Periods and factors of weak model sets

TL;DR

This paper investigates the dynamical systems associated with weak model sets, identifying their maximal equicontinuous factors and conditions for measure-theoretic isomorphisms, with applications to B-free systems.

Contribution

It characterizes the maximal equicontinuous factors of weak model sets and establishes conditions for almost 1:1 extensions and measure-theoretic isomorphisms.

Findings

Identified the maximal equicontinuous factor for certain weak model sets.

Provided conditions under which the system is an almost 1:1 extension.

Established measure-theoretic isomorphism to the Kronecker factor under specific conditions.

Abstract

There is a renewed interest in weak model sets due to their connection to $\mathcal B$-free systems, which emerged from Sarnak's program on the M\"obius disjointness conjecture. Here we continue our recent investigation [arXiv:1511.06137] of the extended hull ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, a dynamical system naturally associated to a weak model set in an abelian group $G$ with relatively compact window $W$. For windows having a nowhere dense boundary (this includes compact windows), we identify the maximal equicontinuous factor of ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ and give a sufficient condition when ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ is an almost 1:1 extension of its maximal equicontinuous factor. If the window is measurable with positive Haar measure and is almost compact, then the system ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ equipped with its Mirsky measure is isomorphic to its Kronecker factor. For general nontrivial ergodic probability measures on ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, we provide a kind of lower bound for the Kronecker factor. All relevant factor systems are natural $G$-actions on quotient subgroups of the torus underlying the weak model set. These are obtained by factoring out suitable window periods. Our results are specialised to the usual hull of the weak model set, and they are also interpreted for ${\mathcal B}$-free systems.

Equations

$${\nu }_W(\hat{x}):=\sum _{y\in (x+\mathcal{L})\cap (G\times W)}{\delta }_y.$$

$${\nu }_W^G:={\pi }_{\ast }^G\circ {\nu }_W:\hat{X}\to {\mathcal{M}}^G.$$

$$H_A:=\left\{h\in H:h+A=A\right\}$$

$$H_W^{Haar}:=\{h\in H:m_H((h+W)△W)=0\}.$$

$${\mathcal{S}}_H(\nu )=\mathrm{supp}({\pi }_{\ast }^H(\nu ))=\overline{{\pi }^H(\mathrm{supp}(\nu ))}\subseteq W.$$

$${\Delta }_{\mathcal{B}}:\text{mathbbm}Z\to \prod _{b\in \mathcal{B}}\text{mathbbm}Z/b\text{mathbbm}Z,{\Delta }_{\mathcal{B}}(n)=(n\text{ mod }b{)}_{b\in \mathcal{B}},$$

$$W=\left\{h\in H:h_b\ne 0\forall b\in \mathcal{B}\right\}.$$

$$(X_{\eta }\times \{0,1{\}}^{\text{mathbbm}Z},\rho ,S\times S)$$

$$\begin{array}{rl}h(P_{max}^G)& =d({\mathcal{F}}_{\mathcal{B}})=Q_W^G\left\{\nu \in {\mathcal{M}}_W^G:\nu (\{0\})>0\right\}=m_{\hat{X}}\left\{\hat{x}\in \hat{X}:{\nu }_W^G(\hat{x})(\{0\})>0\right\}\\ & =m_H\left\{h\in H:\phi (h)(0)=1\right\}=m_H(W).\end{array}$$

$$C_W={\pi }^{\hat{X}}\left(\bigcap _{\ell \in \mathcal{L}}\left(G\times (\partial W{)}^c)-\ell \right)\right),$$

$$\nu \{x+\ell \}=1\iff {\pi }_{\ast }^G\nu \{x_G+{\ell }_G\}=1\iff {\pi }_{\ast }^G{\nu }^{\prime }\{x_G^{\prime }+{\ell }_G\}=1\iff {\nu }^{\prime }\{x^{\prime }+\ell \}=1.$$

$$\nu \{x^{\prime }+\ell -(0,d)\}=1\iff {\nu }^{\prime }\{x^{\prime }+\ell \}=1,$$

$${\mathcal{M}}^{\prime}\subseteq\bigcup_{\ell\in{\mathscr{L}}}\left\{\nu\in{\mathcal{M}}_{\scriptscriptstyle W}:\pi^{\scriptscriptstyle H}(\ell+\pi(\nu))\in B_{r}(h)\cap W_{P}\right\}=\bigcup_{\ell\in{\mathscr{L}}}\pi^{-1}\left(X\cap\bigg{(}\big{(}G\times(B_{r}(h)\cap W_{P})\big{)}-\ell\bigg{)}\right).$$

$$\begin{split}m_{G\times H}\left(X\cap\bigg{(}\big{(}G\times(B_{r}(h)\cap W_{P})\big{)}-\ell\bigg{)}\right)&\leqslant m_{G\times H}\bigg{(}\big{(}G\times(B_{r}(h)\cap W_{P})\big{)}-\ell\bigg{)}\\ &=m_{G\times H}\big{(}G\times(B_{r}(h)\cap W_{P})\big{)}\ ,\end{split}$$

$${\pi }_{\ast }^G\{\nu \in K:{\mathcal{S}}_H(\nu )\subseteq F\}={\pi }_{\ast }^G\{\nu \in K:{\mathcal{S}}_H(\nu )=F\}\cup \left(\bigcup _{n=1}^{\infty }{\pi }_{\ast }^G\{\nu \in K:{\mathcal{S}}_H(\nu )\subseteq F_n\}\right).$$

$${\pi }_{\ast }^G\{\nu \in K:{\mathcal{S}}_H(\nu )=F\}={\pi }_{\ast }^G\{\nu \in K:{\mathcal{S}}_H(\nu )\subseteq F\}\setminus \bigcup _{n=1}^{\infty }{\pi }_{\ast }^G\{\nu \in K:{\mathcal{S}}_H(\nu )\subseteq F_n\}.$$

$$\overline{d}(\nu ):=\underset{n\to \infty }{\mathrm{limsup}}\frac{\nu (A_n\times H)}{m_G(A_n)}\le \mathrm{dens}(\mathcal{L})\cdot m_H(W)$$

$$d(\nu ):=\underset{n\to \infty }{\lim }\frac{\nu (A_n\times H)}{m_G(A_n)}\text{ exists, and if }d(\nu )=\mathrm{dens}(\mathcal{L})\cdot m_H(W).$$

$$Q_W\left\{\nu \in {\mathcal{M}}_W:\text{ν has maximal density}\right\}=1,$$

$$\mathrm{dens}(\mathcal{L})\cdot m_H(W)=d(\nu )⩽\overline{d}({\nu }_{{\mathcal{S}}_H(\nu )}(\hat{x}))⩽\mathrm{dens}(\mathcal{L})\cdot m_H({\mathcal{S}}_H(\nu )),$$

$$0=m_H((d+{\mathcal{S}}_H({\nu }^{\prime }))△{\mathcal{S}}_H(\nu ))=m_H((d+W)△W).$$

$$P={\int }_{{\mathcal{M}}_W}{P}_{\mu }dP(\mu ),$$

$$P^G=P\circ ({\pi }_{\ast }^G{)}^{-1}={\int }_{{\mathcal{M}}_W}{P}_{\mu }\circ ({\pi }_{\ast }^G{)}^{-1}dP(\mu ),$$

$$P_e={\int }_{{\mathcal{M}}_W^G}{p}_{{\nu }^G}d{P}^G({\nu }^G)\text{and}{P}_{\mu }={\int }_{{\mathcal{M}}_W^G}{p}_{{\nu }^G}^{\prime }d{P}^G({\nu }^G).$$

$$P={\int }_{\kappa (H)}\tilde{P}d\tilde{\rho }(\tilde{P}),$$

$$P={\int }_{\kappa (H)}\kappa ({\kappa }^{†}(\tilde{P}))d\tilde{\rho }(\tilde{P})={\int }_H\kappa (h)d\rho (h)={\int }_H{P}_e\circ {\sigma }_h^{-1}d\rho (h).$$

$$h+\mathrm{int}(A)=\mathrm{int}(A)⟹h+\overline{\mathrm{int}(A)}=\overline{\mathrm{int}(A)}⟹h+\mathrm{int}(\overline{\mathrm{int}(A)})=\mathrm{int}(\overline{\mathrm{int}(A)}).$$

$$W^{\prime }=\phi (\overline{\mathrm{int}(W)})\subseteq \overline{\phi (\mathrm{int}(W))}\subseteq \overline{\mathrm{int}(\phi (W))}=\overline{\mathrm{int}(W^{\prime })},$$

$${\nu }_W(x+(0,h)+\mathcal{L})=\sum _{y\in (x+(0,h)+\mathcal{L})\cap (G\times W)}{\delta }_y=\sum _{y\in (x+\mathcal{L})\cap (G\times W)}{\delta }_{y+(0,h)}={\sigma }_h{\nu }_W(x+\mathcal{L}).$$

$${\nu }_{W^{\prime }}(x^{\prime }+{\mathcal{L}}^{\prime })=\sum _{y^{\prime }\in (x^{\prime }+{\mathcal{L}}^{\prime })\cap (G\times W^{\prime })}{\delta }_{y^{\prime }}=\sum _{\iota (y+{\mathcal{H}}_W)\in (x^{\prime }+{\mathcal{L}}^{\prime })\cap (G\times W^{\prime })}{\delta }_{\iota (y+{\mathcal{H}}_W)}=\sum _{y\in (x+\mathcal{L})\cap (G\times W)}{\delta }_{\iota (y+{\mathcal{H}}_W)},$$

Full text

††footnotetext: Key words: cut-and-project scheme, (weak) model set, torus parametrisation, Mirsky measure, ${\mathscr{B}}$-free systems††footnotetext: MSC 2010 classification: 52C23, 37A25, 37B10, 37B50

1 Introduction

Fix two locally compact second countable abelian groups $G$ and $H$. Typically, $G={\mathbbm{Z}}^{d}$ or ${\mathbbm{R}}^{d}$, whereas $H$ will often be a more general group. Take a cocompact lattice ${\mathscr{L}}\subseteq G\times H$ in generic position, i.e., ${\mathscr{L}}$ projects injectively to $G$ and densely to $H$. Consider a relatively compact and measurable subset $W$ of $H$ which is called the window. A weak model set $\Lambda\subset G$ is obtained by projecting all lattice points inside the strip $G\times W$ to $G$. The resulting set $\Lambda=\pi^{\scriptscriptstyle G}({\mathscr{L}}\cap(G\times W))$ is also called “cut-and-project set”, and $H$ is called “internal space”. Any weak model set is uniformly discrete. Model sets additionally require $\operatorname{int}(W)\neq\emptyset$, resulting in a relatively dense point set. They have been introduced by Meyer [26, 27] within a harmonic analysis context and, surprisingly, turned out later to describe physical quasicrystals. By now there is an abundant literature on model sets, see e.g. the list of references in [2]. Weak model sets have been initially studied by Schreiber [35, 36]. Their name was coined by Moody [29], see [15] for further background.

Dynamical systems techniques have turned out to be a powerful tool to analyse model sets [13, 34, 32, 8, 24]. Here one considers the hull of a model set, i.e., its translation orbit closure with respect to a Hausdorff-type metric, and one seeks to infer properties of the model set from its hull. For example, pure point diffraction spectrum of a model set can be inferred from (and is in fact equivalent to) pure point dynamical spectrum of its hull, equipped with its pattern frequency measure, see e.g. [7, Thm. 7]. For general so-called regular model sets, pure point dynamical spectrum of their hull was shown by Schlottmann [34] 111An earlier diffraction result by Hof [14] for regular model sets having Euclidean internal space relied on harmonic analysis techniques. This has been extended beyond the Euclidean setting only recently [31].. It is the aim of this article to perform a dynamical analysis for general weak model sets having a compact or close-to-compact window, thereby refining recent results [4, 22].

A natural dynamical question concerns the relation of the model set hull to its maximal equicontinuous factor and to its Kronecker factor, equipped with the pattern frequency measure. Since model sets inherently display a high degree of regularity, one is tempted to expect almost isomorphisms to these factors, under mild assumptions on the window. One might also expect that these factors are isomorphic to the “torus” ${\hat{X}}=(G\times H)/{\mathscr{L}}$. In this context, previous topological standard assumptions on the window were compactness, topological regularity $W=\overline{\operatorname{int}(W)}$ and aperiodicity, i.e., $h+W=W$ implies $h=0$, compare the discussion in [22, Sec. 4.3]. Note that, in Euclidean internal space, any window is aperiodic. Given these assumptions, the hull is an almost one-to-one extension of its maximal equicontinuous factor. This was shown in [32] for so-called non-singular model sets. For measure-theoretic results, a previous additional assumption was almost vanishing boundary of the window, resulting in a uniquely ergodic hull. In that case, isomorphism to the Kronecker factor has been shown in [34, 32]. In all cases, the relevant factor is indeed the underlying torus ${\hat{X}}$, and a factor map is provided by the so-called torus parametrisation [3, 8], i.e., a natural continuous map which assigns to any element of the hull its torus coordinate.

However, classic model sets such as the Fibonacci chain ($G={\mathbbm{R}}$ and $H=\mathbb{R}$, see [2, Ex. 7.3]) and their generalizations, the Sturmian chains, and also the discrete counterparts of these sets, namely Fibonacci and Sturmian sequences ($G={\mathbbm{Z}}$ and $H={\mathbbm{T}}$, see [30, Ch. 6]) are not subsumed by the above results, as their windows are half-open intervals. Compact internal spaces different from ${\mathbbm{T}}^{d}$ arise for other model sets which are subsets of lattices. Early examples are squarefree integers and visible lattice points [9]. These are not subsumed by the above results either, as their compact aperiodic windows have no interior points. This results in point sets having arbitrarily large holes. Other non-standard examples are non-regular Toeplitz sequences, having their odometer as internal space [6], and certain model sets having a window with fat boundary in Euclidean internal space [17].

The hull of squarefree integers is also of interest in number theory, as it is a factor of the Möbius function flow. It can thus be used to understand elementary properties of the Möbius function. This was made explicit in Sarnak’s influential article [33]. The same idea applies to more general ${\mathcal{B}}$-free systems [10], which were analysed from a dynamical perspective using arguments of arithmetic nature. On the other hand, ${\mathcal{B}}$-free systems are weak model sets with compact windows that may or may not have interior points, see subsection 3.3. Thus one might suspect that results about ${\mathcal{B}}$-free systems admit extensions to a suitable class of weak model sets and that, vice versa, a weak model set analysis could shed some additional light on ${\mathcal{B}}$-free systems from a geometric perspective.

Let us mention two recent approaches along these lines. In [4], squarefree integer hull results from [33] were formulated and proved for visible lattice points. This was extended to a larger class of weak model sets in [5]. Given a condition called maximal density, pure point diffractivity was shown for such weak model sets. As maximal density is satisfied for model sets with windows having an almost vanishing boundary, this generalises previous results about pure point diffraction. This was then used to infer pure point dynamical spectrum for the hull equipped with the pattern frequency measure. The proof used approximation by regular model sets, a technique inspired by [15].

In [22], the torus parametrisation was systematically re-investigated for weak model sets having compact windows. Results for the hull were deduced from a larger dynamical system ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, which one may call the extended hull. It is the translation orbit closure of the set of model sets arising from any window translate. Whereas this approach was already used in [32], properties of the extended hull were inferred from a related extended hull ${\mathcal{M}}_{\scriptscriptstyle W}$, where model sets are viewed as lattice subsets in $G\times H$. Under transparent assumptions on the window, the above almost isomorphisms were quite easily deduced. Properties of ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ could then systematically be deduced from those of ${\mathcal{M}}_{\scriptscriptstyle W}$ via a continuous factor map $\pi^{\scriptscriptstyle G}_{*}$, which describes the projection to $G$. If the above topological standard assumptions on the window are satisfied, then the map $\pi^{\scriptscriptstyle G}_{*}$ is indeed a homeomorphism [22, Prop. 4.8], so no information is lost. Measure-theoretic results were obtained for the Mirsky measure $Q_{\scriptscriptstyle W}^{\scriptscriptstyle G}$ on ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$. This measure is the lift of the Haar measure $m_{\hat{X}}$ on ${\hat{X}}$ to ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ and agrees with the pattern frequency measure [22, Rem. 3.12]. As the extended hull $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},Q_{\scriptscriptstyle W}^{\scriptscriptstyle G})$ is a measure-theoretic factor of the torus, pure point dynamical spectrum follows immediately [22, Thm. 2] 222The corresponding factor map $\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}$ was already systematically used for weighted model sets in [24], where it is continuous.. These results were then transferred to the usual hull of a model set. For measure-theoretic results the above maximal density condition turned out to be a sufficient condition ensuring isomorphism to the extended hull [22, Thm. 5].

In this article, the nature of $\pi^{\scriptscriptstyle G}_{*}$ is studied more systematically. This leads to a rather complete analysis of the maximal equicontinuous and Kronecker factors. For an initial discussion, let us restrict to compact windows. One of our results states that $\pi^{\scriptscriptstyle G}_{*}$ is a homeomorphism whenever $\operatorname{int}(W)$ is aperiodic. Hence for topological results the periods of $\operatorname{int}(W)$ appear to play a central role. Indeed we obtain almost isomorphism to the maximal equicontinuous factor of ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ by factoring from the group ${\hat{X}}$ the periods of $\operatorname{int}(W)$. These results transfer easily to the usual hull, see Theorem A2 and also Theorem A2’ for non-compact windows. If the window has no interior point, the maximal equicontinuous factor is trivial. For measure-theoretic results, we first restrict to the Mirsky measure $Q_{\scriptscriptstyle W}^{\scriptscriptstyle G}$ on ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$. Here a central role is played by the so-called Haar periods of $W$, i.e., by those $h\in H$ satisfying $m_{H}((h+W)\triangle W)=0$. Indeed the dynamical system $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},Q_{\scriptscriptstyle W}^{\scriptscriptstyle G})$ is isomorphic to its Kronecker factor, which is obtained from the torus ${\hat{X}}$ by factoring out the Haar periods of $W$, compare Theorem B2. Given the maximal density condition, these results can be transferred to the usual hull. For any ergodic invariant probability measure $P^{\scriptscriptstyle G}$ which is not supported on the zero configuration, we provide a kind of lower bound for the Kronecker factor of $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},P^{\scriptscriptstyle G})$ in Theorem C2.

Let us interpret the above results in terms of diffraction of a weak model set of maximal density. The Bragg peaks of the weak model set generate a countable discrete group [7, Thm. 9], whose dual is isomorphic to the Kronecker factor of its hull equipped with the Mirsky measure, see [23, Sec. 7] for details. This Kronecker factor is determined in Theorem B2. The amplitudes of the Bragg peaks appear as squared norms of corresponding eigenfunctions [23, Cor. 2]. The maximal equicontinuous factor of the hull, which is determined in Theorem A1, can be identified with the subgroup of eigenvalues which have continuous eigenfunctions. Apparently, the nature of the eigenfunctions is related to the different types of window periods. The amplitude of a Bragg peak can be computed as a certain average over the corresponding eigenvalue [23, Thm. 5(b)]. It would be interesting to analyse how continuity – or weakened versions thereof as in [21] – affect the type of convergence of these averages, compare [23, Thm. 5(c)].

Our paper is organised as follows. After the setting has been explained in Section 2, we give formally precise statements of our results in Section 3. There we also discuss applications of the results to ${\mathcal{B}}$-free systems. A more detailed discussion about non-trivial topological behaviour in ${\mathcal{B}}$-free systems appears in [19]. Section 4 studies the question of reconstructing from a given weak model set a suitable window. This leads to a proof of Theorem A1, assuming that $\operatorname{int}(W)$ is aperiodic. Proofs of measure-theoretic statements for aperiodic windows are then provided in Section 5. As a preparation for the proofs of the remaining statements, period groups and quotient cut-and-project schemes are studied in Section 6. The following section contains the proofs of Theorems A2, B2 and C2. The final section discusses relatively compact windows whose associated dynamical systems behave very similarly to the ones with compact windows.

2 The setting

The following point of view on extended weak model sets was developed in [22].

2.1 Assumptions and notations

Certain spaces and mappings are needed for the construction of weak model sets. As in [22] we make the following general assumptions.

1. (1)

$G$ and $H$ are locally compact second countable abelian groups with Haar measures $m_{G}$ and $m_{H}$. Then the product group $G\times H$ is locally compact second countable abelian as well, and we choose $m_{G\times H}=m_{G}\times m_{H}$ as Haar measure on $G\times H$. 2. (2)

${\mathscr{L}}\subseteq G\times H$ is a cocompact lattice, i.e., a discrete subgroup whose quotient space $(G\times H)/{\mathscr{L}}$ is compact. Thus $(G\times H)/{\mathscr{L}}$ is a compact second countable abelian group. Denote by $\pi^{\scriptscriptstyle G}:G\times H\to G$ and $\pi^{\scriptscriptstyle H}:G\times H\to H$ the canonical projections. We assume that $\pi^{\scriptscriptstyle G}|_{\mathscr{L}}$ is 1-1 and that $\pi^{\scriptscriptstyle H}({\mathscr{L}})$ is dense in $H$.333Denseness of $\pi^{\scriptscriptstyle H}({\mathscr{L}})$ can be assumed without loss of generality by passing from $H$ to the closure of $\pi^{\scriptscriptstyle H}({\mathscr{L}})$. In that case $m_{H}$ must be replaced by $m_{\,\overline{\pi^{\scriptscriptstyle H}({\mathscr{L}})}}$. 3. (3)

$G$ acts on $G\times H$ by translation: $T_{g}x=(g,0)+x$. 4. (4)

Let ${\hat{X}}:=(G\times H)/{\mathscr{L}}$. As we assumed that ${\hat{X}}$ is compact, there is a measurable relatively compact fundamental domain $X\subseteq G\times H$ such that $x\mapsto x+{\mathscr{L}}$ is a bijection between $X$ and ${\hat{X}}$. Elements of $G\times H$ (and hence also of $X$) are denoted as $x=(x_{G},x_{H})$, elements of ${\hat{X}}$ as ${\hat{x}}$ or as $x+{\mathscr{L}}=(x_{G},x_{H})+{\mathscr{L}}$, when a representative $x$ of ${\hat{x}}$ is to be stressed. We normalise the Haar measure $m_{\hat{X}}$ on ${\hat{X}}$ such that $m_{\hat{X}}({\hat{X}})=1$. Thus $m_{\hat{X}}$ is a probability measure. 5. (5)

The window $W$ is a measurable relatively compact subset of $H$. For our topological dynamical results, we first assume that the window $W$ is indeed compact and discuss extensions of the results to certain non-compact windows in Section 8. Our purely measure-theoretic results are first stated and proved for compact windows as well, but they extend easily to windows which agree modulo Haar measure zero with a compact one. Some further measure-theoretic results, which have an additional topological aspect, are only proved for compact windows.

2.2 Consequences of the assumptions

We list a few facts from topology and measure theory that follow from the above assumptions. We will call any neighborhood of the neutral element in an abelian topological group a zero neighborhood.

1. (1)

Being locally compact second countable abelian groups, $G$, $H$ and $G\times H$ are metrisable with a translation invariant metric with respect to which they are complete metric spaces. In particular they have the Baire property. As such groups are $\sigma$-compact, $m_{G}$, $m_{H}$ and $m_{G\times H}$ are $\sigma$-finite. 2. (2)

As $G\times H$ is $\sigma$-compact, the lattice ${\mathscr{L}}\subseteq G\times H$ is at most countable. Note that $G\times H$ can be partitioned by shifted copies of the relatively compact fundamental domain $X$. This means that ${\mathscr{L}}$ has a positive finite point density $\mathrm{dens}({\mathscr{L}})=1/m_{G\times H}(X)$. We thus have $m_{\hat{X}}(\hat{A})=\mathrm{dens}({\mathscr{L}})\cdot m_{G\times H}(X\cap(\pi^{\scriptscriptstyle{\hat{X}}})^{-1}(\hat{A}))$ for any measurable $\hat{A}\subseteq{\hat{X}}$, where $\pi^{\scriptscriptstyle{\hat{X}}}:G\times H\to{\hat{X}}$ denotes the quotient map. As a factor map between topological groups, $\pi^{\scriptscriptstyle{\hat{X}}}$ is continuous and open. 3. (3)

The action $\hat{T}_{g}:{\hat{x}}\mapsto(g,0)+{\hat{x}}$ of $G$ on ${\hat{X}}$ is minimal. This implies that ${\hat{X}}$ with its natural action is uniquely ergodic, see e.g. [29, Prop. 1]. 4. (4)

Denote by ${\mathcal{M}}$ and ${\mathcal{M}}^{\scriptscriptstyle G}$ the spaces of all locally finite measures on the Borel subsets of $G\times H$ and $G$, respectively. They are endowed with the topology of vague convergence and hence compact spaces. As $G$ and $G\times H$ are complete metric spaces, this topology is Polish, see [18, Thm. A.2.3].

2.3 The objects of interest

The pair $({\mathscr{L}},W)$ assigns to any point ${\hat{x}}\in{\hat{X}}$ a discrete point set in $G\times H$. We will identify such point sets $P$ with the measure $\sum_{y\in P}\delta_{y}\,\in{\mathcal{M}}$ and call these objects configurations. More precisely:

1. (1)

For ${\hat{x}}=x+{\mathscr{L}}\in{\hat{X}}$ define the configuration

[TABLE]

It is important to understand $\nu_{\scriptscriptstyle W}$ as a map from ${\hat{X}}$ to ${\mathcal{M}}$. If $W$ is compact, the map $\nu_{\scriptscriptstyle W}$ is upper semicontinuous [22, Prop. 3.3]. The canonical projection $\pi^{\scriptscriptstyle G}:G\times H\to G$ projects measures $\nu\in{\mathcal{M}}$ to measures $\pi^{\scriptscriptstyle G}_{*}\nu$ on $G$ defined by $\pi^{\scriptscriptstyle G}_{*}\nu(A):=\nu((\pi^{\scriptscriptstyle G})^{-1}(A))$. We abbreviate

[TABLE] 2. (2)

Denote by ${\mathcal{M}}_{\scriptscriptstyle W}$ the vague closure of $\nu_{\scriptscriptstyle W}({\hat{X}})$ in ${\mathcal{M}}$, and by ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ the vague closure of $\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}({\hat{X}})$ in ${\mathcal{M}}^{\scriptscriptstyle G}$. The group $G$ acts continuously by translations on all these spaces: $(S_{g}\nu)(A):=\nu(T_{g}^{-1}A)=\nu(T_{-g}A)$. Here we used the same notation $S_{g}$ for translations on ${\mathcal{M}}_{\scriptscriptstyle W}$ and ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, as the meaning will always be clear from the context. 3. (3)

As $\nu_{\scriptscriptstyle W}({\hat{x}})(T_{-g}A)=(S_{g}\nu_{\scriptscriptstyle W}({\hat{x}}))(A)=\nu_{\scriptscriptstyle W}(\hat{T}_{g}{\hat{x}})(A)$, it is obvious that all $\nu_{\scriptscriptstyle W}({\hat{x}})$ are uniformly translation bounded, and it follows from [7, Thm. 2] that all four spaces from item (2) are compact. 4. (4)

$Q_{\scriptscriptstyle W}:=m_{\hat{X}}\circ{\nu_{\scriptscriptstyle W}}^{-1}$ and $Q_{\scriptscriptstyle W}^{\scriptscriptstyle G}:=m_{\hat{X}}\circ(\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G})^{-1}$ are called Mirsky measures on ${\mathcal{M}}_{\scriptscriptstyle W}$ and ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, respectively. Note that $Q_{\scriptscriptstyle W}^{\scriptscriptstyle G}=Q_{\scriptscriptstyle W}\circ(\pi^{\scriptscriptstyle G}_{*})^{-1}$. 444These measures were denoted $Q_{{\mathcal{M}}}$ resp. $Q_{{\mathcal{M}}^{G}}$ in [22].

2.4 Previous results

For compact windows, Mirsky measures on ${\mathcal{M}}_{\scriptscriptstyle W}$ or on ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ were studied in quite some detail in [22]. The following property is immediate from measurability of the map $\nu_{\scriptscriptstyle W}:{\hat{X}}\to{\mathcal{M}}_{\scriptscriptstyle W}$ and from the definition of the Mirsky measure $Q_{\scriptscriptstyle W}$ on ${\mathcal{M}}_{\scriptscriptstyle W}$.

Proposition 2.1**.**

$({\mathcal{M}}_{\scriptscriptstyle W},Q_{\scriptscriptstyle W},S)$* is a measure-theoretic factor of $({\hat{X}},m_{\hat{X}},\hat{T})$. ∎*

In this article, we aim at statements concerning general invariant probability measures on ${\mathcal{M}}_{\scriptscriptstyle W}$ or on ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$. This is achieved using a partial inverse of $\nu_{\scriptscriptstyle W}$: Denote by $\underline{0}\in{\mathcal{M}}$ the zero measure (“empty configuration”). We have $\underline{0}\in{\mathcal{M}}_{\scriptscriptstyle W}$ if and only if $\operatorname{int}(W)=\emptyset$ by [22, Prop. 3.3]. Recall from [22, Lem. 5.4] that, for each $\nu\in{\mathcal{M}}_{\scriptscriptstyle W}\setminus\{\underline{0}\}$, there is a unique $\hat{\pi}(\nu)\in{\hat{X}}$, its “torus parameter”, such that $\operatorname{supp}(\nu)\subseteq\operatorname{supp}(\nu_{\scriptscriptstyle W}(\hat{\pi}(\nu)))$. This yields a continuous map $\hat{\pi}:{\mathcal{M}}_{\scriptscriptstyle W}\setminus\{\underline{0}\}\to{\hat{X}}$, and we have $\hat{\pi}\circ\nu_{\scriptscriptstyle W}=id_{\hat{X}}$ whenever this composition is well defined, compare [22, Lem. 5.6].

The following observation is a measure-theoretic analogue to Theorem 1a in [22]. Its proof, which is already implicit in the proof of [22, Thm. 2a], will be given in Subsection 8.2.

Proposition 2.2**.**

Assume that $P$ is any $S$-invariant probability measure on ${\mathcal{M}}_{\scriptscriptstyle W}$ satisfying $P({\mathcal{M}}_{\scriptscriptstyle W}\setminus\{\underline{0}\})=1$. Then $m_{H}(W)>0$, and $({\mathcal{M}}_{\scriptscriptstyle W},P,S)$ is a measure-theoretic extension of $({\hat{X}},m_{\hat{X}},\hat{T})$.

Specialising to the Mirsky measure, we can combine the above two propositions and recover the following result. For the convenience of the reader, its proof will be given in Subsection 8.2.

Proposition 2.3** (Theorem 2a in [22]).**

Assume that $m_{H}(W)>0$. Then $({\mathcal{M}}_{\scriptscriptstyle W},Q_{\scriptscriptstyle W},S)$ is measure-theoretically isomorphic to $({\hat{X}},m_{\hat{X}},\hat{T})$.

The projection $\pi^{\scriptscriptstyle G}:G\times H\to G$ induces a continuous factor map $\pi^{\scriptscriptstyle G}_{*}:({\mathcal{M}}_{\scriptscriptstyle W},S)\to({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},S)$, which is the object of interest in this article. In order to understand to which extent statements as in the above propositions carry over to the system $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},S)$, one has to understand the degree of (non)invertibility of $\pi^{\scriptscriptstyle G}_{*}$.

Recall that a window $W\subseteq H$ is (topologically) aperiodic or irredundant, if $h+W=W$ implies $h=0$. In particular, any aperiodic window is nonempty. A window $W\subseteq H$ is called topologically regular if $W=\overline{\operatorname{int}(W)}$. Note that if $W\neq\emptyset$ is topologically regular, then $m_{H}(W)>0$. The following fact is the essence of the results in [22, Sec. 4.3]:

Fact A**.**

$\pi^{\scriptscriptstyle G}_{*}:{\mathcal{M}}_{\scriptscriptstyle W}\to{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$* is a homeomorphism, whenever the window $W$ is aperiodic and topologically regular.*

This fact is substantially extended in Theorems A1, A2, A1’ and A2’.

We now turn to measures on ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$. For the Mirsky measure $Q_{\scriptscriptstyle W}^{\scriptscriptstyle G}$ on ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, Proposition 2.1 immediately implies that $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},Q_{\scriptscriptstyle W}^{\scriptscriptstyle G},S)$ is a measure-theoretic factor of $({\hat{X}},m_{\hat{X}},\hat{T})$. This implies pure point dynamical spectrum for the former system and, together with a condition called maximal density, pure point dynamical spectrum transfers to the usual hull [22, Cor. 6], compare Remark 3.6 and the result [5, Cor. 17]. But there is no statement analogous to Proposition 2.2. From Fact A and Proposition 2.3, however, we get the following result for the Mirsky measure $Q_{\scriptscriptstyle W}^{\scriptscriptstyle G}$ on ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$.

Fact B**.**

Suppose that $W$ is aperiodic and topologically regular. Then $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},Q_{\scriptscriptstyle W}^{\scriptscriptstyle G},S)$ is measure-theoretically isomorphic to $({\hat{X}},m_{\hat{X}},\hat{T})$. ∎

Theorems B1 and B2 extend this fact. The following is an immediate consequence of Fact A and Proposition 2.2. It will be extended in Theorems C1 and C2.

Fact C**.**

Suppose that $W$ is aperiodic and topologically regular. If $P^{\scriptscriptstyle G}$ is an $S$-invariant probability measure on ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, then $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},P^{\scriptscriptstyle G},S)$ is a measure-theoretic extension of $({\hat{X}},m_{\hat{X}},\hat{T})$. ∎

3 Main results

3.1 Topological results

In this subsection we assume that the window $W$ is compact. Extensions of the results to certain non-compact windows are discussed in Section 8. Our first main result strengthens Fact A considerably.

Theorem A1**.**

Assume that $W$ is compact and that $\operatorname{int}(W)$ is aperiodic (so in particular non-empty).

1. a)

The topological dynamical systems $({\mathcal{M}}_{\scriptscriptstyle W},S)$ and $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},S)$ are isomorphic, and both are almost 1-1 extensions of their maximal equicontinuous factor $({\hat{X}},\hat{T})$. 2. b)

Denote by $\Gamma:{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}\to{\hat{X}}$ the factor map from a). If $M$ is a non-empty, closed $S$-invariant subset of ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, then $(M,S)$ is an almost 1-1 extension of its maximal equicontinuous factor $({\hat{X}},\hat{T})$ with factor map $\Gamma|_{M}$.

Remark 3.1**.**

1. a)

The above result extends [22, Cor. 1], as aperiodicity of $\overline{\operatorname{int}(W)}$ implies aperiodicity of $\operatorname{int}(W)$ by Lemma 6.1a). In particular, the result applies to $M={\mathcal{M}}^{\scriptscriptstyle G}_{\scriptstyle min}$ the unique minimal subset of ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, see Remark 4.2, and to $M={\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}({\hat{x}}):=\overline{\{S_{g}\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}({\hat{x}}):g\in G\}}$ the so-called hull of $\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}({\hat{x}})\in{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$. 2. b)

Recall that $\operatorname{int}(W)\neq\emptyset$ is aperiodic whenever the only compact subgroup of $H$ is the trivial one, compare the proof of [22, Prop. 4.8]. This holds in particular for $H={\mathbbm{R}}^{d}$. 3. c)

Aperiodicity of $\operatorname{int}(W)$ implies aperiodicity of $W$. Examples of aperiodic $W$ with periodic interior are presented in [19]. 4. d)

For a topologically regular window, aperiodicity of $W$ and $\operatorname{int}(W)$ are equivalent by Lemma 6.1a). This gives the setting of an earlier result of Robinson [32, Th. 5.19, Cor. 5.20], see also [13, Prop. 7.3] in the Euclidean situation.

If $\operatorname{int}(W)$ has non-trivial periods, we still can determine the maximal equicontinuous factor of $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},S)$. Given a subset $A\subseteq H$, we call

[TABLE]

the period group of $A$. The set $A\subseteq H$ is (topologically) aperiodic, if $H_{A}=\{0\}$. The following result extends Theorem A1. We denote by ${\mathcal{H}}_{A}:=\{0\}\times H_{A}\subseteq G\times H$ the canonical embedding of $H_{A}$ into $G\times H$. Note that ${\mathcal{H}}_{\operatorname{int}(W)}$ is compact as $H_{\operatorname{int}(W)}=H_{\overline{\operatorname{int}(W)}}$ is a compact subgroup of $H$, compare Lemma 6.1c).

Theorem A2**.**

Assume that $W$ is compact and that $\operatorname{int}(W)\neq\emptyset$. Let ${\widehat{X^{\prime}}}={\hat{X}}/\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{\operatorname{int}(W)})$ with induced $G$-action $\widehat{T^{\prime}}$, and let $M$ be any non-empty, closed $S$-invariant subset of ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ (thus including the case $M={\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$).

1. a)

$({\widehat{X^{\prime}}},\widehat{T^{\prime}})$* is the maximal equicontinuous factor of the topological dynamical system $(M,S)$.* 2. b)

If $H_{\operatorname{int}(W)}=H_{W}$, then $(M,S)$ is an almost 1-1 extension of $({\widehat{X^{\prime}}},\widehat{T^{\prime}})$.

Remark 3.2**.**

1. a)

We do not know whether $H_{\operatorname{int}(W)}=H_{W}$ is also a necessary condition in part b) of the theorem. The condition is satisfied for topologically regular windows. 2. b)

If $\operatorname{int}(W)=\emptyset$, then the maximal equicontinuous factor of $(M,S)$ is trivial, see Remark 4.2. However statements analogous to the above two theorems hold for maximal equicontinuous generic factors [21].

3.2 Measure-theoretic results

In this subsection we assume that $W\subseteq H$ is relatively compact and measurable, but not necessarily compact. If $\operatorname{int}(W)=\emptyset$, one should not expect that $\pi^{\scriptscriptstyle G}_{*}$ is a homeomorphism. However, the measure-theoretic statements of Facts B and C can still be generalized substantially to windows where $\pi^{\scriptscriptstyle G}_{*}:{\mathcal{M}}_{\scriptscriptstyle W}\to{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ is not necessarily a homeomorphism, but still 1-1 on a sufficiently large subset of ${\mathcal{M}}_{\scriptscriptstyle W}$. This is achieved by replacing topological aperiodicity through a stronger measure-theoretic version. In the following definition, $\triangle$ denotes the symmetric set difference.

Definition 3.3** (Haar aperiodicity).**

A measurable set $A\subseteq H$ is Haar aperiodic, if $m_{H}((h+A)\triangle A)=0$ implies $h=0$.

Remark 3.4**.**

Any Haar aperiodic set $A$ satisfies $m_{H}(A)>0$ and is, in particular, nonempty. Haar aperiodicity implies topological aperiodicity, but the converse holds only for a more restricted class of windows, see Remark 3.12.

Definition 3.5**.**

A measurable subset $A$ of $H$ is compact modulo [math], if there is a compact set $K\subseteq H$ such that $m_{H}(A\triangle K)=0$.

We have the following results for the Mirsky measure $Q_{\scriptscriptstyle W}^{\scriptscriptstyle G}$ on ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ , which generalise Fact B. They are proved in Section 5.

Theorem B1**.**

Suppose that $W$ is compact modulo [math] and Haar aperiodic. Then $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},Q_{\scriptscriptstyle W}^{\scriptscriptstyle G},S)$ is measure-theoretically isomorphic to $({\hat{X}},m_{\hat{X}},\hat{T})$.

Here is an extension of Theorem B1 to windows that are not Haar aperiodic. The proof is provided in Section 7 after some preparations in Section 6. To formulate the statement, we consider the group $H_{W}^{Haar}$ of Haar periods of $W$, i.e.,

[TABLE]

We write ${\mathcal{H}}_{W}^{Haar}=\{0\}\times H_{W}^{Haar}$ for the canonical embedding of $H_{W}^{Haar}$ into $G\times H$.

Theorem B2**.**

Suppose that $W$ is compact modulo [math] and $m_{H}(W)>0$. Let ${\widehat{X^{\prime}}}={\hat{X}}/\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{W}^{Haar})$ with induced $G$-action $\widehat{T^{\prime}}$ and Haar measure $m_{{\widehat{X^{\prime}}}}$. Then the measure-theoretic dynamical system $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},Q_{\scriptscriptstyle W}^{\scriptscriptstyle G},S)$ is isomorphic to $({\widehat{X^{\prime}}},m_{\widehat{X^{\prime}}},\widehat{T^{\prime}})$.

Remark 3.6**.**

In order to transfer these results to hulls ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}({\hat{x}})\subseteq{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, we may assume that $\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}({\hat{x}})$ has maximal density, compare Remark 8.7. In that case $Q_{\scriptscriptstyle W}^{\scriptscriptstyle G}({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}({\hat{x}}))=1$, see [22, Cor. 5], and the two systems $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}({\hat{x}}),Q_{\scriptscriptstyle W}^{\scriptscriptstyle G},S)$ and $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},Q_{\scriptscriptstyle W}^{\scriptscriptstyle G},S)$ are measure-theoretically isomorphic.

The two final results refer to general invariant measures on ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ and generalise Fact C. They are of measure-theoretic nature, but they provide information about arbitrary ergodic measures on ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ - the topological closure of $\nu_{\scriptscriptstyle W}({\hat{X}})$. Therefore they are stated and proved only for compact windows. Hence in the remainder of this subsection we assume that $W\subseteq H$ is compact.

Before we can state the results, we need to introduce one more concept. For each $\nu\in{\mathcal{M}}_{\scriptscriptstyle W}$, $\pi^{\scriptscriptstyle H}_{*}\nu$ is a measure555 Note that generally $\pi^{\scriptscriptstyle H}_{*}\nu$ is not a Borel measure as $\nu$ may be an unbounded configuration and the topological support of $\pi^{\scriptscriptstyle H}_{*}\nu$ lies inside the compact set $W$. on $H$, and we denote the topological support of this measure by ${\mathcal{S}}_{\scriptscriptstyle H}(\nu)$. We thus have

[TABLE]

Remark 3.7**.**

1. a)

Clearly ${\mathcal{S}}_{\scriptscriptstyle H}(\underline{0})=\emptyset$. Recall that there is a unique ${\hat{x}}\in{\hat{X}}$ such that $\nu\leq\nu_{\scriptscriptstyle\overline{W}}({\hat{x}})$ if $\nu\neq\underline{0}$. The set ${\mathcal{S}}_{\scriptscriptstyle H}(\nu)$ is the smallest compact set $W^{\prime}\subseteq W$ such that $\nu\leq\nu_{\scriptscriptstyle W^{\prime}}({\hat{x}})$. 2. b)

We have $\overline{\operatorname{int}(W)}\subseteq{\mathcal{S}}_{\scriptscriptstyle H}(\nu)\subseteq W$ for any $\nu\in{\mathcal{M}}_{\scriptscriptstyle W}$ by Lemma 4.3. The lower bound is attained for any continuity point ${\hat{x}}=x+{\mathscr{L}}$ of the map $\nu_{\scriptscriptstyle W}$, because ${\mathcal{S}}_{\scriptscriptstyle H}(\nu_{\scriptscriptstyle W}({\hat{x}}))=\overline{\operatorname{int}(W)\cap\pi^{\scriptscriptstyle H}(x+{\mathscr{L}})}=\overline{\operatorname{int}(W)}$ for such ${\hat{x}}$ by Eqn. (6) below. 3. c)

In Lemma 4.5 we prove: For each ergodic $S$-invariant probability measure $P$ on ${\mathcal{M}}_{\scriptscriptstyle W}$, there is a compact subset $W_{P}\subseteq W$ of $H$ such that ${\mathcal{S}}_{\scriptscriptstyle H}(\nu)=W_{P}$ for $P$-a.a. $\nu$. It should be no surprise that $m_{H}(W\triangle W_{Q_{\scriptscriptstyle W}})=0$ when $W$ is compact modulo [math], see Corollary 5.2.

The first result is a consequence of Proposition 5.5 below.

Theorem C1**.**

Suppose that $W$ is compact and $m_{H}(W)>0$. Let $P^{\scriptscriptstyle G}$ be an ergodic $S$-invariant probability measure on ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ , and let $P$ be any ergodic $S$-invariant probability measure on ${\mathcal{M}}_{\scriptscriptstyle W}$ satisfying $P^{\scriptscriptstyle G}=P\circ(\pi^{\scriptscriptstyle G}_{*})^{-1}$. (There always exists at least one such measure $P$, see Proposition 5.5.) Suppose that $W_{P}$ is aperiodic. (In particular $W_{P}\neq\emptyset$, i.e. $P({\mathcal{M}}_{\scriptscriptstyle W}\setminus\{\underline{0}\})=P^{\scriptscriptstyle G}({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}\setminus\{\underline{0}\})=1$.) Then $\pi^{\scriptscriptstyle G}_{*}$ is a measure-theoretic isomorphism between $({\mathcal{M}}_{\scriptscriptstyle W},P,S)$ and $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},P^{\scriptscriptstyle G},S)$, and $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},P^{\scriptscriptstyle G},S)$ is a measure-theoretic extension of $({\hat{X}},m_{\hat{X}},\hat{T})$.

Remark 3.8**.**

1. a)

As $W_{Q_{\scriptscriptstyle W}}$ is aperiodic iff $W$ is Haar aperiodic by Remark 3.12 and Corollary 5.2, the above statement is consistent with Theorem B1. Aperiodicity of $W_{P}$ holds in ${\mathbbm{R}}^{d}$, see Remark 3.1. 2. b)

The above result does not depend on the choice of $P$. Indeed we have $H_{W_{P}}=H_{W_{P^{\prime}}}$, whenever $P$ and $P^{\prime}$ are ergodic $S$-invariant measures on ${\mathcal{M}}_{\scriptscriptstyle W}$ with $P\circ(\pi^{\scriptscriptstyle G}_{*})^{-1}=P^{\prime}\circ(\pi^{\scriptscriptstyle G}_{*})^{-1}$. This holds as there is $d\in H$ such that $W_{P}=W_{P^{\prime}}+d$ by Lemma 4.5. Hence, if $h\in H_{W_{P^{\prime}}}$, then $h+W_{P}=h+(W_{P^{\prime}}+d)=(h+W_{P^{\prime}})+d=W_{P^{\prime}}+d=W_{P}$, so that $H_{W_{P^{\prime}}}\subseteq H_{W_{P}}$, and the reverse inclusion follows from interchanging the roles of $P$ and $P^{\prime}$.

Again there is a periodic generalisation of this theorem, which is proved in Section 7.

Theorem C2**.**

Suppose that $W$ is compact and $m_{H}(W)>0$. Let $P^{\scriptscriptstyle G}$ be an ergodic $S$-invariant probability measure on ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}\setminus\{\underline{0}\}$. Take any ergodic $S$-invariant probability measure $P$ on ${\mathcal{M}}_{\scriptscriptstyle W}$ satisfying $P^{\scriptscriptstyle G}=P\circ(\pi^{\scriptscriptstyle G}_{*})^{-1}$. (There always exists at least one such measure $P$, see Proposition 5.5.) Let ${\widehat{X^{\prime}}}={\hat{X}}/\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{W_{P}})$ with induced $G$-action $\widehat{T^{\prime}}$ and Haar measure $m_{{\widehat{X^{\prime}}}}$. Then $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},P^{\scriptscriptstyle G},S)$ is a measure-theoretic extension of $({\widehat{X^{\prime}}},m_{\widehat{X^{\prime}}},\widehat{T^{\prime}})$.

Remark 3.9**.**

The above result does not depend on the choice of $P$, see Remark 3.8b).

For some of the proofs and applications, the following notion of Haar regularity appears to be relevant, compare Lemma 4.5. It is a measure-theoretic substitute for topological regularity.

Definition 3.10** (Haar regularity).**

Consider some compact subset $K$ of $H$.

1. a)

The compact set $K_{reg}$, defined as the topological support of the measure $(m_{H})|_{K}$, is called the Haar regularization of $K$. 2. b)

The set $K$ is Haar regular, if $K=K_{reg}$.

Remark 3.11**.**

1. a)

A compact set $K\subseteq H$ is Haar regular if and only if for every open $U\subseteq H$ such that $U\cap K\neq\emptyset$ we have $m_{H}(U\cap K)>0$. 2. b)

$\overline{\operatorname{int}(K)}\subseteq K_{reg}\subseteq K$ and $m_{H}(K\setminus K_{reg})=0$ by definition. 3. c)

The empty set is Haar regular. If $K\neq\emptyset$ is Haar regular, then $m_{H}(K)>0$. 4. d)

If $K$ is Haar regular, then any translate of $K$ is Haar regular. If $K$ is topologically regular, then $K$ is Haar regular.

Remark 3.12**.**

(Periods and Haar periods)

A compact set $W$ is Haar aperiodic if and only if $W_{reg}$ is aperiodic. More generally, $H_{W}^{Haar}=H_{W_{reg}}$. Indeed, as $W_{reg}\subseteq W$, we have $h+W_{reg}=W_{reg}$ if and only if $m_{H}((h+W)\triangle W)=0$, which implies the claimed equivalence. In particular, every aperiodic Haar regular window is Haar aperiodic. Examples are Haar regular windows in ${\mathbbm{R}}^{d}$ or the windows defining taut ${\mathcal{B}}$-free systems, see Subsection 3.3.

Remark 3.13**.**

(Haar regularity and compactness modulo [math])

If $A\subseteq H$ is compact modulo [math], then there is a compact set $K\subseteq H$ such that $m_{H}(A\triangle K)=0$. Hence also $m_{H}(A\triangle K_{reg})=0$, and $K_{reg}$ is the unique Haar regular set with this property. Therefore we denote it by $A_{reg}$. Observe that $A_{reg}=\operatorname{supp}((m_{H})|_{A})$.

3.3 Applications to ${\mathcal{B}}$-free dynamics

General ${\mathcal{B}}$-free dynamical systems were studied in [10]. They are a special case of our systems $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},S)$, when $G={\mathbbm{Z}}$ and $H$ is a particular compact group constructed from the given set ${\mathcal{B}}\subseteq{\mathbbm{N}}$. In this setting a configuration $\nu^{\scriptscriptstyle G}=\sum_{n\in A}\delta_{n}\in{\mathcal{M}}^{\scriptscriptstyle G}$, where $A$ is a subset of $G={\mathbbm{Z}}$, can be identified with the characteristic function $\chi_{A}$ interpreted as an element of $\{0,1\}^{\mathbbm{Z}}$. Our Theorem B1 reproduces Theorem F of [10] in this context, i.e. the measure-theoretic dynamical systems $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},Q_{\scriptscriptstyle W}^{\scriptscriptstyle G},S)$ and $({\hat{X}},m_{{\hat{X}}},\hat{T})$ are isomorphic.

Since the connection with [10, Theorem F] is not completely obvious, we give some explanation: We assume without loss of generality that the set ${\mathcal{B}}$ is primitive, i.e. that no number from ${\mathcal{B}}$ is a multiple of another number from ${\mathcal{B}}$. The following group homomorphism is associated with the set ${\mathcal{B}}$:

[TABLE]

and $H$ is the topological closure of $\Delta_{\mathcal{B}}({\mathbbm{Z}})$. The lattice is ${\mathscr{L}}=\left\{(n,\Delta_{\mathcal{B}}(n))\in{\mathbbm{Z}}\times H:n\in{\mathbbm{Z}}\right\}$, and a moment’s reflection shows that the group ${\hat{X}}=({\mathbbm{Z}}\times H)/{\mathscr{L}}$ is isomorphic to $H$. The window is the compact set defined as

[TABLE]

With this notation, an integer $n$ is ${\mathcal{B}}$-free, i.e. is not divisible by any number $b\in{\mathcal{B}}$, if and only if $\Delta_{\mathcal{B}}(n)\in W$. Hence $\Delta_{\mathcal{B}}^{-1}(W)$ is precisely the set of ${\mathcal{B}}$-free integers, and $\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}(0)$ coincides with the respective configuration $\sum_{n\in\Delta_{\mathcal{B}}^{-1}(W)}\delta_{n}$. The window $W$ is topologically regular if and only if the set ${\mathcal{B}}$ contains no scaled copy of an infinite pairwise coprime set [19, Theorem B]. A most prominent example of this type is the set ${\mathcal{B}}$ consisting of the squares of all primes, in which case $\Delta_{\mathcal{B}}^{-1}(W)$ is the set of all squarefree integers.

The authors of [10] study, in our notation, the dynamics of $S$ restricted to $\overline{\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}(\Delta_{\mathcal{B}}({\mathbbm{Z}}))}$, and denote this set by $X_{\eta}$. Our $\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}:{\hat{X}}\to{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ corresponds to their $\varphi:H\to\{0,1\}^{\mathbbm{Z}}$ [10, Section 2.7]666Observe that the compact group $H$ is denoted by $G$ in [10]., and so their definition of a Mirsky measure $\nu_{\eta}$ [10, Section 2.9] translates precisely to our $Q_{\scriptscriptstyle W}^{\scriptscriptstyle G}$.

Finally observe that the tautness assumption on ${\mathcal{B}}$ in [10, Thm. F] is equivalent to the Haar regularity of the window $W$ defined in Eqn. (4), see [19, Thm. A], and that $W$ is Haar aperiodic by Remark 3.12, because it is Haar regular and aperiodic [19, Prop. 5.1].

In view of the preceding discussion, also our Theorem C1 applies to ${\mathcal{B}}$-free systems. It complements Theorem I from [10], which we recall here using our notation:777To be precise, [10, Thm. I] is stated for general, not necessarily ergodic measures $P^{\scriptscriptstyle G}$, and it applies also to measures not supported by $X_{\eta}$, but by its hereditary closure.

For any ergodic $S$-invariant probability measure $P^{\scriptscriptstyle G}$ on $X_{\eta}$, there exists an $S$-invariant probability measure $\rho$ on $X_{\eta}\times\{0,1\}^{\mathbbm{Z}}$ whose first marginal is $Q_{\scriptscriptstyle W}^{\scriptscriptstyle G}$ and such that $\rho\circ M^{-1}=P^{\scriptscriptstyle G}$, where $M:X_{\eta}\times\{0,1\}^{\mathbbm{Z}}\to\{0,1\}^{\mathbbm{Z}}$ stands for the coordinatewise multiplication.

Together with our Theorem C1, which adds the lower arrow, this yields the following commutative diagram for measures $P^{\scriptscriptstyle G}$ with an aperiodic associated window $W_{P}$:

[TABLE]

In [10, Thm. 8.2] the authors prove that the system $(X_{\eta},S)$ has a unique invariant measure $P^{\scriptscriptstyle G}_{max}=\rho_{max}\circ M^{-1}$ of maximal entropy, whenever the set ${\mathcal{B}}$ has light tails888This means that for all $\epsilon>0$ there is $k_{0}\in{\mathbbm{N}}$ such that for all $k\geqslant k_{0}$ the asymptotic upper density of the set $\bigcup_{b\in{\mathcal{B}},b>k}b{\mathbbm{Z}}$ does not exceed $\epsilon$. It is known that the asymptotic density $d({\mathcal{F}}_{\mathcal{B}})=\lim_{N\to\infty}N^{-1}\operatorname{card}\left({\mathcal{F}}_{\mathcal{B}}\cap\{1,\dots,N\}\right)$ exists when ${\mathcal{B}}$ has light tails, and hence coincides with the so called logarithmic density of this set [10, Remark 2.29]. and contains an infinite pairwise coprime subset. In fact, $\rho_{max}=Q_{\scriptscriptstyle W}^{\scriptscriptstyle G}\times B(1/2,1/2)$, where $B(1/2,1/2)$ denotes the Bernoulli - $(1/2,1/2)$ measure on $\{0,1\}^{\mathbbm{Z}}$. Here we complement their result by noticing that diagram (5) applies to $P^{\scriptscriptstyle G}_{max}$:

Proposition 3.14**.**

If ${\mathcal{B}}$ has light tails and contains an infinite pairwise coprime subset,999Note added in proof:* These assumptions are only used to show that the subshift $X_{\eta}$ is hereditary, i.e. that $M(X_{\eta}\times\{0,1\}^{\mathbbm{Z}})\subseteq X_{\eta}$ [10, Theorem D]. More recently [20, Theorem 3], the same conclusion was reached when the assumption ,,${\mathcal{B}}$ has light tails” is replaced by the weaker ,,${\mathcal{B}}$ is taut”, which is equivalent to Haar regularity of the window $W$ defined in (4). then the dynamical system $(X_{\eta},P^{\scriptscriptstyle G}_{max},S)$ is a measure-theoretic extension of $({\hat{X}},m_{\hat{X}},\hat{T})$.*

Proof.

We must show that the window $W_{P_{max}}$ is aperiodic: The entropy $h(P^{\scriptscriptstyle G}_{max})$ coincides with the topological entropy of the system, which in turn equals the asymptotic density $d({\mathcal{F}}_{\mathcal{B}})$ of the set ${\mathcal{F}}_{\mathcal{B}}$ of ${\mathcal{B}}$-free numbers, when the logarithm to base $2$ is used to compute the entropy, see [10, Thm. D and Prop. K]. Remark 4.2 of [10] then shows that

[TABLE]

Fix any ergodic invariant measure $P_{max}$ on ${\mathcal{M}}_{\scriptscriptstyle W}$ that projects to $P^{\scriptscriptstyle G}_{max}$ (compare Proposition 5.5a). This measure is supported by the set $\widetilde{{\mathcal{M}}_{\scriptscriptstyle W^{\prime}}}:=\{\nu\in{\mathcal{M}}_{\scriptscriptstyle W}:\exists{\hat{x}}\in{\hat{X}}\text{ s.t. }\nu\leqslant\nu_{\scriptscriptstyle W^{\prime}}({\hat{x}})\}$, where the window $W^{\prime}$ is defined as $W^{\prime}=W_{P_{max}}\subseteq W$, and so $P^{\scriptscriptstyle G}_{max}$ is supported by $\widetilde{{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W^{\prime}}}:=\{\nu^{\scriptscriptstyle G}\in{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}:\exists{\hat{x}}\in{\hat{X}}\text{ s.t. }\nu^{\scriptscriptstyle G}\leqslant\nu_{\scriptscriptstyle W^{\prime}}^{\scriptscriptstyle G}({\hat{x}})\}$.

Let $\epsilon>0$. There is $N\in{\mathbbm{N}}$ such that $\operatorname{card}\{k\in{\mathbbm{N}}:|k|\leqslant n,\nu^{\scriptscriptstyle G}\{k\}=1\}\leqslant n\cdot(m_{H}(W^{\prime})+\epsilon)$ for all $n\geqslant N$ and all $\nu^{\scriptscriptstyle G}\in\nu_{\scriptscriptstyle W^{\prime}}^{\scriptscriptstyle G}({\hat{X}})$, see [28, Thm. 1] or [22, Thm. 3]. Hence the same holds for all $\nu^{\scriptscriptstyle G}\in\widetilde{{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W^{\prime}}}$. Therefore the topological entropy of the subshift $\widetilde{{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W^{\prime}}}\subseteq\{0,1\}^{\mathbbm{Z}}$ is at most $m_{H}(W^{\prime})+\epsilon$. It follows that $m_{H}(W)=h(P^{\scriptscriptstyle G}_{max})\leqslant m_{H}(W^{\prime})$. (For related arguments see also [15].) Hence $m_{H}(W\setminus W^{\prime})=0$. Because sets ${\mathcal{B}}$ with light tails are taut [10, Section 2.6] and give rise to Haar regular windows [19, Thm. A], this implies $W_{P_{max}}=W^{\prime}=W$. As, in the context of ${\mathcal{B}}$-free systems, the window $W$ is always aperiodic [19], we see that $W_{P_{max}}$ is aperiodic. ∎

4 The map ${\mathcal{S}}_{\scriptscriptstyle H}$ and the proof of Theorem A1

Throughout this section we assume that $W$ is compact. Recall from (3) that, for each $\nu\in{\mathcal{M}}_{\scriptscriptstyle W}$, $\pi^{\scriptscriptstyle H}_{*}\nu$ is a measure on $H$, whose topological support is denoted by ${\mathcal{S}}_{\scriptscriptstyle H}(\nu)$. The set ${\mathcal{S}}_{\scriptscriptstyle H}(\nu)\subseteq W$ can be understood as the “minimal” window for $\nu$ in the following sense: Assume that $\nu\in{\mathcal{M}}_{\scriptscriptstyle W}$ satisfies $\nu\leq\nu_{\scriptscriptstyle W}({\hat{x}})$ for some ${\hat{x}}\in{\hat{X}}$. Then the smallest compact set $W^{\prime}\subseteq W$ such that $\nu\leq\nu_{\scriptscriptstyle W^{\prime}}({\hat{x}})$ is given by $W^{\prime}={\mathcal{S}}_{\scriptscriptstyle H}(\nu)$.

Denote $\widetilde{{\mathcal{M}}_{\scriptscriptstyle W}}:=\{\nu\in{\mathcal{M}}:\nu\leqslant\nu_{\scriptscriptstyle W}({\hat{x}})\text{ for some }{\hat{x}}\in{\hat{X}}\}$. Then ${\mathcal{M}}_{\scriptscriptstyle W}=\overline{\nu_{\scriptscriptstyle W}({\hat{X}})}\subseteq\widetilde{{\mathcal{M}}_{\scriptscriptstyle W}}$, because $\nu_{\scriptscriptstyle W}$ is upper semi-continuous. It is advantageous to view ${\mathcal{S}}_{\scriptscriptstyle H}$ as a map from $\widetilde{{\mathcal{M}}_{\scriptscriptstyle W}}\setminus\{\underline{0}\}$ to ${\mathcal{K}}_{\scriptscriptstyle W}$, the space of all non-empty compact subsets of $W$, which is equipped with the topology generated by the Hausdorff distance.

Lemma 4.1**.**

${\mathcal{S}}_{\scriptscriptstyle H}:\widetilde{{\mathcal{M}}_{\scriptscriptstyle W}}\setminus\{\underline{0}\}\to{\mathcal{K}}_{\scriptscriptstyle W}$* is lower semicontinuous, i.e., for each closed $F\subseteq W$ the core of $F$, i.e. the set $\{\nu\in\widetilde{{\mathcal{M}}_{\scriptscriptstyle W}}\setminus\{\underline{0}\}:\ {\mathcal{S}}_{\scriptscriptstyle H}(\nu)\subseteq F\}$ is closed. In particular, ${\mathcal{S}}_{\scriptscriptstyle H}$ is Borel measurable. The same holds for the restriction ${\mathcal{S}}_{\scriptscriptstyle H}|_{{\mathcal{M}}_{\scriptscriptstyle W}\setminus\{\underline{0}\}}$.*

Proof.

The above characterisation of lower semicontinuity is from [1, Prop. 1.4.4]. So let $\nu=\lim_{n\to\infty}\nu_{n}$ with ${\mathcal{S}}_{\scriptscriptstyle H}(\nu_{n})\subseteq F$. Suppose for a contradiction that ${\mathcal{S}}_{\scriptscriptstyle H}(\nu)$ is not contained in $F$. Then, by closedness of $F$, it follows that there is $h\in{\mathcal{S}}_{\scriptscriptstyle H}(\nu)\setminus F$ such that $(\pi^{\scriptscriptstyle H}_{*}\nu)\{h\}=1$. Hence there are $x\in G\times H$ and $\ell\in{\mathscr{L}}$ such that $(x+\ell)_{H}=h$ and $\nu\{x+\ell\}=1$. As $\nu_{n}\to\nu$ vaguely, there are $x_{n}\in G\times H$ such that $x_{n}\to x$ and $\nu_{n}\{x_{n}+\ell\}=1$ for all $n$. But then $(x_{n}+\ell)_{H}\in{\mathcal{S}}_{\scriptscriptstyle H}(\nu_{n})\subseteq F$ for all $n$, and $(x_{n}+\ell)_{H}\to h$, so that $h\in F$, a contradiction. This proves the lower semicontinuity of ${\mathcal{S}}_{\scriptscriptstyle H}$, and its Borel measurability follows from [11, Cor. III.3]. As ${\mathcal{M}}_{\scriptscriptstyle W}$ is a closed subset of $\widetilde{{\mathcal{M}}_{\scriptscriptstyle W}}$, these properties are inherited by the restriction ${\mathcal{S}}_{\scriptscriptstyle H}|_{{\mathcal{M}}_{\scriptscriptstyle W}\setminus\{\underline{0}\}}$. ∎

Denote by $C_{\scriptscriptstyle W}\subseteq{\hat{X}}$ the set of continuity points of the map $\nu_{\scriptscriptstyle W}:{\hat{X}}\to{\mathcal{M}}_{\scriptscriptstyle W}$. It is a dense $G_{\delta}$-set, see Proposition 8.3 below. An explicit characterization of this set is

[TABLE]

see Lemma 8.2 below or e.g. [22, Lem. 6.1].

Remark 4.2**.**

${\mathcal{M}}_{\scriptstyle min}=\overline{\nu_{\scriptscriptstyle W}(C_{\scriptscriptstyle W})}$ is the unique minimal subset of ${\mathcal{M}}_{\scriptscriptstyle W}$ by Lemma 8.4 below, see also [22, Thm. 1a]. Let ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptstyle min}:=\pi^{\scriptscriptstyle G}_{*}({\mathcal{M}}_{\scriptstyle min})\subseteq{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$. Then ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptstyle min}$ is the only minimal subsystem of ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$. Even more, ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptstyle min}=\overline{\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}(C_{\scriptscriptstyle W})}$. The $\subseteq$-inclusion follows, because $\overline{\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}(C_{\scriptscriptstyle W})}$ is non-empty, closed and $S$-invariant. For the reverse inclusion observe that $\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}(C_{\scriptscriptstyle W})=\pi^{\scriptscriptstyle G}_{*}(\nu_{\scriptscriptstyle W}(C_{\scriptscriptstyle W}))\subseteq\pi^{\scriptscriptstyle G}_{*}({\mathcal{M}}_{\scriptstyle min})$. If $\operatorname{int}(W)=\emptyset$, then ${\mathcal{M}}_{\scriptstyle min}=\{\underline{0}\}$ is a singleton which consists only of the zero measure, see Lemma 8.4 or [22, Prop. 3.3].

Lemma 4.3**.**

1. a)

${\mathcal{S}}_{\scriptscriptstyle H}$* is $S$-invariant.* 2. b)

Let $\nu\in\widetilde{{\mathcal{M}}_{\scriptscriptstyle W}}$ and assume that $\nu^{\prime}\in\overline{{\mathcal{O}}(\nu)}:=\overline{\{S_{g}\nu:g\in G\}}\subseteq\widetilde{{\mathcal{M}}_{\scriptscriptstyle W}}$. Then ${\mathcal{S}}_{\scriptscriptstyle H}(\nu^{\prime})\subseteq{\mathcal{S}}_{\scriptscriptstyle H}(\nu)$. 3. c)

$\overline{\operatorname{int}(W)}={\mathcal{S}}_{\scriptscriptstyle H}(\nu)$* for all $\nu\in{\mathcal{M}}_{\scriptstyle min}$.* 4. d)

$\overline{\operatorname{int}(W)}\subseteq{\mathcal{S}}_{\scriptscriptstyle H}(\nu)$* and $\operatorname{int}(W)=\operatorname{int}({\mathcal{S}}_{\scriptscriptstyle H}(\nu))$ for all $\nu\in{\mathcal{M}}_{\scriptscriptstyle W}$.*

Proof.

a) ${\mathcal{S}}_{\scriptscriptstyle H}(S_{g}(\nu))=\operatorname{supp}(\pi^{\scriptscriptstyle H}_{*}(S_{g}(\nu)))=\operatorname{supp}(\pi^{\scriptscriptstyle H}_{*}\nu)={\mathcal{S}}_{\scriptscriptstyle H}(\nu)$ for all $g\in G$.

b) If $\nu=\underline{0}$ or $\nu^{\prime}=\underline{0}$, the claim is trivial. Otherwise, the claim follows immediately from Lemma 4.1.

c) Assertion b) applies to any two $\nu,\nu^{\prime}\in{\mathcal{M}}_{\scriptstyle min}$. Hence ${\mathcal{S}}_{\scriptscriptstyle H}$ is constant on ${\mathcal{M}}_{\scriptstyle min}=\overline{\nu_{\scriptscriptstyle W}(C_{\scriptscriptstyle W})}$. But for any continuity point ${\hat{x}}$ of $\nu_{\scriptscriptstyle W}$ we have ${\mathcal{S}}_{\scriptscriptstyle H}(\nu_{\scriptscriptstyle W}({\hat{x}}))=\overline{\operatorname{int}(W)}$ by Remark 3.7.

d) Let $\nu\in{\mathcal{M}}_{\scriptscriptstyle W}$. As ${\mathcal{M}}_{\scriptstyle min}$ is the unique minimal subset of ${\mathcal{M}}_{\scriptscriptstyle W}$, we have ${\mathcal{M}}_{\scriptstyle min}\subseteq\overline{{\mathcal{O}}(\nu)}$. Let $\nu^{\prime}\in{\mathcal{M}}_{\scriptstyle min}$. Then $\overline{\operatorname{int}(W)}={\mathcal{S}}_{\scriptscriptstyle H}(\nu^{\prime})$ by part c), and ${\mathcal{S}}_{\scriptscriptstyle H}(\nu^{\prime})\subseteq{\mathcal{S}}_{\scriptscriptstyle H}(\nu)$ by part a), because $\nu^{\prime}\in{\mathcal{M}}_{\scriptstyle min}\subseteq\overline{{\mathcal{O}}(\nu)}$. Hence $\operatorname{int}(W)\subseteq\overline{\operatorname{int}(W)}\subseteq{\mathcal{S}}_{\scriptscriptstyle H}(\nu)\subseteq W$, in particular also $\operatorname{int}(W)\subseteq\operatorname{int}({\mathcal{S}}_{\scriptscriptstyle H}(\nu))\subseteq\operatorname{int}(W)$. ∎

Lemma 4.4**.**

Suppose $\nu,\nu^{\prime}\in\widetilde{{\mathcal{M}}_{\scriptscriptstyle W}}$ and $\pi^{\scriptscriptstyle G}_{*}\nu=\pi^{\scriptscriptstyle G}_{*}\nu^{\prime}$. Then there is $d\in H$ such that $\nu^{\prime}=\sigma_{d}\nu$, where $(\sigma_{d}\nu)(A):=\nu(A-(0,d))$ for all Borel subsets $A$ of $G\times H$. In particular, $d+{\mathcal{S}}_{\scriptscriptstyle H}(\nu^{\prime})={\mathcal{S}}_{\scriptscriptstyle H}(\nu)$ and $d+\operatorname{int}(W)=\operatorname{int}(W)$.

Proof.

(inspired by the proof of [22, Lem. 4.5])

If $\pi^{\scriptscriptstyle G}_{*}\nu=\pi^{\scriptscriptstyle G}_{*}\nu^{\prime}=\underline{0}$, then also $\nu=\underline{0}=\nu^{\prime}$, and the claim is obvious. Otherwise, by definition of $\widetilde{{\mathcal{M}}_{\scriptscriptstyle W}}$, there are $x,x^{\prime}\in G\times H$ such that $\nu\leqslant\nu_{\scriptscriptstyle W}(x+{\mathscr{L}})$ and $\nu^{\prime}\leqslant\nu_{\scriptscriptstyle W}(x^{\prime}+{\mathscr{L}})$. Hence, $\pi^{\scriptscriptstyle G}_{*}\nu\leqslant\pi^{\scriptscriptstyle G}_{*}\nu_{\scriptscriptstyle W}(x+{\mathscr{L}})$ and $\pi^{\scriptscriptstyle G}_{*}\nu^{\prime}\leqslant\pi^{\scriptscriptstyle G}_{*}\nu_{\scriptscriptstyle W}(x^{\prime}+{\mathscr{L}})$, and as $\underline{0}\neq\pi^{\scriptscriptstyle G}_{*}\nu=\pi^{\scriptscriptstyle G}_{*}\nu^{\prime}$, there is $\tilde{\ell}\in{\mathscr{L}}$ such that $x_{G}^{\prime}=x_{G}+\tilde{\ell}_{G}$ by [22, Lem. 7.1d]. Let $\tilde{x}=x+\tilde{\ell}$. Then $\tilde{x}_{G}=x^{\prime}_{G}$ and $\nu_{\scriptscriptstyle W}(x+{\mathscr{L}})=\nu_{\scriptscriptstyle W}(\tilde{x}+{\mathscr{L}})$. Hence we can assume without loss of generality that $x_{G}=x_{G}^{\prime}$. As $\pi^{\scriptscriptstyle G}|_{\mathscr{L}}$ is 1-1, we conclude that the following chain of equivalences holds for each $\ell\in{\mathscr{L}}$:

[TABLE]

For $d:=x^{\prime}_{H}-x_{H}$ this can be rewritten as

[TABLE]

and as the measures $\nu$ and $\nu^{\prime}$ are sums of unit point masses supported by the sets $x+{\mathscr{L}}$ and $x^{\prime}+{\mathscr{L}}$, respectively, $\nu^{\prime}=\sigma_{d}\nu$ follows at once. Hence $\operatorname{supp}(\pi^{\scriptscriptstyle H}_{*}(\nu^{\prime}))=\operatorname{supp}(\pi^{\scriptscriptstyle H}_{*}(\nu))-d$, so that ${\mathcal{S}}_{\scriptscriptstyle H}(\nu)=d+{\mathcal{S}}_{\scriptscriptstyle H}(\nu^{\prime})$. Observing Lemma 4.3d, this implies $\operatorname{int}(W)=d+\operatorname{int}(W)$. ∎

Proof of Theorem A1.

a) As $\operatorname{int}(W)$ is aperiodic, it is in particular nonempty. Hence $({\mathcal{M}}_{\scriptscriptstyle W},S)$ is an almost 1-1 extension of its maximal equicontinuous factor $({\hat{X}},\hat{T})$ by [22, Thm. 1a]. As $\pi^{\scriptscriptstyle G}_{*}:({\mathcal{M}}_{\scriptscriptstyle W},S)\to({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},S)$ is a continuous factor map between compact systems, all we have to show is that $\pi^{\scriptscriptstyle G}_{*}$ is $1-1$. So let $\nu,\nu^{\prime}\in{\mathcal{M}}_{\scriptscriptstyle W}$ and suppose that $\pi^{\scriptscriptstyle G}_{*}(\nu)=\pi^{\scriptscriptstyle G}_{*}(\nu^{\prime})$. Then $\nu^{\prime}=\sigma_{d}\nu$ and $d+{\mathcal{S}}_{\scriptscriptstyle H}(\nu^{\prime})={\mathcal{S}}_{\scriptscriptstyle H}(\nu)$ for some $d\in H$ by Lemma 4.4. In particular $d+\operatorname{int}({\mathcal{S}}_{\scriptscriptstyle H}(\nu^{\prime}))=\operatorname{int}({\mathcal{S}}_{\scriptscriptstyle H}(\nu))$, so that $d+\operatorname{int}(W)=\operatorname{int}(W)$ by Lemma 4.3d. By assumption, $\operatorname{int}(W)$ is aperiodic. Therefore $d=0$ and hence $\nu^{\prime}=\sigma_{0}\nu=\nu$.

b) As $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},S)$ and $({\mathcal{M}}_{\scriptscriptstyle W},S)$ are isomorphic by part a), all results for $({\mathcal{M}}_{\scriptscriptstyle W},S)$ from [22] apply to $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},S)$ as well. In particular, $M$ contains the unique minimal invariant subset ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptstyle min}$ of ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, and, just as $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},S)$ itself, $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptstyle min},S)$ is an almost 1-1 extension of $({\hat{X}},\hat{T})$ with factor map $\Gamma\,|_{{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptstyle min}}$ [22, Thm. 1a]. As ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptstyle min}\subseteq M\subseteq{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, the claim of the theorem follows. ∎

For later use we continue with some further lemmas highlighting properties of ${\mathcal{S}}_{\scriptscriptstyle H}$.

Lemma 4.5**.**

Let $P,P^{\prime}$ be ergodic $S$-invariant probability measures on ${\mathcal{M}}_{\scriptscriptstyle W}$.

1. a)

There is a Haar regular subset $W_{P}\subseteq W$ of $H$ such that ${\mathcal{S}}_{\scriptscriptstyle H}(\nu)=W_{P}$ for $P$-a.a. $\nu$. It is empty if and only if $P(\{\underline{0}\})=1$. 2. b)

If $P\circ(\pi^{\scriptscriptstyle G}_{*})^{-1}=P^{\prime}\circ(\pi^{\scriptscriptstyle G}_{*})^{-1}$, then $W_{P}=W_{P^{\prime}}+d$ for some $d\in H$.

Proof.

a) The claim is obvious for $P$ satisfying $P(\{\underline{0}\})=1$, which is equivalent to $W_{P}=\emptyset$. Hence we assume without loss of generality that $P({\mathcal{M}}_{\scriptscriptstyle W}\setminus\{\underline{0}\})=1$.

We need the following preparation. Fix a complete metric $d$ on $H$ that generates the topology, and also a countable dense subset $\{h_{n}:n\in{\mathbbm{N}}\}$ of $H$. Define functions $\delta_{n}:{\mathcal{M}}_{\scriptscriptstyle W}\setminus\{\underline{0}\}\to{\mathbbm{R}}$, $\delta_{n}(\nu):=d(h_{n},{\mathcal{S}}_{\scriptscriptstyle H}(\nu))$. These functions inherit their Borel-measurability from ${\mathcal{S}}_{\scriptscriptstyle H}$ [11, Thm. III.2 and III.9]. Note also that if $(\delta_{n}(\nu))_{n\in{\mathbbm{N}}}=(\delta_{n}(\nu^{\prime}))_{n\in{\mathbbm{N}}}$, then ${\mathcal{S}}_{\scriptscriptstyle H}(\nu)={\mathcal{S}}_{\scriptscriptstyle H}(\nu^{\prime})$. Indeed, otherwise there is $h\in{\mathcal{S}}_{\scriptscriptstyle H}(\nu)\setminus{\mathcal{S}}_{\scriptscriptstyle H}(\nu^{\prime})$ (w.l.o.g.), and for a subsequence $(h_{n_{k}})_{k}$ converging to $h$ one has $\lim_{k\to\infty}\delta_{n_{k}}(\nu)=d(h,{\mathcal{S}}_{\scriptscriptstyle H}(\nu))=0<d(h,{\mathcal{S}}_{\scriptscriptstyle H}(\nu^{\prime}))=\lim_{k\to\infty}\delta_{n_{k}}(\nu^{\prime})$, a contradiction.

Now the $S$-invariance of ${\mathcal{S}}_{\scriptscriptstyle H}$ implies at once that all $\delta_{n}$ are $S$-invariant. As $P$ is ergodic, there are constants $(a_{n})_{n\in{\mathbbm{N}}}$ and a set ${\mathcal{M}}^{\prime}\subseteq{\mathcal{M}}_{\scriptscriptstyle W}\setminus\{\underline{0}\}$ of full $P$-measure such that $\delta_{n}(\nu)=a_{n}$ for all $\nu\in{\mathcal{M}}^{\prime}$ and all $n\in{\mathbbm{N}}$. Hence ${\mathcal{S}}_{\scriptscriptstyle H}(\nu)$ is the same compact subset of $H$, call it $W_{P}$, for all $\nu\in{\mathcal{M}}^{\prime}$. We have $W_{P}\neq\emptyset$, of course.

It remains to prove that $W_{P}$ is Haar regular. Suppose for a contradiction that this is not the case. Then there are $h\in W_{P}$ and $r>0$ such that $m_{H}\left(B_{r}(h)\cap W_{P}\right)=0$, where $B_{r}(h)=\{h^{\prime}\in H:d(h^{\prime},h)<r\}$. In view of (3), $\pi^{\scriptscriptstyle H}\left(\operatorname{supp}(\nu)\right)\cap B_{r}(h)\neq\emptyset$ for each $\nu\in{\mathcal{M}}^{\prime}$. Using the torus map $\hat{\pi}:{\mathcal{M}}_{\scriptscriptstyle W}\setminus\{\underline{0}\}\to{\hat{X}}$, which was explained after Proposition 2.1, we infer $\pi^{\scriptscriptstyle H}\left(\operatorname{supp}(\nu_{\scriptscriptstyle W}(\hat{\pi}(\nu)))\right)\cap B_{r}(h)\neq\emptyset$. Denote by $\pi(\nu)$ the unique representative of $\hat{\pi}(\nu)$ in the fundamental domain $X\subseteq G\times H$ of ${\hat{X}}$. It follows that

[TABLE]

In the remaining part of the proof we will show that $P({\mathcal{M}}^{\prime})=0$, which is the desired contradiction. To that end recall that ${\mathscr{L}}$ is countable and that $P\circ\hat{\pi}^{-1}=m_{\hat{X}}$, compare the proof of Proposition 2.2. Hence $P\circ\pi^{-1}=\mathrm{dens}({\mathscr{L}})\cdot m_{G\times H}|_{X}$, compare Fact 2.2(2). Therefore it suffices to estimate

[TABLE]

and to observe that the latter expression evaluates to [math], because $m_{H}(B_{r}(h)\cap W_{P})=0$.

b) Assume now that $P\circ(\pi^{\scriptscriptstyle G}_{*})^{-1}=P^{\prime}\circ(\pi^{\scriptscriptstyle G}_{*})^{-1}$. In view of part a) of the lemma and of Lemma 4.4, there are $\nu,\nu^{\prime}\in{\mathcal{M}}_{\scriptscriptstyle W}$ and $d\in H$ such that $W_{P}={\mathcal{S}}_{\scriptscriptstyle H}(\nu)=d+{\mathcal{S}}_{\scriptscriptstyle H}(\nu^{\prime})=d+W_{P^{\prime}}$. ∎

Lemma 4.6**.**

Suppose that $W$ is aperiodic. Then $\pi^{\scriptscriptstyle G}_{*}:{\mathcal{M}}_{\scriptscriptstyle W}\to{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ is 1-1 at $\nu\in{\mathcal{M}}_{\scriptscriptstyle W}$ whenever ${\mathcal{S}}_{\scriptscriptstyle H}(\nu)=W$.

Proof.

If ${\mathcal{S}}_{\scriptscriptstyle H}(\nu)=W$ and $\pi^{\scriptscriptstyle G}_{*}\nu=\pi^{\scriptscriptstyle G}_{*}\nu^{\prime}$ for $\nu,\nu^{\prime}\in{\mathcal{M}}_{\scriptscriptstyle W}$, then $\nu^{\prime}=\sigma_{d}\nu$ and $W-d={\mathcal{S}}_{\scriptscriptstyle H}(\nu)-d={\mathcal{S}}_{\scriptscriptstyle H}(\nu^{\prime})\subseteq W$ by Lemma 4.4. Let $W_{n}=W-n\cdot d$. Then $W_{0}\supseteq W_{1}\supseteq\dots$ is a nested sequence of compact sets. Suppose for a contradiction that there exists some $h_{0}\in W_{0}\setminus W_{1}$ and let $h_{n}:=h_{0}-n\cdot d\in W_{n}\subseteq W$ for $n\in{\mathbbm{N}}$. Then, for $k>n$, $d(h_{n},h_{k})\geqslant d(h_{n},W_{k})\geqslant d(h_{n},W_{n+1})=d(h_{0},W_{1})>0$, which is impossible, because $W$ is compact. Hence $W=W_{0}=W_{1}=W-d$, so that $d=0$ because $W$ is aperiodic. ∎

Lemma 4.7**.**

Suppose that $F\subseteq W$ is Haar regular and $K\subseteq{\mathcal{M}}_{\scriptscriptstyle W}$ is closed. Then $\pi^{\scriptscriptstyle G}_{*}\{\nu\in K:{\mathcal{S}}_{\scriptscriptstyle H}(\nu)=F\}$ is a Borel subset of ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$.

Proof.

The case $F=\emptyset$ is trivial. So we may assume that $F\neq\emptyset$. Denote by $V_{1},V_{2},\ldots\subseteq H$ those elements of a base of the second countable space $H$, for which $F_{n}:=F\setminus V_{n}$ is a proper subset of $F$. Then any compact proper subset $F^{\prime}$ of $F$ is contained in some $F_{n}$. We thus can write

[TABLE]

Next, let $\nu,\nu^{\prime}\in{\mathcal{M}}_{\scriptscriptstyle W}$ with ${\mathcal{S}}_{\scriptscriptstyle H}(\nu)=F\neq\emptyset$, ${\mathcal{S}}_{\scriptscriptstyle H}(\nu^{\prime})\subseteq F$ and $\pi^{\scriptscriptstyle G}_{*}(\nu^{\prime})=\pi^{\scriptscriptstyle G}_{*}(\nu)$. In particular, $\nu,\nu^{\prime}\neq\underline{0}$. Then $m_{H}({\mathcal{S}}_{\scriptscriptstyle H}(\nu^{\prime}))=m_{H}({\mathcal{S}}_{\scriptscriptstyle H}(\nu))=m_{H}(F)$ by Lemma 4.4. As $F$ is Haar regular, this implies ${\mathcal{S}}_{\scriptscriptstyle H}(\nu^{\prime})=F$. Therefore the union of the two sets on the rhs of Eqn. (7) is disjoint, and we have

[TABLE]

As all sets involved in the rhs of (8) are continuous images of sets which are compact by Lemma 4.1, the lhs of (8) is in particular Borel measurable. ∎

5 Proofs of Theorems B1 and C1

We first prove both theorems for compact windows, and discuss the extension of Theorem B1 to windows which are compact modulo [math] in Subsection 8.2.

Fix any tempered van Hove sequence $(A_{n})_{n\in\mathbb{N}}$ of subsets of $G$, compare [22, Footnote 5]. We always have the upper bound

[TABLE]

on the upper density of any configuration $\nu\in{\mathcal{M}}_{\scriptscriptstyle W}$, see [22, Eqn. (14)]. We say that $\nu\in{\mathcal{M}}_{\scriptscriptstyle W}$ has maximal density if

[TABLE]

Recall from [29, Thm. 5a] that

[TABLE]

where $Q_{\scriptscriptstyle W}=m_{\hat{X}}\circ(\nu_{\scriptscriptstyle W})^{-1}$ denotes the Mirsky measure on ${\mathcal{M}}_{\scriptscriptstyle W}$.

Lemma 5.1**.**

$W_{reg}\subseteq{\mathcal{S}}_{\scriptscriptstyle H}(\nu)\subseteq W$* for each $\nu\in{\mathcal{M}}_{\scriptscriptstyle W}$ with maximal density.*

Proof.

We have ${\mathcal{S}}_{\scriptscriptstyle H}(\nu)\subseteq W$ for all $\nu\in{\mathcal{M}}_{\scriptscriptstyle W}$ by definition. If $m_{H}(W)=0$, then $W_{reg}=\emptyset\subseteq{\mathcal{S}}_{\scriptscriptstyle H}(\nu)$. Assume now that $m_{H}(W)>0$ and that $\nu$ has maximal density $d(\nu)=\mathrm{dens}({\mathscr{L}})\cdot m_{H}(W)>0$. In particular, $\nu\in{\mathcal{M}}_{\scriptscriptstyle W}\setminus\{\underline{0}\}$. There is a unique ${\hat{x}}=x+{\mathscr{L}}\in{\hat{X}}$ such that $\nu\leqslant\nu_{\scriptscriptstyle W}({\hat{x}})$ by [22, Lem. 5.4]. We thus get $\operatorname{supp}(\nu)\subseteq(x+{\mathscr{L}})\cap(G\times{\mathcal{S}}_{\scriptscriptstyle H}(\nu))$, which implies $\nu\leqslant\nu_{\scriptstyle{\mathcal{S}}_{\scriptscriptstyle H}(\nu)}({\hat{x}})$. Hence

[TABLE]

which yields $m_{H}(W)=m_{H}({\mathcal{S}}_{\scriptscriptstyle H}(\nu))$. As ${\mathcal{S}}_{\scriptscriptstyle H}(\nu)$ is a compact subset of $W$, this implies $W_{reg}\subseteq{\mathcal{S}}_{\scriptscriptstyle H}(\nu)$. ∎

Corollary 5.2**.**

$W_{Q_{\scriptscriptstyle W}}=W_{reg}$.

Proof.

Observe that $W_{Q_{\scriptscriptstyle W}}={\mathcal{S}}_{\scriptscriptstyle H}(\nu)\subseteq W$ for $Q_{\scriptscriptstyle W}$-a.a. $\nu$ by Lemma 4.5. Hence $W_{reg}\subseteq W_{Q_{\scriptscriptstyle W}}$ in view of (9) and Lemma 5.1. Haar regularity of $W_{Q_{\scriptscriptstyle W}}$, which holds by definition, implies $W_{reg}\supseteq W_{Q_{\scriptscriptstyle W}}$. Indeed, let $w\in W_{Q_{\scriptscriptstyle W}}$ and let $U$ any neighborhood of $w$. Then $m_{H}(U\cap W)=m_{H}(U\cap W_{Q_{\scriptscriptstyle W}})>0$, because $m_{H}(W\setminus W_{Q_{\scriptscriptstyle W}})\leqslant m_{H}(W\setminus W_{reg})=0$ and as $W_{Q_{\scriptscriptstyle W}}$ is Haar regular. ∎

Corollary 5.3**.**

Denote by ${\mathcal{M}}_{\scriptscriptstyle W}^{\prime}\subseteq{\mathcal{M}}_{\scriptscriptstyle W}$ the set of configurations of maximal density. If $W$ is Haar aperiodic, then $\pi^{\scriptscriptstyle G}_{*}|_{{\mathcal{M}}_{\scriptscriptstyle W}^{\prime}}:{\mathcal{M}}_{\scriptscriptstyle W}^{\prime}\to{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ is 1-1.

Proof.

Note that $W_{reg}\subseteq{\mathcal{S}}_{\scriptscriptstyle H}(\nu)\subseteq W$ for any configuration $\nu\in{\mathcal{M}}_{\scriptscriptstyle W}^{\prime}$ by Lemma 5.1, where $m_{H}(W\setminus W_{reg})=0$. Now assume that $\pi^{\scriptscriptstyle G}_{*}(\nu)=\pi^{\scriptscriptstyle G}_{*}(\nu^{\prime})$ for $\nu,\nu^{\prime}\in{\mathcal{M}}_{\scriptscriptstyle W}^{\prime}$. Then $\nu^{\prime}=\sigma_{d}\nu$ and $d+{\mathcal{S}}_{\scriptscriptstyle H}(\nu^{\prime})={\mathcal{S}}_{\scriptscriptstyle H}(\nu)$ for some $d\in H$ by Lemma 4.4, which implies

[TABLE]

As $W$ is Haar aperiodic, we conclude $d=0$ and thus $\nu=\nu^{\prime}$. ∎

Proof of Theorem B1.

$\pi^{\scriptscriptstyle G}_{*}$ is 1-1 at $Q_{\scriptscriptstyle W}$-a.a. $\nu\in{\mathcal{M}}_{\scriptscriptstyle W}$ by Corollary 5.3 and Eqn. (9). Hence $\pi^{\scriptscriptstyle G}_{*}:({\mathcal{M}}_{\scriptscriptstyle W},Q_{\scriptscriptstyle W},S)\to({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},Q_{\scriptscriptstyle W}^{\scriptscriptstyle G},S)$ is a measure-theoretic isomorphism, and thus $\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}:({\hat{X}},m_{\hat{X}},\hat{T})\to({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},Q_{\scriptscriptstyle W}^{\scriptscriptstyle G},S)$ is a measure-theoretic isomorphism by Proposition 2.3. Here we use $m_{H}(W)>0$, which follows from Haar aperiodicity of $W$. ∎

We now turn to general $S$-invariant probability measures on ${\mathcal{M}}_{\scriptscriptstyle W}$.

Corollary 5.4**.**

Fix an ergodic $S$-invariant probability measure $P$ on ${\mathcal{M}}_{\scriptscriptstyle W}$ and consider the Haar regular set $W_{P}\subseteq W$ from Lemma 4.5. Then ${\mathcal{M}}_{P}:={\mathcal{S}}_{\scriptscriptstyle H}^{-1}\{W_{P}\}\subseteq{\mathcal{M}}_{\scriptscriptstyle W}$ has $P$-measure one. If $W_{P}$ is aperiodic, then $\pi^{\scriptscriptstyle G}_{*}|_{{\mathcal{M}}_{P}}$ is 1-1.

Proof.

If $P=\delta_{\underline{0}}$, then $W_{P}=\emptyset$ and ${\mathcal{M}}_{P}=\{\underline{0}\}$, and the claim is trivial. Otherwise we may assume that $P({\mathcal{M}}_{\scriptscriptstyle W}\setminus\{\underline{0}\})=1$. Then by Lemma 4.1, the set ${\mathcal{M}}_{P}$ is measurable. By Lemma 4.5 we have $P({\mathcal{M}}_{P})=1$. The injectivity of $\pi^{\scriptscriptstyle G}_{*}|_{{\mathcal{M}}_{P}}$ follows from Lemma 4.4, where we use that $W_{P}$ is aperiodic. ∎

In order to infer results on ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ from ${\mathcal{M}}_{\scriptscriptstyle W}$, we need to “lift” invariant probability measures from ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ to ${\mathcal{M}}_{\scriptscriptstyle W}$. This is the content of the following proposition.

Proposition 5.5**.**

Let $P^{\scriptscriptstyle G}$ be an ergodic $S$-invariant probability measure on ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, and denote by ${\mathcal{P}}(P^{\scriptscriptstyle G})$ the family of all $S$-invariant probability measures $P$ on ${\mathcal{M}}_{\scriptscriptstyle W}$ that project to $P^{\scriptscriptstyle G}$, i.e., for which $P\circ(\pi^{\scriptscriptstyle G}_{*})^{-1}=P^{\scriptscriptstyle G}$. Then the following hold.

1. a)

${\mathcal{P}}(P^{\scriptscriptstyle G})\neq\emptyset$. 2. b)

Each $P\in{\mathcal{P}}(P^{\scriptscriptstyle G})$ has an ergodic decomposition $P=\int P_{e}\circ\sigma_{h}^{-1}\ d\rho(h)$ for some compactly supported probability measure $\rho$ on $H$ and some ergodic $S$-invariant probability measure $P_{e}\in{\mathcal{P}}(P^{\scriptscriptstyle G})$. 3. c)

If $P\in{\mathcal{P}}(P^{\scriptscriptstyle G})$ is ergodic and if $W_{P}\subseteq W$ is aperiodic, then $\pi^{\scriptscriptstyle G}_{*}:{\mathcal{M}}_{\scriptscriptstyle W}\to{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ is a measure-theoretic isomorphism between $({\mathcal{M}}_{\scriptscriptstyle W},P,S)$ and $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},P^{\scriptscriptstyle G},S)$, and both systems are extensions of $({\hat{X}},m_{\hat{X}},\hat{T})$.

Proof.

a) Denote by $\Gamma$ the set valued map (also called multifunction) from ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ to compact subsets of ${\mathcal{M}}_{\scriptscriptstyle W}$ defined by $\Gamma(\nu^{\scriptscriptstyle G})=(\pi^{\scriptscriptstyle G}_{*})^{-1}\{\nu^{\scriptscriptstyle G}\}$. It is measurable in the sense of [11, Thm. III.2], because $\Gamma^{-}(C):=\{\nu^{\scriptscriptstyle G}\in{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}:(\pi^{\scriptscriptstyle G}_{*})^{-1}\{\nu^{\scriptscriptstyle G}\}\cap C\neq\emptyset\}=\pi^{\scriptscriptstyle G}_{*}(C)$ is compact and hence Borel measurable for any closed $C\subseteq{\mathcal{M}}_{\scriptscriptstyle W}$. Hence, by the measurable selection theorem [11, Thm. III.6], there is a Borel measurable map $\psi_{*}:{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}\to{\mathcal{M}}_{\scriptscriptstyle W}$ such that $\pi^{\scriptscriptstyle G}_{*}\circ\psi_{*}=\operatorname{id}_{{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}}$. In particular, $P:=P^{\scriptscriptstyle G}\circ(\psi_{*})^{-1}$ is a well defined probability measure on ${\mathcal{M}}_{\scriptscriptstyle W}$ that projects to $P^{\scriptscriptstyle G}$. The measure $P$ is not necessarily $S$-invariant, but a Krylov-Bogoliubov construction on $P$, see e.g. [12, Thm. 8.10], provides an $S$-invariant probability measure in ${\mathcal{P}}(P^{\scriptscriptstyle G})$. The latter holds since also $(P\circ S_{g}^{-1})$ projects to $P^{\scriptscriptstyle G}$ for every $g\in G$ as $(P\circ S_{g}^{-1})\circ(\pi^{\scriptscriptstyle G}_{*})^{-1}=P\circ(\pi^{\scriptscriptstyle G}_{*}\circ S_{g})^{-1}=P\circ(S_{g}\circ\pi^{\scriptscriptstyle G}_{*})^{-1}=P\circ(\pi^{\scriptscriptstyle G}_{*})^{-1}\circ S_{g}^{-1}=P^{\scriptscriptstyle G}\circ S_{g}^{-1}=P^{\scriptscriptstyle G}$, and since the measure transport by $\pi^{\scriptscriptstyle G}_{*}$ is continuous w.r.t. the weak topology.

b) For any $P\in{\mathcal{P}}(P^{\scriptscriptstyle G})$, its ergodic decomposition [12, Thm. 8.20] can be written as

[TABLE]

where the probability measures $P_{\mu}$ on ${\mathcal{M}}_{\scriptscriptstyle W}$ are ergodic and where $\mu\mapsto P_{\mu}$ is Borel measurable. Then

[TABLE]

where all $P_{\mu}\circ(\pi^{\scriptscriptstyle G}_{*})^{-1}$ are ergodic $S$-invariant measures on ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$. As $P^{\scriptscriptstyle G}$ itself is ergodic, $P^{\scriptscriptstyle G}=P_{\mu}\circ(\pi^{\scriptscriptstyle G}_{*})^{-1}$ for $P$-a.a. $\mu$, so that $P_{\mu}\in{\mathcal{P}}(P^{\scriptscriptstyle G})$ for $P$-a.a. $\mu$.

Fix any measure $P_{e}$ from the ergodic decomposition. Let $P_{\mu}$ be any other measure from this decomposition. $P_{e}$ and $P_{\mu}$ can be disintegrated over $P^{\scriptscriptstyle G}$, namely there are systems $\{p_{\nu^{\scriptscriptstyle G}}:\nu^{\scriptscriptstyle G}\in{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}\}$ and $\{p_{\nu^{\scriptscriptstyle G}}^{\prime}:\nu^{\scriptscriptstyle G}\in{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}\}$ of probability measures on ${\mathcal{M}}_{\scriptscriptstyle W}$, such that

[TABLE]

Hence, for $P^{\scriptscriptstyle G}$-a.a. $\nu^{\scriptscriptstyle G}\in{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ and $(p_{\nu^{\scriptscriptstyle G}}\otimes p_{\nu^{\scriptscriptstyle G}}^{\prime})$-a.a. $(\nu,\nu^{\prime})\in{\mathcal{M}}_{\scriptscriptstyle W}\times{\mathcal{M}}_{\scriptscriptstyle W}$, we have $\pi^{\scriptscriptstyle G}_{*}\nu=\nu^{\scriptscriptstyle G}=\pi^{\scriptscriptstyle G}_{*}\nu^{\prime}$, so that $\nu^{\prime}=\sigma_{d}\nu$ for some $d=d(\nu,\nu^{\prime})\in H$. Since we may assume that $\nu$ and $\nu^{\prime}$ are generic for $P_{e}$ and $P_{\mu}$, respectively, we can conclude $P_{\mu}=P_{e}\circ\sigma_{d}^{-1}$.

Consider the map $\kappa:H\to{\mathcal{P}}(P^{\scriptscriptstyle G}),\ h\mapsto P_{e}\circ\sigma_{h}^{-1}$. It is continuous so that $\kappa(H)$ is a compact subset of the space of probability measures on ${\mathcal{M}}_{\scriptscriptstyle W}$. Observe that the set $\kappa(H)$ does not depend on the choice of a particular $P_{e}$ in the definition of $\kappa$. As $P_{\mu}\in\kappa(H)$ for $P$-a.a. $\mu$, we can rewrite the ergodic decomposition (10) as

[TABLE]

where $\tilde{\rho}$ is the distribution of the random measures $P_{\mu}$ under $P$.

The set valued map $\Gamma:P\mapsto\kappa^{-1}(\{P\})$ from ${\mathcal{P}}(P^{\scriptscriptstyle G})$ to compact subsets of $H$ is Borel measurable by the same arguments as in part a) of the proof. Hence, by the measurable selection theorem [11, Thm. III.6], there is a Borel measurable map $\kappa^{\dagger}:\kappa(H)\to H$ such that $\kappa\circ\kappa^{\dagger}=\operatorname{id}_{\kappa(H)}$. In particular, $\rho:=\tilde{\rho}\circ(\kappa^{\dagger})^{-1}$ is a well defined probability measure on $H$, and

[TABLE]

c) If $P\in{\mathcal{P}}(P^{\scriptscriptstyle G})$ is ergodic and if $W_{P}$ is aperiodic, then $\pi^{\scriptscriptstyle G}_{*}:({\mathcal{M}}_{\scriptscriptstyle W},P,S)\to({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},P^{\scriptscriptstyle G},S)$ is a measure-theoretic isomorphism in view of Corollary 5.4, and both systems are extensions of $({\hat{X}},m_{\hat{X}},\hat{T})$ by Proposition 2.2. Here we use $m_{H}(W)\geq m_{H}(W_{P})>0$, as the aperiodic Haar regular set $W_{P}$ is Haar aperiodic. ∎

6 Periodic windows and quotient cut-and-project schemes

For a given a subset $A\subseteq H$, recall its period group $H_{A}=\left\{h\in H:h+A=A\right\}$. The set $A\subseteq H$ is (topologically) aperiodic, if $H_{A}=\{0\}$.

Lemma 6.1**.**

1. a)

$H_{A}\subseteq H_{\bar{A}}\cap H_{\operatorname{int}(A)}$. 2. b)

If $A$ is closed, then also $H_{A}$ is closed. 3. c)

If $\operatorname{int}(\bar{A})=\operatorname{int}(A)$ (e.g. if $A$ is closed), then $H_{\operatorname{int}(A)}=H_{\overline{\operatorname{int}(A)}}$. 4. d)

If $A$ is compact and nonempty, then $H_{A}$ is compact.

Proof.

a) For each $h\in H$, the translation by $h$ is a homeomorphism on $H$.

b) Let $h_{n}\in H_{A}$, $h=\lim_{n}h_{n}$. If $w\in A$, then $\pm h+w=\lim_{n}(\pm h_{n}+w)\in A$, because all $\pm h_{n}+w$ are in $A$ and $A$ is closed. This shows $\pm h+A\subseteq A$, i.e., $h+A=A$.

c) The assumption implies $\operatorname{int}(\overline{\operatorname{int}(A)})=\operatorname{int}(A)$. Indeed, $\operatorname{int}(A)\subseteq\overline{\operatorname{int}(A)}$ implies $\operatorname{int}(A)\subseteq\operatorname{int}(\overline{\operatorname{int}(A)})$, and $\operatorname{int}(A)\subseteq A$ implies $\operatorname{int}(\overline{\operatorname{int}(A)})\subseteq\operatorname{int}(\overline{A})\subseteq\operatorname{int}(A)$. Now the result follows from the implications

[TABLE]

d) By definition we have $H_{A}+A=A$. As $A$ is compact nonempty, $H_{A}$ must be compact, too. ∎

Remark 6.2**.**

The condition $\operatorname{int}(\overline{A})\subseteq\operatorname{int}(A)$ in part c) of the lemma implies that the topological boundary $\partial A=\overline{A}\setminus\operatorname{int}(A)$ is nowhere dense. Indeed, nowhere denseness is equivalent to the condition $\operatorname{int}(\overline{A})\subseteq\overline{\operatorname{int}(A)}$. These two conditions are however not equivalent, as is easily seen by looking at open dense sets $A$.

For a given cut-and-project scheme with window $W\subseteq H$, an important example is the period group $H_{W}$ of the window. Some structural results for model sets rely on the assumption of an aperiodic window. Aperiodicity may however be assumed without loss of generality by passing to an associated quotient cut-and-project scheme, where the periods of the window have been factored out, compare [8, Section 9]. As this construction has not been fully described before, we present it here in some detail.

Let $(G,H,{\mathscr{L}})$ and ${\hat{X}}=(G\times H)/{\mathscr{L}}$ be as before, with quotient map $\pi^{\scriptscriptstyle{\hat{X}}}:G\times H\to{\hat{X}}$. Fix any compact subgroup $H_{0}\subseteq H$ and consider $H^{\prime}:=H/H_{0}$ with factor map $\varphi:H\to H^{\prime}$. Consider ${\mathcal{H}}_{0}:=\{0\}\times H_{0}\subseteq G\times H$ and note that ${\mathscr{L}}\cap{\mathcal{H}}_{0}=\{(0,0)\}$ as $\pi^{\scriptscriptstyle G}$ is 1-1 on ${\mathscr{L}}$. Observe next that $\iota:(G\times H)/{\mathcal{H}}_{0}\to G\times H^{\prime}$, $\iota\left((g,h)+{\mathcal{H}}_{0}\right):=(g,h+H_{0})$, is a (rather trivial) isomorphism of topological groups. Denote by $\Phi$ the quotient map $\Phi:G\times H\to(G\times H)/{\mathcal{H}}_{0}$, and let ${\mathscr{L}}^{\prime}:=\iota(\Phi({\mathscr{L}}))$.

Lemma 6.3**.**

${\mathscr{L}}^{\prime}=\iota(\Phi({\mathscr{L}}))$* is a discrete subgroup of $G\times H^{\prime}$.*

Proof.

As $\iota\circ\Phi$ is a group homomorphism, ${\mathscr{L}}^{\prime}$ is a subgroup of $G\times H$. We prove that ${\mathscr{L}}^{\prime}$ is discrete: Take a compact zero neighborhood $\mathcal{U}\subseteq G\times H$ such that ${\mathscr{L}}\cap\mathcal{U}=\{0\}$, which is possible since ${\mathscr{L}}$ is discrete in $G\times H$. Now ${\mathscr{L}}^{\prime}\cap\iota(\mathcal{U}+{\mathcal{H}}_{0})$ contains [math] and is finite as $\mathcal{U}+{\mathcal{H}}_{0}\subseteq G\times H$ is compact. Hence there is a zero neighborhood $\mathcal{V}\subseteq\mathcal{U}$ such that ${\mathscr{L}}^{\prime}\cap\iota(\mathcal{V}+{\mathcal{H}}_{0})=\{0\}$. ∎

This lemma allows to consider the locally compact abelian quotient group ${\widehat{X^{\prime}}}=(G\times H^{\prime})/{\mathscr{L}}^{\prime}$. Now $(G,H^{\prime},{\mathscr{L}}^{\prime})$ is a cut-and-project scheme with associated torus ${\widehat{X^{\prime}}}$, compare [8, Section 9], which we call a quotient cut-and-project scheme. For the convenience of the reader, we give a proof which is based on the following general facts about quotient groups.

Lemma 6.4**.**

[16, Proposition III.20]

$(G\times H)/({\mathscr{L}}+{\mathcal{H}}_{0})$* is isomorphic (as a topological group) to each of the following groups:*

1. a)

${\hat{X}}/\pi^{\scriptscriptstyle{\hat{X}}}({\mathscr{L}}+{\mathcal{H}}_{0})={\hat{X}}/\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{0})$** 2. b)

$((G\times H)/{\mathcal{H}}_{0})/\Phi({\mathscr{L}}+{\mathcal{H}}_{0})=((G\times H)/{\mathcal{H}}_{0})/\Phi({\mathscr{L}})$**

Corollary 6.5**.**

In the above setting, $(G,H^{\prime},{\mathscr{L}}^{\prime})$ is a cut-and-project scheme, in particular ${\mathscr{L}}^{\prime}$ is cocompact. The topological quotient group ${\widehat{X^{\prime}}}=(G\times H^{\prime})/{\mathscr{L}}^{\prime}$ is isomorphic to ${\hat{X}}/\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{0})$.

Proof.

Projection properties are inherited: Assume that $0=\pi^{\scriptscriptstyle G}((\iota\circ\Phi)(\ell))=\ell_{G}$ for some $\ell\in{\mathscr{L}}$. As $\pi^{\scriptscriptstyle G}$ is 1-1 on ${\mathscr{L}}$, we infer $\ell=0$, which implies $(\iota\circ\Phi)(\ell)=0$. Note also that $\overline{\pi^{\scriptscriptstyle H^{\prime}}({\mathscr{L}}^{\prime})}=\overline{\varphi(\pi^{\scriptscriptstyle H}({\mathscr{L}}))}\supseteq\varphi(\overline{\pi^{\scriptscriptstyle H}({\mathscr{L}})})=\varphi(H)=H^{\prime}$. Here we used continuity and surjectivity of the projection map, together with the assumption that $\pi^{\scriptscriptstyle H}({\mathscr{L}})$ is dense in $H$.

As $G\times H^{\prime}=\iota\left((G\times H)/{\mathcal{H}}_{0}\right)$ and ${\mathscr{L}}^{\prime}=\iota(\Phi({\mathscr{L}}))$, where $\iota$ is an isomorphism, we see that $(G\times H^{\prime})/{\mathscr{L}}^{\prime}$ is isomorphic to $((G\times H)/{\mathcal{H}}_{0})/\Phi({\mathscr{L}})$. Combining this with a) and b) of Lemma 6.4, we conclude that $(G\times H^{\prime})/{\mathscr{L}}^{\prime}$ is isomorphic to ${\hat{X}}/\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{0})$. As ${\mathcal{H}}_{0}$ is compact, $\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{0})$ is compact, so that ${\hat{X}}/\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{0})$ is compact [16, Theorem III.11]. In particular, ${\mathscr{L}}^{\prime}$ is cocompact in $G\times H^{\prime}$. ∎

Remark 6.6**.**

The factor map $\iota\circ\Phi:G\times H\to G\times H^{\prime}$ carries over to a factor map $\widehat{\iota\Phi}:{\hat{X}}\to{\widehat{X^{\prime}}}$, because $\iota(\Phi(x+{\mathscr{L}}))=\iota(\Phi(x))+{\mathscr{L}}^{\prime}$. It pushes the Haar measure $m_{\hat{X}}$ to the Haar measure $m_{{\widehat{X^{\prime}}}}$ on ${\widehat{X^{\prime}}}$.

If $W\subseteq H$ is a window in $(G,H,{\mathscr{L}})$ and $H_{0}$ is a closed subgroup of $H$, we can consider the quotient cut-and-project scheme $(G,H^{\prime},{\mathscr{L}}^{\prime})$ with window $W^{\prime}:=\varphi(W)$, because $W^{\prime}$ inherits the basic topological properties from $W$:

Lemma 6.7**.**

1. a)

$W^{\prime}$* is compact.* 2. b)

If $W$ is topologically regular, then so is $W^{\prime}$. 3. c)

If $W$ is Haar regular, then so is $W^{\prime}$.

Proof.

a) Recall that $\varphi:H\to H^{\prime}$ is continuous and open [16, Theorem III.10]. In particular, $W^{\prime}=\varphi(W)$ is compact.

b) If $W$ is topologically regular, i.e. if $W=\overline{\operatorname{int}(W)}$, then

[TABLE]

where we used continuity of $\varphi$ for the first inclusion and openness for the second one.

c) Suppose now that $W$ is Haar regular, and let $U\subseteq H^{\prime}$ be open with $m_{H^{\prime}}(U\cap W^{\prime})=0$. We must show that $U\cap W^{\prime}=\emptyset$. By definition of the quotient topology, $\varphi^{-1}(U)$ is open in $H$. As $\varphi^{-1}(U)\cap W\subseteq\varphi^{-1}(U\cap W^{\prime})$, we have $m_{H}(\varphi^{-1}(U)\cap W)\leqslant m_{H}(\varphi^{-1}(U\cap W^{\prime}))=m_{H^{\prime}}(U\cap W^{\prime})=0$, so that $\varphi^{-1}(U)\cap W=\emptyset$, because $W$ is Haar regular. This implies $U\cap W^{\prime}=U\cap\varphi(W)=\emptyset$. ∎

An important example is $H_{0}:=H_{W}$, the period group of a window, because $W^{\prime}$ is aperiodic in this case. We will study the relations between the two cut-and-project schemes $(G,H,{\mathscr{L}})$ and $(G^{\prime},H^{\prime},{\mathscr{L}}^{\prime})$ with associated windows $W$ and $W^{\prime}$, respectively. For clarity we write ${\mathcal{H}}_{W}$ instead of ${\mathcal{H}}_{0}$ and add the index $W$ also to the quotient maps. Recall that the quotient map $\iota\circ\Phi_{W}:G\times H\to G\times H^{\prime}$ is given by $x\mapsto(\iota\circ\Phi_{W})(x)=\iota(x+{\mathcal{H}}_{W})=(x_{G},x_{H}+H_{W})$.

Lemma 6.8**.**

The window $W^{\prime}=\varphi_{W}(W)$ is an aperiodic subset of $H^{\prime}$.

Proof.

Suppose that $W^{\prime}+h^{\prime}=W^{\prime}$ for some $h^{\prime}=\varphi_{W}(h)\in H^{\prime}$. Then $\varphi_{W}(W+h)=\varphi_{W}(W)$, in particular $W+h+H_{W}=W+H_{W}$. As $W+H_{W}=W$, this shows that $h\in H_{W}$, so that $h^{\prime}=\varphi_{W}(h)$ is the neutral element of $H^{\prime}$. ∎

Lemma 6.9**.**

Let $x\in G\times H$ and $x^{\prime}\in G\times H^{\prime}$ be such that $x^{\prime}=(\iota\circ\Phi)(x)$, and let $W^{\prime}=\varphi_{W}(W)$.

1. a)

The sets $(x+{\mathscr{L}})\cap(G\times W)$ and $(x^{\prime}+{\mathscr{L}}^{\prime})\cap(G\times W^{\prime})$ are in $1-1$ correspondence via the quotient map $\iota\circ\Phi$. In particular we have $\pi^{\scriptscriptstyle G}((x+{\mathscr{L}})\cap(G\times W))=\pi^{\scriptscriptstyle G}((x^{\prime}+{\mathscr{L}}^{\prime})\cap(G\times W^{\prime}))$. 2. b)

For every $h\in H_{W}$ we have $\nu_{W}(x+(0,h)+{\mathscr{L}})=\sigma_{h}\nu_{\scriptscriptstyle W}(x+{\mathscr{L}})$, where $\sigma_{h}\nu(A)=\nu(A-(0,h))$. 3. c)

We have $\nu_{\scriptscriptstyle W^{\prime}}(x^{\prime}+{\mathscr{L}}^{\prime})=\sum_{y\in(x+{\mathscr{L}})\cap(G\times W)}\delta_{\iota(y+{\mathcal{H}}_{W})}$, which implies $\pi^{\scriptscriptstyle G}_{*}(\nu_{\scriptscriptstyle W^{\prime}}(x^{\prime}+{\mathscr{L}}^{\prime}))=\pi^{\scriptscriptstyle G}_{*}(\nu_{\scriptscriptstyle W}(x+{\mathscr{L}}))$, i.e. $\nu_{\scriptscriptstyle W^{\prime}}^{\scriptscriptstyle G}(x^{\prime}+{\mathscr{L}}^{\prime})=\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}(x+{\mathscr{L}})$.

Proof.

a) To show injectivity of the quotient map, assume that $(\iota\circ\Phi)(x+\ell_{1})=(\iota\circ\Phi)(x+\ell_{2})$, that is $x+\ell_{1}+{\mathcal{H}}_{W}=x+\ell_{2}+{\mathcal{H}}_{W}$. We thus can conclude $\ell_{1}-\ell_{2}\in\{0\}\times H_{W}$. Hence $\ell_{1,G}-\ell_{2,G}=0$, and as $\pi^{\scriptscriptstyle G}$ is 1-1 on ${\mathscr{L}}$, we infer $\ell_{1,H}-\ell_{2,H}=0$. Hence $\ell_{1}=\ell_{2}$, and we get $x+\ell_{1}=x+\ell_{2}$. To show that the quotient map is onto, assume without loss of generality that $W$ is nonempty. Take arbitrary $y^{\prime}\in(x^{\prime}+{\mathscr{L}}^{\prime})\cap(G\times W^{\prime})$. Then $y^{\prime}=\iota(x+\ell+{\mathcal{H}}_{W})$ for some $\ell\in{\mathscr{L}}$, which implies $x_{H}+\ell_{H}+H_{W}\in W^{\prime}=\{w+H_{W}:w\in W\}$. This implies $x_{H}+\ell_{H}\in W+H_{W}=W$, which means that $y:=x+\ell\in(x+{\mathscr{L}})\cap(G\times W)$. The remaining statement is now obvious, since we have $\pi^{\scriptscriptstyle G}(x)=\pi^{\scriptscriptstyle G}((\iota\circ\Phi)(x))$.

b) Let $h\in H_{W}$ be given. A direct calculation yields

[TABLE]

c) A direct calculation yields

[TABLE]

where we use a) for the third equality. The remaining statement is now obvious. ∎

Proposition 6.10**.**

Let $W^{\prime}=\varphi_{W}(W)$.

1. a)

${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}=\pi^{\scriptscriptstyle G}_{*}({\mathcal{M}}_{\scriptscriptstyle W})=\pi^{\scriptscriptstyle G}_{*}({\mathcal{M}}_{\scriptscriptstyle W^{\prime}}^{\prime})={\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W^{\prime}}$. 2. b)

$Q_{\scriptscriptstyle W}^{\scriptscriptstyle G}=m_{\hat{X}}\circ(\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G})^{-1}=m_{{\widehat{X^{\prime}}}}\circ(\nu_{\scriptscriptstyle W^{\prime}}^{\scriptscriptstyle G})^{-1}=Q_{\scriptscriptstyle W^{\prime}}^{\scriptscriptstyle G}$**

Proof.

a) Note the following chain of equivalences:

[TABLE]

where we used Lemma 6.9 c). This means that $\pi^{\scriptscriptstyle G}_{*}(\nu_{\scriptscriptstyle W}({\hat{X}}))=\pi^{\scriptscriptstyle G}_{*}(\nu_{\scriptscriptstyle W^{\prime}}({\widehat{X^{\prime}}}))$. Now a) of the proposition follows from continuity of $\pi^{\scriptscriptstyle G}_{*}$ and compactness of ${\mathcal{M}}_{\scriptscriptstyle W}=\overline{\nu_{\scriptscriptstyle W}({\hat{X}})}\subseteq{\mathcal{M}}$ and ${\mathcal{M}}_{\scriptscriptstyle W^{\prime}}^{\prime}=\overline{\nu_{\scriptscriptstyle W^{\prime}}({\widehat{X^{\prime}}})}\subseteq{\mathcal{M}}^{\prime}$, the space of locally finite measures on the Borel subsets of $G\times H^{\prime}$.

b) In view of Lemma 6.9c, $(\nu_{\scriptscriptstyle W^{\prime}}^{\scriptscriptstyle G}\circ\widehat{\iota\Phi})(x+{\mathscr{L}})=\nu_{\scriptscriptstyle W^{\prime}}^{\scriptscriptstyle G}(\iota(\Phi(x))+{\mathscr{L}}^{\prime})=\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}(x+{\mathscr{L}})$ for all $x\in G\times H$. Hence, observing Remark 6.6, $m_{{\widehat{X^{\prime}}}}\circ(\nu_{\scriptscriptstyle W^{\prime}}^{\scriptscriptstyle G})^{-1}=m_{\hat{X}}\circ\widehat{\iota\Phi}^{-1}\circ(\nu_{\scriptscriptstyle W^{\prime}}^{\scriptscriptstyle G})^{-1}=m_{\hat{X}}\circ(\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G})^{-1}$. ∎

7 Proofs of Theorems A2, B2 and C2

In this section, $W$ is again a compact window. We begin with a technical lemma that will be used at several places.

Lemma 7.1**.**

Let $\nu,\nu^{\prime}\in{\mathcal{M}}_{\scriptscriptstyle W}$ and $W_{0}\subseteq W$ be such that $\pi^{\scriptscriptstyle G}_{*}(\nu^{\prime})=\pi^{\scriptscriptstyle G}_{*}(\nu)$ and ${\mathcal{S}}_{\scriptscriptstyle H}(\nu)={\mathcal{S}}_{\scriptscriptstyle H}(\nu^{\prime})=W_{0}$. Then $\gamma(\nu^{\prime})-\gamma(\nu)\in\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{W_{0}})$.

Proof.

As $\pi^{\scriptscriptstyle G}_{*}(\nu)=\pi^{\scriptscriptstyle G}_{*}(\nu^{\prime})$, Lemma 4.4 implies that $\nu^{\prime}=\sigma_{d}\nu$ for some $d\in H_{W_{0}}$. Recall that $\gamma(\nu)$ equals $x+{\mathscr{L}}$ for any point $x\in G\times H$ with $\nu\{x\}=1$. But $\nu\{x\}=\nu^{\prime}\{x+(0,d)\}$, so $\gamma(\nu^{\prime})=x+(0,d)+{\mathscr{L}}$. Hence $\gamma(\nu^{\prime})-\gamma(\nu)=(0,d)+{\mathscr{L}}\in\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{W_{0}})$. ∎

Suppose that $\operatorname{int}(W)\neq\emptyset$. Denote by $\gamma$ the factor map from ${\mathcal{M}}_{\scriptscriptstyle W}$ onto its maximal equicontinuous factor ${\hat{X}}$ 101010This is the map $\pi^{\scriptscriptstyle{\hat{X}}}_{*}\circ(\pi^{\scriptscriptstyle G\times H}_{*})^{-1}$ from [22, Thm. 1a]., and by $\rho$ the factor map from ${\hat{X}}$ onto ${\hat{X}}/\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{\operatorname{int}(W)})$. We define a factor map $\Gamma$ from ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ to ${\hat{X}}/\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{\operatorname{int}(W)})$ as follows: for $\nu^{\scriptscriptstyle G}\in{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ pick any $\nu\in(\pi^{\scriptscriptstyle G}_{*})^{-1}\{\nu^{\scriptscriptstyle G}\}$ and let $\Gamma(\nu^{\scriptscriptstyle G})=\rho(\gamma(\nu))$.

Lemma 7.2**.**

Suppose that $\operatorname{int}(W)\neq\emptyset$.

1. a)

$\Gamma$* is well defined.* 2. b)

$\Gamma$* is continuous and commutes with the dynamics.* 3. c)

If $H_{\operatorname{int}(W)}=H_{W}$ and $\Gamma(\nu^{\scriptscriptstyle G})=\rho({\hat{x}})$ for some $\nu^{\scriptscriptstyle G}\in{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ and ${\hat{x}}\in C_{\scriptscriptstyle W}$, then $\nu^{\scriptscriptstyle G}=\pi^{\scriptscriptstyle G}_{*}(\nu_{\scriptscriptstyle W}({\hat{x}}))$. 4. d)

If $H_{\operatorname{int}(W)}=H_{W}$, then $\Gamma$ is almost 1-1.

Proof.

a) Suppose that $\pi^{\scriptscriptstyle G}_{*}(\nu)=\pi^{\scriptscriptstyle G}_{*}(\nu^{\prime})$. Then $\gamma(\nu^{\prime})-\gamma(\nu)\in\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{\operatorname{int}(W)})$ by Lemma 7.1, so that $\rho(\gamma(\nu^{\prime}))=\rho(\gamma(\nu)+\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{\operatorname{int}(W)}))=\rho(\gamma(\nu))$. This shows that $\Gamma$ is well defined.

b) Let $D\subseteq{\hat{X}}/\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{\operatorname{int}(W)})$ be closed. Then $E:=\gamma^{-1}(\rho^{-1}(D))$ is closed in ${\mathcal{M}}_{\scriptscriptstyle W}$, and

[TABLE]

As $E$ is also compact, this shows that $\Gamma^{-1}(D)=\pi^{\scriptscriptstyle G}_{*}(E)$ is closed. Hence $\Gamma$ is continuous. We show that it commutes with the dynamics: Let $\nu^{\scriptscriptstyle G}\in{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, $g\in G$, and denote $D:=\{\Gamma(\nu^{\scriptscriptstyle G})\}$ and $E:=\gamma^{-1}(\rho^{-1}(D))$. Then $\nu^{\scriptscriptstyle G}\in\Gamma^{-1}(D)=\pi^{\scriptscriptstyle G}_{*}(E)$ by (13), and

[TABLE]

Hence $S_{g}\nu^{\scriptscriptstyle G}\in S_{g}(\pi^{\scriptscriptstyle G}_{*}(E))=\pi^{\scriptscriptstyle G}_{*}(S_{g}(\gamma^{-1}(\rho^{-1}(D))))=\pi^{\scriptscriptstyle G}_{*}(\gamma^{-1}(\rho^{-1}(\hat{T}_{g}(D))))=\Gamma^{-1}(\hat{T}_{g}(D))$, where we used again (13) for the last identity. Therefore, $\Gamma(S_{g}\nu^{\scriptscriptstyle G})\in\hat{T}_{g}(D)=\{\hat{T}_{g}(\Gamma(\nu^{\scriptscriptstyle G}))\}$.

c) Let $h\in H_{W}$. Then ${\hat{x}}\in C_{\scriptscriptstyle W}$ if and only if ${\hat{x}}+(0,h)\in C_{\scriptscriptstyle W}$. This follows immediately from

[TABLE]

see Lemma 6.9b).

Suppose now that ${\hat{x}}\in C_{\scriptscriptstyle W}$ and $\Gamma(\nu^{\scriptscriptstyle G})=\rho({\hat{x}})$ for some $\nu^{\scriptscriptstyle G}\in{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$. There is $\nu\in{\mathcal{M}}_{\scriptscriptstyle W}$ such that $\pi^{\scriptscriptstyle G}_{*}(\nu)=\nu^{\scriptscriptstyle G}$ and $\rho(\gamma(\nu))=\rho({\hat{x}})$. Hence $\gamma(\nu)\in{\hat{x}}+{\mathcal{H}}_{\operatorname{int}(W)}={\hat{x}}+{\mathcal{H}}_{W}$, i.e., there is $h\in H_{W}$ such that $\gamma(\nu)={\hat{x}}+(0,h)\in C_{\scriptscriptstyle W}$. This implies $\nu=\nu_{\scriptscriptstyle W}({\hat{x}}+(0,h))$, see [22, Prop. 3.3b]. Hence

[TABLE]

where we used (14) for the third identity.

d) In view of assertion c), $\Gamma^{-1}\{\rho({\hat{x}})\}$ is a singleton for each $\rho({\hat{x}})\in\rho(C_{\scriptscriptstyle W})$. For countable acting groups $G$ it is well known that this implies that $\Gamma$ is almost 1-1. For uncountable groups we could not locate such a statement in the literature. So we provide a proof for the convenience of the reader: In view of c) we only need to show that $\rho(C_{\scriptscriptstyle W})$ is a dense $G_{\delta}$-set in ${\hat{X}}/\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{\operatorname{int}(W)})$: $C_{\scriptscriptstyle W}$ is a dense $G_{\delta}$-set in ${\hat{X}}$ by [22, Prop. 3.3c]. Hence denseness of $\rho(C_{\scriptscriptstyle W})$ follows as $\rho$ is continuous and onto. In (14) we argued that $C_{\scriptscriptstyle W}$ - and hence also ${\hat{X}}\setminus C_{\scriptscriptstyle W}$ - are invariant under translations by elements from the subgroup $H_{W}$. Hence $\rho({\hat{X}}\setminus C_{\scriptscriptstyle W})=({\hat{X}}/\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{\operatorname{int}(W)}))\setminus\rho(C_{\scriptscriptstyle W})$. Indeed, suppose $\rho({\hat{x}})=\rho({\hat{y}})$ for some ${\hat{x}}\in C_{\scriptscriptstyle W}$ and ${\hat{y}}\in{\hat{X}}\setminus C_{\scriptscriptstyle W}$. Then ${\hat{y}}\in{\hat{x}}+{\mathcal{H}}_{\operatorname{int}(W)}={\hat{x}}+{\mathcal{H}}_{W}\subseteq C_{\scriptscriptstyle W}$, a contradiction. To show that $\rho(C_{\scriptscriptstyle W})$ is a $G_{\delta}$-set, it now suffices to show that $\rho({\hat{X}}\setminus C_{\scriptscriptstyle W})$ is an $F_{\sigma}$-set. But this is obvious since ${\hat{X}}\setminus C_{\scriptscriptstyle W}$ is an $F_{\sigma}$-set by Eqn. (6) and hence a countable union of compact sets, and since $\rho$ is continuous.

∎

Proof of Theorem A2.

a) Let $M$ be any non-empty, closed $S$-invariant subset of ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$. $({\widehat{X^{\prime}}},\widehat{T^{\prime}})$ is a factor of $(M,S)$ by Lemma 7.2. We prove that it is the maximal equicontinuous factor of $(M,S)$.

Let $W_{0}:=\overline{\operatorname{int}(W)}$ and $H_{0}:=H_{W_{0}}=H_{\operatorname{int}(W)}$ (see Lemma 6.1 for the second identity). Then $W^{\prime}=\varphi_{W_{0}}(W_{0})$ is an aperiodic subset of $H^{\prime}=H/H_{W_{0}}$ by Lemma 6.8. As $W_{0}$ is topologically regular, also $W^{\prime}$ is topologically regular (Lemma 6.7). Hence also $\operatorname{int}(W^{\prime})$ is aperiodic.

Let ${\widehat{X^{\prime}}}={\hat{X}}/\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{\operatorname{int}(W)})$ with induced $G$-action $\widehat{T^{\prime}}$, and recall from Corollary 6.5 that ${\widehat{X^{\prime}}}$ is isomorphic to $(G\times H^{\prime})/{\mathscr{L}}^{\prime}$. Theorem A1 implies that $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W^{\prime}},S)$ is an almost 1-1 extension of $({\widehat{X^{\prime}}},\widehat{T^{\prime}})$. As ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W^{\prime}}=\pi^{\scriptscriptstyle G}_{*}({\mathcal{M}}_{\scriptscriptstyle W^{\prime}})=\pi^{\scriptscriptstyle G}_{*}({\mathcal{M}}_{\scriptscriptstyle W_{0}})={\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W_{0}}$ by Proposition 6.10 (applied to $W_{0}$ instead of $W$), also $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W_{0}},S)$ is an almost 1-1 extension of $({\widehat{X^{\prime}}},\widehat{T^{\prime}})$. It follows that also the unique minimal subsystem $(\overline{\nu_{\scriptscriptstyle W_{0}}^{\scriptscriptstyle G}(C_{\scriptscriptstyle W_{0}})},S)$ of $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W_{0}},S)$ is an almost 1-1 extension of $({\widehat{X^{\prime}}},\widehat{T^{\prime}})$.

Note next that $C_{\scriptscriptstyle W}\subseteq C_{\scriptscriptstyle W_{0}}$ and $\nu_{\scriptscriptstyle W_{0}}^{\scriptscriptstyle G}|_{C_{\scriptscriptstyle W}}=\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}|_{C_{\scriptscriptstyle W}}$, because $\partial W_{0}\subseteq\partial W$. As $C_{\scriptscriptstyle W}$ is dense in ${\hat{X}}$ (and as $C_{\scriptscriptstyle W_{0}}$ is the set of continuity points of $\nu_{\scriptscriptstyle W_{0}}^{\scriptscriptstyle G}$), this implies $\overline{\nu_{\scriptscriptstyle W_{0}}^{\scriptscriptstyle G}(C_{\scriptscriptstyle W_{0}})}=\overline{\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}(C_{\scriptscriptstyle W})}={\mathcal{M}}^{\scriptscriptstyle G}_{\scriptstyle min}$. Hence the minimal system $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptstyle min},S)$ is an almost 1-1 extension of $({\widehat{X^{\prime}}},\widehat{T^{\prime}})$, so that $({\widehat{X^{\prime}}},\widehat{T^{\prime}})$ is the maximal equicontinuous factor of $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptstyle min},S)$.

Suppose now that $(\tilde{X},\tilde{T})$ is an equicontinuous factor of $(M,S)$ and observe that ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptstyle min}\subseteq M$. Then the restriction of the factor map to ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptstyle min}$ defines a factor map from $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptstyle min},S)$ to $(\tilde{X},\tilde{T})$. It follows that $(\tilde{X},\tilde{T})$ is a factor of $({\widehat{X^{\prime}}},\widehat{T^{\prime}})$. As this holds for any equicontinuous factor $(\tilde{X},\tilde{T})$ of $(M,S)$, the system $({\widehat{X^{\prime}}},\widehat{T^{\prime}})$ is in fact the maximal equicontinuous factor of $(M,S)$.

b) This is Lemma 7.2d. ∎

Remark 7.3**.**

A proof of Theorem A2 only for the case when $H_{\operatorname{int}(W)}=H_{W}$ is much simpler: In that case $H^{\prime}=H/H_{W}$, and $W^{\prime}=\varphi_{W}(W)$ is aperiodic by Lemma 6.8. As ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}={\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W^{\prime}}$ by Proposition 6.10a, all assertions of Theorem A2 follow from Theorem A1 applied to the cut-and-project scheme $(G,H^{\prime},{\mathscr{L}}^{\prime})$ with window $W^{\prime}$.

Proof of Theorem B2.

Assume first that the window $W$ is Haar regular. Then $W=W_{reg}=W_{Q_{\scriptscriptstyle W}}$ by Corollary 5.2. Let $W^{\prime}=\varphi_{W}(W)$. This set is Haar regular by Lemma 6.7 and aperiodic by Lemma 6.8. Hence $W^{\prime}$ is also Haar aperiodic, see Remark 3.12. As $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},Q_{\scriptscriptstyle W}^{\scriptscriptstyle G},S)=({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W^{\prime}},Q_{\scriptscriptstyle W^{\prime}}^{\scriptscriptstyle G},S)$ by Proposition 6.10, the claim of the theorem follows now from Theorem B1.

In the general case, note that $({\mathcal{M}}_{\scriptscriptstyle W},Q_{\scriptscriptstyle W},S)$ is measure-theoretically isomorphic to $({\mathcal{M}},Q_{\scriptscriptstyle W},S)$. As the present theorem applies to the Haar regularized window $W_{reg}$, we must only show that $Q_{\scriptscriptstyle W}=m_{\hat{X}}\circ(\nu_{\scriptscriptstyle W})^{-1}$ equals $Q_{\scriptscriptstyle W_{reg}}=m_{\hat{X}}\circ(\nu_{\scriptscriptstyle W_{reg}})^{-1}$ on ${\mathcal{M}}$. But this follows from the observation that

[TABLE]

and this is a set of $m_{\hat{X}}$-measure zero, because ${\mathscr{L}}$ is countable and $m_{H}(W\setminus W_{reg})=0$. ∎

Proof of Theorem C2.

Let $P^{\scriptscriptstyle G}$ be an ergodic $S$-invariant probability measure on ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}\setminus\{\underline{0}\}$. Take any ergodic $S$-invariant probability measure $P$ on ${\mathcal{M}}_{\scriptscriptstyle W}\setminus\{\underline{0}\}$ satisfying $P^{\scriptscriptstyle G}=P\circ(\pi^{\scriptscriptstyle G}_{*})^{-1}$. In particular we have $W_{P}\neq\emptyset$. Let ${\widehat{X^{\prime}}}={\hat{X}}/\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{W_{P}})$ with induced $G$-action $\widehat{T^{\prime}}$ and Haar measure $m_{{\widehat{X^{\prime}}}}$. Denote

[TABLE]

where $W_{P}\subseteq W$ is the Haar regular set from Lemma 4.5, for which ${\mathcal{S}}_{\scriptscriptstyle H}(\nu)=W_{P}$ for $P$-a.a. $\nu$. Hence $P({\mathcal{A}}_{P})=1$, and ${\mathcal{A}}_{P}$ is Borel measurable and $S$-invariant, because ${\mathcal{S}}_{\scriptscriptstyle H}$ is (Lemmas 4.1 and 4.3a). Then also $\pi^{\scriptscriptstyle G}_{*}({\mathcal{A}}_{P})\subseteq{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}\setminus\{\underline{0}\}$ is $S$-invariant, it is Borel measurable by Lemma 4.7, and $P^{\scriptscriptstyle G}(\pi^{\scriptscriptstyle G}_{*}({\mathcal{A}}_{P}))=P\left((\pi^{\scriptscriptstyle G}_{*})^{-1}(\pi^{\scriptscriptstyle G}_{*}({\mathcal{A}}_{P}))\right)\geqslant P({\mathcal{A}}_{P})=1$.

Denote by $\gamma$ the factor map from ${\mathcal{M}}_{\scriptscriptstyle W}\setminus\{\underline{0}\}$ onto ${\hat{X}}$ 111111This is the map $\pi^{\scriptscriptstyle{\hat{X}}}_{*}\circ(\pi^{\scriptscriptstyle G\times H}_{*})^{-1}$ from [22, Thm. 1a]., and by $\rho$ the factor map from ${\hat{X}}$ onto ${\hat{X}}/\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{W_{P}})$. We define now a map

[TABLE]

In order to see that $\Gamma$ is well defined, observe first the cardinality is at least $1$, because $\nu^{\scriptscriptstyle G}\in\pi^{\scriptscriptstyle G}_{*}({\mathcal{A}}_{P})$. On the other hand, if $\nu,\nu^{\prime}\in(\pi^{\scriptscriptstyle G}_{*})^{-1}\{\nu^{\scriptscriptstyle G}\}\cap{\mathcal{A}}_{P}$, then $\gamma(\nu^{\prime})-\gamma(\nu)\in\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{W_{P}})$ by Lemma 7.1, i.e. $(\rho\circ\gamma)(\nu^{\prime})=(\rho\circ\gamma)(\nu)$.

Next observe that $\Gamma$ commutes with the dynamics: For each $g\in G$,

[TABLE]

It remains to show that $\pi^{\scriptscriptstyle G}_{*}({\mathcal{A}}_{P})\cap\Gamma^{-1}(K^{\prime})$ is Borel measurable for each closed subset $K^{\prime}$ of ${\widehat{X^{\prime}}}$. Then it follows that $\Gamma$ is Borel measurable and $({\widehat{X^{\prime}}},m_{{\widehat{X^{\prime}}}},\widehat{T^{\prime}})$ is a measure-theoretic factor of $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},P^{\scriptscriptstyle G},S)$ as claimed in Theorem C2.

So let $K:=(\rho\circ\gamma)^{-1}(K^{\prime})\subseteq{\mathcal{M}}_{\scriptscriptstyle W}$. Then $K$ is closed, and

[TABLE]

The $\subseteq$-inclusion is trivial. To see the reverse inclusion, let $\nu\in{\mathcal{A}}_{P}$ and assume that there exists $\nu^{\prime}\in K\cap{\mathcal{A}}_{P}$ such that $\pi^{\scriptscriptstyle G}_{*}(\nu)=\pi^{\scriptscriptstyle G}_{*}(\nu^{\prime})$. Then $\gamma(\nu)-\gamma(\nu^{\prime})\in\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{W_{P}})$ by Lemma 7.1, so that $(\rho\circ\gamma)(\nu)\in(\rho\circ\gamma)(\nu^{\prime})+\rho(\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{W_{P}}))=(\rho\circ\gamma)(\nu^{\prime})\in(\rho\circ\gamma)(K)\subseteq K^{\prime}$, i.e. $\nu\in(\rho\circ\gamma)^{-1}(K^{\prime})=K$.

Now let $\nu^{\scriptscriptstyle G}\in\pi^{\scriptscriptstyle G}_{*}({\mathcal{A}}_{P})$. Then

[TABLE]

The last equivalence is seen as follows:

,,$\Rightarrow$”: As $\nu^{\scriptscriptstyle G}\in\pi^{\scriptscriptstyle G}_{*}({\mathcal{A}}_{P})$, there exists some $\nu\in(\pi^{\scriptscriptstyle G}_{*})^{-1}\{\nu^{\scriptscriptstyle G}\}\cap{\mathcal{A}}_{P}\subseteq F\cap{\mathcal{A}}_{P}$, so that $\nu^{\scriptscriptstyle G}=\pi^{\scriptscriptstyle G}_{*}(\nu)\in\pi^{\scriptscriptstyle G}_{*}(F\cap{\mathcal{A}}_{P})$.

,,$\Leftarrow$”: Let $\nu\in(\pi^{\scriptscriptstyle G}_{*})^{-1}\{\nu^{\scriptscriptstyle G}\}\cap{\mathcal{A}}_{P}$. Then $\pi^{\scriptscriptstyle G}_{*}(\nu)=\nu^{\scriptscriptstyle G}\in\pi^{\scriptscriptstyle G}_{*}(F\cap{\mathcal{A}}_{P})$, so that in view of (15),

[TABLE]

Hence $\pi^{\scriptscriptstyle G}_{*}({\mathcal{A}}_{P})\cap\Gamma^{-1}(K^{\prime})=\pi^{\scriptscriptstyle G}_{*}(K\cap{\mathcal{A}}_{P})$, and this set is Borel measurable by Lemma 4.7. ∎

8 Relatively compact windows

For suitable relatively compact windows $W\subseteq H$, dynamical properties are the same as in the compact case. This has already been observed in [22, Rem. 3.16].

8.1 Topological results

For some topological results, we assume that the boundary $\partial W$ is nowhere dense. This condition characterises denseness of the set $C_{\scriptscriptstyle W}$ of continuity points of the map $\nu_{\scriptscriptstyle W}:{\mathcal{M}}_{\scriptscriptstyle W}\to{\hat{X}}$, see Proposition 8.3. As any compact window has nowhere dense boundary, this condition generalises the compact case. The next lemma shows that this condition also generalises topological regularity.

Lemma 8.1**.**

For each $W\subseteq H$, $\partial W$ is nowhere dense if and only if $\operatorname{int}(\overline{\operatorname{int}(W)})=\operatorname{int}(\overline{W})$.

Proof.

As $\partial W=\overline{W}\cap\overline{W^{c}}$, we have $\operatorname{int}(\partial W)=\operatorname{int}(\overline{W})\cap\operatorname{int}(\overline{W^{c}})$. Hence $\partial W$ is nowhere dense if and only if $\operatorname{int}(\overline{W})\subseteq(\operatorname{int}(\overline{W^{c}}))^{c}=\overline{\operatorname{int}(W)}$. But this is obviously equivalent to $\operatorname{int}(\overline{\operatorname{int}(W)})=\operatorname{int}(\overline{W})$. ∎

The next lemma generalises slightly Lemma 6.1 in [22].

Lemma 8.2**.**

Let $W\subseteq H$ be relatively compact. Then the set $C_{\scriptscriptstyle W}\subseteq{\hat{X}}$ of continuity points of the map $\nu_{\scriptscriptstyle W}:{\hat{X}}\to{\mathcal{M}}_{\scriptscriptstyle W}$ is given by

[TABLE]

Proof.

This is seen by reinspecting the proof of [22, Lem. 6.1]. If $x_{H}\not\in\bigcap_{\ell\in{\mathscr{L}}}((\partial W)^{c}-\ell_{H})$, then there is $\ell\in{\mathscr{L}}$ such that $x_{H}+\ell_{H}\in\partial W$. In the case $x_{H}+\ell_{H}\in W$ we take $x^{n}_{H}\in(W)^{c}-\ell_{H}$ such that $x_{H}^{n}\to x_{H}$ as $n\to\infty$ and let $x^{n}=(x_{G},x_{H}^{n})$. Then, for each sufficiently small open neighbourhood $U$ of $x+\ell$ in $G\times H$ and sufficiently large $n$ we have $\nu_{\scriptscriptstyle W}(x^{n}+{\mathscr{L}})(U)=1_{W}(x_{H}^{n}+\ell_{H})\cdot\delta_{x_{n}+\ell}(U)=0$ while $\nu_{\scriptscriptstyle W}(x+{\mathscr{L}})(U)=1_{W}(x_{H}+\ell_{H})\cdot\delta_{x+\ell}(U)=1$ so that $\nu_{\scriptscriptstyle W}(x^{n}+{\mathscr{L}})\not\to\nu_{\scriptscriptstyle W}(x+{\mathscr{L}})$. In particular, $(x+{\mathscr{L}})\not\in C_{\scriptscriptstyle W}$. If $x_{H}+\ell_{H}\in(W)^{c}$, we may argue analogously with a sequence $x^{n}_{H}\in W-\ell_{H}$ such that $x_{H}^{n}\to x_{H}$ as $n\to\infty$.

Conversely, let $x+{\mathscr{L}},x^{n}+{\mathscr{L}}\in{\hat{X}}$, $\lim_{n\to\infty}(x^{n}+{\mathscr{L}})=(x+{\mathscr{L}})$ and take a sufficiently small open neighbourhood $U$ of $x+\ell$ in $G\times H$. Assume that $x_{H}\in\bigcap_{\ell\in{\mathscr{L}}}((\partial W)^{c}-\ell_{H})$, i.e., $(x+\ell)_{H}\not\in\partial W$ for all $\ell\in{\mathscr{L}}$. Now consider $\ell\in{\mathscr{L}}$ such that $x_{H}+\ell\in W$. Then by assumption $x_{H}+\ell_{H}\in\operatorname{int}(W)$, which implies $1=\nu_{\scriptscriptstyle W}(x+{\mathscr{L}})(U)=\nu_{\scriptscriptstyle W}(x^{n}+{\mathscr{L}})(U)$ for sufficiently large $n$. If $\ell\in{\mathscr{L}}$ such that $x_{H}+\ell_{H}\notin W$, then by assumption $x_{H}+\ell\in(\overline{W})^{c}$, which implies $0=\nu_{\scriptscriptstyle W}(x+{\mathscr{L}})(U)=\nu_{\scriptscriptstyle W}(x^{n}+{\mathscr{L}})(U)$ for sufficiently large $n$. This implies $\lim_{n\to\infty}\nu_{\scriptscriptstyle W}(x^{n}+{\mathscr{L}})=\nu_{\scriptscriptstyle W}(x+{\mathscr{L}})$ and hence $x+{\mathscr{L}}\in C_{\scriptscriptstyle W}$. ∎

The following proposition collects some further properties of the set $C_{\scriptscriptstyle W}$.

Proposition 8.3**.**

Let $W\subseteq H$ be relatively compact. Then the set $C_{\scriptscriptstyle W}\subseteq{\hat{X}}$ of continuity points of the map $\nu_{\scriptscriptstyle W}:{\hat{X}}\to{\mathcal{M}}_{\scriptscriptstyle W}$ is a $G_{\delta}$-set. The set $C_{\scriptscriptstyle W}$ is dense in ${\hat{X}}$ if and only if $\partial W$ is nowhere dense in $H$. Otherwise $C_{\scriptscriptstyle W}=\emptyset$.

Proof.

121212For compact windows this was claimed in [22, Prop. 3.3c]. As the proof of that proposition contains a mistake, we provide a full proof for the more general case treated here. (Indeed, the argument given in footnote $\dagger$ of the proof of that proposition is wrong, because, with the sets $U_{\ell}$ and $X$ defined there, it is not true that $\pi^{\scriptscriptstyle{\hat{X}}}(X\cap U_{\ell})=\pi^{\scriptscriptstyle{\hat{X}}}(U_{\ell})$.)

Let $A:=\bigcap_{\ell\in{\mathscr{L}}}\left((G\times(\partial W)^{c})-\ell\right)$ and $B:=(G\times H)\setminus A=\bigcup_{\ell\in{\mathscr{L}}}G\times(\partial W-\ell_{H})$, and recall that ${\hat{X}}=(G\times H)/{\mathscr{L}}$. As both sets, $A$ and $B$, are invariant under translations by elements from the lattice ${\mathscr{L}}$, $\pi^{\scriptscriptstyle{\hat{X}}}(B)={\hat{X}}\setminus\pi^{\scriptscriptstyle{\hat{X}}}(A)$. Indeed, suppose $\pi^{\scriptscriptstyle{\hat{X}}}(a)=\pi^{\scriptscriptstyle{\hat{X}}}(b)$ for some $a\in A$ and $b\in B$. Then $b\in a+{\mathscr{L}}\subseteq A$, a contradiction. To show that $\pi^{\scriptscriptstyle{\hat{X}}}(A)$ is a $G_{\delta}$-set, it suffices to show that $\pi^{\scriptscriptstyle{\hat{X}}}(B)$ is a $F_{\sigma}$-set. To that end recall that $G$ is $\sigma$-compact, i.e. there are compact $K_{1},K_{2},\ldots\subseteq G$ such that $G=\bigcup_{j\in\mathbbm{N}}K_{j}$. Hence $\pi^{\scriptscriptstyle{\hat{X}}}(B)=\bigcup_{\ell\in{\mathscr{L}}}\bigcup_{j\in\mathbbm{N}}\pi^{\scriptscriptstyle{\hat{X}}}\left(K_{j}\times(\partial W-\ell_{H})\right)$ is a countable union of compact sets, hence $F_{\sigma}$.

Now assume that $\partial W$ is nowhere dense. Then $A_{H}:=\bigcap_{\ell\in{\mathscr{L}}}\left((\partial W)^{c}-\ell_{H}\right)$ is dense in $H$ as $\partial W$ is nowhere dense and as $H$ is a Baire space. This readily implies that $A=(\pi^{\scriptscriptstyle H})^{-1}(A_{H})=G\times A_{H}$ is dense in $G\times H$. Now denseness of $\pi^{\scriptscriptstyle{\hat{X}}}(A)=C_{\scriptscriptstyle W}$ in ${\hat{X}}$ follows as $\pi^{\scriptscriptstyle{\hat{X}}}$ is continuous and onto.

Conversely, assume that $\partial W$ is not nowhere dense. Then there is some nonempty open set $O\subseteq\partial W$, which implies $A_{H}^{c}\supseteq\bigcup_{\ell\in{\mathscr{L}}}(O-\ell_{H})=H$ because of the denseness of $\pi^{\scriptscriptstyle H}({\mathscr{L}})$ in $H$. Thus $A_{H}=\emptyset$ and $C_{\scriptscriptstyle W}=\pi^{\scriptscriptstyle{\hat{X}}}((\pi^{\scriptscriptstyle H})^{-1}(A_{H}))=\emptyset$, in particular $C_{\scriptscriptstyle W}$ is not dense. ∎

Let us denote by ${{\mathcal{G}}\nu_{\scriptscriptstyle W}}:=\{({\hat{x}},\nu_{\scriptscriptstyle W}({\hat{x}})):{\hat{x}}\in{\hat{X}}\}\subseteq{\hat{X}}\times{\mathcal{M}}_{\scriptscriptstyle W}$ the graph of the map $\nu_{\scriptscriptstyle W}$, and by ${\mathcal{G}}{\mathcal{M}}_{\scriptscriptstyle W}$ its closure in the vague topology. Likewise, denote by ${\mathcal{G}}(\nu_{\scriptscriptstyle W}|_{C_{\scriptscriptstyle W}})$ the restriction of ${{\mathcal{G}}\nu_{\scriptscriptstyle W}}$ to its continuity points. We have the following general result on minimal subsets of ${\mathcal{G}}{\mathcal{M}}_{\scriptscriptstyle W},{\mathcal{M}}_{\scriptscriptstyle W}$ and ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$.

Lemma 8.4**.**

Let $W\subseteq H$ be relatively compact. Then

1. a)

The set $\overline{{\mathcal{G}}(\nu_{\scriptscriptstyle W}|_{C_{\scriptscriptstyle W}})}$ is the unique minimal subset of ${\mathcal{G}}{\mathcal{M}}_{\scriptscriptstyle W}$. 2. b)

The set $\overline{\nu_{\scriptscriptstyle W}(C_{\scriptscriptstyle W})}$ is the unique minimal subset of ${\mathcal{M}}_{\scriptscriptstyle W}$. 3. c)

The set $\overline{\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}(C_{\scriptscriptstyle W})}$ is the unique minimal subset of ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$. 4. d)

If $\operatorname{int}(W)=\emptyset$, then $\overline{\nu_{\scriptscriptstyle W}(C_{\scriptscriptstyle W})}$ and $\overline{\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}(C_{\scriptscriptstyle W})}$ are singletons consisting of the zero measure only.

Proof.

This can be seen by re-inspecting the proofs in [22], but we give a simple direct argument for the ease of the reader.

a) Let $\emptyset\neq A\subseteq{\mathcal{G}}{\mathcal{M}}_{\scriptscriptstyle W}$ be any closed invariant set. Then $\emptyset\neq\pi^{\scriptscriptstyle{\hat{X}}}(A)\subseteq{\hat{X}}$ is closed invariant. Hence $\pi^{\scriptscriptstyle{\hat{X}}}(A)={\hat{X}}\supseteq C_{\scriptscriptstyle W}$, since $({\hat{X}},\hat{T})$ is minimal. As $(\pi^{\scriptscriptstyle{\hat{X}}})^{-1}\{{\hat{x}}\}\cap{\mathcal{G}}{\mathcal{M}}_{\scriptscriptstyle W}=\{({\hat{x}},\nu_{\scriptscriptstyle W}({\hat{x}}))\}$ for each ${\hat{x}}\in C_{\scriptscriptstyle W}$, this implies $A\supseteq{\mathcal{G}}(\nu_{\scriptscriptstyle W}|_{C_{\scriptscriptstyle W}})$, which means $A\supseteq\overline{{\mathcal{G}}(\nu_{\scriptscriptstyle W}|_{C_{\scriptscriptstyle W}})}=:A_{min}$.

b) Let $\emptyset\neq B\subseteq{\mathcal{M}}_{\scriptscriptstyle W}$ be any closed invariant set. Then $\emptyset\neq(\pi^{\scriptscriptstyle G\times H}_{*})^{-1}(B)\subseteq{\mathcal{G}}{\mathcal{M}}_{\scriptscriptstyle W}$ is closed invariant. By the previous result, we infer $(\pi^{\scriptscriptstyle G\times H}_{*})^{-1}(B)\supseteq A_{min}$. Hence $B\supseteq\pi^{\scriptscriptstyle G\times H}_{*}(A_{min})\supseteq{\nu_{\scriptscriptstyle W}(C_{\scriptscriptstyle W})}$, so that $\overline{\nu_{\scriptscriptstyle W}(C_{\scriptscriptstyle W})}\subseteq\overline{B}=B$.

c) This follows using the same argument as in b).

d) If $\operatorname{int}(W)=\emptyset$, then $(\partial W)^{c}\subseteq(W)^{c}$. Hence ${\hat{x}}=x+{\mathscr{L}}\in C_{\scriptscriptstyle W}$ implies $x_{H}\in\bigcap_{\ell\in{\mathscr{L}}}\left((\partial W)^{c}-\ell_{H}\right)\subseteq\bigcap_{\ell\in{\mathscr{L}}}\left((W)^{c}-\ell_{H}\right)$. But the latter condition means $(x+{\mathscr{L}})\cap(G\times W)=\emptyset$. Hence the claim follows. ∎

Consider now the window $\overline{W}$, as well. We infer from [22, Lem. 5.4] that, for each $\nu\in{\mathcal{M}}_{\scriptscriptstyle W}\setminus\{\underline{0}\}$, there is a unique $\hat{\pi}(\nu)\in{\hat{X}}$ such that $\operatorname{supp}(\nu)\subseteq\operatorname{supp}(\nu_{\scriptscriptstyle\overline{W}}(\hat{\pi}(\nu)))$. Thus the map $\hat{\pi}:{\mathcal{M}}_{\scriptscriptstyle W}\setminus\{\underline{0}\}\to{\hat{X}}$ is still well-defined and continuous in our more general setting, and it satisfies $\hat{\pi}=\pi^{\scriptscriptstyle{\hat{X}}}_{*}\circ(\pi^{\scriptscriptstyle G\times H}_{*})^{-1}$. We have the following version of [22, Thm. 1a].

Proposition 8.5**.**

Let $W\subseteq H$ be relatively compact and such that $\partial W$ is nowhere dense in $H$. Assume that $\operatorname{int}(\overline{W})$ is nonempty (being equivalent to $\operatorname{int}(W)$ nonempty, in this case). Then

1. a)

$\pi^{\scriptscriptstyle{\hat{X}}}_{*}:({\mathcal{G}}{\mathcal{M}}_{\scriptscriptstyle W},S)\to({\hat{X}},\hat{T})$* is a topological almost 1-1-extension of its maximal equicontinuous factor.* 2. b)

$\hat{\pi}:({\mathcal{M}}_{\scriptscriptstyle W},S)\to({\hat{X}},\hat{T})$* is a topological almost 1-1-extension of its maximal equicontinuous factor.*

Proof.

a) We can argue as in the proof of [22, Prop. 3.5c]. The assumption $\partial W$ nowhere dense guarantees that $C_{\scriptscriptstyle W}$ is a dense $G_{\delta}$-set by Proposition 8.3.

b) This follows from a) by noting that the statements and proofs of [22, Prop. 3.5b] and [22, Prop. 3.3e] still apply to the present situation. ∎

If $\partial W$ is nowhere dense, then ${\mathcal{M}}_{\scriptscriptstyle W}$ and ${\mathcal{M}}_{\scriptscriptstyle\overline{W}}$ have the same unique minimal subset, and a similar result holds for the $G$-projections.

Lemma 8.6**.**

(See also [22, Cor. 1b and Remark 3.16]) Let $W\subseteq H$ be relatively compact and such that $\partial W$ is nowhere dense in $H$. Then $\overline{\nu_{\scriptscriptstyle W}(C_{\scriptscriptstyle W})}=\overline{\nu_{\scriptscriptstyle\overline{W}}(C_{\scriptscriptstyle\overline{W}})}$ and $\overline{\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}(C_{\scriptscriptstyle W})}=\overline{\nu_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}(C_{\scriptscriptstyle\overline{W}})}$.

Proof.

As $\partial\overline{W}\subseteq\partial W$, we have $C_{\scriptscriptstyle W}\subseteq C_{\scriptscriptstyle\overline{W}}$ and $\nu_{\scriptscriptstyle W}|_{C_{\scriptscriptstyle W}}=\nu_{\scriptscriptstyle\overline{W}}|_{C_{\scriptscriptstyle W}}$, because $1_{\overline{W}}(h)=1_{W}(h)$ for all $h\in H\setminus\partial W$. Hence $\overline{\nu_{\scriptscriptstyle W}(C_{\scriptscriptstyle W})}=\overline{\nu_{\scriptscriptstyle\overline{W}}(C_{\scriptscriptstyle W})}\subseteq\overline{\nu_{\scriptscriptstyle\overline{W}}(C_{\scriptscriptstyle\overline{W}})}$. On the other hand, as $C_{\scriptscriptstyle W}$ is dense in ${\hat{X}}$ and as $\nu_{\scriptscriptstyle\overline{W}}$ is continuous on $C_{\scriptscriptstyle\overline{W}}$, we have $\nu_{\scriptscriptstyle\overline{W}}(C_{\scriptscriptstyle\overline{W}})\subseteq\overline{\nu_{\scriptscriptstyle W}(C_{\scriptscriptstyle W})}$. This proves the first identity. The second identity follows at once, because $\nu_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}=\pi^{\scriptscriptstyle G}_{*}\circ\nu_{\scriptscriptstyle\overline{W}}$ with a continuous $\pi^{\scriptscriptstyle G}_{*}$. ∎

Now we are ready to state and prove the following extensions of Theorems A1 and A2.

Theorem A1’****.

Let $W\subseteq H$ be relatively compact and $\partial W$ be nowhere dense. Assume that $\operatorname{int}(\overline{W})$ is aperiodic (so in particular non-empty).

1. a)

The topological dynamical systems $({\mathcal{M}}_{\scriptscriptstyle W},S)$ and $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},S)$ are isomorphic, and both are almost 1-1 extensions of their maximal equicontinuous factor $({\hat{X}},\hat{T})$. 2. b)

Denote by $\Gamma:{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}\to{\hat{X}}$ the factor map from a). If $M$ is a non-empty, closed $S$-invariant subset of ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, then $(M,S)$ is an almost 1-1 extension of its maximal equicontinuous factor $({\hat{X}},\hat{T})$ with factor map $\Gamma|_{M}$.

Sketch of proof of Theorem A1’.

In Section 4, the space $\widetilde{{\mathcal{M}}_{\scriptscriptstyle\overline{W}}}$ was defined such that ${\mathcal{M}}_{\scriptscriptstyle W}\subseteq\widetilde{{\mathcal{M}}_{\scriptscriptstyle\overline{W}}}$. Hence Lemma 4.1 applies also to ${\mathcal{S}}_{\scriptscriptstyle H}|_{{\mathcal{M}}_{\scriptscriptstyle W}\setminus\{\underline{0}\}}$. In view of Lemma 8.6, Lemma 4.3c) and d) and Lemma 4.4 remain valid. Keeping in mind the above results, one readily checks that the proof of Theorem A1 also applies under the above assumptions. ∎

Theorem A2’****.

Let $W\subseteq H$ be relatively compact and $\partial W$ be nowhere dense. Assume that $\operatorname{int}(\overline{W})\neq\emptyset$. Let ${\widehat{X^{\prime}}}={\hat{X}}/\pi^{\scriptscriptstyle{\hat{X}}}({\mathcal{H}}_{\operatorname{int}(\overline{W})})$ with induced $G$-action $\widehat{T^{\prime}}$, and let $M$ be any non-empty, closed $S$-invariant subset of ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ (thus including the case $M={\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$).

1. a)

$({\widehat{X^{\prime}}},\widehat{T^{\prime}})$* is the maximal equicontinuous factor of the topological dynamical system $(M,S)$.* 2. b)

If $H_{\operatorname{int}(\overline{W})}=H_{\overline{W}}$, then $(M,S)$ is an almost 1-1 extension of $({\widehat{X^{\prime}}},\widehat{T^{\prime}})$.

Sketch of proof of Theorem A2’.

Here we note that $H_{\operatorname{int}(\overline{W})}$ is compact due to Lemma 6.1c) and d). This ensures that all arguments in the proof of Theorem A2 for compact windows directly apply to the present situation. ∎

8.2 Measure-theoretic results

For measure-theoretic results, let us assume that $W\subseteq H$ is relatively compact and measurable. In that situation, the map $\nu_{\scriptscriptstyle W}:{\hat{X}}\to{\mathcal{M}}_{\scriptscriptstyle W}\subseteq{\mathcal{M}}$ is still measurable such that the Mirsky measure is well defined, compare [22, Rem. 3.16]. In fact Propositions 2.1, 2.2 and 2.3 continue to hold. In particular, $({\mathcal{M}}_{\scriptscriptstyle W},Q_{\scriptscriptstyle W},S)$ is a measure-theoretic factor of $({\hat{X}},m_{\hat{X}},\hat{T})$, and thus the same holds for $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},Q_{\scriptscriptstyle W}^{\scriptscriptstyle G},S)$. Hence both systems have pure point dynamical spectrum.

Whereas the statement of Proposition 2.1 is obvious from measurability of $\nu_{\scriptscriptstyle W}$, we give proofs of the other propositions for the convenience of the reader.

Proof of Proposition 2.2.

Note that $P\circ\hat{\pi}^{-1}$ is a probability measure on ${\hat{X}}$ by assumption on $P$. As $P\circ\hat{\pi}^{-1}$ is $\hat{T}$-invariant and the $\hat{T}$-action is uniquely ergodic, it thus equals $m_{\hat{X}}$, compare Fact 2.2 (3). In particular, m_{\hat{X}}(\nu_{\scriptscriptstyle W}^{-1}\{\underline{0}\})=P\,\big{(}(\nu_{\scriptscriptstyle W}\circ\hat{\pi})^{-1}\{\underline{0}\}\big{)}=0, because $\operatorname{supp}(\nu)\subseteq\operatorname{supp}(\nu_{\scriptscriptstyle W}(\hat{\pi}(\nu))$ for all $\nu\in{\mathcal{M}}_{\scriptscriptstyle W}\setminus\{\underline{0}\}$. But this implies $m_{H}(W)>0$ [22, Prop. 3.6b], which continues to hold in the non-compact setting. The assertion of the proposition now follows as the desired factor map [12, Def. 2.7] is provided by $\hat{\pi}$. ∎

Proof of Proposition 2.3.

As $m_{H}(W)>0$, we have $m_{\hat{X}}(\nu_{\scriptscriptstyle W}^{-1}\{\underline{0}\})=0$ by [22, Prop. 3.6b], which continues to hold in the non-compact setting. Hence we have $Q_{\scriptscriptstyle W}({\mathcal{M}}_{\scriptscriptstyle W}\setminus\{\underline{0}\})=m_{\hat{X}}\circ(\nu_{\scriptscriptstyle W})^{-1}({\mathcal{M}}_{\scriptscriptstyle W}\setminus\{\underline{0}\})=1$. We thus can combine the statements of Proposition 2.2 and Proposition 2.1 to get the result. ∎

For general relatively compact measurable windows $W$, it might be difficult to give an isomorphism between $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W},Q_{\scriptscriptstyle W}^{\scriptscriptstyle G},S)$ and an explicit group rotation. But the previous results for compact windows, i.e., Theorems B1 and B2, continue to hold for windows $W$ that are compact modulo [math]. Consider a window $W$ that is compact modulo [math] and its Haar regularization $W_{reg}=\operatorname{supp}((m_{H})|_{W})$. Then the Mirsky measures $Q_{\scriptscriptstyle W}$ and $Q_{\scriptscriptstyle W_{reg}}$ coincide on $\mathcal{M}$, because

[TABLE]

which is a set of $m_{\hat{X}}$-measure zero as ${\mathscr{L}}$ is countable. This implies that $({\mathcal{M}}_{\scriptscriptstyle W},Q_{\scriptscriptstyle W},S)$ is measure-theoretically isomorphic to $({\mathcal{M}}_{\scriptscriptstyle W_{reg}},Q_{\scriptscriptstyle W_{reg}},S)$. As the Haar periods of $W_{reg}$ coincide with those of $W$, Theorems B1 and B2, which apply to $W_{reg}$, continue to hold for $W$.

In oder to better understand the passage from the extended hull ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ to the usual hull ${\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}({\hat{x}})$ for model sets without compact windows, we discuss the maximal density condition in that case.

Remark 8.7**.**

(Generic configurations)

Assume that some configuration $\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}({\hat{x}})$ has maximal density in the sense that $d(\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}({\hat{x}}))=\mathrm{dens}({\mathscr{L}})\cdot m_{H}(\overline{W})$ along some given tempered van Hove sequence $(A_{n})_{n}$, compare also Section 5. In that case $\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}({\hat{x}})$ is generic for the Mirsky measure $Q_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}$ on $\mathcal{M}^{\scriptscriptstyle G}$, as will be shown below. This implies that $({\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}({\hat{x}}),Q_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G},S)$ is isomorphic to $(\widetilde{{\mathcal{M}}_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}},Q_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G},S)$, where $\widetilde{{\mathcal{M}}_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}}=\{\nu\in{\mathcal{M}}^{\scriptscriptstyle G}:\nu\leqslant\nu_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}({\hat{x}})\text{ for some }{\hat{x}}\in{\hat{X}}\}$, compare the proof of [22, Thm. 5].

To see genericity, note that for any ${\hat{x}}\in{\hat{X}}$ we have $0\leqslant\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}({\hat{x}})\leqslant\nu_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}({\hat{x}})$, so in particular the density of $\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}({\hat{x}})$ along $(A_{n})_{n}$ is bounded by that of $\nu_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}({\hat{x}})$ and hence by $\mathrm{dens}({\mathscr{L}})\cdot m_{H}(\overline{W})$, see [22, Thm. 3]. If $\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}({\hat{x}})$ achieves this maximal density, then the density of $\nu_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}({\hat{x}})-\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}({\hat{x}})$ is clearly zero. Now consider for $\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}({\hat{x}})\in{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ its empirical measures

[TABLE]

and likewise for $\nu_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}({\hat{x}})\in{\mathcal{M}}_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}$ its empirical measures $Q^{G,n}_{\overline{W},{\hat{x}}}$ . The previous reasoning shows that the empirical measures $Q_{W,{\hat{x}}}^{G,n}$ are asymptotically equivalent to the measures $Q_{\overline{W},{\hat{x}}}^{G,n}$ in the sense that both sequences do have the same weak limit points, and [22, Thm. 5] implies that the measures $Q_{W,{\hat{x}}}^{G,n}$ converge weakly to $Q_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}$. It follows that statistical properties of $\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}({\hat{x}})$ and $\nu_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}({\hat{x}})$, like pattern frequencies and especially their autocorrelation coefficients, coincide for such ${\hat{x}}$, and that they are determined by the Mirsky measure $Q_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}$, to which Theorems B1 and B2 apply.

Note, however, that for $m_{{\hat{X}}}$-a.a ${\hat{x}}$ the configuration $\nu_{\scriptscriptstyle W}({\hat{x}})$ has density $\mathrm{dens}({\mathscr{L}})\cdot m_{H}({W})$, see [29, Thm. 1]. Thus the above reasoning applies to $Q_{\scriptscriptstyle W}^{\scriptscriptstyle G}$-Mirsky-typical configurations only, if $m_{H}(W)=m_{H}(\overline{W})$, i.e., if $W$ is compact modulo [math].

Remark 8.8**.**

(Autocorrelation measure, diffraction spectrum and generic configurations)

For $P^{\scriptscriptstyle G}\in\mathcal{M}_{S}(\widetilde{{\mathcal{M}}_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}})$, the set of $S$-invariant probability measures on $\widetilde{{\mathcal{M}}_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}}$, the so-called autocorrelation measure $\gamma_{P^{\scriptscriptstyle G}}$ is a positive definite Radon measure on $G$ that is naturally associated to $P^{\scriptscriptstyle G}$. In particular, $\gamma_{P^{\scriptscriptstyle G}}$ is Fourier transformable. We recall its definition from [7, Prop. 6], [24, Sec. 2.3] and [25, Lemma 4.1]. Take any $\psi\in C_{c}(G)$ such that $m_{G}(\psi)=1$ and define the Radon measure $\gamma_{P^{\scriptscriptstyle G}}$ via its associated linear functional $\gamma_{P^{\scriptscriptstyle G}}:C_{c}(G)\to\mathbb{C}$ by requiring

[TABLE]

for every $\varphi\in C_{c}(G)$. The measure $\gamma_{P^{\scriptscriptstyle G}}\in{\mathcal{M}}^{\scriptscriptstyle G}$ is independent of the choice of $\psi$, and it is a rather direct consequence of the above definition that the map $\gamma:\mathcal{M}_{S}(\widetilde{{\mathcal{M}}_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}})\to{\mathcal{M}}^{\scriptscriptstyle G}$, defined by $P^{\scriptscriptstyle G}\mapsto\gamma_{P^{\scriptscriptstyle G}}$, is continuous with respect to the vague topologies. For $P^{\scriptscriptstyle G}:=Q_{\scriptscriptstyle W}^{\scriptscriptstyle G}$, the Mirsky measure on $\widetilde{{\mathcal{M}}_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}}$ , the Fourier transform $\widehat{\gamma_{Q_{\scriptscriptstyle W}^{\scriptscriptstyle G}}}$ is a pure point measure as $(\widetilde{{\mathcal{M}}_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}},Q_{\scriptscriptstyle W}^{\scriptscriptstyle G},S)$ has pure point dynamical spectrum [7, Thm. 7]. One says that $(\widetilde{{\mathcal{M}}_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}},Q_{\scriptscriptstyle W}^{\scriptscriptstyle G},S)$ has pure point diffraction spectrum.

Now fix any configuration $\nu^{\scriptscriptstyle G}\in\widetilde{{\mathcal{M}}_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}}$ and consider its empirical measures $Q^{n}_{\nu^{\scriptscriptstyle G}}$ on $\widetilde{{\mathcal{M}}_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}}$, given as

[TABLE]

along some fixed tempered van Hove sequence $(A_{n})_{n}$ in $G$. As $\widetilde{{\mathcal{M}}_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}}$ is compact, we can assume that the associated sequence $(\gamma_{Q_{\nu^{\scriptscriptstyle G}}^{n}})_{n}$ of empirical autocorrelations converges to a limit $\gamma_{\nu^{\scriptscriptstyle G}}$, possibly after passing to some subsequence of $(A_{n})_{n}$. For sufficiently large $n$, the Fourier transform of $\gamma_{\nu^{\scriptscriptstyle G}}$ describes the outcome of a diffraction experiment on a physical realisation of $\nu^{\scriptscriptstyle G}$ restricted to $A_{n}$. A standard tedious calculation which we omit yields the explicit expression $\gamma_{\nu^{\scriptscriptstyle G}}=\sum_{\ell\in{\mathscr{L}}}\eta(\ell)\delta_{\ell_{G}}$, where

[TABLE]

are the autocorrelation coefficients of $\gamma_{\nu^{\scriptscriptstyle G}}$. In particular, if $\nu^{\scriptscriptstyle G}$ is generic for the Mirsky measure $Q_{\scriptscriptstyle W}^{\scriptscriptstyle G}$ on $\widetilde{{\mathcal{M}}_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}}$, i.e., if $(Q_{\nu^{\scriptscriptstyle G}}^{n})_{n}$ converges to $Q_{\scriptscriptstyle W}^{\scriptscriptstyle G}$, then by continuity of $\gamma$ we have $\gamma_{\nu^{\scriptscriptstyle G}}=\gamma_{Q_{\scriptscriptstyle W}^{\scriptscriptstyle G}}$. By the above argument, this implies that $\nu^{\scriptscriptstyle G}$ is pure point diffractive. In particular, from [22, Eqn. (18)] we obtain explicit expressions for the autocorrelation coefficients, namely $\eta(\ell)=\mathrm{dens}({\mathscr{L}})\cdot m_{H}(W\cap(W+\ell_{H}))$ for all $\ell$ such that $\eta(\ell)\neq 0$.

The above result alternatively follows by combining Theorem 5 and Proposition 6 in [7]. That there is a full $Q_{\scriptscriptstyle W}^{\scriptscriptstyle G}$-measure set of configurations $\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}({\hat{x}})\in(\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G})^{-1}({\hat{X}})\subseteq{\mathcal{M}}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ such that $\gamma_{\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}({\hat{x}})}=\gamma_{Q_{\scriptscriptstyle W}^{\scriptscriptstyle G}}$ on any given tempered van Hove sequence $(A_{n})_{n}$ has been shown by Moody [29, Cor. 1], by constructing a full $m_{\hat{X}}$-measure set of pure point diffractive $\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}({\hat{x}})$ via repeated applications of Birkhoff’s ergodic theorem.

A sufficient criterion for pure point diffractiveness has been observed in [5, 22]: The configuration $\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}({\hat{x}})$ is pure point diffractive if $\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}({\hat{x}})$ has maximal density $d(\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}({\hat{x}}))=\mathrm{dens}({\mathscr{L}})\cdot m_{H}(\overline{W})$. Indeed, in that case $\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}({\hat{x}})$ is generic for the Mirsky measure $Q_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}$ on $\widetilde{{\mathcal{M}}_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}}$ by Remark 8.7 above. Note however that such $\nu_{\scriptscriptstyle W}^{\scriptscriptstyle G}({\hat{x}})$ is typically not generic for the Mirsky measure $Q_{\scriptscriptstyle W}^{\scriptscriptstyle G}$ on $\widetilde{{\mathcal{M}}_{\scriptscriptstyle\overline{W}}^{\scriptscriptstyle G}}$.