On substitution tilings and Delone sets without finite local complexity

We consider substitution tilings and Delone sets without the assumption of finite local complexity (FLC). We first give a sufficient condition for tiling dynamical systems to be uniquely ergodic and a formula for the measure of cylinder sets. We then obtain several results on their ergodic-theoretic properties, notably absence of strong mixing and conditions for existence of eigenvalues, which have number-theoretic consequences. In particular, if the set of eigenvalues of the expansion matrix is totally non-Pisot, then the tiling dynamical system is weakly mixing. Further, we define the notion of rigidity for substitution tilings and demonstrate that the result of [Lee-Solomyak (2012)] on the equivalence of four properties: relatively dense discrete spectrum, being not weakly mixing, the Pisot family, and the Meyer set property, extends to the non-FLC case, if we assume rigidity instead.


Introduction
Delone sets and tilings of the Euclidean space R d have been used for a long time to model physical structures. It is often fruitful, moreover, to consider not just an individual object (tiling or Delone set), but rather its "hull," defined as a closure of the collection of all its translates in a natural topology. The group R d acts on the hull by translations, and we obtain a dynamical system, whose properties have a direct connection with the physical properties of the material. Some of the references to this approach are [38,17,7,30], see also the recent book [2]. One of the central issues for the resulting dynamical system is whether it has unique ergodicity, which allows one to use powerful ergodic theorems that are important for applications. Another much studied aspect of a system is its spectrum, and it has been realized for a long time that there is a close link between the "dynamical" and the "diffraction" spectrum. The list of references here is too long, so we refer the reader to the recent survey [5] and its bibliography.
Here we investigate dynamical systems associated with substitution tilings and Delone sets in R d , without the assumption of finite local complexity (FLC). Whereas the theory of FLC tilings is well-developed, much less is known about non-FLC tilings, which are sometimes called ILC (for "infinite local complexity"). We consider primitive substitution tilings, with finitely many prototiles up to translation. In this class, the ILC phenomenon (at least, in the planar case) is generally caused by tilings with tiles whose edges are "sliding" (in the literature, this phenomenon has been referred to as an "earthquake" or a "fault line"). Thus our setting does not cover pinwheel type tilings in which prototiles appear with infinitely many rotations. For dynamical properties of the latter, we refer the reader to [37,11]. First examples of ILC substitution tilings were constructed by Danzer [9] and Kenyon [18], and a large class of such tilings was studied by Frank and Robinson [13]. More recently, investigation of ILC tilings appeared in the work of Frank and Sadun [14,16], see also [12], in a general framework of "fusion tilings". In particular, they established unique ergodicity under some assumptions and obtained a number of results on the topological and ergodic-theoretic properties of ILC tilings.
In this paper, we revisit the theory developed in [40,27,42,28,29] and extend parts of it to include the ILC case. We work in parallel in the frameworks of Delone κ-sets and tilings, which are essentially equivalent. The notion of Delone κ-set formalizes the idea of a "coloured Delone set," in which to each point a particular color is assigned, from a finite list. This way we can model structures with atoms of different kind. The first issue that we address is unique ergodicity. In the FLC case unique ergodicity is equivalent to the existence of uniform cluster frequencies (UCF) [26]. In the ILC case the UCF property is no longer sufficient; however, we show that under an additional technical condition, the unique ergodicity follows, and this is our first main result, Theorem 3.2. Although this theorem is general, it is adapted for substitution tilings with finitely many prototiles up to translation, but does not apply, for instance, to the pinwheel tiling. Along the way we also obtain a formula for the measure of cylinder sets in terms of patch frequencies.
In the FLC case the property of "linear repetitivity" is known to imply unique ergodicity, see [24,Thm 6.1], [26,Thm 2.7], and [8,Cor 4.6]. Frettlöh and Richard [11] developed versions of this property suitable for the ILC case. The two main ones, the "linear wigglerepetitivity" and "almost linear repetitivity" were shown in [11] to imply unique ergodicity.
The former property is satisfied, in particular, for the pinwheel tiling. It is an open question whether for tilings with fault lines, like in [13], the corresponding point sets are almost linearly repetitive. We should note that the results of Frank and Sadun [16] on unique ergodicity are more general and cover substitution systems of both kinds; however, the framework that we develop is better suited for our purposes in the sequel.
Next we turn to ILC substitution systems under our assumptions, and to their ergodictheoretic and spectral properties. We prove that these systems are not strongly mixing and obtain a necessary condition for eigenvalues, generalizing results of [40,42]. If we also assume repetitivity and recognizability, as well as an algebraic property of the expansion map, this condition becomes sufficient, and all eigenfunctions may be chosen continuous.
As in the FLC case, this leads to number-theoretic considerations. If the expansion is a pure dilation, then the existence of non-trivial eigenfunctions (equivalently, absence of weak mixing) implies that the expansion constant is a Pisot (or PV) number. More generally, if the set of eigenvalues of the expansion map Q is "totally non-Pisot" [39], then the tiling dynamical system is weakly mixing. Interestingly, if the set of eigenvalues of the dynamical system form a relatively dense subset of R d , this forces the set of translation vectors between equivalent tiles to be Meyer, and this implies FLC. This phenomenon is also related to an important property of the tiling which we call "rigidity". Rigidity was established in [29] for FLC primitive substitution tilings, under some assumptions of algebraic nature. Actually, rigidity is easy to check directly in examples, and it holds for many ILC tilings, specifically, for those considered in [13]. However, it does not hold for Kenyon's tiling [18] and its relatives. For these tilings, the dynamical system may have non-trivial discrete spectrum; however, it will not be relatively dense. We demonstrate that the result of [29] on the equivalence of four properties: relatively dense discrete spectrum, being not weakly mixing, the Pisot family, and the Meyer set property, extends to the non-FLC case, if we assume rigidity instead.
The structure of the paper is as follows. In section 2, we introduce notation and definitions used in the paper. In section 3, we introduce a special class of patches and cylinder sets which generate topology and the Borel σ-algebra of the space under consideration. We then provide a sufficient condition for unique ergodicity in terms of frequencies of these patches. In section 4, we consider substitution Delone κ-sets which are representable in terms of substitution tilings. In fact, it is much easier to compute patch frequencies for tilings than cluster frequencies for Delone κ-sets. We show that the sufficient condition for unique ergodicity holds for ILC primitive substitution tilings, so it is valid for representable primitive substitution Delone κ-sets as well. In section 5, we prove Lemma 5.1 which implies absence of strong mixing and provides the key tool for the characterization of dynamical eigenvalues in Theorem 5.3. In section 6 we define the rigidity and consider its implications.
In particular, we give an answer to a question of Lagarias from [23] in a more general setting than in [28] (without the FLC assumption), namely, a primitive substitution tiling which is pure point diffractive must have the Meyer property. We also include several examples.

Preliminaries
where Λ i ⊂ R d and κ is the number of colours. We also write Λ = (Λ 1 , . . . , Recall that a Delone set is a relatively dense and uniformly discrete subset of R d . We say Many of the clusters that we consider have the form Λ ∩ A : We say that a Delone κ-set Λ has finite local complexity(FLC) if for every R > 0 there The Delone κ-set Λ is called repetitive if every Λ-cluster occurs relatively dense in space, up to translation. More precisely, this means that for any cluster G ⊂ Λ there exists M > 0 such that every ball of radius M = M (G) contains a translated copy of G. (In some papers, see e.g. [11], this property is called weak repetitivity, to distinguish it from the condition that for every R > 0 there exists M = M (R) such that every ball of radius M contains a translated copy of every Λ-cluster of radius R. However, the latter property can only hold in the FLC case, and then the two repetitivity properties are equivalent.) We will make use of the following notation: Vol((∂F n ) +r )/Vol(F n ) = 0, for all r > 0. (2.2) For any Delone κ-set Γ = (Γ i ) i≤κ and B ⊂ R d , we define Suppose that a Delone κ-set Λ is given. For a cluster G of Λ and a bounded set where the symbol # stands for cardinality of a set. 1 In the literature [25], there is also the notion of Delone multisets; they differ from the Delone κ-sets. In a Delone multiset, points with the same colour may be counted with multiplicity.

2.2.
Tilings. We begin with a set of types (or colours) {1, . . . , κ}, which we fix once and for all. A tile in R d is defined as a pair T = (A, i) where A = supp(T ) (the support of T ) is a compact set in R d , which is the closure of its interior, and i = l(T ) ∈ {1, . . . , κ} is the type of T . We let g + T = (g + A, i) for g ∈ R d . We say that a set P of tiles is a patch if the number of tiles in P is finite and the tiles of P have mutually disjoint interiors. The support of a patch is the union of the supports of the tiles that are in it. The translate of a patch P by g ∈ R d is g + P := {g + T : T ∈ P }. We say that two patches P 1 and P 2 are translationally equivalent if P 2 = g + P 1 for some g ∈ R d . A tiling of R d is a set T of tiles such that R d = {supp(T ) : T ∈ T } and distinct tiles have disjoint interiors. Given a tiling T , a finite set of tiles of T is called T -patch. We define FLC and repetitivity for tilings in the same way as the corresponding properties for Delone κ-sets. The types (or colours) of tiles for tilings have the meaning analogous to the colours of points for Delone κ-sets.
We always assume that any two T -tiles with the same colour are translationally equivalent (hence there are finitely many T -tiles up to translations).

Substitutions.
Definition 2.1. Λ = (Λ i ) i≤κ is called a substitution Delone κ-set if Λ is a Delone κ-set and there exist an expansive map Q : R d → R d and finite sets D ij for i, j ≤ κ such that where the unions on the right-hand side are disjoint.
we will call them prototiles. Denote by P A the set of patches made of tiles each of which is a translate of one of T i 's. We say that ω : A → P A is a tile-substitution (or simply substitution) with expansive map Q if there exist finite sets D ij ⊂ R d for i, j ≤ κ, such that Here all sets in the right-hand side must have disjoint interiors; it is possible for some of the D ij to be empty. The substitution κ × κ matrix S is defined by S(i, j) = #D ij .
The tile-substitution is extended to translated prototiles by The equations (2.5) allow us to extend ω to patches in P A by ω(P ) = T ∈P ω(T ). It is similarly extended to tilings all of whose tiles are translates of the prototiles from A. A tiling T satisfying ω(T ) = T is called a fixed point of the tile-substitution, or a substitution tiling with expansion map Q. We avoid the term "self-affine tiling," since these are usually assumed to have FLC.
When a substitution Delone κ-set Λ is given, one can set up an adjoint system of equations (2.5) for the system (2.3). By the theory of graph-directed iterated function systems [32] (see also [6]), it follows that the system of equations (2.5) has a unique solution for which A = {A 1 , . . . , A κ } is a family of non-empty compact subsets of R d . The following result [25] of J. Lagarias and Y. Wang will be important for us; it is a special case of [25, Th. 2.3 and Th. 5.5]. We emphasize that in their paper the FLC condition is not assumed.  (ii) all the sets A i have non-empty interiors and each A i is the closure of its interior; Note that for substitution tilings (without the FLC assumption) the property (iii) was shown in [35]. We say that Λ is representable (by tiles) if Λ+A := {x+T i : x ∈ Λ i , i ≤ κ} is a tiling of R d , where T i = (A i , i), i ≤ κ, and A i 's arise from the solution to the adjoint system (2.5) and A = {T i : i ≤ κ}. One can define a tile-substitution ω satisfying ω(Λ+A) = Λ+A from (2.5). If Λ is representable, then T := Λ + A is a substitution tiling with expansion Q and we call it the associated substitution tiling of Λ. Observe that there is a one-to-one correspondence between Λ-clusters and T -patches: if G = (G i ) i≤κ is a Λ-cluster, then P = G + A is the associated T -patch, and this procedure can be reversed. Not every substitution Delone κ-set is representable, but there is a checkable necessary and sufficient condition, see [27] for details.
Given a van Hove sequence {F n } n≥1 , we define the property of uniform patch frequencies for a tiling T , by analogy with uniform cluster frequencies(UCF) for Delone κ-sets. It turns out that this property holds for a fixed point of a primitive tile substitution, even without the FLC assumption, see Section 5 for more details.
2.4. Dynamical system from a point set. Let Λ = (Λ i ) i≤κ be a Delone κ-set in R d and let X be the collection of all Delone κ-sets each of whose clusters is a translate of some Λ-cluster. We equip X with a metric, which comes from the "local rubber topology" (see [34,30,4] and references therein). This is, essentially, the "local" topology used already in [10]; in the tiling setting it is generated by the Hausdorff metric on the patch boundaries.
Formally, this metric is defined as follows: For Λ 1 , Λ 2 ∈ X, It is a standard fact that this is indeed a metric. For the reader's convenience, we include the proof in the Appendix.
We consider the topology induced by this metric. For any g ∈ R d consider the translation This defines a continuous action of the group R d on X. The set

Unique ergodicity
In the case of Delone κ-set with FLC, it is shown in [26] that unique cluster frequencies (UCF) is an equivalent property for the dynamical system to be uniquely ergodic. However, without assuming FLC, the relation between the two is less clear. A simple observation shows that UCF does not imply unique ergodicity, without extra assumptions. Indeed, consider a "random" Delone κ-set Λ (say, on R). With probability one, every patch will occur only once (up to translation) in X, hence it has uniform frequency equal to zero, but there is no reason why the associated dynamical system should be uniquely ergodic.
So, in addition to UCF, we impose another, technical condition, and show that together they imply the unique ergodicity of the dynamical system corresponding to the Delone κ-set.
Let us define cylinder sets in X Λ . In order to do this, we first introduce the sets ∆ m , ∆ m,α , and U m,α below. Let η(Λ) > 0 be chosen so that every ball of radius η(Λ) 2 so that each cube of the subdivision of side length 2 −m contains at most one point of supp(Λ). We will assume that m ≥ m 0 .
Let ∆ m be the set of all small cubes from this grid, with colours assigned from 1 to κ. Formally, an element of ∆ m is C = (B, i), where B is a cube and i ∈ {1, . . . , κ}. We write B = supp(C). Observe that the cardinality of ∆ m is |∆ m | = κ · 2 (2m+1)d . Next we consider the collection of subsets of ∆ m , characterized by the property that each cube may be chosen at most once. We denote these subsets by ∆ m,α , with α = 1, . . . , N m . Formally, their property is that . . , κ}, finite set of coloured points, such that, In words, U m,α is the family of κ-sets Q of coloured points of cardinality |∆ m,α | such that each coloured cube of ∆ m,α contains exactly one point of Q of its color. Let us consider all the clusters in X Λ which arise from Γ ∩ [−2 m , 2 m ) d and belong to U m,α , for some Γ ∈ X Λ , that is, This is a cylinder set in X Λ that we define. Clearly, for any m ∈ N, where denotes disjoint union.
Let Λ be a Delone κ-set. It has countably many clusters (finite subsets). For every (m, α) we define We identify clusters in G m,α that are translationally equivalent and choose one representative from each equivalence class; denote the resulting set of clusters by (G m,α ) ≡ . Enumerate elements of this set arbitrarily and write For each cluster G of Λ and V ⊂ R d , consider the cylinder set defined in the usual way: Recall that for V ⊂ R d , we use the notation For every j ≥ 1 and G is the maximal set of all possible "wiggle vectors x" such that all the points of −x + G (m,α) j remain inside their small grid boxes. This way, we get for all (m, α), is the set of coloured Delone κ-sets whose clusters in [−2 m , 2 m ) d are only "admitted in the limit", according to the terminology of [16]. Observe that, by construction, choosing The following is a standard result in topological dynamics; see e.g. [44, 6.19]  for all continuous functions f : X → C (notation f ∈ C(X)), where C f is a constant depending on f , for all x ∈ X, in which case the convergence is necessarily uniform in x ∈ X.
Now we are ready to state our first result.
Theorem 3.2. Let Λ be a Delone κ-set and let {F n } n≥1 be a van Hove sequence. Suppose that Λ has UCF, and in addition, for all m ≥ m 0 and all α = 1, . . . , N m we have Proof. We are going to use the sufficient condition from Theorem 3.1. Since X Λ is the orbit closure of Λ and continuous functions on the compact space X Λ are uniformly continuous, subsets whose diameters tend to zero as m → ∞, we can approximate any f ∈ C(Λ) by linear combinations of characteristic functions of X(U m,α ). Thus it is enough to show that for all (m, α), Fix m ≥ m 0 and α ∈ {1, . . . , N m }. To simplify the notation, we will drop the superscript (m, α) for the rest of the proof, writing where x j,ν are all the vectors such that x j,ν + G j ⊂ Λ. In fact, Observe that the set X m,α from (3.4), which has patches admissible in the limit, does not enter into consideration, since we are integrating over the orbit of Λ.
Recall that every m-grid box contains at most one point of supp(Λ), hence for −x + Λ ∈ X(U m,α ) there is a unique j ∈ N and a unique t ∈ V j such that (x−t)+G j ⊂ Λ. Thus the sets where F ± are defined in (2.1). In view of (3.6), we deduce Notice that the infinite sums above have finitely many non-zero terms, since there are finitely many different clusters in a finite domain, hence there is no problem with convergence.
It is clear that for any (m, α), the total number of Λ-clusters from U m,α contained in a set A is not greater than the cardinality of supp(Λ) ∩ A. It follows that for any bounded Since {F n } n≥1 is a van Hove sequence, we conclude that Denote the right-hand side of (3.8) by C m,α . The equation (3.8), together with (3.12), shows the desired convergence of The following sufficient condition for unique ergodicity will be useful in the next section.
Proof. We only need to check the equality (3.8). Observe that if we replace the infinite sum by any partial sum in (3.8), convergence claimed holds by the UCF property. Therefore, where h is allowed to vary with n. Thus it remains to estimate the lim sup from above.
Given > 0, find the corresponding k 0 from (3.13). There exists n 0 such that for all n ≥ n 0 and all h ∈ R d , Then for n ≥ max{n 0 , k 0 }, for all h ∈ R d , we have by the above, in view of (3.13), and the desired claim follows.

Substitution Delone κ-sets and tilings
In order to show unique ergodicity for representable primitive substitution Delone κ-sets, we need to investigate frequencies of clusters. This is much more convenient to do in the tiling setting. We first discuss additional background and preliminaries, part of which will be used in the following section.
Recall that we have a primitive substitution tiling T = T Λ associated to the substitution Delone κ-set. The tiling has a finite prototile set A on which the tile-substitution ω is defined. We consider the tiling space X T and the associated dynamical system (X T , R d ).
The metric can be defined directly, using the Hausdorff distance between large patches around the origin, see e.g. [16]. This metric is equivalent to the one obtained from the "local rubber topology" on the associated Delone κ-sets. The following is immediate from the definitions (see also [27,Lemma 3.10] in the FLC case). The tile-substitution ω extends to a continuous map ω : X T → X T . This is sometimes called the inflate-and-subdivide map. It follows from (2.6) that By definition of the substitution tiling, we have ω(T ) = T . Thus we can compose the tiles of T into supertiles of order k, for any k ≥ 1. Formally we define The prototiles of Q k T are {(Q k A i , i), i = 1, . . . , κ}. One can see from the definition of the topology and the tiling space that for every tiling S ∈ X T we can also compose its tiles into supertiles, so in fact, ω is a surjection. Denote by K(T ) the group (a subgroup of R d ) of translational periods of T : By the definition of X T as the orbit closure of T we have that K(S) ⊃ K(T ) for all S ∈ X T .
It follows from (4.1) that Since Q is expanding and K(T ) is discrete, we see that, in the case when the group of translational periods is non-trivial, K(T ) = {0}, the tile-substitution ω is not 1-to-1 on X T .
The following notions will be needed in the next section (characterization of eigenvalues), but it is convenient to introduce them here.
Definition 4.2. The tile-substitution ω is said to be recognizable if ω is one-to-one, in which case ω is a homeomorphism of X T . The tile-substitution ω is called recognizable modulo periods if ω(S) = ω(S ) implies S = S − Q −1 x for some x ∈ K(T ).
Next we discuss the existence of patch/cluster frequencies and unique ergodicity. About the proof. This was proved under the standing FLC assumption in [27]; however, the argument did not use the FLC. Note that for any prototile is van Hove. This follows from the property Vol(∂A i ) = 0, see Theorem 2.3(iii). The main part of the proof is to show UCF with respect to {Q n A i } n≥0 (for any fixed i). Ultimately, this is a consequence of the Perron-Frobenius Theorem and similar to the proof of uniform word frequencies for primitive symbolic substitution systems [36].
It follows that primitive representative substitution Delone κ-sets have uniform cluster frequencies.
As mentioned in the introduction, the following theorem can also be deduced from [16]; we present a proof using our approach. Proof. We are going to verify the hypothesis of Corollary 3.3. Again, it is easier to prove this in the setting of substitution tilings. Let T be a primitive substitution tiling. It has countably many patches. Fix r > 0 and enumerate all the patches P j whose supports have diameter ≤ r. (In the FLC case this will be a finite set.) Fix also a van Hove sequence {F n } n≥1 . In view of Corollary 3.3, translated to the tiling setting, it suffices to show that for any > 0 there exists k 0 such that for all h ∈ R d and n ≥ k 0 , Say that a patch P ⊂ T is k-special and write sp(P ) = k if P occurs (up to translation) as a subpatch of ω k (T j ) for some j and k ∈ N is minimal with this property. Note that the hierarchical structure for T is fixed, even if recognizability does not hold. We write sp(P ) = ∞ if sp(P ) ≥ k for all k ∈ N. This will happen if all occurrences of the patch P in T cross the boundary of a supertile, for any level supertiling composed from T . Clearly, for any k there are finitely many k-special patches. We will next estimate the total number of patches P j of diameter ≤ r, with sp(P j ) > k, that are contained in F n + h. By definition, these patches must cross (hence intersect) the boundary of one of the k-level supertiles. Therefore, One can see that the number of T -patches of diameter ≤ r contained in any given set is bounded above by C r times the volume of that set, with a constant C r depending only on the tiling and on r > 0. An easy (although inefficient) estimate is as follows: let V min be the minimal volume of a T -prototile. Then any patch of diameter ≤ r contains at most Vol(B r )/V min tiles, hence any given tile belongs to at most 2 Vol(Br)/V min patches of diameter ≤ r. Thus we can take

and (4.3) yields
Let δ > 0. Recall that {Q k A} k≥1 is a van Hove sequence, hence there exists k 1 such that for any tile support A and any k ≥ k 1 . Thus P j : sp(P j )>k Since F n is van Hove as well, there exists n 1 such that Vol (F n ) +r ≤ (1 + δ)Vol(F n ), for all n ≥ n 1 . So finally, we obtain P j : sp(P j )>k and since the right-hand side can be made arbitrarily small, (4.2) follows.
We will need formulae for the measure of cylinder sets X(U m,α ). Although it seems natural that the constant C m,α in (3.9) must be X Λ χ m,α dµ = µ(X(U m,α )), this is not immediately clear, since the characteristic function is not continuous. The usual approach is to approximate the characteristic function by continuous functions, but this procedure is less trivial in the ILC case. Thus we first investigate frequencies of clusters, and this is much more convenient to do in the tiling setting.
Following [16] we use the terminology: a patch P ⊂ T − x for some x ∈ R d is called literally admitted; patches that appear in tilings S ∈ X T , but are not literally admitted are called admitted in the limit. A patch P is called legal if it is a subpatch of ω k (T i ) for some k ∈ N and a prototile T i . Note that a literally admitted patch is not necessarily legal: it may happen that the fixed point of the substitution T is not repetitive. Recall that a tiling (resp. Delone κ-set) is said to be repetitive, whenever its every patch (resp. cluster) appears relative dense in space.
In [16,12] a slightly different definition of the substitution tiling space is used: it is X ω consisting of all tilings whose every patch is legal. We have X ω = X T whenever there exists a legal T -patch whose support contains the origin in its interior, and then T is repetitive.
One can show that if ω is a primitive substitution, then the topological dynamical system  Legal patches have positive frequency, whereas the non-legal ones, including the patches that are admitted in the limit, have zero frequency.
We note that zero frequency of patches that are admitted in the limit has been established in [16] in the more general context of fusion tilings (but assuming recognizability, which we do not need).
Proof. We only give a sketch, since this is standard. By the Perron-Frobenius Theorem, for any legal patch we have Here L P (Q n A i ) refers to the number of patches equivalent to P in the supertile ω n (T i ), subdivided n times according to the subdivision rule. Then we pass to estimates of the number of patches in the van Hove sets F n , as in [27, Appendix 1]. Since every S ∈ X T can be composed into supertiles of order n, for any n ∈ N (even in the absence of recognizability), the same argument yields the desired equality. As for the patches that are non-legal, they necessarily have to intersect the boundary of supertiles for every order n, hence their number is negligible, because the sets Q n A i have the van Hove property.
The following is now immediate.
Proposition 4.7. Let Λ be a representable primitive substitution Delone κ-set, and let µ be the unique invariant Borel probability measure for the associated dynamical system.
(i) For any cluster G ⊂ Λ and a Borel set V ⊂ R d we have Moreover, where X m,α is the set of Delone κ-sets from X Λ that are admitted in the limit, see (3.4).
Proof. (i) Consider the characteristic function of X(G, V ), denoted by χ X(G,V ) . By the Birkhoff Ergodic Theorem, we have µ(X(G, V )) = lim for µ-a.e. Γ ∈ X Λ . The right-hand side equals Vol(V ) · freq(G, Γ); this is shown using standard estimates based on the van Hove property, similarly to the proof of Theorem 3.2 above. In view of Corollary 4.6, the desired claim follows.
(ii) In view of the disjoint union decomposition (3.4) and the already proved (4.4),

Ergodic-theoretic properties; eigenvalues
Let Λ be a representable primitive substitution Delone κ-set, and let µ be the unique invariant Borel probability measure for the associated dynamical system. Further, fix m ≥ m 0 and let X(U m,α ) be the cylinder sets from (3.1).
be the set of translation vectors between points of the κ-set of the same color. We will also need the corresponding substitution tiling T = T Λ . For a tiling T we define Clearly, Ξ(T Λ ) = Ξ(Λ). Further, we need Of course, we have Ξ legal (T ) = Ξ(T ) if T is repetitive. We also let Ξ legal (Λ) = Ξ legal (T Λ ).
Let µ be the unique ergodic translation-invariant measure on X Λ .
We fix (m, α) and drop the superscripts, writing G j := G (m,α) j and V j := V (m,α) j to simplify the notation. Notice that in view of (4.4).
As before, it is easier to estimate frequencies for the corresponding substitution tiling T = T Λ . The argument is similar to that of [40], but we show the details for completeness.
Let P j = G j + A be the T -patch corresponding to the Λ-cluster G j . Fix T = (A k , k), a T -tile of (any) type k. We have Since z ∈ Ξ legal (Λ) = Ξ legal (T ), there exist T -tiles T , T in the same legal patch, such that T = T + z. Fix another T -tile, T i of type i. By legality and primitivity, there Then for any n ∈ N we have ω n+k 0 (T i ) ⊃ ω n (T ) ∩ (ω n (T ) + Q n z). Thus for any T -patch P in ω n (T ) equivalent to P j , the patch P ∪ (Q n z + P ) is in ω n+k 0 (T i ). It follows that there are at least L P j (Q n A ) patches in ω n+k 0 (T i ) which are equivalent to P j ∪ (Q n z + P j ); in other words, Therefore, for N > n + k 0 , By the definition of the substitution matrix S we have So for any n ∈ N and N > n + k 0 , It follows from the Perron-Frobenius Theorem (see [40,Cor. 2.4]) that we obtain lim inf n→∞ freq(P j ∪ (Q n z + P j ), T ) freq(P j , T ) ≥ r Vol(A ) · | det Q| −k 0 .
Let δ = 1 4 r Vol(A )·| det Q| −k 0 , which is independent of P j , as well as of (m, α). It follows from the above that for n = n(P j ) ∈ N sufficiently large, freq(P j ∪ (Q n z + P j ), T ) > 2δ · freq(P j , T ) .
Corollary 5.2. Let Λ be a primitive representable substitution Delone κ-set. Then the associated uniquely ergodic measure-preserving system is not strongly mixing.
Recall that K(T ) = {x ∈ R d : T − x = T } is the group of periods of a tiling T . (i) If α ∈ R d is an eigenvalue of the measure-preserving system (X T , R d , µ) then lim n→∞ e 2πi Q n z,α = 1 for all z ∈ Ξ legal (T ), (5.5) and e 2πi g,α = 1 for all g ∈ K.
(5.6) (ii) Suppose, in addition, that T is repetitive, the tile-substitution ω is recognizable modulo periods, and all the eigenvalues of Q are algebraic integers. If (5.5) and (5.6) hold, then α is an eigenvalue, and the eigenfunction can be chosen continuous.
The following is an immediate consequence of the theorem.
Corollary 5.4. Suppose that T is a repetitive primitive non-periodic substitution tiling with recognizable tile-substitution and linear expansion map Q, all of whose eigenvalues are algebraic integers. Then α ∈ R d is an eigenvalue of the measure-preserving system (X T , R d , µ) if and only if the property (5.5) holds.
Recognizability in the non-periodic primitive substitution FLC case and recognizability modulo periods in the general FLC framework was proved in [41], as a generalization of [33] from the 1-dimensional symbolic substitution case. Recognizability in specific non-FLC examples is usually easy to verify by inspection.
Question. Let Q be the expansion map of a substitution tiling with a finite set of prototiles up to translation. Is it necessarily true that all the eigenvalues of Q are algebraic integers?
In the FLC case the answer is positive, and the proof is not hard [43] (see also [28,Cor. 4.2]). Observe that even without FLC we have that | det(Q)| is an algebraic integer (a Perron number in the primitive substitution case), since it is the dominant eigenvalue of the substitution matrix, corresponding to the left eigenvector whose components are the volumes of the prototiles. Thus, in particular, if Q is a pure dilation Q(x) = θx, then θ is necessarily an algebraic integer.
Before proceeding with the proof of Theorem 5.3, we need the following.
In the course of the proof of Theorem 5.3, we will prove the following.
Corollary 5.6. Let T be a primitive substitution tiling with expansion map Q, whose eigenvalues are algebraic integers.
(i) Suppose that α ∈ R d is an eigenvalue of the measure-preserving system (X T , R d , µ). Let Θ = {θ 1 , . . . , θ r } be the set of eigenvalues of Q (real and complex) such that the corresponding eigenvectors satisfy e j , α = 0.
Then Θ is a Pisot family.
(ii) Suppose that the set of eigenvalues of Q is totally non-Pisot. Then (X T , R d , µ) is weakly mixing.
In the case of Q being a pure dilation we obtain the following. (i) If the system (X T , R d , µ) is not weakly mixing (i.e., there exists a non-trivial eigenvalue), then |θ| is a Pisot number.
(ii) If, in addition, T is repetitive and the substitution is recognizable, then every measuretheoretic eigenvalue is also a topological eigenvalue, i.e., every eigenfunction may be chosen continuous.
Proof sketch of Theorem 5.3. (i) For the necessity of (5.5), the argument is similar to the proof of [40,Thm. 4.3], based on Lemma 5.1. We use that any f ∈ L 2 (X T , µ) may be approximated by a linear combination of characteristic functions of the cylinder sets X(U m,α ). The necessity of (5.6) is straightforward, proved as in [42, §4].
(ii) The argument is similar to the proof of [42,Theorem 3.13], but since the latter relied on FLC in several places, we will sketch it in more detail. Suppose that (5.5) and (5.6) hold, and define The orbit {T − x : x ∈ R d } is dense in X T by definition. It suffices to show that f α is uniformly continuous on this orbit; then we can extend f α to X T by continuity, and this extension will satisfy the eigenvalue equation.
The idea is, roughly, as follows. Suppose that T − x is very close to T − y in the tiling metric. Without loss of generality, we can assume that they agree exactly on a large neighborhood of the origin, say, B R (0), with R 1. Since T − x = ω(T − Q −1 x) and T − y = ω(T − Q −1 y), it follows from recognizability modulo periods (see Definition 4.2) that T − Q −1 x agrees with T − Q −1 y − Q −1 g 1 for some period g 1 , on a smaller, but still large neighborhood. We can repeat this argument a number of times, say n, depending on how large R is, until we can only say that the resulting tilings: T − Q −n x and T − Q −n y − Q −n g 1 − Q −n+1 g 2 − · · · − Q −1 g n agree on at least one tile near the origin, where g 1 , . . . , g n are translational periods of T .
But this means that Repetitivity implies that Ξ(T ) = Ξ legal (T ), so z ∈ Ξ legal (T ), and we have x − y, α = Q n z, α (mod Z), using condition (5.6) and the fact that the group of periods is mapped by Q into itself. From (5.5), we have that e 2πi Q n z,α ≈ 1 for large n, and hence This is, of course, far from a proof, and significant amount work is needed to realize this scheme.
The actual proof in [42] proceeded with four lemmas. The first one, [42,Lemma 4.1] claimed that the eigenvalues of the expansion map are algebraic integers; now this is an assumption. Next we need the extension of [42, Lemma 4.2]: Lemma 5.8. Suppose that Q is a linear expansion map whose eigenvalues are algebraic integers. Then there exists ρ ∈ (0, 1), depending only on Q, such that, if (5.5) holds for z ∈ Ξ(T ), then e 2πi Q n z,α − 1 < C z ρ n , n ∈ N. where P i (n) are non-zero polynomials and θ 1 , . . . , θ r are all the eigenvalues of Q (real and complex) for which z has a non-zero coefficient and for which there is an eigenvector e i satisfying e i , α = 0. The degree of the polynomial P i is the maximal size of a Jordan block for θ i which contributes non-trivially to the decomposition. We can now apply a generalization of the classical Pisot's Theorem, due to Körnei [22,Theorem 1] and/or a similar result of Mauduit [31], which asserts that if (5.8) holds, then Θ = {θ 1 , . . . , θ r } is a Pisot family and (5.7) is satisfied, with any ρ ∈ (0, 1) that is larger in absolute value than all the Galois conjugates of θ j 's from Θ that do not appear in Θ.
The rest of the proof of Theorem 5.3 proceeds exactly as in [42], after noting that [28,Lem. 4.5] was obtained without the FLC assumption. The repetitivity was needed there in the same place as in the rough scheme above, namely, to claim that Ξ(T ) = Ξ legal (T ).
Proof of Corollary 5.6. (i) This follows from Theorem 5.3(i), following the argument in the proof of Lemma 5.8. The only additional observation needed is that the set of return vectors Ξ(T ) is relatively dense in R d , hence if e j , α = 0, we can find z ∈ Ξ(T ) whose decomposition into a linear combination of eigen-and root vectors of Q will have a non-zero coefficient with respect to e j .
(ii) This follows from part (i) by the definition of totally non-Pisot family.
Proof of Corollary 5.7. The first claim is immediate from Corollary 5.6. The second claim follows from Theorem 5.3(i) and the remark about | det(Q)| = |θ| d being an algebraic integer. 6. Rigidity for substitution tilings. Examples.
There are many notions of "rigidity" in mathematics. Ours originates from the work of Kenyon [19,20]. In [19,Theorem 1] it is shown that any sufficiently small perturbation of a planar tiling with finitely many prototiles (not necessarily a substitution tiling) must have an "earthquake", or "fault line" discontinuity (we refer to [19], as well as to [12,Section 5], for details). In [19,Cor. 3] Kenyon derives from this that a tiling of R 2 with a finite number of prototiles, up to translation, either has FLC, or the union of tile boundaries contains arbitrarily long line segments. In the planar FLC case, with the expansion Q given by a complex multiplication z → λz, with C ∼ = R 2 , the impossibility of a small perturbation led Kenyon [21,Section 5,p. 484] to conclude, with a reference to the method of [20], that for such tilings holds the inclusion Ξ(T ) ⊂ Z[λ]ξ, for some ξ ∈ C. In the paper [29] we gave a careful proof of this, as well as a generalization to the case of R d , under some algebraic assumptions, as stated below. We start with a definition. Definition 6.1. Let d = mJ, for some J ≥ 1, and let T be a substitution tiling in R d with Assume that Q is diagonalizable over C, and all the eigenvalues of Q are algebraic conjugates with the same multiplicity J. The tiling T is said to be rigid if there exists a linear isomorphism ρ : R d → R d such that where α j ∈ H j , 1 ≤ j ≤ J, are such that for each 1 ≤ n ≤ d, Here Z[Q] := { N n=0 a n Q n : a n ∈ Z, N ∈ N}. The most basic case is when Q(x) = θx is a pure dilation in R d . Then T is rigid if and only if there exists a basis {x 1 , . . . , We say that a representable substitution Delone κ-set Λ is rigid, if the corresponding tiling T is rigid, that is, (6.1) holds for Ξ(Λ) instead of Ξ(T ). In the special case when Q(x) = θx this was proved in [42, §5].
On the other hand, let α = (α 1 , α 2 ) ∈ R 2 be an eigenvalue. By the condition (5.5), since (1, 0), (0, 1) ∈ Ξ legal (T ), we have that 3 n α 1 → 0 mod 1 and 3 n α 2 → 0 mod 1, hence α 1 and α 2 are 3-adic rationals. On the other hand, (1, a) ∈ Ξ legal (T ) as well, hence 3 n α 1 + 3 n aα 2 → 0 mod 1, and we conclude that aα 2 is a 3-adic rational. But since a is irrational, we obtain that α 2 = 0. We can modify the Kenyon's example to make it non-periodic. First take a constant length substitution tiling and a Fibonacci substitution tiling in R and consider a direct product substitutin, then slide the last column relative to the first column. We give a precise example of this kind below. where a ∈ R is irrational, such that a ∈ Q(τ ). Note that Since (τ, 0), (0, τ ), (2τ, a) cannot be linearly independent over R, the tiling T does not have the rigidity property. We observe that the tiling is non-periodic. However note that lim n→∞ (τ, 0), (x, y)Q n = 0 mod Z for any (x, y) ∈ Ξ(T ). Moreover, one can check by inspection that it is repetitive and recognizable. So the tiling dynamical system has non-trivial eigenvalue and is not weakly mixing.
In [29], we showed under the rigidity assumption the equivalence between the Pisot family property and the existence of a relatively dense set of eigenvalues for (X T , R d , µ). However, we note that the equivalence can be proved without assuming FLC. So we revisit the theorem under the assumption of the rigidity. Kenyon's construction [21] where λ is not a complex Pisot number. (We should correct that example slightly, as follows: θ(a) = b, θ(b) = c, θ(c) = a −3 b −1 and λ is the non-real root of the equation λ 3 + λ + 3 = 0.) Lagarias [23,Problem 4.11] asked whether a primitive substitution Delone set, which is pure-point diffractive, is necessarily Meyer. In [28] we gave a positive answer under the FLC assumption. Here we show that the FLC assumption may be dropped if we assume that all the eigenvalues of the expansion map are algebraic integers. For the notion of diffraction spectrum and its connection with dynamical spectrum we refer the reader to [5] and references therein. Briefly, pure point diffraction implies that there is a relatively dense set of Bragg peaks, which in turn implies that there is a relatively dense set of eigenvalues.
Now the next proposition provides the answer.
Proposition 6.6. Let Λ be a repetitive primitive substitution Delone κ-set with expansion map Q. Suppose that all the eigenvalues of Q are algebraic integers. Assume that the set of eigenvalues of (X Λ , R d , µ) is relatively dense. Then supp(Λ) is a Meyer set.
Proof. Under the assumption that all the eigenvalues of Q are algebraic integers, the only place that requires FLC in the proof of [ negative: there exists a repetitive pure point diffractive set, which is not Meyer. An example (the "scrambled Fibonacci tiling") was constructed by Frank and Sadun [15].
Theorem 6.7. Let Λ be a repetitive primitive substitution Delone κ-set with expansion map Q. Suppose that Q is diagonalizable and all the eigenvalues of Q are algebraic conjugates with the same multiplicity. Then the following are equivalent: (i) the set of eigenvalues of (X Λ , R d , µ) is relatively dense; (ii) supp(Λ) is a Meyer set; (iii) Λ is rigid and the set of eigenvalues of Q forms a Pisot family.
Proof. (i) ⇒ (ii) This is a special case of Proposition 6.6. So, in particular, we obtain that a relatively dense set of eigenvalues for the dynamical system (X Λ , R d , µ) is impossible in the case of infinite local complexity.
As we saw from Kenyon's example, weak mixing is not equivalent to the Pisot family condition in general. However, we recover this if we assume rigidity.
Corollary 6.8. Let Λ be a repetitive primitive substitution Delone κ-set with expansion map Q, which is diagonalizable over C, and all the eigenvalues of Q are algebraic conjugates with the same multiplicity. Assume that Λ is rigid. Then the following are equivalent: (i) the set of eigenvalues of (X Λ , R d , µ) is relatively dense; (ii) (X Λ , R d , µ) is not weakly mixing; (iii) the set of eigenvalues of Q forms a Pisot family; (iv) supp(Λ) is a Meyer set.

Acknowledgement
We would like to thank to M. Baake, D.