Conjugacies of model sets

Let $M$ be a model set meeting two simple conditions: (1) the internal space $H$ is a product of $R^n$ and a finite group, and (2) the window $W$ is a finite union of disjoint polyhedra. Then any point pattern with finite local complexity (FLC) that is topologically conjugate to $M$ is mutually locally derivable (MLD) to a model set $M'$ that has the same internal group and window as $M$, but has a different projection from $H \times R^d$ to $R^d$. In cohomological terms, this means that the group $H^1_{an}(M,R)$ of asymptotically negligible classes has dimension $n$. We also exhibit a counterexample when the second hypothesis is removed, constructing two topologically conjugate FLC Delone sets, one a model set and the other not even a Meyer set.


Introduction and statement of results.
A substantial part of the analysis of Delone sets (or tilings) is based on the study of their associated dynamical systems.This includes characterizing certain classes of Delone sets by ergodic and topological properties of their dynamical systems.For instance, the dynamical system of a repetitive finite local complexity (FLC) Delone set of R d is topologically conjugate to that of a repetitive Meyer set if and only if it has d independent topological eigenvalues [KS].Similar characterizations are known for model sets (see [ABKL] for a review).
This suggest a natural definition and a question.Two Delone sets Λ and Λ ′ are called topologically conjugate whenever their associated dynamical systems are topologically conjugate.We call them pointed topologically conjugate if the conjugacy maps Λ to Λ ′ .Which properties of FLC Delone sets are preserved under topological conjugacy or pointed topological conjugacy?
In [KS] we showed that the Meyer property is not always preserved under topological conjugacy.In view of this, we call a Meyer set rigid if every FLC Delone set topologically conjugate to Λ is a Meyer set.Our aim in this article is to study rigity for model sets.
Recall that a Meyer set is a Delone set Λ such that the set of difference vectors Λ − Λ is uniformly discrete, and that model sets are Meyer sets arising from a particular construction.This construction involves a cut & project scheme, that is, an internal (locally compact abelian) group H, a lattice Γ ⊂ H × R d and a strip S = W × R d where W ⊂ H, the so-called window, is a compact subset that is the closure of its interior.The projection set for these data is the set of points arising by projecting the points of S ∩ Γ onto R d along H. Generically (but not always) the projection set is repetitive in which case we call any element of its hull, that is, any point set which has the same local patches up to translation, a model set.In particular, all model sets are repetitive according to this definition.
We say that a model set M ′ is a reprojection of a model set M if it arises from the same setup, except that the projection of S ∩ Γ onto R d is not along H, but is along H ′ ⊂ H × R d with H ′ = {(h, g(h))|h ∈ H} where g : H → R d is a continuous group homomorphism.
We pay particular attention to polyhedral model sets, by which we mean model sets satisfying two additional assumptions: H1.The internal space H is the product of the vector space R n with a finite (discrete, pure torsion) group C. H2.The window W is a finite union of polyhedra.
Our main result states that such model sets are extremely rigid: Theorem 1.1.If M is a polyhedral model set and Λ is a Delone set of finite local complexity that is pointed topologically conjugate to M, then Λ is mutually locally derivable (MLD) to a reprojection of M.
Without hypothesis H2, this theorem is false.In Section 8 we exhibit a one dimensional model set M satisfying H1 but not H2, and a Delone set Λ of finite local complexity that is pointed topologically conjugate to M, with Λ not being a Meyer set, much less a model set or a reprojection of M.
A first result in the direction of studying topological conjugacies can be found in [KS].It says that any pointed topological conjugacy between repetitive FLC Delone sets is the composition of an MLD transformation followed by a shape conjugacy (defined below) which can be chosen arbitrarily close to the identity.Given that MLD transformations are wellunderstood, this reduces the task of understanding topological conjugacies to a study of shape conjugacies.Shape conjugacies modulo MLD transformations are parametrized (at least infinitesimally) by a subgroup of the first tiling cohomology with R d -coefficients H 1 (Λ, R d ), called the asymptotically negligible group in [CS], which we denote by H 1 an (Λ, R d ).If Λ is a Meyer set, then within H 1 an (Λ, R d ) there is a subgroup of shape deformations that preserves the Meyer property.We call these nonslip, and denote the subgroup H 1 ns (Λ, R d ).If Λ is a model set, there is a further subgroup, denoted H 1 repr (Λ, R d ), corresponding to reprojections.One can similarly define H 1 an (Λ, R), H 1 ns (Λ, R), and H 1 repr (Λ, R). (See Sections 2 and 4, below.)In cohomological terms, Theorem 1.1 can be restated as follows: Another cohomological restatement is as follows: There is a natural map from the cohomology of H × R d /Γ to the cohomology of a model set constructed from the data (Γ, H, R d ).Let us denote its image by H 1 max (M, R) Theorem 1.3.If M is a polyhedral model set then H 1 an (M, R) is n-dimensional and is contained in H 1 max (M, R).
The organization of the paper is as follows.In Section 2 we review the machinery of Delone dynamical systems that is needed in the remainder of the paper.In Section 3 we review the theory of model sets, identifying how different model sets with the same parameter can differ.In Section 4 we introduce the notion of nonslip generators of shape conjugacies and show (Theorem 4.7) that all nonslip generators are, up to local deformation, generators of reprojections.In Section 5 we show (Theorem 5.1) that asymptotically negligible classes are represented by coboundaries of nonslip generators.Taken together, this proves Theorem 1.1.In Section 6 we interpret these results in terms of cohomology and prove Theorems 1.2 and 1.3.
Nonslip generators are introduced as a means of proving Theorem 1.1, but we believe that they have independent interest.In Section 7 we explore the significance of the nonslip property for model sets that do not necessarily satisfy hypotheses H1 and H2, and for more general Meyer sets.We prove Theorem 1.4.Let Λ be a repetitive Meyer set and F a generator of a shape deformation.If F is not nonslip, then the deformed set Λ F is not Meyer.
In Section 8 we exhibit a model set that does not satisfy H2 and a generator of shape conjugacies that is not nonslip, and hence a deformation of a model set that is not Meyer.We do not know whether it is ever possible to construct a nonslip class that is not a reprojection.

Preliminaries on point sets and their dynamical systems
In this section we review some of the necessary background on Delone sets and their dynamical systems.Apart from the notion of reprojection for model sets, this is all wellestablished in the literature.
2.1.Dynamical system of a Delone set.A Delone set is a set Λ ⊂ R d that is uniformly discrete and relatively dense.That is, there exists an r > 0 such that every ball of radius r contains at most one point of Λ, and there exists an R > 0 such that every ball of radius R contains at least one point of Λ.
A Meyer set is a Delone set Λ for which Λ − Λ (i.e., the set of displacement vectors between points of Λ) is uniformly discrete.Equivalently, Λ is Meyer if 0 is not a limit point of ( We denote it by (P, B) or simply by P .An R-patch of Λ at x is the intersection of Λ with B = B R (x), the ball of radius R at x.
A Delone set has finite local complexity, or FLC, if for each R > 0 the set {B R (0) ∩ (Λ − x)|x ∈ Λ} is finite, or stated differently, the number of R-patches occuring at points of Λ and counted up to translation is finite.A Delone set is repetitive if for every patch P of Λ, there exists an R such that every R-patch of Λ contains at least one translated copy of P .Henceforth all Delone sets in this paper will be assumed to have FLC and to be repetitive.
Delone sets are associated with dynamical systems as follows.We pick a metric on the space of Delone sets with given inner and outer radii r and R such that two Delone sets are close if their restriction to a large ball around the origin are close in the Hausdorff metric.If the Delone sets have FLC, this means that they agree exactly on a large ball, up to a small translation.R d acts on the space of Delone sets by translation.The closure of the orbit of a Delone set Λ is called the continuous hull of Λ, and is denoted Ω Λ , or just Ω when there is no ambiguity about which Delone set is being considered.If Λ is a repetitive FLC Delone set, then (Ω Λ , R d ) is a minimal dynamical system.We will also consider the canonical transversal Ξ Λ (or simply Ξ) of Ω Λ which is given by the closure of the set {Λ − x : x ∈ Λ}.Ξ Λ consists of all point patterns of Ω Λ that contain the origin.
A Delone set Λ ′ is locally derived from Λ if there exists a radius R > 0 such that, whenever Λ − x 1 and Λ − x 2 agree to radius R around the origin, Λ ′ − x 1 and Λ ′ − x 2 agree to radius 1 around the origin.If Λ ′ is locally derived from Λ and Λ is locally derived from Λ ′ , we say that Λ and Λ ′ are mutually locally derivable, or MLD.
A local derivation of Λ ′ from Λ extends to a factor map from Ω Λ to Ω Λ ′ .If Λ and Λ ′ are MLD, then this factor map is a topological conjugacy called an MLD map.
2.2.Maximal equicontinuous factor.An important factor of the dynamical system (Ω, R d ) is the largest factor (up to conjugacy) on which the action is equicontinuous.We denote this so-called maximal equicontinuous factor by Ω max and the factor map by π max .
The equivalence relation can be alternatively described using the (regional) proximality relation and its strong version.This description can be used to show the following two results results needed later on: • Any topological conjugacy preserves R max .
Proofs can be found in [BK].

Model sets
A model set (or cut & project set) is a Meyer set that is obtained by a particular construction.
3.1.Cut and project scheme.To construct a model set one needs a cut and project scheme (Γ, H, R d ) and a subset W ⊂ H called the window.
The cut & project scheme consists of the space R d in which the model set lives (the parallel or physical space), a locally compact abelian group H (called the internal group or perpendicular space) and a lattice (a cocompact discrete subgroup) Γ ⊂ H × R d .The set S := W × R d is called the strip.As usual, we require three further assumptions: (1) The projection onto the second factor π : We also use the notation Γ = π (Γ), Γ ⊥ = π ⊥ (Γ), x = π (x) and x ⊥ = π ⊥ (x).We write π ⊥ Γ for the restriction of π ⊥ to Γ.The point set ξ (W ) := {π (γ) : γ ∈ S ∩ (Γ + ξ)} is called the projection set of the cut & project scheme with window W and parameter ξ ∈ H × R d /Γ.Note that any element of ker π ⊥ Γ is a period of the projection set ξ (W ).We are interested in windows W that are compact and the closure of their interiors.In this case, the projection set ξ (W ) is repetitive if the parameter ξ is such that π ⊥ (Γ + ξ) ∩∂W is empty.We call such parameters non-singular and denote the set of non-singular parameters by NS.Definition 3.1.A model set (with window W ) is an element in the hull of a projection set ξ (W ) whose parameter is non-singular.We always assume that W is compact and the closure of its interior.We call the model set polyhedral if H is the product of R n with a finite group and the window is a finite disjoint union of polyhedra.
Model sets are repetitive Meyer sets.It is well-known that MLD maps send model sets to model sets, and send Meyer sets to Meyer sets.
Remark 1.Our use of the term model set is slightly different than that of some other authors.First, a model set in our sense need not to be a projection set with closed window, but might be what elsewhere is called an inter-model set, containing some but not all points x for which x ⊥ lies on the boundary of W . Second, the requirement that the parameter be non-singular automatically makes our model sets repetitive.So model sets in our sense might elsewhere be called repetitive inter-model sets (or complete Meyer sets).We should also mention that the window being the closure of its interior is not required by all authors.
For instance [BG] demand only that the window be relatively compact and have non-empty interior.
3.2.Reprojection.The projection π is along H (onto R d ).If we change the direction along which we project, that is, if we tilt the space H, but otherwise leave the strip S and the parameter ξ fixed, then this affects the projection set ξ (W ) = {π (x) : Here π ξ is the restriction of π to Γ + ξ which is injective by our assumption.We can think of H ′ as being the image of H under a group isomorphism which corresponds to a shear transformation in the case H is a vector space: there exists a continuous group homomorphism g : 3.3.Non-singular parameters and model sets.A model set is called non-singular if it is a projection set ξ (W ) with non-singular parameter ξ.Since the window is the closure of its interior, this occurs for a dense G δ -set of ξ.Therefore these model sets are also called generic.
For a fixed window W , the hull of a non-singular model set ξ 0 (W ) contains all other model sets ξ (W ) with ξ ∈ NS.In other words, the hull of a non-singular model set with window W depends not on the choice of the non-singular parameter ξ but only on the window.We therefore denote the hull with Ω(W ).Hence a model set with window W is an element of Ω(W ).
It is not difficult to see that It turns out that this torus is also the maximal equicontinuous factor, Ω max = (H ×R d )/Γ, the action being given by left translation on the second factor R d [BK].The factor map π max : Ω(W ) → (H × R d )/Γ is the continuous extension of the map ξ (W ) → ξ (for ξ ∈ NS) which associates to a non-singular model set ξ (W ) its parameter ξ.This map is injective precisely on its pre-image of NS.The elements of the hull that are mapped by π max to singular points are called singular model sets.
3.4.The star map.The star map sends points in R d to points in H.There are several related star maps to be considered: where [•] Γ denotes an equivalence class mod Γ.We denote the general star map also simply with a star, σ(x) = x * .The general star map is not continuous if one gives π (Γ) the relative topology induced by R d .• The general star map σ ξ with parameter ξ sends π (Γ + ξ) to H, where (σ ξ (x), x) is the unique lift of x ∈ π (Γ + ξ) to Γ + ξ.Like the general star map without parameter (or equivalently, with parameter 0), this has a dense domain in R d , dense range in H, and is not continuous.• If Λ is a Delone subset of π (Γ + ξ) we denote the restriction of σ ξ to Λ also by σ Λ and call it the star map of Λ.Note that the support of σ Λ is uniformly discrete.This will allow us later to talk about weak pattern equivariance of σ Λ .If Λ is a model set with window W then the image of σ Λ is a dense subset of W .
Note that if x 1 and x 2 are points of M and π max (M) = ξ, then both (σ M (x 1 ), x 1 ) and (σ M (x 2 ), x 2 ) are in Γ + ξ, so x 2 − x 1 ∈ Γ and The factor map π max is related to the star map of a pattern as follows.If M is a model set and ξ = π max (M) then (σ M (x), x) ∈ Γ + ξ for all points x ∈ M. Thus In particular, if 0 ∈ M, then By assumption ı H is injective.Furthermore, by equivariance of π max we have 3.5.Acceptance domains of patches.Let M be a model set with window W let B be a compact set and let P = M ∩ B. We call P the "B-patch of M".More generally we say that a finite set P ∈ R d is a patch for a model set with window W if there is M ∈ Ω(W ) and a compact set We wish to determine a condition for another model set M ′ in the hull of M to have M ′ ∩ B = P .It will be sufficient for our applications to consider the case that P contains the origin which we will assume troughout.
Let D be the preimage σ −1 ((W −W )∩Γ ⊥ ).This gives the set of all possible displacements between points in the same pattern, since if x 1 , x 2 ∈ M, then σ M (x 2 )−σ M (x 1 ) = σ(x 2 −x 1 ) ∈ W − W .Let P ′ = (D ∩ B)\P .P ′ the set of points that can appear in a B-patch of some model set which contains 0 but are not in the B-patch P .Note that P ′ is a finite set.Let W P := W o P is called the acceptance domain of P .By construction, it is the closure of an open set and hence the closure of its interior.Proposition 3.2.Let P be a B-patch for a model set with window W .We assume that P contains the origin.Let M ′ ∈ Ω(W ) be a possibly different model set which contains the origin.If M ′ is non-singular then, for all x ∈ M, σ M ′ (x) ∈ Int(W P ) if and only if The condition M ′ ∩ B = P has two parts: (1) All the points of P should be in M ′ .This is equivalent to having σ ξ (x) ∈ W for all , where W c denotes the complement of W . Thus Therefore and since M was assumed non-singular, σ M ′ (0) cannot lie on the boundary of W o P .Thus we can replace the right-hand side with the closed set W P .
Lemma 3.3.Suppose that M is a model set containing the origin and satisfying H1 and H2.The acceptance domain of every (non-empty) patch of M containing the origin can be written as a finite union of closed convex sets that have non-empty interior.
Proof.Since W is a finite union of connected polyhedra this is also the case for W P .We can decompose W P into a finite union of connected polyhedra and each of those into a finite union of convex polyhedra.Since W P is the closure of its interior the convex polyhedra can be taken to have non-empty interior.
We formulate the result of the above lemma as a hypothesis on the window W that is weaker than H2.H2 ′ .The acceptance domain of every patch containing the origin can be written as a finite union of closed convex sets that have non-empty interior.
Lemma 3.4.Consider a cut & project scheme (Γ, H, R d ) and a compact subset W ⊂ H that is the closure of its interior.For any neighborhood U of 0 ∈ H there exists a finite set Proof.Without loss of generality we can assume that U is contained in the compact set W − u contains a point x i ∈ K then, by compactness, there is a limit point W − u = {0}, which is a contradiction.
Corollary 3.5.Let M be a model set containing the origin.Any open subset of the window of M contains the acceptance domain for a patch of M that contains the origin.
Proof.Let M be a model set with window W and let U ⊂ W be an open subset.Pick x ∈ M such that σ M (x) ∈ U, and let which is an open neighborhood of 0. We apply Lemma 3.4 to obtain a finite set Furthermore, P contains the origin.
3.6.Singular model sets.We wish to describe the singular model sets, that is, the elements in the hull Ω(W ) that are not themselves projection sets with non-singular parameter.
Proposition 3.6.Let M be a model set with window W and ξ = π max (M).Then By continuity of π max we thus have ξ = lim ξ n and so the sequence (ξ n ) n lifts to a sequence ( ξn ) n ⊂ H × R d which converges to a lift ξ of ξ.We may quickly restrict to the case that ξ n = 0, because otherwise we can replace x n by x n − ξ n and η by η − [0, ξ ] Γ to reduce to that situation.Then M and all ξn are subsets of Γ .In particular, Γ is the common domain of all functions σ ξn , and the sequence of functions (σ ξn ) n converges uniformly to σ ξ , since ( ξn ) n converges to ξ. Hence if σ ξ (x) ∈ Int(W ) then an open neighborhood of σ ξ (x) in Int(W ) contains all σ ξn (x) for n sufficiently large which shows that σ ξn (x) ∈ W for all n sufficiently large.The latter means that x belongs to ξn for all n sufficiently large and thus also to M. This shows that To obtain the other inclusion we use the same kind of argument but for the complement W c instead of Int(W ).Indeed, if σ ξ (x) ∈ W c then an open neighborhood of σ ξ (x) in W c contains all σ ξn (x) for n sufficiently large which shows that x cannot belong to M.
We can now generalize one direction of Proposition 3.2 to singular model sets.
Corollary 3.7.Let P be a B-patch for a model set with window W and M ′ ∈ Ω(W ) a possibly singular model set.We assume that P and M ′ contain the origin.Let Proof.Set ξ = π max (M ′ ).We see from the description of M ′ given in Theorem 3.8 that the projection set ξ (Int(W P )) is contained in M ′ .Hence condition (2) also applies to a singular M ′ , as long as σ M ′ (0) does not lie on the boundary of We now describe the potential difference between two singular model sets with the same parameter.For that we assume that the window W is polyhedral and decompose it as a polyhedral complex.Let F (W ) be the set of open faces of W . Thus W is the disjoint union of its interior Int(W ) with the f ∈ F (W ) (where vertices are considered open 0-cells).Let V (f ) be the vector space parallel to f , that is, the space spanned by f − f .
Let Hξ (f ) be the closure of ) is a sublattice and we let E ξ (f ) be its real span.We define E ξ (f ) to be empty if Hξ (f ) is empty.
Furthermore, given that f is bounded, there is a finite subset Φ ξ (f ) Theorem 3.8.Let M be an arbitrary model set with polyhedral window W and let ξ = Proof.By Proposition 3.6 we have Lemma 3.9.E f has codimension at least 1.
and so we see that ξ (V ) is also the projection set with window V and parameter 0 for the cut & project scheme [BG] and thus has strictly positive lower density in E ξ (f ).
Since π ⊥ (Γ) is dense in H we can find a subset V ⊂ f η , open in the topology of H ξ (f ), and an infinite subset Ψ ⊂ π ⊥ (Γ) such that V + ψ ⊂ W for all ψ ∈ Ψ and such that the sets V + ψ have pairwise empty intersection.Hence ξ (W ) contains the disjoint union of all ξ (V + ψ), ψ ∈ Ψ.Now the lower density of ξ (V + ψ) in E f is independent of ψ.Therefore, if E f has dimension d and hence is all of R d than the lower density of ξ (W ) must be infinite, which is a contradiction.
Corollary 3.10.Consider two singular elements M 1 , M 2 of the hull of a model set.If is not finite that it contains a period which must be contained in E ξ (W ).There are only finitely many affine hyperplanes, because F (W ) and Φ ξ (f ) are finite.
The affine hyperplanes making up A(M 1 , M 2 ) depend on the pair M 1 , M 2 .However, their number is uniformly bounded: Lemma 3.11.Given the hull Ω(W ) of a model set.There is a finite number N such that for all M 1 , M 2 ∈ Ω(W ) with π max (M 1 ) = π max (M 2 ) the number of hyperplanes in the arrangement A(M 1 , M 2 ) is bounded by N.
Proof.There is a finite number of faces and each face f gives rise to a set Φ ξ (f ) which can vary with ξ, but the number of its elements is bounded from above since f is bounded and ∆ f discrete.Proposition 3.12.Given the hull Ω(W ) of a model set.Given r > 0 there exists a ρ > 0 such that for all M 1 , M 2 ∈ Ω(W ) which satisfy π max (M 1 ) = π max (M 2 ) and all x ∈ R d the ball B ρ (x) contains at least one point at which M 1 and M 2 agree out to distance r.
Proof.Let ξ = π max (M 1 ) = π max (M 2 ).Pick a patch (P, B) where B contains a ball of radius r.Since the interior of the acceptance domain W P is in the interior of W , ξ (W 0 P ) is a subset of both M 1 and M 2 .This is a relatively dense subset of M, so there exists a ρ such that for all x ∈ R d the ball B ρ (x) contains at least one point of ξ (Int(W P )), and around this point M 1 and M 2 agree out to distance r.
3.7.Pattern equivariant functions.Let Λ ⊂ R d be a FLC-Delone set and Y some set.A function f : R d → Y is called strongly pattern equivariant if there exists an R > 0 (called the radius) such that, whenever x 1 , x 2 ∈ R d are such that Λ − x 1 and Λ − x 2 agree exactly on the ball B R (0), then f (x 1 ) = f (x 2 ).In other words, each function value f (x) is determined exactly by the pattern of Λ in a ball of radius R around x.For most of our purposes we need functions only defined on Λ, or the CW -complex it defines.So we call a function φ : Λ → Y strongly pattern equivariant if there exists an R > 0 such that, whenever x 1 , x 2 ∈ Λ are such that Λ − x 1 and Λ − x 2 agree exactly on the ball B R (0), then φ(x 1 ) = φ(x 2 ).It can be shown that if Y is a finite dimensional real vector space then any strongly pattern equivariant function on Λ is the restriction of a smooth strongly pattern equivariant function on R d [K2].
Any locally constant function f : Ξ Λ → Y defines a strongly pattern equivariant function on is a finite dimensional vector space then φ is weakly pattern equivariant if and only if it is the uniform limit of strongly pattern equivariant functions.
The following is a very important example.A direct proof not using the maximal equicontinuous factor map can obtained from Corollary 3.5, see also [BL].
Lemma 3.13.The star map σ M : M → H of a model set is weakly pattern equivariant.
Proof.Recall that the image of σ M lies in W .We can hence rewrite (1) as A point pattern Λ can always be realized as the vertex set of a polygonal tiling of R d .Such a tiling gives a cell decomposition of R d .Functions on Λ with values in an abelian group A can then be viewed as 0-cochains on this CW complex.Likewise, we can consider 1-cochains, 2-cochains, etc.These cochains are considered strongly pattern equivariant if their values on a cell are determined exactly by the pattern of Λ in some fixed finite radius around that cell, and weakly pattern equivariant if they can be uniformly approximated by strongly pattern equivariant cochains.
The coboundary of a strongly pattern equivariant cochain is strongly pattern equivariant.The cohomology of the complex of strongly pattern equivariant cochains with values in A is called the A-valued cohomology of Λ and denoted H * (Λ, A).It is naturally isomorphic to the Čech cohomology Ȟ * (Ω Λ , A) [S1].
There is an equivalent description of the cohomology of Λ provided that A = R (or R k ).We can consider de Rham forms on R d which are strongly pattern equivariant for Λ.These form a sub complex of the usual de Rham complex for R d and H * (Λ, R) can be seen as the cohomology of this sub complex [K1].

Shape conjugacies and nonslip generators
A shape conjugation is a particular shape deformation in the sense of [CS,K2], and shape deformations arise as follows: Consider a function F : Λ → R d , defined on an FLC Delone set Λ.It defines a new set We assume that the coboundary δF of F is strongly pattern equivariant.This means that the elements of Λ F − Λ F can be locally derived from Λ − Λ and implies in particular that Λ F has FLC.We think of Λ F as a deformation of Λ (as if we had turned on F slowly) and call the function F the generator of the deformation.
One possible way for δF to be strongly pattern equivariant is for F already to be strongly pattern equivariant.This is the case precisely if Λ F can be locally derived from Λ.
A deformation is a shape semi-conjugacy if the map Λ − x → Λ F − x extends from the orbit of Λ in Ω Λ to a topological semi-conjugacy s F : Ω Λ → Ω Λ F .
Our ultimate aim is to understand the extent to which the dynamical system of a Delone set determines the Delone set.More specifically, given an FLC-Delone set Λ, we ask which FLC-Delone sets are topologically equivalent to it?To investigate this question we recall the following theorem.
Theorem 4.1 ([KS], Theorem 5.1).Let Λ and Λ ′ be FLC Delone sets that are pointed topologically conjugate.For each ǫ > 0 there exists a FLC Delone set Λ ǫ that is MLD with Λ and a function F ǫ : Λ ǫ → R d whose coboundary is strongly pattern equivariant such that Λ ′ = Λ Fǫ and s Fǫ : Ω Λǫ → Ω Λ ′ is a topological conjugacy; in other words F ǫ is a generator of a shape conjugation mapping Λ ǫ to Λ ′ .Moreover sup Remark 2. The term "deformation of a model set" has been used in the literature (see [BL, BD]) also for other kinds of deformations for which, in particular, the deformed set is no longer necessarily FLC.These could be achieved by functions F whose co-boundaries are not strongly pattern equivariant.Without the FLC requirement there are many more possible deformations and our rigidity results do not apply.
4.1.Auto-conjugacy.Theorem 4.1 is about pointed topological conjugacy.To understand what it means for the two patterns to be merely topologically conjugated we need to understand when, given two elements Λ 1 , Λ 2 of the same hull, the map Λ 1 → Λ 2 extends to a topological conjugacy.This is then an auto-conjugacy.
A first observation to make is that an auto-conjugacy ϕ must preserve the equivalence relation given by π max , i.e. if π max (Λ 1 ) = π max (Λ 2 ) then π max (ϕ(Λ 1 )) = π max (ϕ(Λ 2 )).Furthermore, the map induced by ϕ on Ω max must be the rotation by η := π max (Λ 2 ) − π max (Λ 1 ), since this is a homeomorphism (and hence the only one) on Ω max mapping π max (Λ 1 ) − x to π max (Λ 2 ) − x for all x ∈ R d .It follows then that for model sets the rotation by η must leave the set NS of nonsingular points invariant.But for a generic choice of the window the only translations leaving NS invariant are the elements of ({0}×R d +Γ)/Γ.It follows that on the orbit of a non-singular model set ϕ is given by a global translation.By continuity ϕ must then be everywhere this global translation.Thus for generic model sets auto-conjugacies are global translations and hence in particular MLD transformations.The more general case in which NS admits symmetries is presently under investigation.4.2.Asymptotically negligible co-chains.It turns out that the map Λ − x → Λ F − x extends to a topological semi-conjugacy s F : Ω Λ → Ω Λ F if and only if F is weakly pattern equivariant [CS,K2].So the generator of a shape semi-conjugacy is a weakly pattern equivariant function F : Λ → R d whose coboundary is strongly pattern equivariant.If F were strongly pattern equivariant, then the shape deformation would be a local derivation.If F is small enough, then this procedure can be inverted [K2], so that s F is a shape conjugacy.
Shape semi-conjugacies (and small shape conjugacies), up to MLD shape conjugacies, are thus parametrized by the sub-group of the first cohomology group consisting of strongly pattern equivariant co-cycles which are coboundaries of weakly pattern equivariant functions.Such co-cycles are called asymptoticaly negligible and we denote the subgroup by H1 an (Λ, R d ).
Proposition 4.2.The coboundary of the star map σ M of a model set is an asymptotically negligible 1-cocycle.
Remark 3. The above proposition can be understood as a generalization of a result from Boulmezaoud's thesis [B].
Proof.By Lemma 3.13 σ M : M → H is weakly pattern equivariant.That its coboundary is strongly pattern equivariant is a direct consequence of its additivity.In fact, if e is an edge between points x 1 , x 2 in M then δσ M (e) = σ M (∂e) = σ M (x 2 ) − σ M (x 1 ) = (x 2 − x 1 ) * , and so depends only on the displacement between x 1 and x 2 .δσ M (e) is thus strongly pattern equivariant for any radius R greater than the maximal distance between two neighboring points.
Corollary 4.3.Consider a model set M with internal group H. Let L : H → R d be a continuous group homomorphism.Then ) is a generator of a shape conjugation.The model set M F L resulting from the shape change is the reprojection along Proof.Continuity of L implies that F L is weakly pattern equivariant.Additivity of L implies that F L has strongly pattern equivariant coboundary.Finally, we have 4.3.Nonslip generators.We will introduce a property for generators of shape conjugations which characterizes those we have discussed above in the context of model sets.We state the definition for arbitrary weakly pattern equivariant functions, but the primary application is for vector-valued functions.Recall that F being weakly pattern equivariant means that there exists a continuous function F : Definition 4.4.Let Λ be a Meyer set.A weakly pattern equivariant function F : Λ → Y is nonslip if there exists an ǫ > 0 such that, for all (Λ 1 , Λ 2 ) ∈ R Ξ max we have F (Λ 1 ) = F (Λ 2 ) whenever Λ 1 and Λ 2 agree out to a radius of ǫ −1 .We call R = ǫ −1 the nonslip radius of F .
A strongly pattern equivariant function is manifestly nonslip.The star map is not strongly pattern equivariant, but we shall see below that it is nonslip.
Lemma 4.5.The star map of M is nonslip.
2 the star map is non-slip for every positive radius.
Corollary 4.6.Every generator of a shape conjugation of the form F L is nonslip.
4.4.Nonslip = reprojection.We have just seen that continuous linear functions L : H → R d define nonslip generators of shape conjugations, and hence that reprojections are generated by nonslip generators.We next show the converse, that under hypotheses H1 and H2 ′ , all nonslip generators are essentially of this form.The following theorem is the main result of this section, and proves half of Theorem 1.1.
Theorem 4.7.Let M be a model set satisfying hypotheses H1 and H2 ′ .Let F : M → R be a weakly pattern equivariant function that is nonslip, and whose coboundary δF is strongly pattern equivariant.Then F can be written as F (x) = L(σ M (x)) + ψ(x), where L : H → R is a continuous linear map and ψ : M → R is strongly pattern equivariant.
Proof.Note that since all elements of C have finite order, {0} × C must lie in the kernel of L and L is determined on R n alone.
To view the nonslip property from a different angle we consider an inverse limit construction for the canonical transversal Ξ.Let M 1 ∼ ǫ M 2 if π max (M 1 ) = π max (M 2 ) and the patterns M 1 , M 2 agree out to radius ǫ −1 .This is the intersection of two closed equivalence relations and therefore itself a closed equivalence relation; the quotient space Ξ/ ∼ ǫ is a compact Hausdorff space.Denote by π ǫ : Ξ → Ξ/ ∼ ǫ the canonical projection.Then Ξ = lim 0←ǫ Ξ/ ∼ ǫ and a weakly pattern equivariant F is nonslip iff F is the pullback by π ǫ of a continuous function on some approximant Ξ/ ∼ ǫ .
Let F : M → R be a nonslip, weakly pattern equivariant function, whose coboundary δF is strongly pattern equivariant.Since the star map σ M is also nonslip, there exists an ǫ > 0 such that we have the commutative diagram where F ǫ and σ M ǫ are the pullbacks of the continuous maps F and σ M induced by F and σ M .We need to show that there exists a 0 < η ≤ ǫ, a strongly pattern equivariant function ψ and a continuous group homomorphism L such that the right side of the diagram can be completed to a commutative diagram is hence at least as large as the strongly pattern equivariant radius of ψ and the nonslip radius of F ).Given that a global translation of a point set is an MLD transformation which may be absorbed in the definition of ψ we may assume that M contains the origin.
By FLC there are a finite number of possible ǫ −1 -patches at 0 which contain the origin.By H2 ′ the acceptance domain of each ǫ −1 -patch can be written as a finite union of closed convex sets that have non-empty interior.We call these convex sets sectors and let I index the sectors of all ǫ −1 -patches.Thus α ∈ I denotes both a patch (located for reference at the origin) and a closed convex subset W α ⊂ W of non-empty interior.For each such α, let Ξ α be the corresponding subset of Ξ, i.e. the set of point patterns that (a) contain the origin, (b) have the correct ǫ −1 patch around the origin, and (c) are mapped to sector W α by σ M .
The crucial observation is that on Ξ α the equivalence relation ∼ ǫ coincides with the relation defined by π max , and hence for It follows that for each α there is a unique function We consider also the restriction ∆ Since F has strongly pattern equivariant coboundary, and since strongly pattern equivariant functions are locally constant on Ξ, ∆ y F is locally constant on Ξ ∩ (Ξ + y).Hence by ( 4) There are finitely many sectors and each sector has non-empty interior.So there is an open neighborhood U ⊂ H of 0 such that for all α and all v ∈ U we have The sectors have non-empty interior, so by Corollary 3.5 for each sector α we can find a patch P α of M whose acceptance domain is contained in the interior of W α .By repetitivity, there is a radius such that every ball of that radius contains at least one copy of each patch P α .Let P be a patch of M of that radius, so that P contains translates of all the patches P α .That is, there are x α ∈ R d such that P α + x α is a subpatch of P .It follows that σ M (M − x α ) lies in the interior of W α .By Corollary 3.7 the set {σ M (x) : P = (M − x) ∩ B} is dense in the acceptance domain W P .If P = (M − x 1 ) ∩ B = (M − x 2 ) ∩ B we call x 2 − x 1 a return vector of P .The possible values of y * for return vectors y of P are thus dense in W P − W P .
Pick a y such y * is a return vector of P .We then have ) is obtained by adding the δF (e i ) over the edges e i along a path in P which joins summing the values of δF (e i ) over the corresponding path in P + y.Since y is a return vector to P and δF is strongly pattern equivariant, the result is the same.Hence To summarize, we have established that for all y * ∈ (W P − W P ) ∩ Γ ⊥ the value of ∆ α,α y * f (u) is the same for all α and all u ∈ W α ∩ (W α − y * ).Moreover, for fixed u in the interior of Ũ is an open neighborhood of the identity in H.We define L : We saw that L is continuous.We claim that L is additive where sums are defined.Indeed, if u 1 , u 2 , u 1 + u 2 ∈ U then there is u ∈ W α such that also u + u 1 and u + u 1 + u 2 lie in W α .It follows that, for all u ∈ Ũ A continuous additive function on a neighborhood of the origin is necessarily linear.That is, L equals its derivative.We may then extend L to a group homomorphism on the group generated by Ũ and thus obtain a linear function L : H → R that is trivial on the torsion factor.Now let ψ(x) = F (x) − L(σ M (x)) for x ∈ M.This then defines a function ψ : by the construction above and so ψ is a continuous function which is constant on Ξ α .Moreover, if W α ∩ W β = ∅ then continuity implies that ψ takes the same value on Ξ α and Ξ β .It follows that ψ is constant on the the pre-images under σ M of the connected components of the acceptance domains of the ǫ −1 -patches at 0. However, different components of the same central ǫ −1 patch are separated by a nonzero distance in W , and so can be distinuished by the R-patches at 0 for some (possibly large) fixed R > 0. Different central patches are distinguished by their patterns out to distance ǫ −1 .Thus ψ(x) is in fact strongly pattern equivariant with radius η −1 where η = min(R −1 , ǫ).
Corollary 4.8.If F is a nonslip generator of a shape conjugation for a model set M satisfying H1 and H 2 ′ then, up to MLD transformations, M F is a reprojection of M. In particular M F is a model set.
Proof.Theorem 4.7 applied to vector valued functions and Corollary 4.3 imply that F is the generator of a reprojection plus a strongly pattern equivariant function.Hence, up to an MLD transformation, M F is a reprojection of M, and is a model set.

Asymptotically negligible = nonslip
We now turn to the question of when an asymptotically negligible cocycle is nonslip.Here we need the stronger assumption H2.
Theorem 5.1.Let M be a model set satisfying assumptions H1 and H2, and let F : M → R be a weakly pattern equivariant function whose coboundary is strongly pattern equivariant.Then F is nonslip.
Proof.Let F : M → R be weakly pattern equivariant with strongly pattern equivariant δF .We need to show that there exists R > 0 such that, for any choice of pair M 1 , M 2 in the hull of M and point We denote by R 0 the radius of pattern equivariance of δF .Fix R > R 0 and consider doubly pointed double R-patches.These are double-R-patches (P, , together with two points x, y ∈ P ∩ Q which are at least distance R 0 away from z. We denote such an object by P (2) (x, y) = (x, y; P, Q; B R (z)).By FLC there are finitely many up to translation.Since δF is strongly pattern equivariant with radius R 0 the expression depends only on the translational congruence class of P (2) (x, y).Hence the set D R of possible values F can take on doubly pointed double R-patches is finite.We now need the following lemma.
Lemma 5.2.There exists Ñ and R 1 > 0 such that for all pairs Proof of the lemma.Let V be the collection of subspaces E f of R d associated to the faces of W .Let ν ∈ R d be a vector of length one and pick ω > 0 such that the cone intersects a vectorspace of V only trivially.Therefore C intersects each hyperplane from A(M 1 , M 2 ) at most once and so, by Lemma 3.11, intersects in total at most N hyperplanes.Now let M 1 , M 2 ∈ Ω M satisfy π max (M 1 ) = π max (M 2 ) and consider a point x ∈ M 1 ∩ M 2 .Application of Proposition 3.12 with r > 0 guaranties that there is ρ > 0 so that we may choose a point x n ∈ B ρ (x + nρν) at which M 1 and M 2 agree, and this for all n ∈ N. We choose x 0 = x and thus obtain a sequence ( are distinct and so for at most Ñ values of n we have F (P (2)  n ) = 0. Once all hyperplanes are crossed, M 1 − x n and M 2 − x n agree out to distance approximately nρω and hence lim n→∞ We continue the proof of Theorem 5.1.
Proof of Theorem 1.1.If M ′ is pointed topologically conjugate to M, then M ′ is MLD to a pattern M ′′ that is shape conjugate to M. Let F be the generator of that shape conjugacy.By Theorem 5.1 applied to vector-valued functions, F is nonslip.By Theorem 4.7 applied to vector-valued functions, F is then the sum of a linear map L : H → R d and a strongly pattern equivariant function ψ ǫ .Since L induces a reprojection and ψ ǫ induces an MLD transformation, M ′′ is MLD to a reprojection of M. Since M ′ is MLD to M ′′ , M ′ is also MLD to a reprojection of M.

Cohomological interpretation
We have already described how the coboundary δF of a generator of a shape deformation defines an element in the first cohomology H 1 (Λ, R d ) of Λ with coefficients in R d .Here the cohomology is the cohomology of strongly pattern equivariant cochains on the CW -complex defined by Λ. Alternatively we can describe H 1 (Λ, R d ) as the cohomology of the complex of strongly pattern equivariant R d -valued de Rham 1-forms on R d .Its elements then arise as differentials dF of smooth functions F : R d → R d and the relation between the two description is given by integration: Given a 1-chain defined by Λ, that is, essentially an oriented edge e in a tiling whose vertices are given by Λ, δF (e) = e dF = F (y) − F (x), where x, y ∈ Λ are the source and the range vertex of the edge.
There are a number of canonical subgroups of H 1 (Λ, R d ), and our results can be viewed as saying when these subgroups are, and aren't, equal.
One subgroup, denoted H 1 lin (Λ, R d ) is given by generators F that are the the restriction to Λ of linear maps L : R d → R d .For each such F , the deformed set is simply the result of applying the linear transformation id + L to the points of Λ.If L is non-zero then the deformation cannot be a local derivation, so For model sets, reprojections give another subgroup, The asymptotically negligible classes [CS] H 1 an (Λ, R d ) are represented by the coboundaries of shape semi-conjugacies.That is, by strongly pattern equivariant 1-cochains that are the coboundaries of weakly pattern equivariant functions.Recall that a generator of a shape semi-conjugacy is weakly pattern equivariant and hence bounded.(In fact, a generator F is weakly pattern equivariant iff it is bounded [KS]).Since linear maps are unbounded, it follows that We also considered nonslip generators.Recall that any strongly pattern equivariant generator is nonslip and a nonslip generator is asymptotically negligible.Hence the classes of nonslip generators define a subgroup Thus we have a sequence of inclusions A natural question is whether these groups coincide.
6.1.Reinterpretation of Theorems 4.7 and 5.1.We can also consider functions and cochains with values in R rather than with values in R d , with Proof.By Theorem 4.7, each nonslip generator is the sum of a linear function on H and a strongly pattern equivariant function, so the 1-cohomology class of its coboundary is in Corollary 6.2.If M is a model set satisfying H1 and H2, then Proof.Theorem 5.1 gives this result on the level of cochains.Every generator of a shape semi-conjugacy is nonslip, period.
The last two corollaries together give Theorem 1.2.
6.2.Image of the first cohomology of the maximal equicontinuous torus.The maximal equicontinuous factor map π max : Ω → Ω max induces an injective map in cohomology π * max : H 1 (Ω max , Z) → H 1 (Ω, Z).We have thus a fourth subgroup which is worth comparing with the other, namely the image under π * max of H 1 (Ω max , R d ) which we denote by To do that we need a better understanding of the image of π * max .
We recall from [BKS] that Ω max can be alternatively described with the help of the topological eigenvalues of the action.Let Rd = Hom(R d , U(1)) be the Pontryagin dual of R d , where we require each homomorphism to be continuous on R d .An element χ ∈ Rd is a topological eigenvalue if there exists a non-vanishing continuous function f : Ω → C such that f (Λ − t) = χ(t)f (Λ).Topological eigenvalues form a countable subgroup E of Rd and Ω max can be identified with the dual Ê = Hom(E, U(1)), where E is given the discrete topology.
For a general topological space X, H 1 (X, Z) is isomorphic to [X, S 1 ], the homotopy classes of continuous maps X → S 1 .Furthermore, if ϕ : X → Y is a continuous map then ϕ * : We apply this to π max : Ω → Ω max .Since Ω max ∼ = Ê the elements of [Ω max , S 1 ] are the homotopy classes of characters on Ê. Hence [Ω max , S 1 ] ∼ = E and the image of χ ∈ E under π * max in [Ω, S 1 ] is given by the homotopy class of an eigenfunction f χ of χ (f χ is normalized so as to have modulus 1).This describes the image of π * max (in degree one) in [Ω, S 1 ] [BKS].To obtain the image of π * max in pattern equivariant cohomology we consider the restriction of a representative f of an element of [Ω, S 1 ] to the orbit of Λ and define f : is strongly pattern equivariant.
Proof.[Ω, S 1 ] can be seen as the direct limit of [G n , S 1 ] where G n is the nth approximant in the Gähler complex [S2].Each element thus comes from some [G n , S 1 ] and the latter elements produce strongly pattern equivariant functions when considered on the orbit.Since R d is simply connected we can lift f to a continuous function τ : R d → R such that f (x) = exp 2πiτ (x).We define F to be the restriction of τ to Λ. Then δF is strongly pattern equivariant and so we have a map [Ω, Proof.Using Gähler's approximation this statement boils down to consider the map between [G n , S 1 ] and H 1 (G n , R) where the latter can be considered as de Rham cohomology on a branched manifold.If [f ] ∈ [G n , S 1 ] then (assuming without restriction of gen- R).This is well-known (see also [KP]).Now in order to obtain the map on cellular cohomology of G n one just needs to integrate over 1-chains.The result is a 1-cochain which, when interpreted as strongly pattern equivariant 1-cochain on Λ coincides presicely with δF .
If we combine the two arguments we see that the image of H 1 (Ω max , Z) ∼ = E in pattern equivariant cohomology can be described as follows.
Corollary 6.5.Upon the above identification of H 1 (Ω max , Z) with E and the identification of H 1 (Ω, Z) with H 1 (Λ, Z) the map π * max becomes the map where β | Λ is the restriction to Λ of a continuous function β : R d → R such that exp(2πi β) is strongly pattern equivariant and homotopic to χ.
Each eigenvalue χ can be lifted, that is there exists a β ∈ R d * such that χ(x) = exp 2πiβ(x).Therefore β − β must be bounded and hence weakly pattern equivariant.It follows that an (Λ, R) and, since β is linear, we see that Stated differently, the image of can be identified with a subspace of the vector space of linear deformations.Indeed let ψ : Lemma 6.6.Let Λ be a repetitive FLC Delone set.We have whenever Λ is topologically conjugate to a Meyer set.
Proof.The above shows that ψ(H ) and the r.h.s. is clearly isomorphic to the vector space of all linear deformations.Thus equality holds precisely if the real span of {β ∈ R d * : exp 2πiβ ∈ E} has dimension d.By the results of [KS] this is equivalent to saying that Λ is topologically conjugate to a Meyer set.6.3.Case of model sets.In the case of model sets which satisfy H1 we can say more, because we have a more explicit model for the maximal equicontinuous factor, namely the torus parametrization.Indeed, if H = R n ×C then, by cocompactness of Γ, Ω max = H ×R d /Γ is an n+d-torus and so we can identify H 1 (Ω max , Z) ∼ = E ∼ = Ωmax with the so-called recicrocal lattice Γ rec which is given by those continuous group homomorphisms α : H × R d → R which satisfy α(γ) ∈ Z for all γ ∈ Γ.Again, α must be trivial on the torsion component C and restricts to a linear map R n × R d → R. Proposition 6.7.Let M a model set satisfying H1.Upon the identification of H 1 (Ω max , Z) with Γ rec and the identification of H 1 (Ω, Z) with H 1 (M, Z) the map π * max becomes the map Recall that σ M : M → H is weakly pattern equivariant and has strongly pattern equivariant coboundary.We may therefore extend it to a weakly pattern equivariant function on all of R d whose differential is strongly pattern equivariant [K2].We denote the extension also by σ M .As {σ M (x) : x ∈ M} lies in a compact subset of H, χ t (x) = exp 2πiα(tσ M (x), x) is a homotopy between χ and χ 1 .Furthermore, if x, y ∈ M then (σ M (y), y) − (σ M (x), x) = ((y − x) * , y − x) ∈ Γ and so M ∋ x → exp 2πiα(σ M (x), x)) is constant and hence a strongly pattern equivariant function on M. It follows that χ 1 is a strongly pattern equivariant function on R d and thus we may apply Corollary 6.5 to obtain the statement.Proposition 6.8.Let M be a model set satisfying H1, that is, Proof.We have This also shows that the dimension of H 1 repr (M, R) is n.
With this proposition at hand we see that Theorem 1.2 is equivalent to Theorem 1.3.Indeed, it shows that for polyhedral model sets the statement dim H

Nonslip sets and the Meyer property
In this section we explore a little further the concept of nonslip generators.Our hope is that this may turn out useful later for the study of shape conjugations of Delone sets which are not model.We consider an analogous property of sets, and show that a generator of a shape conjugation is nonslip if an only its associated shape conjugacy preserves that property.
Proof.Recall that for Meyer sets π max (Λ 1 ) = π max (Λ 2 ) implies that 0 ∈ Λ 1 −Λ 2 .But Λ 1 −Λ 2 is uniformly discrete by the Meyer property.Hence if Λ 1 and Λ 2 are close enough they have to coincide on a ball of radius equal to the inverse of their distance.
Recall that a generator of shape conjugacy of an FLC Delone set Λ ⊂ R d is a function We saw that in this case F extends to a continuous map F : Ω Λ → R d and so we may define Nonslip generators of shape conjugacies and nonslip sets are closely related.For a more general (not necessarily Meyer) Delone set we generalize the concept of a nonslip weakly pattern equivariant function as follows: Definition 7.3.Let Λ be an FLC Delone set.A weakly pattern equivariant function Note that if Λ is nonslip, then the above definition reduces to the definition we previously gave for Meyer sets.This follows as in the proof of the last lemma form the fact that π max (Λ 1 ) = π max (Λ 2 ) implies that Λ 1 and Λ 2 agree on balls once they are close.
Proof."⇒" We suppose that Λ F is nonslip.Hence, given R there exists δ such that for all x ∈ Λ i } we see that, if R is large enough, this implies F Λ 1 (0) = F Λ 2 (0)."⇐" We suppose that F is nonslip, hence there exists δ such that for all (Λ 1 , Λ 2 ) ∈ R Ξ max and all x ∈ Λ 1 ∩ Λ 2 we have that d(Λ 1 − x, Λ 2 − x) < δ implies F Λ 1 (x) = F Λ 2 (x).By definition of the metric there exists ǫ 1 such that if d(Λ 1 , Λ 2 ) < ǫ 1 then d(Λ 1 −x, Λ 2 −x) < δ for all x of size smaller or equal to the radius of relative denseness.Let R > 0 and F = sup The following corollary is just a special case: Corollary 7.5.Let Λ be a Meyer set and F be a generator of a shape conjugacy.If Λ F is a Meyer set then F must be nonslip.
The contrapositive says that if F is not nonslip, then Λ F is not Meyer.That is Theorem 1.4.
Applying these observations to model sets we obtain: Corollary 7.6.Given a model set M which satisfies H1.If H 1 ns (M, R) ⊂ H 1 max (M, R) then any shape conjugation of M which is a Meyer set is a reprojection of M.
Proof.Let F be the generator of a shape conjugation of M such that M F is a Meyer set.By Corollary 7.5, F must be nonslip.By Proposition 6.8, H 1 ns (M, R) ⊂ H 1 max (M, R) is equivalent to the equality H 1 ns (M, R) = H 1 repr (M, R).Hence M F is a reprojection.

8.
A model set that is not rigid Theorem 1.1 states that most common examples of model sets, constructed by direct application of the cut & project method with polyhedral windows, are rigid.In this example we exhibit a model set with a Euclidean internal space (so satisfying H1) that is not.The example is constructed by a substitution; and model sets arrising from substitutions may have very complicated (fractal) windows.
Consider the 1-dimensional substitution σ on four letters: Its substitution matrix is and has eigenvalues φ 2 , −φ, φ −1 and φ −2 where φ = (1 + √ 5)/2 is the golden mean.We choose tile length proportional to the right Perron Frobenius eigenvalue, namely b 1 and b 2 tiles to have unit length, and a 1 and a 2 tiles to have length equal to φ = (1 + √ 5)/2.This is then a geometric primitive aperiodic unimodular Pisot substitution (which is not irreducible).By the results of [BSW] the dynamical spectrum is pure point, indeed the balanced pair (a 1 b 1 , b 1 a 1 ) terminates with coincidence (see [BSW] for explanations on this notion).A lot is known about such substitution tilings.The following can be found more or less implicit in, for instance, [BM, BBK, Si].
• Since the spectrum is pure point, the tilings are MLD to (possibly colored) regular model sets.Moreover, the set M of left boundary points of a 1 -tiles in a tiling is MLD to the tiling and hence also a regular model set.• Since the substitution is unimodular, the maximal equicontinuous factor Ω max of the associated dynamical system is a torus of dimension J where J is the algebraic degree of the Perron Frobenius eigenvalue.This eigenvalue is here φ 2 and hence J = 2. This, in turn implies that M has a cut & project scheme in which the internal space H is R.
One readily computes using the technique of [AP] that H 1 (Ω M , R) = R 4 , and that substitution acts on H 1 (Ω M , R) by the transpose A T σ of the substitution matrix.For substitution tilings, H 1 an (Ω M , R) is the span of all of the generalized eigenspaces of this action with eigenvalues strictly inside the unit circle [CS].In our case this means that H 1 an (Ω M , R) = R 2 is the span of the φ −1 and φ −2 eigenvectors of A T σ .The generator of a shape conjugacy corresponding to the φ −2 eigenvector induces a reprojection and hence is nonslip.It corresponds to a shape conjugacy in which all of the a tiles are lengthened and the b tiles are shortened (or vice-versa), while maintaining The generator F of a shape conjugacy corresponding to the φ −1 eigenvector is not nonslip, and results in a Delone set M F that is not Meyer.To see this, let A n i = σ n (a i ) and B n i = σ n (b i ) be n-th order supertiles, and let |A i | and |B i | be the Euclidean lengths of these supertiles after deformation.This time 1 appears in our tiling and A n 1 and B n 1 start with a 1 .Likewise, |A n 2 | ∈ M F − M F , since A n 2 B n 1 appears in the tiling and also A n 2 starts with a 1 .However, since |A n 1 | − |A n 2 | is proportional to φ −n the set of differences M F − M F cannot be uniformly discrete, so M F is not Meyer.
The key feature of this example is that the substitution matrix is reducible.The maximal equicontinuous factor is determined by the dynamical spectrum, which for self-similar tile lengths is determined by the Perron-Frobenius eigenvalue λ P F .If λ P F is a Pisot number, then a basis for the nonslip generators of shape conjugacy is given by the eigenvectors with eigenvalues algebraically conjugate to λ P F .However, a basis for H 1 an is given by all the eigenvectors with eigenvalue strictly smaller than 1.The eigenvectors whose small eigenvalues are not conjugate to λ P F correspond to weakly pattern equivariant functions that are not nonslip.
this defines a bijective correspondence.Now let Y be a topological space.Any continuous function f : Ω Λ → Y is uniquely determined by the function f : R d → Y , f (x) = f (Λ − x).We call a function f : R d → Y arising in such a way from a continuous function f : Ω Λ → Y weakly pattern equivariant.Likewise φ : Λ → Y is weakly pattern equivariant if it arises in the above way from a corresponding function φ : Ξ Λ → Y .Equivalently we may say that φ : Λ → Y is weakly pattern equivariant for Λ iff the function {Λ − R)⊗R d .The following are immediate corollaries of Theorem 4.7 and Theorem 5.1.Corollary 6.1.If M is a model set satisfying H1 and H 2 ′ , then H 1 repr (M, R) = H 1 ns (M, R).
be the composition of π * max with the canonical projection.It is induced by [exp 2πiβ] → [δβ].