertible examples for such systems with a non-trivial maximal equicontinuous factor, see [Kolyada2001, Theorem 3.3]. Therefore, there are examples in which \crefthm:mefequi provides meaningful information that cannot be deduced from \crefcor:mefequi.

In preparation for the next section, we collect several properties of monotone subalgebras.

Proposition 2.15**.**

Let $K$ be a locally connected compact space and $\mathcal{A}\subset\uC(K)$ a unital $\uC^{*}$-subalgebra.

1. (a)

If $(f_{n})_{n\in\N}$ is a sequence in $\uC(K)$ converging to $f\in\uC(K)$ and $f_{n}=h_{n}+g_{n}$ is an orthogonal decomposition for each $n\in\N$, then there is a subsequence $(f_{n_{k}})_{k\in\N}$ such that $(g_{n_{k}})_{k\in\N}$ and $(h_{n_{k}})_{k\in\N}$ converge uniformly. If $f$ is irreducible, one of the sequences converges to 0 uniformly. 2. (b)

If $(f_{n})_{n\in\N}$ is a positive increasing sequence in $\uC(K)$ converging uniformly to $f\in\uC(K)$, then for each $g\in\operatorname{irr}(f)$ there is a positive increasing sequence $(g_{n})_{n\in\N}$ such that $g_{n}\uparrow g$ and for each $n\in\N$ either $g_{n}=0$ or $g_{n}\in\operatorname{irr}(f_{n})$. 3. (c)

If $f\in\mathcal{A}$, then $\operatorname{irr}(f)\subset\mathcal{A}$ if and only if $\operatorname{irr}(|f|)\subset\mathcal{A}$. 4. (d)

If $D\subset\mathcal{A}$ is a dense $\Q$-vector sublattice containing $\1_{K}$, then $\operatorname{irr}(D)\subset D$ implies $\operatorname{irr}(\mathcal{A})\subset\mathcal{A}$. 5. (e)

Let $\mathcal{S}$ be a system of unital $\uC^{*}$-subalgebras of $\uC(K)$ closed under arbitrary intersections and containing $\uC(K)$. Then there exists a smallest monotone subalgebra $\mathcal{A}_{\mathrm{mon}}^{\mathcal{S}}\in\mathcal{S}$ containing $\mathcal{A}$, called the monotone hull of $\mathcal{A}$ in $\mathcal{S}$. 6. (f)

Furthermore, suppose that for every separable $\uC^{*}$-subalgebra $\mathcal{B}\subset\uC(K)$ there is a separable $\uC^{*}$-subalgebra $\mathcal{B}^{\mathcal{S}}\in\mathcal{S}$ satisfying $\mathcal{B}\subset\mathcal{B}^{\mathcal{S}}$ and that if $(\mathcal{B}_{n})_{n\in\N}$ is an increasing sequence in $\mathcal{S}$, then $\overline{\bigcup_{n\in\N}\mathcal{B}_{n}}\in\mathcal{S}$. Then the monotone hull $\mathcal{A}_{\mathrm{mon}}^{\mathcal{S}}$ is separable if and only if $\mathcal{A}$ is.

Proof 2.16**.**

For 3a, it follows from the Arzelà-Ascoli theorem that the set

[TABLE]

is relatively compact in $\uC(K)$ which implies the existence of the sequence $(n_{k})_{n\in\N}$. The limits $g$ and $h$ of $(g_{n_{k}})_{k\in\N}$ and $(h_{n_{k}})_{k\in\N}$ satisfy $f=g+h$ and $g\perp h$ and hence yield an orthogonal decomposition of $f$. Therefore, if $f$ is irreducible, $g=0$ or $h=0$ which proves 3a.

For 3b, one can pass to $(\1_{[g\neq 0]}f_{n})_{n\in\N}$ and $\1_{[g\neq 0]}f=g$ and hence assume that $f$ itself is already irreducible. Pick $x\in[f\neq 0]$. Without loss of generality, we may assume that $f_{1}(x)>0$. Then for each $n\in\N$, there is a unique $g_{n}\in\operatorname{irr}(f_{n})$ with $g_{n}(x)=f_{n}(x)>0$. To see that $(g_{n})_{n\in\N}$ is increasing, note that for each $n\in\N$

[TABLE]

This implies

[TABLE]

and since this union is disjoint, $g_{n}$ is irreducible, and $x\in[g_{n}\neq 0]\cap[g_{n+1}\neq 0]$, $[g_{n}\neq 0]\subset[g_{n+1}\neq 0]$. Since $(f_{n})_{n\in\N}$ is increasing, this yields that $(g_{n})_{n\in\N}$ is increasing. Now consider the orthogonal decomposition $f_{n}=g_{n}+(f_{n}-g_{n})$. Since $g_{n}(x)\geq g_{1}(x)>0$ for each $n\in\N$, 3a yields that every subsequence of $(f_{n}-g_{n})_{n\in\N}$ has a subsequence converging to 0, showing that $f_{n}-g_{n}\to 0$. Therefore, $g_{n}\uparrow f$.

For 3c, note that if $f\in\mathcal{A}$, the absolute value yields a bijection $\operatorname{irr}(f)\to\operatorname{irr}(|f|)$ and that $\operatorname{irr}(f)\subset\mathcal{A}$ hence implies $\operatorname{irr}(|f|)\subset\mathcal{A}$. So suppose that, conversely, $\operatorname{irr}(|f|)\subset\mathcal{A}$ and let $g\in\operatorname{irr}(f)$. Then $|g|\in\operatorname{irr}(|f|)\subset\mathcal{A}$ and so $f|g|^{\frac{1}{n}}\in\mathcal{A}$ for each $n\in\N$. Without loss of generality, we may assume that $|g|\leq 1$, in which case $f|g|^{\frac{1}{n}}$ converges uniformly to $g$ as $n\to\infty$ and so $\operatorname{irr}(f)\subset\mathcal{A}$.

Now let $D$ be as in 3d and $f\in\mathcal{A}$. We need to show that $\operatorname{irr}(f)\subset\mathcal{A}$ and by 3c we may assume that $f$ is positive. We want to use \crefprop:mongen_b and therefore claim that $D$ contains a positive, increasing sequence $(f_{n})_{n\in\N}$ converging to $f$. To see this, let $(h_{n})_{n\in\N}$ be a sequence in $D$ converging to $f$. By passing to $(|h_{n}|)_{n\in\N}$, we may assume that the sequence is positive. Moreover, we can arrange that $\|f-h_{n}\|<\frac{1}{n}$ for each $n\in\N$ and by passing to $g_{n}^{\prime}=(g_{n}-\frac{1}{n}\1_{K})_{+}=\sup(g_{n}-\frac{1}{n}\1_{K},0)$ we may therefore also assume that for each $n\in\N$

[TABLE]

Since $(f-\frac{2}{n}\1_{K})_{+}$ converges uniformly to $f$ as $n\to\infty$, we can set $f_{n}\defeq\bigvee_{k=0}^{n}g_{k}$ for $n\in\N$ and obtain a sequence $(f_{n})_{n\in\N}$ in $D$ such that $f_{n}\uparrow f$. Now pick $g\in\operatorname{irr}(f)$. Then by 3b, there is a sequence $(g_{n})_{n\in\N}$ with $g_{n}\uparrow g$ and $g_{n}\in\operatorname{irr}(f_{n})\cup\{0\}$ for each $n\in\N$. Since by assumption $\operatorname{irr}(D)\subset D$, $(g_{n})_{n\in\N}$ lies in $D\subset\mathcal{A}$ and so $g\in\mathcal{A}$.

\Cref

prop:mongen_e immediately follows from \crefcor_algebrachar by taking the intersection of all monotone $\uC^{*}$-subalgebras in $\mathcal{S}$ that contain $\mathcal{A}$. For 3f, it suffices to find a separable, monotone subalgebra in $\mathcal{S}$ that contains $\mathcal{A}$. To this end, define increasing $\Q$-vector sublattices $D_{n}\subset\uC(K)$ and subalgebras $\mathcal{B}_{n}\subset\uC(K)$ as follows: Let $\mathcal{B}_{0}\in\mathcal{S}$ be a separable subalgebra containing $\mathcal{A}$ and let $D_{0}\subset\mathcal{B}_{0}$ be a countable dense $\Q$-vector sublattice containing $\1_{K}$. For $n\in\N_{0}$ then define $\mathcal{B}_{n+1}$ to be a separable $\uC^{*}$-algebra in $\mathcal{S}$ containing the $\uC^{*}$-algebra generated by $\operatorname{irr}(D_{n})$ and $\mathcal{B}_{n}$ and let $D_{n+1}\subset\mathcal{B}_{n}$ be a countable dense $\Q$-vector sublattice containing $D_{n}$ and $\operatorname{irr}(D_{n})$. Then $\mathcal{B}\defeq\overline{\bigcup_{n\in\N}\mathcal{B}_{n}}$ lies in $\mathcal{S}$ and $D\defeq\bigcup_{n\in\N}D_{n}$ is a countable dense vector sublattice of $\mathcal{B}$ satisfying $\operatorname{irr}(D)\subset D$. Since $\mathcal{B}$ is therefore separable, contains $\mathcal{A}$, and is monotone by 3d, $\mathcal{A}_{\mathrm{mon}}^{\mathcal{S}}\subset\mathcal{B}$ is separable.

Remark 2.17**.**

Let $p\colon(K,G)\to(L,G)$ be a factor map of flows on locally connected spaces. Then the family $\mathcal{S}$ of $G$-invariant $\uC^{*}$-subalgebras of $\uC(K)$ satisfies the condition in \crefprop:mongen3e and so we can consider the monotone $G$-invariant hull of $\mathcal{A}_{L}=T_{p}(\uC(L))$ in $\mathcal{S}$ which we denote by $T_{p}(\uC(L))^{G}_{\mathrm{mon}}$. This subalgebra corresponds to a monotone factor map $q\colon(K,G)\to(L_{\mathrm{mon}},G)$ of $(K,G)$ and $p$ factorizes over $q$:

[TABLE]

Moreover, since $(L_{\mathrm{mon}},G)$ corresponds to the monotone hull of $T_{p}(\uC(L))$, it is the smallest monotone factor of $(K,S)$ over which $p$ factorizes. It is not difficult to see that $(L_{\mathrm{mon}},G)$ is therefore isomorphic to the quotient of $(K,G)$ by the $G$-invariant equivalence relation

[TABLE]

that was apparently first considered in [McMahonWu1976, Definition 2.2] and is closed by [McMahonWu1976, Proposition 2.3] or the more general [HauserJaeger2017, Proposition 2.3]. We note for the next section that for separable $G$, \crefprop:mongen3f implies that $L_{\mathrm{mon}}$ is metrizable if and only if $L$ is. Combined with \crefthm:mefequi, this means that if $(L,G)$ is an equicontinuous metrizable factor of $(K,G)$, then so is $(L_{\mathrm{mon}},G)$.

3. Equicontinuous factors

Given a quotient map $p\colon X\to Y$ of topological spaces, it is generally very difficult to relate geometric properties of $X$ to those of $Y$. The Hahn-Mazurkiewicz theorem illustrates how hopeless the situation is in general: It shows that every non-empty, connected, locally connected, compact metric space is the quotient of the unit interval. Considering that this includes, in particular, all compact manifolds, it is clear that additional properties of $p$ are needed in order to relate the geometric structure of $X$ to that of $Y$. The following theorem shows that monotonicity is such a property.

Theorem 3.1** ([Calcut2012, Theorem 1.1]).**

Let $f\colon(X,x_{0})\to(Y,y_{0})$ be a quotient map of pointed topological spaces, where $X$ is locally path-connected and $Y$ is semi-locally simply-connected. If each fiber $f^{-1}(y)$ is connected, then the induced homomorphism $f_{*}\colon\pi_{1}(X,x_{0})\to\pi_{1}(Y,y_{0})$ of the fundamental groups is surjective.

Combining this with the previous discussion, we obtain our main representation result for equicontinuous factors.

Theorem 3.2**.**

Let $(K,G)$ be a flow such that $K$ is locally path-connected with finite first Betti number $\ub_{1}(K)$, $G$ is abelian, and such that $K$ is metrizable or $G$ is separable. If all $G$-invariant functions $f\in\uC(K)$ are constant, then every equicontinuous factor of $(K,G)$ is isomorphic to a minimal flow on some compact abelian Lie group of dimension less than $\nicefrac{\ub_{1}(K)}{\ub_{0}(K)}$. More precisely, for every equicontinuous factor $(L,G)$ of $(K,G)$, there are a finite abelian group $F$ of order $|F|\leq\ub_{0}(K)$ and an $m\leq\nicefrac{\ub_{1}(K)}{\ub_{0}(K)}$ such that $(L,G)$ is isomorphic to a minimal action of $G$ on $F\times\T^{m}$ via rotations.

Proof 3.3**.**

We assume that $(M,G)$ is a monotone equicontinuous factor of $(K,G)$ which will imply the claim for every other equicontinuous factor. Denote by $\theta\colon(K,G)\to(M,G)$ the corresponding factor map and note that $(M,G)$ is minimal since an equicontinuous system is minimal if and only if every continuous $G$-invariant continuous function on it is constant, see \crefrem:mtf. Since $G$ is abelian and acts equicontinuously on $M$, the Ellis group $\uE(M,G)$ is a compact abelian group and it is well-known that a minimal equicontinuous flow $(M,G)$ with an abelian group $G$ is isomorphic to the minimal action $(\uE(M,G),G)$ of $G$ on the Ellis group $\uE(M,G)$ via rotation (see [Auslander1988, Theorem 3.6]). Since $\uE(M,G)\cong M$ is the quotient of a locally connected space, it follows by \creflem:locallyconnected that $\uE(M,G)$ is locally connected too.

First, assume $M$ and hence $\uE(M,G)$ to be metrizable. It then follows from the classification of locally connected, second-countable, compact abelian groups that $\uE(M,G)\cong F\times\T^{I}$ for a finite group $F$ and an (at most countable) set $I$, see [Hofmann2013, Theorem 8.34] and [Hofmann2013, Theorem 8.46]. We again denote the induced map from $K$ to $F\times\T^{I}$ by $\theta$. Since $\theta$ is monotone by \crefthm:mefequi, $K$ and $F\times\T^{I}$ have the same number of connected components, and so $F$ is of order $\ub_{0}(K)$. Since $G$ acts minimally on $F\times\T^{I}$ via the isomorphism $\uE(M,G)\cong F\times\T^{I}$, it follows that $G$ acts transitively on $F$ and hence on the connected components of $K$. Therefore, if we fix the connected component $K_{0}\defeq\theta^{-1}(\{0\}\times\T^{I})$, then $\ub_{1}(K)=\ub_{0}(K)\ub_{1}(K_{0})$.

Next, we show that $I$ is finite by using \crefthm:calcut to show that $|I|\leq\nicefrac{\ub_{1}(K)}{\ub_{0}(K)}$, though we need to be careful since $\T^{I}$ is semi-locally simply connected if and only if $I$ is finite. We therefore proceed by considering monotone finite-dimensional quotients: For $k\in\N$ with $k\leq|I|$, let $i_{1},\dots,i_{k}\in I$ be pairwise different and denote by $p_{k}\colon\T^{I}\to\T^{k}$ the canonical projection induced by the isomorphism $\T^{k}\cong\T^{\{i_{1},\dots,i_{k}\}}$. Moreover, let $\theta_{0}\colon K_{0}\to\T^{I}$ be the map canonically induced by $\theta$. Then $p_{n}\circ\theta_{0}\colon K_{0}\to\T^{k}$ is monotone by \crefprop:monotonechar3b, being the composition of monotone maps. Since $\T^{k}$ is semi-locally simply connected, \crefthm:calcut shows that $p_{k}\circ\theta_{0}$ induces a surjective morphism $(p_{k}\circ\theta_{0})_{*}\colon\pi_{1}(K_{0})\to\pi_{1}(\T^{n})$. Since $\pi_{1}(\T^{n})\cong\Z^{k}$ is abelian, this morphism factorizes through the abelianization of $\pi_{1}(K_{0})$, which is canonically isomorphic to $H_{1}(K_{0})$ by the Hurewicz theorem. If we denote by $\eta\colon H_{1}(K_{0})\to\pi_{1}(\T^{k})$ the induced surjective group homomorphism, then

[TABLE]

Since $k\in\N$ was arbitrary with $k\leq|I|$, this shows that $|I|\leq\nicefrac{\ub_{1}(K)}{\ub_{0}(K)}$.

Now we show that $(M,G)$ is necessarily metrizable which we only need to check for the maximal equicontinuous factor $(K_{\mathrm{eq}},G)$. By \crefexample:factor, $K_{\mathrm{eq}}$ is metrizable if and only if the subalgebra

[TABLE]

is separable. If $K$ is metrizable, this is the case for every $\uC^{*}$-subalgebra of $\uC(K)$, so assume instead that $G$ is separable. If $\mathcal{A}_{\mathrm{eq}}$ is not separable, \crefprop:mongen3f and \crefrem:mongen yield a sequence $(\mathcal{A}_{j})_{j\in\N}$ of strictly (!) increasing, separable, monotone, $G$-invariant $\uC^{*}$-subalgebras of $\mathcal{A}_{\mathrm{eq}}$ which induces the following commutative diagram of factor maps:

[TABLE]

Since each of the systems $(L_{j},G)$ is metrizable and the factor maps $\rho_{j}\circ\theta$ are monotone, the above discussion applies and we can therefore replace the diagram with

[TABLE]

where $m_{j}\in\{1,\dots,\nicefrac{\ub_{1}(K)}{\ub_{0}(K)}\}$ and $F_{j}$ is a finite abelian group for each $j\in\N$. Since each of the systems $(F_{j}\times\T^{m_{j}},G)$ is minimal and $G$ acts via rotations, each $r_{j}$ is a surjective group homomorphism and so \creftorus below implies that $m_{j-1}\leq m_{j}$. Moreover, since $\rho_{j-1}\circ\theta$ is monotone by the choice of $\mathcal{A}_{j-1}$, it follows that $\rho_{j-1}$ is monotone for each $j\in\N$. But if $\rho_{j-1}=r_{j}\circ\rho_{j}$ is monotone, it also follows that $r_{j}$ is monotone for each $j\in\N$.

Since $m_{j}\leq\nicefrac{\ub_{1}(K)}{\ub_{0}(K)}$ for each $j\in\N$, there can be only finitely many $j$ such that $m_{j}<m_{j+1}$. In particular, there is a $J\in\N$ such that $m_{j}=m_{j+1}$ for each $j>J$. However, if $m_{j}=m_{j+1}$, then $\mathcal{A}_{j}=\mathcal{A}_{j+1}$: A surjective group homomorphism on $F\times\T^{m_{j+1}}$ must have finite kernel by \creftorus and can therefore only be monotone if its kernel is trivial, i.e., if it is an isomorphism. This contradicts the strict inclusion $\mathcal{A}_{j}\subsetneq\mathcal{A}_{j+1}$ and shows that $\mathcal{A}_{\mathrm{eq}}$ must be separable. Hence, $K_{\mathrm{eq}}$ is metrizable.

Now suppose $(L,G)$ is an arbitrary equicontinuous factor of $(K,G)$ and let $(K_{\mathrm{eq}},G)$ be the maximal equicontinuous factor of $(K,G)$. Then as in the monotone case, it follows that $(L,G)\cong(\uE(L,G),G)$ and since the factor map $(K_{\mathrm{eq}},G)\to(L,G)$ induces a surjective group homomorphism $r\colon\uE(K_{\mathrm{eq}},G)\to\uE(L,G)$, the claim follows from the monotone case via \creftorus.

Lemma 3.4**.**

Let $m\in\N$, $F$ be a finite abelian group, and $r\colon F\times\T^{m}\to H$ be a continuous, surjective group homomorphism onto a compact group $H$. Then $H\cong F^{\prime}\times\T^{n}$ for some finite abelian group $F^{\prime}$ of order $|F^{\prime}|\leq|F|$ and $n\leq m$. Moreover, $n=m$ if and only if the kernel of $r$ is finite.

Proof 3.5**.**

If $G$ is a Lie group and $K\subset G$ is a closed normal subgroup, then $G/K$ carries a canonical differentiable structure turning $G/K$ into a Lie group and $\pi\colon G\to G/K$ into a submersion, see [Lee2012, Theorems 21.17 and 21.26]. Therefore, $H\cong F\times\T^{m}/\ker(r)$ is a Lie group of dimension less than $m$. Being the quotient of $F\times\T^{m}$, $H$ is a compact abelian Lie group and it is well-known (see [Sepanski2007, Theorem 5.2]) that this implies that $H\cong F^{\prime}\times\T^{n}$ for some finite abelian group $F^{\prime}$ and $n\in\N$, proving the first statement. If $n=m$, $r\colon F\times\T^{m}\to F^{\prime}\times\T^{n}$ is a submersion between manifolds of equal dimension and it thus follows from the inverse function theorem that it is in fact a local diffeomorphism. Therefore, the kernel of $r$ is discrete and since $F\times\T^{m}$ is compact, the kernel of $r$ must be finite. Conversely, if $\ker(r)$ is finite, $r$ is a local diffeomorphism and so $n=m$.

Many spaces satisfy the conditions of \crefmthm, including compact manifolds and finite CW complexes for which it follows from the Seifert-van Kampen theorem that their first Betti number is finite. In particular, we obtain the following generalization of [HauserJaeger2017, Theorem 3.1] to quotients of connected, simply connected Lie groups by discrete, cocompact subgroups. Important examples for such spaces are given by nilmanifolds, see [HostKra2018, Chapter 10].

Corollary 3.6**.**

Let $H$ be a connected, simply connected Lie group, $\Gamma\subset H$ a discrete, cocompact subgroup, $G$ an abelian group, and $(H/\Gamma,G)$ a dynamical system on $H/\Gamma$ such that every $G$-invariant function $f\in\uC(H/\Gamma,G)$ is constant. Then every equicontinuous factor of $(H/\Gamma,G)$ is isomorphic to a minimal action of $G$ on some torus $\T^{m}$, $m\leq\rank(\Gamma/[\Gamma,\Gamma])$, via rotations.

Proof 3.7**.**

The canonical map $H\to H/\Gamma$ is the universal cover of $H/\Gamma$ and it is well-known that its kernel is thus isomorphic to the fundamental group of $H/\Gamma$, i.e., $\pi_{1}(H/\Gamma)\cong\Gamma$. Since, $H$ being connected, $\ub_{0}(H/\Gamma)=1$ and

[TABLE]

the claim follows by \crefmthm.

In the special case $H=\R^{n}$ and $\Gamma=\Z^{n}$, this yields the following.

Corollary 3.8**.**

Let $(\T^{n},G)$ be a flow such that $G$ is abelian and all $G$-invariant functions $f\in\uC(\T^{n})$ are constant. Then every equicontinuous factor of $(\T^{n},G)$ is isomorphic to a minimal action of $G$ on some torus $\T^{m}$, $m\leq n$, via rotations.

Note, however, that in the case of the two-torus, [HauserJaeger2017, Theorem 3.8] is a lot more general: They show that each monotone, minimal quotient of a strongly effective flow on $\T^{2}$ is isomorphic to a flow on $\T^{2}$, $\T$, or a point. They require neither that the acting group $G$ be abelian nor that the factor under consideration be equicontinuous.

If $\ub_{1}(K)=0$, we also obtain the following for the maximal distal factor of a minimal homeomorphism.

Corollary 3.9**.**

Let $\phi\colon K\to K$ be a minimal homeomorphism on a locally path-connected space with $\ub_{1}(K)=0$. Then the maximal distal factor of $(K,\phi)$ is trivial.

Proof 3.10**.**

The maximal equicontinuous factor and maximal distal factor of $(K,\phi)$ are trivial as a consequence of \crefmthm and the Furstenberg structure theorem for distal minimal flows.

In particular, there are no minimal distal transformations on such a space $K$. This includes simply connected manifolds such as $S^{n}$ for $n\geq 2$ but also spaces for which $\pi_{1}(K)$ or $H_{1}(K)$ are torsion groups, e.g., $\R\uP^{n}$ for $n>1$ as $H_{1}(\R\uP^{n})\cong\Z/2$ for $n>1$. Note that a very similar result to \crefcor:b1zero using Čech cohomology exists in [KeynesRobertson1969, Theorem 3.5].

Corollary 3.11**.**

Let $\phi\colon K\to K$ be a homeomorphism on a locally path-connected, compact space with finite first Betti number $\ub_{1}(K)$ and suppose that $\dim\fix(T_{\phi})=1$. Then the point spectrum $\sigma_{\up}(T_{\phi})$ of the Koopman operator $T_{\phi}$ on $\uC(K)$ is a subgroup of $\T$ generated by at most $\ub_{1}(K)$ elements.

Proof 3.12**.**

Let $(M,\psi)$ be the maximal equicontinuous factor of $(K,\phi)$. By the discussion in \crefexample:factor, the point spectrum of $T_{\phi}$ on $\uC(K)$ is the same as that of $T_{\psi}$ on $\uC(M)$, so we only need to consider the system $(M,\psi)$. By \crefmthm, $(M,\psi)$ is isomorphic to a minimal rotation $(F\times\T^{n},a)$ for an abelian group $F$ of order $|F|\leq\ub_{0}(K)$, a torus $\T^{n}$ of dimension $n\leq\nicefrac{\ub_{1}(K)}{\ub_{0}(K)}$, and an $a\in F\times\T^{n}$. If we denote by $L_{a}\colon\uC(F\times\T^{n})\to\uC(F\times\T^{n})$ the Koopman operator corresponding to this rotation, then

[TABLE]

where $\widehat{F\times\T^{n}}$ denotes the Pontryagin dual of $F\times\T^{n}$ (see [EFHN2015, 14.24]). Since

[TABLE]

the claim follows from the inequalities $|F|\leq\ub_{0}(K)$ and $n\leq\nicefrac{\ub_{1}(K)}{\ub_{0}(K)}$.

Remark 3.13**.**

\cref

mthm imposes constraints on the maximal equicontinuous factor if it is minimal. In the case of $K=\T^{n}$, these constraints are sharp: Every torus $\T^{m}$ of dimension $m\leq n$ can be realized as the maximal equicontinuous factor of an invertible system $(\T^{n},\phi)$. To see this, let $(\T^{m},\phi_{a})$ be a minimal rotation with $a\in\T^{m}$ and let $\psi\colon\T^{n-m}\to\T^{n-m}$ be the map which is, after the identification $\T^{n-m}\cong[0,1)^{n-m}$, given by

[TABLE]

Then $(\T^{m},\phi_{a})$ is the maximal equicontinuous factor of $(\T^{n},\phi_{a}\times\psi)$.

A natural question now is whether the constraints on the maximal equicontinuous factor listed in \crefmthm are sharp in general. The answer is negative: Consider the wedge sum $K\defeq\bigvee_{i=1}^{n}\T$. Then $K$ is connected, locally connected, and $\ub_{1}(K)=n$. However, if $k>1$, there cannot be any monotone surjective map $\rho\colon K\to\T^{k}$ since $\T^{k}$ is the disjoint union of uncountably many connected nonsingleton sets whereas $K$ is not.

In light of this example, one might look for other topological constraints on the maximal equicontinuous factor and the covering dimension $\dim(K)$ of $K$ presents itself as a possible candidate. Unfortunately, monotonicity by itself cannot yield such a bound: As observed by Hurewicz in [Hurewicz1930], every compact metric space embeds into a monotone image of $S^{3}$. In particular, $S^{3}$ has monotone quotients of infinite dimension. Therefore, one cannot, in general, conclude that if $p\colon K\to L$ is a monotone quotient map, that $\dim(L)\leq\dim(K)$. Positive results exist for factors of distal minimal flows for which this estimate does hold as shown in [Rees1977, Theorem 1.1]. Without distality, results only exist in low dimensions: If $p\colon K\to L$ is a monotone quotient map of compact spaces and $K$ is a two-manifold, then $\dim(L)\leq\dim(K)$, see [Zemke1977]. In higher dimensions, though, one cannot hope for dimension inequalities for factors without additional structural assumptions.

Remark 3.14**.**

The commutativity of $G$ cannot be dropped in \crefmthm since any compact Lie group acts equicontinuously on itself. However, as mentioned in the introduction, the maximal equicontinuous factor of a distal minimal flow on a compact manifold is isomorphic to a compact abelian Lie group in the case of abelian $G$ and to a flow on a homogeneous space of some compact Lie group if $G$ is nonabelian. One could therefore ask whether \crefmthm generalizes to nonabelian groups $G$ in an anlogous way. Unfortunately, the proof given above hinges on the fact that the dimension of a compact abelian Lie group $H$ is precisely $\nicefrac{\ub_{1}(H)}{\ub_{0}(H)}$ and thus encoded in the first two homology groups, which is false for general compact Lie groups.