Johannes Kellendonk

Daniel Lenz
email: daniel.lenz@uni-jena.de

Robert V Moody

EQUICONTINUOUS DELONE DYNAMICAL SYSTEMS

We characterize equicontinuous Delone dynamical systems as those coming from Delone sets with strongly almost periodic Dirac combs. Within the class of systems with finite local complexity the only equicontinuous systems are then shown to be the crystalline ones. On the other hand, within the class without finite local complexity, we exhibit examples of equicontinuous minimal Delone dynamical systems which are not crystalline. Our results solve the problem posed by Lagarias whether a Delone set whose Dirac comb is strongly almost periodic must be crystalline.

Introduction

The study of disordered systems is one of the most prominent issues in mathematics and physics today. A special focus concerns aperiodic order i.e. the very 'border' between order and disorder (see e.g. the monographs and conference proceedings [START_REF]Directions in Mathematical Quasicrystals[END_REF][START_REF] Janot | Quasicrystals -A Primer[END_REF][START_REF] Moody | The Mathematics of Long-Range Aperiodic Order[END_REF][START_REF]Quasicrystals and Discrete Geometry[END_REF][START_REF]Quasicrystals and Geometry[END_REF]). This topic is particularly relevant for various reasons: On the one hand, the actual discovery of physical substances [START_REF] Ishimasa | New ordered state between crystalline and amorphous in Ni-Cr particles[END_REF][START_REF] Shechtman | Metallic phase with long-range orientational order and no translational symmetry[END_REF], later called quasicrystals, lying at this border has triggered enormous research activities in the experimental and theoretical description of low complexity systems. On the other hand, aperiodic order has come up in disguise in various contexts in purely conceptually motivated studies.

As for such conceptual studies of almost periodic order (in disguise), we mention work of Hedlund/Morse on Sturmian systems and on complexity of aperiodic sequences [START_REF] Morse | Symbolic Dynamics[END_REF][START_REF] Morse | Symbolic dynamics II. Sturmian trajectories[END_REF]. Geometric analogues have recently been studied by Lagarias [START_REF] Lagarias | Meyer's concept of quasicrystal and quasiregular sets[END_REF][START_REF] Lagarias | Geometric models for quasicrystals I. Delone sets of finite type[END_REF] and Lagarias/Pleasants [START_REF] Lagarias | Repetitive Delone sets and quasicrystals[END_REF][START_REF] Lagarias | Local complexity of Delone sets and crystallinity Canad[END_REF]. Very loosely speaking, the corresponding results show that in terms of suitable complexity notions there is a gap between the ordered world and the disordered world. Another approach to (dis)order in a spirit of Fourier analysis has been given in Meyer's work on harmonious sets [START_REF] Meyer | Algebraic Numbers and Harmonic Analysis[END_REF]. There, a basic aim is to find and study a class of sets allowing for a Fourier type expansion. Meyer's considerations have been taken up by Moody [START_REF] Moody | Meyer sets and their duals[END_REF][START_REF] Moody | Model sets: A Survey[END_REF]. The corresponding results have become a cornerstone in the study of diffraction aspects of aperiodic order initiated by Hof [START_REF] Hof | On diffraction by aperiodic structures[END_REF].

Of course, there are connections between the geometry and complexity based approach to aperiodic order and the Fourier analytic side of the picture. For example, pure pointedness on the Fourier side implies zero entropy [START_REF] Baake | Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies[END_REF]. A crucial link is provided by concepts of almost periodicity. On the one side almost periodicity is by its very definition a geometric concept about how the systems 'repeats itself'. On the other side, it is an important ingredient in all Fourier analytic considerations concerning pure point measures. Almost periodicity has always played a role in the theory, see e.g. Solomyak [START_REF] Solomyak | Spectrum of dynamical systems arising from Delone sets[END_REF] and Queffelec [START_REF] Quéffélec | Substitution dynamical systems-spectral analysis[END_REF]. Still, it seems that only recently with the work of Baake/Moody [START_REF] Baake | Weighted Dirac combs with pure point diffraction[END_REF] and subsequent work as e.g. [START_REF] Moody | Point sets and dynamical systems in the autocorrelation topology[END_REF][START_REF] Gouéré | Diffraction and Palm measure of point processes[END_REF][START_REF] Gouéré | Quasicrystals and almost periodicity[END_REF][START_REF] Lenz | Pure point spectrum for measure dyamical systems on locally compact Abelian groups[END_REF][START_REF] Lenz | Pure point diffraction and cut and project schemes for measures: The smooth case[END_REF][START_REF] Strungaru | Almost periodic measures and long-range order in Meyer sets[END_REF] the role of almost periodicity in diffraction theory begins to be properly ascertained.

A most basic question in this context is whether it is possible to characterize order by suitably strong forms of almost periodicity. An affirmative answer to one version of this question is given in the so-called Córdoba Theorem [START_REF] Córdoba | La formule sommatoire de Poisson[END_REF] (see [START_REF] Lagarias | Mathematical quasicrystals and the problem of diffraction[END_REF] for a generalization as well). Another version of this question has been posed by Lagarias in [START_REF] Lagarias | Mathematical quasicrystals and the problem of diffraction[END_REF]. More precisely, (with notions to be explained later), Problem 4.4 in [START_REF] Lagarias | Mathematical quasicrystals and the problem of diffraction[END_REF] asks whether a Delone set whose associated Dirac comb is strongly almost periodic is necessarily completely periodic.

The main aim in this paper is to answer this question. More specifically, we will show the following two points:

• For Delone sets with finite local complexity complete periodicity is indeed equivalent to strong almost periodicity of the associated Dirac comb (Corollary 5.6). • There exist Delone sets (without finite local complexity) which are strongly almost periodic but not completely periodic (Corollary 6.1).

So, in some sense we show that the answer to the problem is both yes and no.

In order to avoid confusion let us mention that our notation differs from the notation of [START_REF] Lagarias | Mathematical quasicrystals and the problem of diffraction[END_REF] in the following way: In line with [START_REF] Gil De Lamadrid | Almost Periodic Measures, Memoirs of the AMS[END_REF] and recent work on almost periodicity such as [START_REF] Baake | Weighted Dirac combs with pure point diffraction[END_REF][START_REF] Moody | Point sets and dynamical systems in the autocorrelation topology[END_REF][START_REF] Lenz | Pure point spectrum for measure dyamical systems on locally compact Abelian groups[END_REF][START_REF] Lenz | Pure point diffraction and cut and project schemes for measures: The smooth case[END_REF] we use the term strongly almost periodic measure for measures called uniformly almost periodic in [START_REF] Lagarias | Mathematical quasicrystals and the problem of diffraction[END_REF]. We use the term crystalline or completely periodic for what is called ideal crystal in [START_REF] Lagarias | Mathematical quasicrystals and the problem of diffraction[END_REF].

The result given in the second point above is treated in Section 6 by providing a class of models based on the Kronecker flow on the two dimensional torus (Corollary 6.1).

As for the result given in the first point, it is quite clear that complete periodicity implies both finite local complexity and strong almost periodicity of the associated Dirac comb. Thus, the main work is to show the converse. Under the stronger assumption of Meyer property (instead of only finite local complexity) the result is already known. It can be found in the recent work [START_REF] Strungaru | On the spectrum of a Meyer set[END_REF] and could also rather directly be inferred from the earlier [START_REF] Baake | A characterization of model sets by dynamical systems[END_REF] (see [START_REF] Lenz | Pure point diffraction and cut and project schemes for measures: The smooth case[END_REF] as well for results in a very similar spirit).

Our approach to the second point relies on three ingredients: The first ingredient is the theory of equicontinuous systems discussed in Section 2. We say that a Delone set is equicontinuous if its associated dynamical system is equicontinuous. Equicontinuity of a dynamcial system is equivalent to the system being a rotation on a compact abelian group (Ellis theorem, see Theorem 2.4). A second ingredient are results of [START_REF] Lenz | Pure point diffraction and cut and project schemes for measures: The smooth case[END_REF] discussed in Section 3 giving that a measure dynamical system is a rotation on a compact abelian group if and only if the measures are strongly almost periodic (Theorem 3.3). Combining these two pieces for Delone dynamical systems, we find that strong almost periodicity of the associate Dirac comb is equivalent to equicontinuity of the system (Theorem 4.2). Now, the third ingredient is a variant of a recent reasoning of Barge/Olimb [START_REF] Barge | Asymptotic structure in substitution tiling spaces[END_REF] presented in Section 5. It gives that equicontinuity of a Delone dynamical system with finite local complexity is in fact equivalent to complete periodicity (Theorem 5.5). Put together these considerations easily give the desired result.

We present both, the theory of equicontinuous systems as well as the considerations on strongly almost periodic Delone dynamical systems in somewhat more detail than needed for the actual answer to the question of Lagarias. The reason is that we believe that these considerations may be of importance for future study as well.

Preliminaries on dynamical systems

We deal with compact dynamical systems whose acting group is locally compact abelian. An important special case are dynamical systems consisting of measures on the group. The necessary notation is introduced in this section.

The space of continuous functions on a topological space X is denoted by C(X) and the subspace of continuous functions with compact support by C c (X) and the space of continuous bounded functions by C b (X). We will deal with spaces X which are locally compact σ-compact Hausdorff spaces.

Then, X carries the Borel σ-algebra generated by all closed subsets of X and by the Riesz-Markov representation theorem (see e.g. [START_REF] Pedersen | Analysis Now[END_REF]), the set M(X) of all complex regular Borel measures on X can be identified with the dual space C c (X) * of complex valued, linear functionals on C c (X) which are continuous with respect to a suitable topology (see e.g. [START_REF] Pedersen | Analysis Now[END_REF]Ch. 6.5] for details). We then write

X f dµ = µ(f ) for f ∈ C c (X)
. The space M(X) carries the vague topology, which is the weakest topology that makes all functionals µ → µ(ϕ), ϕ ∈ C c (X), continuous.

The group operation of an locally compact abelian group will mostly be written additively as +. The neutral element will be denoted by e. Now, let G be a σ-compact locally compact abelian group. Whenever G acts on the compact space X by a continuous action

α : G × X -→ X , (t, ω) → α t ω ,
where G × X carries the product topology, the pair (X, G) is called a topological dynamical system over G. We will mostly suppress the action α in our notation and write

t • ω := α t ω. A dynamical system (X, G) is called minimal if, for all ω ∈ X, the G-orbit {t • ω : t ∈ G} is dense in X.
Definition 1.1. (Factor) Let two topological dynamical systems (X, G) and (Y, G) under the action of G be given. Then, (Y, G) is called a factor of (X, G), with factor map ̺, if ̺ : X -→ Y is a continuous surjection with ̺(t • ω) = t • (̺(ω)) for all ω ∈ X and t ∈ G.

We will be concerned with special dynamical systems in which X is a compact group. In order to simplify the notation, we introduce the following notation for these systems. Definition 1.2. (Rotation on a compact abelian group) A dynamical system (X, G) is called a rotation on a compact abelian group if X is a compact abelian group and the action of G on X is induced by a homomorphism ι : G -→ X such that t • x = ι(t) + x for all t ∈ G and x ∈ X.

Equicontinuous systems and proximality

In this section we recall some aspects of the theory of equicontinuous systems and proximality. The importance of the proximality relation for the dynamical systems defined by tilings or Delone sets seems only to have recently been emphazised. In [START_REF] Barge | Proximality in Pisot tiling spaces Fund[END_REF] it is shown that topological closure of the proximality relation is a necessary and sufficient condition for (strong) Pisot substitution tilings to have pure point dynamical spectrum. This has been generalized in [START_REF] Barge | Proximality in tiling spaces[END_REF]. Related to proximality are the notions of asymptotic tilings and augmentation which have led to finer topological invariants for substitution tilings [START_REF] Barge | Augmented dimension groups and ordered cohomology[END_REF]. Whereas the above mentionned work concerns the case in which the proximality relation is non-trivial, we consider here the opposite case, namely when this relation is trivial.

We refer the reader to Auslander's book [START_REF] Auslander | Minimal Flows and their Extensions[END_REF] for more detailled information. The theory is cast within the framework of uniform structures instead of more conventional (but more restrictive) metric topologies.

A uniform structure U , see [START_REF] Kelley | General Topology[END_REF] for details, is a family of subsets of X × X satisfying

(1) the diagonal ∆ = {(x, x) : x ∈ X} is contained in all U ⊂ U , (2) with U any superset V ⊃ U is contained in U , (3)
with U and V also U ∩ V belong to U , (4) with U also {(y, x) : (x, y) ∈ U } belongs to U , (5) for all U ∈ U exists V ∈ U such that, whenever (x, y), (y, z) ∈ V , then (x, z) ∈ U .

A uniform structure U generates as unique toplogy such that for each x ∈ X the sets V x := {y : (x, y) ∈ V } (for V ∈ U ) are a neighborhood basis for x.

A base of a uniformity structure U is any subfamily U ′ ⊂ U such that all members of U are supersets of some menber of U ′ .

Uniform structures have already been used in tiling theory. In fact, Schlottmann develops basic theory of points sets (with finite local complexity) on locally compact abelian groups in the framework of uniform structures in [START_REF] Schlottmann | Generalized model sets and dynamical systems[END_REF]. For arbitrary point sets a discussion can be found in [START_REF] Baake | Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra[END_REF] (see [START_REF] Baake | A characterization of model sets by dynamical systems[END_REF] as well). Note also that the construction of a cut and project scheme out of a suitable autocorrelation measure in [START_REF] Baake | Weighted Dirac combs with pure point diffraction[END_REF] is based on uniform structures.

The reader who is not familiar with uniform structures may at first consider the case of a metric space whose uniformity structure has a base given by the ǫ-diagonals ∆ ǫ := {(x, y) ∈ X × X : d(x, y) < ǫ}. A map f : X → X is uniformly continuous (with respect to a given uniformity structure U of X) if for all V ∈ U we find U ∈ U such that (f × f )(U ) ⊂ V . A family of maps F from X to itself is called equicontinuous if for all V ∈ U we find U ∈ U such that for all f ∈ F: (f × f )(U ) ⊂ V . These notions reduce to the conventional ones in a metric space provided one used the fundamental system given by the ǫ-diagonals. On a compact space all uniformity structures coincide.

Consider a minimal dynamical system (X, G) where X is a compact Hausdorff space and G a locally compact abelian group acting by α on X. If the action is free then X can be seen as a compactification of G: it is the completion of one orbit and this orbit is a copy of G. One might ask when is X a group compactification, that is, when does X carry a group structure such that the orbit is a subgroup isomorphic to G, or, in other words, when is (X, G) a rotation on a compact abelian group? The question of when this happens is to the equicontinuity of the action.

Definition 2.1. The dynamical system (X, G, α) is called equicontinuous if the family of homeomorphisms {α t } t∈G is equicontinuous.

This reduces to the usual definition if (X, d) is a metric space:

∀ǫ∃δ∀x, y ∈ X : d(x, y) < δ ⇒ ∀t ∈ G : d(α t (x), α t (y)) < ǫ.
An equicontinuous metrizable system admits always an invariant metric which induces the same topology. Indeed one can just take

d(x, y) := sup t∈G d(t • x, t • y).
Likewise, any compact metrizable abelian group T admits a left invariant metric: Whenever d is a metric then d(x, y) := sup t∈T d(x + t, y + t) is a metric on T which induces the same topology and is G-invariant.

We next provide a useful necessary condition for a system to be equicontinuous.

Definition 2.2 (Proximality)

. Consider a G action on a compact Hausdorff space X. Two points x, y ∈ X are proximal if for all U ∈ U there exists a t ∈ G such that (t • x, t • y) belongs to U .

In the context of compact metric spaces (X, d) this translates into saying that x, y are proximal whenever

inf t∈G d(t • x, t • y) = 0. Corollary 2.3. If (X, G) is equicontinuous then the proximal relation is trivial.
Moreover, a system with non-trivial proximality relation cannot carry an invariant metric which is compatible with the topology. An equicontinuous dynamical system need not be minimal but a transitive equicontinuous dynamical system is always minimal. So in the context of Delone and tiling dynamical systems equicontinuous systems are always minimal. The following theorem due to Ellis is of great importance for our results. Recall that we suppose that (X, G) is minimal.

Theorem 2.4. (X, G) is conjugate to a minimal rotation on a compact abelian group if and only if it is equicontinuous.

If (X, G) is equicontinuous the group structure on X arises as follows: given any point x 0 ∈ X the operation

t 1 • x 0 + t 2 • x 0 := (t 1 + t 2 ) • x 0
extends to an addition in X so that X becomes a group with x 0 as neutral element. Conversely any rotation on a compact abelian group is obviously equicontinuous.

To present the following characterization of equicontinuity in the context of metrizable dynamical systems we need some further notation: Let (X, G) be a dynamical system and d a metric on X. Then, the ǫ-ball around x ∈ X is denoted by B ǫ (x). The elements of

R(x, ǫ) := {t ∈ G : t • x ∈ B ǫ (x)}
are called return vectors to B ǫ (x). Now, (X, G) is called uniformly almost periodic if, for any ǫ > 0 the joint set of return vectors to ǫ-balls, given by

A = x∈X R(x, ǫ)
is relatively dense (i.e. there exists a compact K with A + K = G).

Theorem 2.5 ([1]). (X, G) is equicontinuous if an only if it is uniformly almost periodic.

Almost periodic measures on locally compact abelian groups

We will be concerned with dynamical systems built from measures. We will show that equicontinuity of such a system is equivalent to almost periodicity of the underlying measure (Theorem 3.3). This provides a characterization of equicontinuity in this framework. The considerations of this section can be understood as a (slight) reformulation of results obtained in [START_REF] Lenz | Pure point diffraction and cut and project schemes for measures: The smooth case[END_REF]. Alternatively, they could -at least partly -be based on Theorem 2.5. A thorough study of measure dynamical systems in the framework of aperiodic order and diffraction can be found in [START_REF] Baake | Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra[END_REF][START_REF] Baake | Deformation of Delone dynamical systems and pure point diffraction[END_REF][START_REF] Lenz | Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks[END_REF] to which we refer for further details.

Let G be a locally compact σ-compact abelian group. A measure ν ∈ M(G) is translation bounded if for some (and then all) open non empty relatively compact set V in G there exists a C = C V ≥ 0 with

|ν|(t + V ) ≤ C
for every t ∈ G. Here, |ν| is the total variation measure of ν. The set of all translation bounded measures is denoted by M ∞ (G). As a subset of M(G), it carries the vague topology. There is an obvious action of G on M ∞ (G), given by

G× M ∞ (G) -→ M ∞ (G) , (t, ν) → α t ν with (α t ν)(ϕ) := ν(δ -t * ϕ) for ϕ ∈ C c (G). Here, δ t denotes the unit point mass at t ∈ G and the convolution ω * ϕ between ϕ ∈ C c (G) and ω ∈ M ∞ (G) is defined by ω * ϕ(s) := ϕ(s -u) dω(u).
It is not hard to see that this action is continuous when restricted to a compact subset of M ∞ (G) (see e.g. [START_REF] Baake | Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra[END_REF]).

Definition 3.1. (X, G) is called a dynamical system on the translation bounded measures on G (TMDS) if X is a compact α-invariant subset of M ∞ (G).
Every translation bounded measure ν gives rise to a (TMDS) (X(ν), G), where X(ν) := {α t ν : t ∈ G}. In fact, if ν satisfies the inequality |ν|(t + V ) ≤ C for some nonempty open relatively compact V in G and all t ∈ G, this inequality will be true for all µ ∈ X(ν) and this shows the desired compactness by Theorem 2 of [START_REF] Baake | Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra[END_REF].

As

usual ϕ ∈ C b (G) is called almost periodic (in the sense of Bohr) if, for every ǫ > 0, the set of t ∈ G with δ t * ϕ -ϕ ∞ ≤ ǫ is relatively dense in G.
By standard reasoning this is equivalent to {δ t * ϕ : t ∈ G} being relatively compact in in C b (G) (see e.g. [START_REF] Zaidman | Almost Periodic Functions in Abstract Spaces[END_REF]). Definition 3.2. A translation bounded measure ν is called strongly almost periodic if ν * ϕ is almost periodic (in the Bohr sense) for every ϕ ∈ C c (G).

To deal with almost periodic functions it is useful to introduce the strong topology. To do so, we recall that for ν ∈ M ∞ (G) and ϕ ∈ C c (G), the convolution ν * ϕ belongs to C b (G). The space C b (G) is equipped with the supremum norm. The strong topology is the weakest topology on M ∞ (G) such that all maps

M ∞ (G) -→ C b (G), ν → ν * ϕ, are continuous. Theorem 3.3. Let G be a locally compact σ-compact abelian group. Let ν ∈ M ∞ (G) be given.

The following assertions are equivalent:

(i) The measure ν is strongly almost periodic (i.e. ν * ϕ is Bohr almost periodic for every ϕ ∈ C c (G)). (ii) {α t ν : t ∈ G} is relatively compact in the strong topology. (iii) The topological space X(ν) is a topological group with addition ∔ satisfying α s ν ∔ α t ν = α s+t ν for all s, t ∈ G. (iv) The dynamical system (X(ν), G) is a rotation on a compact abelian group. (v) The hull (X(ν), G) is an equicontinuous dynamical system.

Proof. The equivalence of (i), (ii) and (iii) is shown in Lemma 4.2 of [START_REF] Lenz | Pure point diffraction and cut and project schemes for measures: The smooth case[END_REF]. The equivalence of (iii) and (iv) is immediate from the definition of rotation on a compact abelian group. The equivalence of (iv) and (v) is Theorem 2.4.

Factors of equicontinuous dynamical systems are equicontinuous as well.

In our context the following variant will be of use in Section 6.

Corollary 3.4. Let ν be a translation bounded measure on G. Then, the following assertions are equivalent:

(i) The measure ν is strongly almost periodic.

(ii) There exists a rotation of a compact abelian group (T, G) together with a continuous map

π : T -→ M ∞ (G) with π(e) = ν and π(t•ξ) = t • π(ξ) for all ξ ∈ T and t ∈ G.
In this case π(T) = X(ν) and π is a factor map as well as a group homomorphism.

Proof. In the situation of (ii) we must have that π is a factor map with π(T) = X(ν) as T is compact and π is continuous. Now the equivalence of (i) and (ii) follows from Theorem 5.1 of [START_REF] Lenz | Pure point diffraction and cut and project schemes for measures: The smooth case[END_REF]. The fact, that π is a group homomorphism now follows from Lemma 5.2 of [START_REF] Lenz | Pure point diffraction and cut and project schemes for measures: The smooth case[END_REF].

Remark. The almost periodic measure dynamical systems considered above are of interest in the study of diffraction. In fact, they exhibit pure point diffraction and pure point dynamical spectrum with continuous eigenfunctions. Details can be found in [START_REF] Lenz | Pure point diffraction and cut and project schemes for measures: The smooth case[END_REF] (see [START_REF] Baake | A characterization of model sets by dynamical systems[END_REF][START_REF] Lenz | Pure point spectrum for measure dyamical systems on locally compact Abelian groups[END_REF][START_REF] Lenz | Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks[END_REF] for related arguments on diffraction as well). For a further study of a particular subclass of systems and a characterization in the context of cut and project schemes we refer the reader to [START_REF] Strungaru | [END_REF].

Application to Delone dynamical systems

The previous sections dealt with first general dynamical systems and then measure dynamical systems. In this section we specialize even further and consider Delone dynamical systems. These systems are made of suitable uniformly discrete subsets of the underlying group. Such subsets can be considered as translation bounded measures in a canonical way. We will introduce the necessary notation to deal with these systems and to state Lagarias question. We will then go on and characterize what it means for such a system to be equicontinuous in Theorem 7.1. This provides a main step in our dealing with the Lagarias question. Our strategy to view Delone sets as measures has the advantage that it allows us to treat colored Delone sets with no extra effort. This is discussed at the end of the section.

Let G be a locally compact abelian group. We will deal with subsets L of G. A subset L of G is called uniformly discrete if there exists an open neighborhood U of the identity in G such that

(x + U ) ∩ (y + U ) = ∅ for all x, y ∈ L with x = y. A subset L of G is called relatively dense if there exists a compact neighborhood K of the identity of G such that G = x∈L (x + K).
A subset L of G is called a Delone set if it is both uniformly discrete and relatively dense. Any uniformly discrete set L (and hence any Delone set) in G can naturally be identified with the translation bounded measure

δ L := x∈L δ x ,
where δ x denotes the unit point may at x ∈ G. The hull of X(δ L ) in the vague topology then consists of measures of the form δ M with M ⊂ G uniformly discrete. In the sequel we will identify such measures with the underlying sets and then also write X(L) instead of X(δ L ). We call (X(L), G) a Delone dynamical system if L is a Delone set. As discussed e.g. in [START_REF] Baake | Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra[END_REF] the topology on X inherited in this way from the vague topology on the measures can also be obtained from a uniform structure on the set P of all uniformly discrete point sets on G. Namely, for K ⊂ G compact and V a neighbourhood of e in G, we set

U K,V := {(L 1 , L 2 ) ∈ P × P : L 1 ∩ K ⊂ L 2 + V and L 2 ∩ K ⊂ L 1 + V }.
Then, the set U consisting of all U K,V with K ⊂ G compact and V a neighborhood of e ∈ G, will provide a uniform structure. The induced topology is called the local rubber topology in [START_REF] Baake | Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra[END_REF] and shown to agree with the topology induced by the vague topology on measures.

Besides the hull of L we will also need the canonical transversal given by

Ξ(L) = {L ′ ∈ X : e ∈ L}.
This set can easily be seen to be the closure of the set {Lx : x ∈ L}.

For this reason it is also sometimes denoted as the discrete hull of L. Of course, the discrete hull of L is a subset of the hull X and as such inherits a topology. Whenever L is a subset of G a set of the form (-

x + L) ∩ K with x ∈ L and K compact is called a patch of L. A set L in G is said to have finite local complexity (FLC) if for any compact K in G the set {(-x + L) ∩ K : x ∈ L}
is finite. This just means that there are only finitely many patches for fixed 'size' K. The relevance of sets with finite local complexity in our setting comes from the following feature of the topology. Lemma 4.1. Let G be a locally compact abelian group. Let L be a Delone set with finite local complexity in G, X = X(L) its hull and Ξ(L) its discrete hull. The unique uniform structure on the discrete hull compatible with the relative topology has a fundamental system given by the subsets

U C := {(L ′′ , L ′ ) ∈ Ξ × Ξ : L ′′ ∩ C = L ′ ∩ C} for C ⊂ G compact. In particular, if (L ι
) is a net in Ξ and L ′ belongs to X, then the following assertions are equivalent:

(i) L ι → L ′ . (ii) For any compact C in G there exists an ι C with L ι ∩ C = L ′ ∩ C for all ι > ι C .
Proof. This is well known and appears -at least implicitly -in e.g. [START_REF] Schlottmann | Generalized model sets and dynamical systems[END_REF][START_REF] Baake | Weighted Dirac combs with pure point diffraction[END_REF][START_REF] Baake | A characterization of model sets by dynamical systems[END_REF]. We sketch a proof for completeness: Let a compact C in G be given. By definition of Ξ, we can assume without loss of generality that C contains the point e. Then, by finite local complexity, there are only finitely many possibilities for L ′ ∩ C (for L ′ ∈ Ξ). This implies that for any sufficiently small neighbourhood V of e ∈ G, the inclusions

L 1 ∩ C ⊂ L 2 + V and L 2 ∩ C ⊂ L 1 + V can only hold if L 1 ∩ C = L 2 ∩ C
. This shows that the U C , C ⊂ G compact, form indeed a basis of the uniformity. Now, the last statement characterizing convergence is immediate.

An occurrences of the patch (-x + L) ∩ K in a Delone set L is an element of {y ∈ L : (-x + L) ∩ K ⊂ (-y + L)}. A Delone set L is called repetitive if for any patch the set of occurrences is relatively dense. For FLC Delone sets this condition is equivalent to minimality of the associated system (X(L), G), but for non-FLC this is no longer the case.

For any uniformly discrete subset L of G and any ϕ ∈ C c (G) we can define the function

f L,ϕ : G -→ C, f L,ϕ (t) = x∈L ϕ(t -x).
From these definitions and Theorem 3.3, we immediately obtain the following theorem. (ii) The measure δ L is strongly almost periodic.

(ii) The hull (X(L), G) is a compact abelian group with neutral element L and group addition satisfying (t + L) + (s + L) = (s + t + L) for all t, s ∈ G. (iv) The dynamical system (X(L), G) is equicontinuous.

We finish this section by discussing how the above above considerations can be carried over to colored subsets of groups: Fix a finite set C. We think of elements of C as colors. A colored set (with colors from C) on the locally compact abelian group G is a pair (L, c) consisting of a subset L of G and a map c : L -→ C. The set L is called the support of the colored set. A colored set is called uniformly discrete, relatively dense, and Delone respectively if and only if its support has the corresponding property. A colored patch (with colors from C) in a colored set (L, c) is a pair (P, c P ) consisting of a patch P in L and the restriction c P of c to P . The notion of finite local complexity then carries directly over to colored sets. As there are only finitely many colors a colored set (L, c) has finite local complexity if and only if L has finite local complexity. A colored set whose support is uniformly discrete can be identified with a translation bounded measure on the following way: Chose an injective function f : C -→ C. Then, we can associate to (L, c) the measure

δ (L,c) := x∈L f (c(x))δ x .
This allows one to transfer results from measures to colored Delone sets in the same way as they were transferred from measures to Delone sets. In particular, the direct analogue of Theorem 4.2 for colored Delone sets can be shown in this way. We refrain from an explicit statement.

An affirmative answer to the question of Lagarias in the case of finite local complexity

For tilings of Euclidean space Barge and Olimb have given a beautiful argument showing that in any FLC aperiodic repetitive tiling system there exist two distinct tilings which are proximal and hence their dynamical systems are not equicontinuous. We will present (a variant of) their reasoning for Delone sets on rather general locally compact abelian groups.

In short, the arguement of Barge and Olimb contains two steps: First they show that aperiodicity, and repetitivity imply that no finite patch of a tiling can uniquely determine this tiling. We reproduce this step literally in Lemma 5.2. From this it is concluded that there exist distinct tilings which agree on arbitrarily large patches and hence are proximal. We will follow a somewhat different argumentation which applies to Delone sets on compactly generated locally compact abelian groups (Lemma 5. The Delone set L is called completely periodic or crystalline if the set of its periods is relatively dense in G. A Delone set is called aperiodic if it is not completely periodic.

We say that a patch of a Delone set forces the whole Delone set if any Delone set in its hull which contains the patch at the same place is equal to the original Delone set. Lemma 5.2. Let L be a repetitive Delone set in a locally compact abelian group G, which is not completely periodic. Then, it is not forced by any of its patches.

Proof. Assume the contrary. Then, there exists a patch forcing the Delone set L. This means that all occurrences of this patch are periods of the Delone sets. As the Delone set is repetitive, this means that the set of its periods is relatively dense. This is a contradiction (as L is not completely periodic).

So far, we did not need any restrictions on the geometry of the underling group. For the next lemma we will need a further restriction viz the group G needs to be compactly generated. This means that there exists a compact neighborhood W of the identity, such that the smallest subgroup of G containing W is in fact G. This condition is clearly met for G = R n . Obviously, any compactly generated group is σ-compact.

Lemma 5.3. Let G be a locally compact, compactly generated group. Let L be a Delone set in G. Then, there exists a compact set C in G with the following property: For all

L 1 , L 2 in X(L) with e ∈ L 1 ∩ L 2 and L 1 = L 2 there exists an v ∈ L 1 ∩ L 2 with (-v + L 1 ) ∩ C = (-v + L 2 ) ∩ C.
Proof. By the main structure theorem on locally compact, compactly generated abelian groups, the group G has the form G = R n × Z f × H with a compact group H and non-negative integer numbers n and f . For r ≥ 0 let B r and U r be the closed and open balls respectively with radius r around the origin in R n × Z f (with respect to the usual Euclidean distance). As L is a Delone set, there exists an R > 0 such that

L ′ ∩ ((t + B R ) × H) = ∅ for any t ∈ R n × Z f and L ′ ∈ X(L).
To L 1 = L 2 in X(L) with e ∈ L 1 ∩ L 2 we can now define

r M := inf{r ≥ 0 : L 1 ∩ (B r × H) = L 2 ∩ (B r × H)}.
From L 1 = L 2 we obtain r M < ∞.

We distinguish two cases:

Case 1: r M = 0: Then, L 1 ∩ ({0} × H) = L 2 ∩ ({0} × H).
Here, 0 is the origin of R n × Z f . Thus, any compact set C containing {0} × H will do in this case (with v = e ∈ G).

Case 2: r M > 0: In particular, n or f are non-zero. Without loss of generality we can assume r M > 2R + ǫ for some ǫ > 0 which we determine later (as otherwise, any compact C containing B 2R+ǫ × H will do (with v = e ∈ G)). Now, by definition of r M we can find a t = (s, h) ∈ (B r M \U r M )×H which belongs to only one of L 1 and L 2 . Without loss of generality we assume that t belongs to L 1 but not to L 2 . On the other hand, (U

r M × H) ∩ L 1 = (U r M × H) ∩ L 1 ∩ L 2 . So the idea is to find a point v = (s ′ , h ′ ) ∈ (U r M × H) ∩ L 1 such
that the distance between s and s ′ is bounded by a constant c which depends only on R, n and f and then choose C = B c × H.

To see that this is possible suppose first that f = 0. Let ŝ be the unit vector in the direction of s. The R ball around

x = (r M -R -ǫ)ŝ lies in U r M . By definition of R it contains a point s ′ such that v = (s ′ , h ′ ) ∈ L 1 .
Clearly ss ′ ≤ 2R + ǫ and so we can take any ǫ > 0. Now if f = 0 then (r M -Rǫ)ŝ might not lie in R n × Z f . So we take the nearest point to it in R n × Z f such that its R-ball is contained in U r M . This requires ǫ to be larger than √ f .

The above considerations show that we can choose

C = B 2R+ √ f +1 × H. This finishes the proof.
Remark. Note that the preceding lemma neither assumes that the Delone set in question in repetitive nor that it has finite local complexity.

Corollary 5.4. Let G be a compactly generated group and L an repetitive aperiodic Delone set in G with finite local complexity. Then, (X(L), G) is not equicontinuous.

Proof. Assume the contrary i.e. assume that (X(L), G) is equicontinuous. Then the induced partial action of G on Ξ(L) is equicontinuous. By Lemma 4.1 this means that for any compact C ⊂ G there exists a compact C ′ such that for all

L 1 , L 2 in Ξ(L), L 1 ∩ C ′ = L 2 ∩ C ′ implies (-v + L 1 ) ∩ C = (-v + L 2 ) ∩ C for all v ∈ L 1 ∩ L 2 .
Thus, if we chose C as in the previous lemma we obtain that

L 1 = L 2 whenever L 1 ∩ C ′ = L 2 ∩ C ′ .
In particular, the patch P := L 1 ∩ C ′ forces the Delone set L. By Lemma 5.2, we then obtain that L is completely periodic. This is a contradiction.

As a consequence of this corollary, we obtain the following characterization of equicontinuous Delone dynamical systems with finite local complexity.

Theorem 5.5. Let G be a compactly generated group and L a Delone set in G with finite local complexity. Then, the following assertions are equivalent:

(i) L is completely periodic. (ii) (X(L), G) is equicontinuous.
Proof. The implication (i)=⇒ (ii) is clear (and does not depend on FLC). It remains to show the implication (ii)=⇒ (i). By the previous Corollary 5. [START_REF] Baake | A characterization of model sets by dynamical systems[END_REF], it suffices to show that L is repetitive. This, in turn, follows from general principles as each transitive equicontinuous system must be minimal. We include the short proof for the convenience of the reader: By general principles, there will exist a minimal component (Y, G) of (X(L), G). Let L ′ be an element of Y . Now, as L ′ belongs to the hull of L, there exists a net (t i ) in G with t i + L converging to L ′ . By the equicontinuity assumption (ii), we infer that then -t i + L ′ must converge to L. This shows that L belongs to Y and hence is repetitive.

Remarks. (a)

The theorem can also be phrased in terms of the proximality relation.

(b) The preceding theorem can be combined with Theorem 4.2 to give various further characterizations of complete periodicity of a Delone set with finite local complexity. We single out one of these in the next corollary.

The previous theorem allows us to give an affirmative answer to a question of Lagarias (under the additional assumption of finite local complexity):

Corollary 5.6. Let G be a compactly generated group and L a Delone set in G with finite local complexity. Then, the following assertions are equivalent:

(i) L is completely periodic.

(ii) δ L is strongly almost periodic.

Proof. Combine the previous theorem with Theorem 4.2.

Remark. It is not difficult to see that Lemma 5.2 and Lemma 5.3 remain true for colored Delone sets. This allows one to obtain an analogue of the preceding corollary for colored Delone sets. We leave the details to the reader.

A negative answer to the question of Lagarias for systems without finite local complexity

In this section we show that Lagarias question does not have an affirmative answer if the assumption of finite local complexity is dropped. More precisely, we provide examples of strongly almost periodic Delone sets on the real line which are not crystalline. The class of examples we give can clearly be generalized to arbitrary Euclidean spaces (and even to more general locally compact abelian groups). It may be of interest in further contexts as well. For this reason we give a characterization in the one dimensional situation discussed below.

Our examples arise as factors of the Kronecker flow on the two-dimensional torus. By Corollary 3.4 such a factor automatically yields strongly almost periodic measures. These measures will be given by Dirac combs of certain Delone sets. Thus, it suffices to show that the Delone sets in question are not crystalline. This will be clear from the construction. Here are the details: Let T : R 2 /Z 2 be the two dimensional torus and p : R 2 -→ T, p((x, y)) := (x, y) + Z 2 be the canonical projection. Fix an irrational θ ∈ (0, 1) and define

h = h θ : R -→ R 2 , h(t) = (t, θt) and ι = ι θ : R -→ T, ι = p • h θ .
Then, ι induces an action of R on T via t • ξ := ι(t) + ξ and in this way (T, R) becomes a rotation on a compact abelian group, known as Kronecker flow (with slope θ). Let now

g : [0, 1] -→ R 2
be a continuous curve with the properties:

(1) g is everywhere transversal to the direction defined by the line im(h). More precisely, there exists ǫ > 0 such that for all (x, y) ∈ im(g) we have (x, y) + h(t) ∈ im(g) only if t = 0 or |t| > ǫ.

(2) g(1)g(0) ∈ Z 2 . In particular Γ := Z 2 + im(g) is the image of a closed connected curve on the torus which is transversal to the Kronecker flow. We also suppose that (3) Γ does not intersect itself. For example, any continuous strictly monotone g : [0, 1] -→ [0, 1] with g(0) = 1 and g(1) = 0 will satisfy these assumptions (1), ( 2) and ( 3). Now, we define the set

L Γ,θ = {t ∈ R : (t, θt) ∈ Γ } = h -1 (Γ ).
Corollary 6.1. Let g be non-linear and θ ∈ (0, 1) irrational. Then, L Γ,θ is Delone set with a strongly almost periodic Dirac comb which is not completely periodic.

Proof. The construction directly gives that L Γ,θ is a Delone set: Since Γ is a closed curve transversal to the Kronecker flow the set L Γ,θ is relatively dense and since Γ does not intersect itself L Γ,θ is uniformly discrete.

We now show that complete periodicity of L Γ,θ implies that g is linear: Assume without loss of generality that 0 belongs to L Γ,θ . Then, the orbit P of 0 under the set of periods of L Γ,θ is just this set of periods and hence a group. Let S ′ be the image of P under the map ι. Then, S ′ is a subgroup of T. Furthermore by construction S ′ is contained in the compact image of im(g) under p. Denote the closure of S ′ by S. Then, S is again is a subgroup and contained in the image of im(g) under p. Hence, S can not be T. On the other hand, as θ is irrational, the subgroup S of T must contain an accumulation point. Thus, S can not be discrete. Thus, S must be a circle. The image of g under p must then contain this circle. As this image is connected it must then agree with this circle. Hence, g is linear.

It remains to show the claim on strong almost periodicity: Define the map π from R 2 to the measures on R via π(x, y) := t:(x,y)+h(t)∈Γ δ t .

In particular, π(0, 0) = δ L Γ,θ . Then, π can easily be seen to have the following three properties:

• π(x, y) is translation bounded (as Γ is transversal to the Kronecker flow). • π is continuous (as Γ is a continuous closed curve).

• π((x, y)) = π((x, y) + (n, m)) for all (x, y) ∈ R 2 and (n, m) ∈ Z 2 (as our construction is invariant under shifts by Z 2 ). • π(h(t) + (x, y)) = α -t π(x, y) (as follows from a direct computation).

Thus, π induces a continuous map π : T → M ∞ (R) with π(t • ξ) = α -t π(ξ) for all ξ ∈ T and t ∈ R. By Corollary 3.4, the measure π((0, 0) + Z 2 ) = δ L Γ,θ is then strongly almost periodic. In this way, we have therefore constructed a Delone set L Γ,θ whose Dirac comb is strongly almost periodic.

To complete the above discussion we determine the dynamical system (X, R) of L Γ,θ , or, what is the same, δ L Γ,θ . By Corollary 3.4 π is a group homomorphism onto its image and this image is X. Let K = ker π. Then (X, R) is the rotation on T/K induced by the Kronecker flow.

Clearly π((x, y) + (k 1 , k 2 )) = π((x, y)) if and only if (k 1 , k 2 ) + Γ = Γ i.e. K is the stabiliser of Γ . Given that K is a closed subgroup of T 2 we can distinguish essentially two possibilities depending on whether it is discrete or not. As Γ is connected and non intersecting K must be a cyclic sub-group of the torus in the discrete case, or a circle in the other case.

If K is not discrete it actually coincides with a translate of Γ . It then must intersect the line im(h) more then once. It follows that the restriction of π to the line (which corresponds to the orbit of ν) is not injective. This implies that ν (and L Γ,θ ) are completely periodic and so π maps its orbit to S 1 . Since S 1 is closed, X = S 1 in this case. If K is discrete then X is a two dimensional torus again and consequently ν is aperiodic.

Remarks. (a) Since the Kronecker flow is dense in T 2 distinct subgroups K yield systems which are not topologically conjugate. The finite cyclic subgroups of T 2 thus classify the possible hulls of aperiodic Delone dynamical systems up to conjugacy which arise as factors of the Kronecker flow in the above manner. Likewise this classification is given for periodic Delone sets by the sub-groups of T 2 which are circles.

(b) If one drops the requirement that g is continuous or allows for g(1)g(0) = Z 2 , then the above analysis completely breaks down. In fact, the Sturmian sequences associated to θ are obtained if one takes im(g) to be the closed interval in the orthocomplement of im(h) which is obtained by projecting (orthogonally) the unit cube onto that orthocomplement. The resulting Delone set is aperiodic and has FLC, and is thus not uniformly almost periodic!

A further look at Delone sets in Euclidean space

Delone sets of Euclidean spaces are of particular importance for the theory both from the point of view of geometry and of physics. Thus, we have a closer look at these in this section. The main advantage is that the space of all Delone sets of a given Euclidean space carries a well-known metric and with the help of this metric we can formulate more directly what it means for such a Delone set to be equicontinuous.

Let ρ be the standard Euclidean metric on R n i.e.

ρ(x, y) =   n j=1 |x j -y j | 2   1/2 .
Denote the closed ball (with respect to this metric) around the origin with radius r by B r and the open ball with U r . Then, a subset L of R n is a uniformly discrete if there exists an > 0 with (x + U r ) ∩ (y + U r ) = ∅ for all x, y ∈ L with x = y. The largest such r is denoted by r min and called the packing radius of L. A subset L of R n is relatively dense if there exists an r > 0 with

R n = x∈L (x + B r ).
The smallest such r is denoted by r max and called the covering radius of L.

The intersection of B r with a set L is denoted by B r [L] and called an r-patch. The Hausdorff distance on compact subsets of R n is denoted by d H . It is not hard to see that then

(7.1) d(L 1 , L 2 ) := sup{r > 0 : d H (B r [L 1 ] ∪ ∂B r , B r [L 2 ] ∪ ∂B r ) ≤ 1 r }
gives a metric on the set of all Delone sets and that the induced topology agrees with the topology induced by considering the Delone sets as measures [START_REF] Baake | Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra[END_REF][START_REF] Baake | Deformation of Delone dynamical systems and pure point diffraction[END_REF][START_REF] Bellissard | Hulls of aperiodic solids and gaplabelling theorems[END_REF][START_REF] Forrest | Topological invariants for projection method patterns[END_REF][START_REF] Lenz | Delone dynamical systems and associated random operators[END_REF]]. We will be concerned with dynamical systems (X, R n ) with X = X(L) arising as the hull of a Delone set L. Recall from Section 4 that the canonical transversal or discrete hull of L denoted by Ξ := Ξ(L) is defined by

Ξ := {L ′ ⊂ X(L) : 0 ∈ L ′ }.
The metric d allows us now to use some notions introduced in Section 2 for such dynamical systems (X(L), R n ). This is discussed next: For a Delone set L, the set of r return vectors is given by

R(L, r) = {a ∈ R n : d(L, -a + L) ≤ 1 r } = {a ∈ R n : d H (B r [L] ∪ ∂B r , B r [L -a] ∪ ∂B r ) ≤ 1 r }.
(If the set R(L, r) is is relatively dense for all r > 0 the set L is called rubber repetitive. For Delone sets with finite local complexity, of course, rubber repetitivity coincides with repetitivity. In general, rubber repetitive is equivalent to minimality of the dynamical system.) A Delone set L is now uniformly almost periodic if for all r > 0 the set

A := L ′ ∈X(L) R(L ′ , r)
is relatively dense. Here comes our theorem: (i) ⇔ (ii). Clearly it suffices to consider large r. If |x| ≤ r max and r is large then R(Lx, r) ⊂ R(L, rr max ). It follows that L ′ ∈X(L) R(L ′ , r) is relatively dense (for all r > 0) if and only if L ′ ∈Ξ(L) R(L ′ , r) is relatively dense (for all r > 0).

(ii) ⇒ (iii). This is clear, since {Lx : x ∈ L} is a subset of Ξ(L).

(iii) ⇒ (ii). As the discrete hull Ξ(L) is the closure of the set {Lx :

x ∈ L}, we can find for any r > 0, L ′ ∈ Ξ(L) and c > 1 an x ∈ L with

d H (B cr [L -x], B cr [L ′ ]) ≤ 1 cr .
Let a ∈ R(Lx, r) and |a| ≤ (c -1)r. Then, for L ′ and x as above,

d H (B r [L ′ -a], B r [L ′ ]) ≤ d H (B r [L ′ -a], B r [L -x -a]) +d H (B r [L -x -a], B r [L -x]) +d H (B r [L -x], B r [L ′ ]) ≤ 3 r .
Thus a ∈ R(L ′ , r 3 ). It follows that x∈L R(Lx, r) ∩ B (c-1)r (0) is contained in L ′ ∈Ξ(L) R(L ′ , r 3 ). Letting c → ∞ allows to conclude the desired statement.

(iii) ⇒ (iv). Let a ∈ R(L, r), that is d(La, L) ≤ 1 r . From (7.1) we conclude that a ∈ L + B 1 holds and the desired statement (iv) follows.

(iv) ⇒ (v). We show that A := x∈L Lx + B ǫ (0) ∩ x∈L -L + x + B ǫ (0) has the desired properties: Let a ∈ A be given. Then, for each x ∈ L there exists an y ∈ L such that a + x ∈ y + B ǫ (0).

If ǫ > 0 is small enough, for instance smaller than r min 2 , then this y is unique and this defines a function f a : L → L, f a (x) = y with |f a (x) -(a + x)| ≤ ǫ. Clearly, for small enough ǫ > 0 such as e.g. 0 < ǫ < r min 3 the function f a is injective. Since A = -A we can also construct f -a which is easily seen to be the inverse of f a .

(v) ⇒ (iii). Let ǫ > 0. Given any r-patch P ⊂ L we have f a (P ) ⊂ L and d H (P + a) ∩ ∂B r (0), f a (P ) ∪ ∂B r (0)) ≤ ǫ. It follows that, if P is an r = 1 ǫpatch of L at x then d H (Px) ∩ ∂B r (0), (f a (P )xa) ∪ ∂B r (0) ≤ 1 r . Now since f a is bijective and ǫ-close to the translation by a it is, perhaps up to an irrelevant difference near ∂B r (f a (x)), the r-patch of L at f a (x) we have d H (f a (P )xa) ∪ ∂B r (0), B r [Lxa] ∪ ∂B r (0) ≤ 1 r and thus a ∈ R(Lx, r 2 ). Remarks. (a) Note that the theorem above has some overlap with Theorem 4.2. Indeed, (v) of the above theorem easily gives the uniform almost periodicity of all functions of the form x∈L ϕ(•x) with ϕ ∈ C c (R n ) and this is statement (i) of Theorem 4.2 in the more restrictive case G = R n .

(b) The theorem can be used to provide a weak version of the so-called Córdoba theorem [START_REF] Córdoba | La formule sommatoire de Poisson[END_REF]. This theorem states that for a uniformly discrete set L of the form L = ∪ m j=1 L m and w 1 , . . . , w m ∈ C the Fourier transform (taken in the sense of tempered distributions) of the generalized Dirac comb m j=1 x∈L j w j δ x is a translation bounded pure point measure if and only if each L j is a lattice. In our context, we can show that for L with finite local complexity the Fourier transform of δ L (taken as a tempered distribution) is a pure point measure if and only if L is completely periodic. Here, the 'if' part is clear from the Poisson summation formula and the 'only if' part follows from the theorem as pure pointedness of the Fourier transform of a measure implies almost periodicity of the measure [START_REF] Gil De Lamadrid | Almost Periodic Measures, Memoirs of the AMS[END_REF].

Theorem 4 . 2 .

 42 Let L be a Delone set in the locally compact σ-compact abelian group G. Then, the following assertions are equivalent:(i) The function f L,ϕ is Bohr-almost periodic for any ϕ ∈ C c (G).

  3 and Corollary 5.4). This will allow us to give a characterization of equicontinuous Delone dynamical systems which have finite local complexity (Theorem 5.5) and yield an affirmative answer to the a question of Lagarias for Delone dynamical systems with finite local complexity (Corollary 5.6). Definition 5.1. Let G be locally compact abelian group and L a Delone set in G. The elements of P := {t ∈ G : t + L = L} are called periods of L.

Theorem 7 . 1 .

 71 Let L be a Delone set in R n . Then, the following assertions are equivalent:(i) L is uniformly almost periodic.(ii) For all r > 0 the setL ′ ∈Ξ(L) R(L ′ , r) is relatively dense. (iii) For all r > 0 the set x∈L R(Lx, r) is relatively dense. (iv) For all ǫ > 0 the set x∈L Lx + B ǫ (0) ∩ x∈L -L + x + B ǫ (0)is relatively dense. (v) For all ǫ > 0 there exists a relatively dense set A such that for all a ∈ A there is a bijectionf a : L → L satisfying |f a (x) -(x + a)| ≤ ǫ. (vi) The dynamical system (X(L), R n ) is equicontinuous.Proof. Denote the covering radius of L by r max and the packing radius by r min .The equivalence (i) ⇔ (vi) is just Theorem 2.5.

r( 0 L -x + B 1 r

 01 ) and hence R(L, r) ⊂ L + B 1 r (0). In particular there exist an ã ∈ L such that |a -ã|< 1 r . From d(La, L) ≤ 1 r we then obtain d(Lã + a, Lã) ≤ |ã -a| + d(L, La) ≤ 2 r . Thus, a ∈ R(L, r) implies -a ∈ R(Lã, r2). Hence x∈L R(Lx, r) ⊂ x∈L

Acknowledgments. Daniel Lenz would like to thank Nicolae Strungaru for most illuminating discussions on almost periodicity for measures and Meyer sets. Johannes Kellendonk would like to thank Marcy Barge for explaining to him the importance of proximality for tiling systems.


Vatsala Nundloll
email: nundloll@comp.lancs.ac.uk

Paul Grace
email: gracep@comp.lancs.ac.uk

Gordon S Blair
email: gordon@comp.lancs.ac.uk

The Role of Ontologies in Enabling Dynamic Interoperability

Keywords: Interoperability, Ontology, Vehicular Ad-Hoc Networks

Advances in the middleware paradigm has enabled applications to be integrated together thus enabling more reliable distributed systems. Although every middleware tries to solve interoperability issues among a given set of applications, nonetheless there still remains interoperability challenges across various middlewares. Interoperability enables diverse systems to work in accordance and extend the scope of services that are provided by individual systems. During an interoperability process, it is imperative to interpret the information exchanged in a correct and accurate manner in order to maintain coherence of data. Hence, the aim of this paper is to tackle this issue of semantic interoperability through an experimental approach using the domain of vehicular ad-hoc networked systems.

Introduction

Middleware technologies have proven a successful solution in ensuring interoperability within distributed systems. However, due to the heterogeneity of applications and environments, a number of different middleware systems have emerged. Due to conflicting standards, communication styles and heterogeneous protocols are unable to interoperate with one another. Solutions to this problem of middleware heterogeneity include bridging [START_REF]COM/CORBA Interworking Specification Part A & B[END_REF]15,[START_REF] Brueckne | SOAP2CORBA[END_REF] and interoperability frameworks [START_REF] Bromberg | INDISS: Interoperable Discovery System for Networked Services[END_REF][START_REF] Flores | An Adaptive Middleware to Overcome Service Discovery Heterogeneity in Mobile Ad Hoc Environments[END_REF][START_REF] Grace | A Reflective Framework for Discovery and Interaction in Heterogeneous Mobile Environments[END_REF]. However, these are insufficient where interoperability is required 'on-the-fly', i.e., where two heterogeneous systems spontaneously encounter one another at runtime, there is a need to automatically learn the middleware solutions employed and then generate a dynamic bridge between the two systems.

CONNECT 1 is a software framework that dynamically generates these middleware mediators and hence provides emergent middleware solutions, which can encounter different networked systems dynamically, enable them to connect together, understand one another and be able to exchange data. Here, a fundamental requirement is the ability to discover and understand the meaning and behaviour of middleware technologies and standards in order to learn and generate the required software. This entails matching or comparison between different middleware in order to identify how they are different, and based upon this synthesize an appropriate adaptive mapping mechanism to underpin the interoperability solution. To achieve such understanding, we advocate the novel use of ontologies that crosscut middleware solutions; this allows us to obtain semantic knowledge of the middleware in order to allow two different systems to effectively be able to interoperate with each other.

In this paper, we present a dynamic interoperability framework that leverages ontologies in order to provide the following two important capabilities:  Classifying Protocols. This involves discovering and defining the type of messaging protocols from each system.

 Matching Protocols. We can observe where the fields of two messages are the same and different to provide the information required to build a bridge. We use a case-study based evaluation that shows we can understand and match communication protocols from Vehicular Ad-hoc Networks (VANETS) domain.

The paper is structured as follows: Section 2 briefly presents a background on ontologies. Section 3 explains the dynamic framework, and Section 4 details the case study. Section 5 then presents an experiment that validates and evaluates the case study results. Finally, Section 6 discusses related work and then Section 7 concludes the paper and briefly outlines our future work.

Background on Ontologies

An ontology is a formal descriptive notation given to concepts that constitute a particular domain. It is a simple but powerful notion used to classify concepts of a domain in the form of a superclass-subclass model and also to define the relationships that exist among these different objects. In addition to making domain assumptions explicit, ontologies also permit reuse and analysis of this domain knowledge. Enabling the classified information regarding a domain to be shared as a common vocabulary across people and applications, ontologies consequently pave the way towards building a supportive infrastructure for information exchange and discovery.

Any kind of domain can be modelled using ontologies, ranging from concrete such as defining a type of bread, to abstract such as defining an organization. Fig. 1 elicits the components that make up an ontology. The Primitive Concepts, for instance, denote the different constituents of a domain. On the other hand, the Defined Concepts set the criteria to determine whether an object is a member of a certain class. The Axioms define the restrictions that are laid on the domain, for example, a particular object O cannot have more than x number of subordinates. These assets can define a domain by structuring its different constituents and defining the relationships that exist among them. Furthermore, the ontologies can also classify this information in order to infer additional meaning to the domain. This is possible through the use of reasoners such as Hermit [1], RacerPro [2] and Fact++ [3].

As an example, referring to the food ontology presented in [START_REF] Cantais | An example of food ontology for diabetes control[END_REF], let us suppose we need to classify the foods into categories such as Healthy Foods and Non-Healthy Foods. Through the means of defined concepts these two classes can thus be created and stored in the ontology. Moreover, through the usage of relations, as explained in Fig. 1, the different foods can be defined in terms of their amount of fat, food energy, cholesterol, weight and saturated fat. Then, through the use of a reasoner, these various classes of food can be classified as either Healthy Foods or Non-Healthy Foods. The power of the reasoner is that it infers all meanings and facts based upon the semantic meaning provided by the ontology about the concepts of a domain.

Fig. 1. Ontology Concepts

The OWL language (Ontology Web Language) is the language designed by the W3C in order to build ontologies and is mainly devised for Semantic Web applications. The language formulates the domain information in terms of instances of OWL classes and enables the use of axioms to interpret and manipulate this information.

The software mostly employed to develop ontologies is Protégé, a free, opensource ontology editor [4]. Active development is being carried out to improve the software and its two latest versions are Protégé Version 3.x and Version 4.x. P3.x is shown to be in more stable state to use inference rules within the ontology and hence supports the SWRL rule language and SQWRL query language. SWRL stands for Semantic Web Rule Language and can add more expressivity to the OWL language through the creation of rules. These rules are expressed in terms of OWL concepts (classes, properties, instances). A rule-engine bridge mechanism is provided to embed a rule engine into the Protégé-OWL in order to execute the SWRL rules. One such bridge is the Jess rule engine which can be embedded in P3.x to execute rules and add more expressivity to the OWL language. In addition, the SWRL language has been extended to a query language called SQWRL (Semantic Query Web Rule Language) in order to enable extraction of information from the ontology. The SQWRL library is packaged with SQL-like built-ins that are used within the SWRL rules in order to execute SQL-like queries to pull out required information. The library further provides new sets of operators classified as Core and Collection operators, which enable basic as well as advanced operations such as select, counting, difference and data aggregation to execute in the rules. An example eliciting the use of a SQWRL query is given below, where a query retrieves all Breads having a price less than £2 from a given ontology:

On the other hand, P4.x supports the latest version of OWL language (OWL 2.0) and is tailored to handle large and complex ontologies. It can produce very expressive ontologies, but it yet cannot provide full support for the creation and execution of inference rules through SWRL and SQWRL. Since the features of P3.x suit more the requirements of our experiment in view of performing queries against our vehicular ontology and enabling matching of different concepts, we have resorted to the usage of Protégé Version 3.4.4 for the purpose of our experiment.

Framework for Dynamic Interoperability

Our framework for dynamic interoperability provides mechanisms to achieve emergent middleware (Fig. 2); in this the crosscutting role of ontologies is depicted as central to achieving the objectives. The framework offers three phases of behaviour:  In the first phase, which regards the discovery and learning phase, the ontology is used to give a semantic meaning to the different concepts that are involved in a system. Based on Fig. 1, a system can be defined using the primary and defined concepts together with the axioms available from the ontology. Learning this system involves classifying these defined concepts within the ontology through the use of a reasoner, hence identifying related concepts.

 The second phase, which involves enabling matching between any two systems, is achieved through use of semantic rules defined within an ontology. These rules compare the definition of any two concepts classified by the ontology and generate the difference that emanates from the given definitions. This step is crucial as it also shows the degree of similarity/difference that exists between two systems, thus determining the possibility of mapping from one system to another.

 The third step involves the dynamic synthesis of a mapping mechanism in order to enable a system A to operate as another system B. The ontology is helpful here to list the missing requirements in A for it to perform as B and vice versa. Once this information is available, the mapping determines how to provide A with the adequate and absent requirements so that it can adapt itself to perform as B. The role of ontologies spans all the phases required to enable interoperability. We hence advocate and emphasize the importance of using ontologies to define the role or behaviour of a system; these define the types of protocols being deployed by the system and can help bridge the gap between any two different systems trying to interoperate with each other.

Bread(?b) ^ hasPrice(?b, ?p) ^ swrlb:lessThan(?p, 2) sqwrl:select(?b, ?p)

Case Study on Vehicular Ontology

To enable interoperability between any two systems means dealing with the low-level message exchange between them. Since different systems deal with different message formats, it is imperative to interpret these message formats in a way so that a solution can be devised regarding message exchange between them. The case study that we present in this section is based on the framework explained in section 3 and aims at tackling the interoperability problem at the level of message formats. The main hindrance in exchanging data packets stems from the difference in the packet formats themselves. In this respect, our case study is based on the role played by ontologies in facilitating some level of dynamic semantic interoperability among different packet formats. It shows how we can use ontologies to interpret and enable some level of comparison between message formats from different protocols.

Motivation of the application of Ontologies to VANETs. We chose VANETs as a case-study for our framework, because it is a domain of protocols with heterogeneity of message formats and routing strategies as shown in Fig. 3. Each particular VANET can only interpret the packet formats it has defined for itself. Hence, if we intend to make two different VANETs interoperate with each other, we need a way to be able to interpret the format of incoming packets to a VANET system. To enable this, we define a vehicular ontology to create a vocabulary of the various routing strategies defining their set of requirements. The main idea is to use this ontology to classify unknown incoming packets under the appropriate routing scheme and deduce how to enable the packet to interoperate with the current VANET.

Fig 3. Routing Strategies in VANETs

In Fig. 4 we show the application of our interopeability framework to VANETs. An important element is the Domain-Component based Model for VANETs; this is a dynamic middleware for sending, receiving and routing VANET packets-each distinct component serves one specific function within a VANET protocol. This is leveraged to create the emergent middleware between two VANET protocols. We now in turn describe the phases of the interoperability framework.

Fig. 4 Applying the dynamic interoperability framework to VANETs

Phase 1, Discover & Learn: Defining the VANET domain in the ontology. The first step is to define the VANET domain within the ontology, which is part of the first phase regarding discovery and learning. This ontology contains all of the meanings of the different routing strategies applicable to VANETs, together with the definition of known packet formats. As can be seen in Fig. 5, which shows part of the vehicular ontology, existing packet formats are defined and stored within the ontology. In this case, they are stored as subclasses of a class called NamedPackets. For instance, one of these is BBRPacket which is a protocol performing Broadcast and is derived from the protocol BBR [START_REF] Zhang | Border Node Based Routing Protocol for VANETs in Sparse and Rural Areas[END_REF]. Referring to Fig. 4, let us assume that our VANET system sends packets P 1 , suppose a broadcast-based packet, the format of which is already defined by the ontology. Upon receiving packet P 2 with a new unknown format, suppose a trajectory-based packet, the system enables this new format to be defined and stored within the ontology repository.

Fig. 5 The VANET Ontology

Phase 1, Discover & Learn: Identifying a VANET protocol. The presence of a reasoner engine embedded within the ontology tool enables to infer the meaning of a packet. As a result, the packet is classified under the most appropriate routing strategy. This classification is an important step as it helps to establish a ground for comparison between packets belonging to different routing categories. Part of the inferred ontology is shown in Fig. 6, where the class BBRPacket has been properly classified as an IdentifiedPacket and also as an MFRBroadcastPacket. The requirements for MFRBroadcastPacket are the fields CommonNeighbourNo and NeighbourList. These fields form part of the format of a BBRPacket and hence the reasoner is able to classify the latter as being of type MFRBroadcast packet (Most Forwarding Broadcast). On the other hand, the class IdentifiedPacket denotes that the incoming packets contain fields that are known. It is possible that an incoming packet does not correspond to any of the routing strategies defined within the ontology, yet contains fields that have been already defined by the ontology. In this case, the packet is an IdentifiedPacket and this classification is enough to show that information can be extracted from the packet using SQWRL mechanisms (as shown later in the section). Phase 2, Match: Dynamic Bridging between P 1 and P 2 . Once the classification process is done, the packet P 2 (Fig. 4) can be compared to the existing packet P 1 through an intuitive mechanism which makes use of SWRL rules and SQWRL query rules within the ontology itself. These rules enable to deduce the difference that lies between the packet formats P 1 and P 2 . For instance, let us assume P 1 to be a BBR packet [START_REF] Zhang | Border Node Based Routing Protocol for VANETs in Sparse and Rural Areas[END_REF] (designed for performing Broadcast-based routing) and P 2 to be a Broadcomm [START_REF] Durresi | Emergency Broadcast Protocol for Inter-Vehicle Communications[END_REF] packet (designed for performing cluster-based routing). At this stage, both packets P 1 and P 2 have already been classified under the appropriate routing scheme by the ontology. As can be seen in Fig. 7, which details out the packet format of both BBR (P 1 ) and Broadcomm (P 2 ), there is no direct mapping possible. In order to enable some kind of comparison between them, a rule-based mechanism needs to be deployed within the ontology in order to provide the adequate reasoning to enable matching. We need to make use of SQWRL like queries to retrieve the required information from the ontology. The following SQWRL rule formulates a comparison between BBR and Broadcomm in order to find out which fields are different between them:

The SQWRL query states that if b is a BBR packet and has fields represented by f, create a set of all these fields called bag. On the other hand, if p is a Broadcomm packet and has fields denoted by pf, create a set of such fields called bagt. Then find the difference between these two bags and in such a case, select those fields that have been found to be in Broadcomm packet p but not in BBR packet b. The result of this query is the set of fields missing from BBR for it to function as a Broadcomm packet. As shown in Fig. 7, the fields lacking in BBR are: LocationCoordinates, TargetRoute and ClusterHead and this information is retrieved via the SQWRL query.

The OWL language enhanced with the use of SWRL and SQWRL results in creating an expressive vehicular ontology, which determines the nature of a packet given the field descriptions. Furthermore, it enables a comparison of any two particular packets and thus provides the difference between them in terms of the fields that are missing. Once the matching of the packets has been achieved, this leads to the next step, which is to perform the mapping between these two packets.

Phase 3, Synthesize Mapping: Mapping P 2 onto P 1 . Once the difference in the two packet formats P 1 and P 2 have been provided via the ontology, this final step entails engineering an adaptive mapping mechanism to enable P 1 to function as P 2 . For example, if we need to enable BBR to function as Broadcomm, the step will determine how to provide the missing fields in the BBR packet, which are Location Coordinates, Cluster Head and Target Route. These are among the set of fields required for Broadcomm to perform cluster-based routing and are lacking from BBR. To detail how to enable this mapping mechanism is not within the scope of this paper as it is included in part of our future work regarding the interoperability process.

Dynamic Interoperability Experiments

Methodology

The case study above explains how interoperability can be tackled between two specific VANETs, which are cluster-based (Broadcomm) and broadcast-based (BBR). In order to validate this case study, we have conducted an experiment to enable the same interoperability procedures (i.e. discovery/learning and matching) to execute at run-time. The experiment consists of tackling interoperability between other systems and our VANET system at run time through the use of our vehicular ontology. In order to enable the experiment at run time, we have made use of java-based programs and Protege-owl API [10] in order to manipulate the ontology at run time. The version of the Protege ontology tool that we have used is Protege3.4.4 [4] which provides full support to apply SWRL rules and SQWRL-based queries.

In order to interpret incoming packets, we read those incoming packets and extract their field labels from their format at run time. These field labels are then stored in a text file. Another java program loads and manipulates the vehicular ontology at run time. The field names, stored in the text file, are then fed as input to the ontology which creates a new packet based on these values. We have used the reasoner Pellet [11] in order to classify the packets defined within the ontology. If the new packet contains fields which are identified by the ontology, then it is classified under a class called IdentifiedPacket, as shown in Fig. 8. Otherwise, the packet is ranked under UnIdentifiedPacket. Moreover, if the packet corresponds to the requirements of a given routing scheme, the reasoner classifies the packet under the appropriate routing class. However, if the packet partly corresponds to these requirements, it is classified as partially fulfilling the role of that routing scheme by the reasoner. Part of the resulting inferred version of the ontology is displayed in Fig. 6.

We carried out test runs with different packet formats and these are displayed in Table 1. For each new incoming packet, the java program creates a new packet class in the ontology at runtime, displayed in the ObjectName column in Table 1. All the new packet objects are initially created as subclasses of the class UnNamedPackets, pictured in Fig. 8. The reasoner then classifies them as identified or non-identified packets and also ranks them under the appropriate routing scheme.

Table 1. Test Cases

Experiment Results

Fig. 8 portrays the resulting ontology after the generation and classification of these test cases by the reasoner at run time. The first test run, UnIdentifiedPacketRecv0, consists of only the fields CommonNeighbourNo and DestinationIP, whereby CommonNeighbourNo is among the set of fields required for performing the MFRBroadcast routing. Therefore, the reasoner classifies the packet UnIdentifiedPacketRecv0 under the class PartialMFRBroadcast, which implies that this packet can partially provide requirements for performing MFRBroadcast routing. It is also classified as an IdentifiedPacket since it contains known fields. In the second test case, UnIdentifiedPacketRecv1, the fields Longitude, Latitude and TargetRoute are required for performing a Position-based routing. On the other hand, TargetRoute is also required among other set of fields to perform a Cluster-based routing. Therefore, the packet UnIdentifiedPacketRecv1 is classified under ClusterBasedPacket which, in turn, is a subclass of PositionBasedPacket and is also ranked as an IdentifiedPacket. In test case 3, although the packet, UnIdentifiedPacketRecv2, contains known fields such as BroadcastMeter, Distance and DestinationIP, the packet is not classified under any routing strategy since these fields are not defined as critical in the running of any routing strategy. However, because the packet has been identified as containing existing fields, it has been categorized under the IdentifiedPacket class. In addition to containing the same fields as in test case 1, the fourth test case, UnIdentifiedPacketRecv3, also contains 2 unknown fields. Consequently, it is classified both under PartialMFRBroadcast and UnIdentifiedPacket. Finally, the fifth test case, UnIdentifiedPacketRecv4, contains all unknown fields and hence, is classified as UnIdentifiedPacket.

Evaluation

When using BBR, which requires fields such as CommonNeighbourNo, NeighbourList, and DestinationIP address to perform broadcast-based routing, then test case 0 will require NeighbourList to operate as BBR and the system will then be able to route the packet. The matching mechanism through the use of SQWRL queries as explained earlier indicates that the latter field is required to enable the existing VANET to route the packet. The mapping mechanism will eventually determine how to enable test case 0 to function as BBR. On the other hand, if test case 0 is compared against Broadcomm, there are more missing fields since Broadcomm requires more fields to operate. Furthermore, if we take test case 3, although a few of the fields have been identified to enable this packet to be partially classified under MFRBroadcast routing scheme, the lack of information about the unidentified fields acts as a hindrance to properly identify the format. We may need additional mechanisms to interpret the fields that are unknown, which we consider as part of our future work.

In this experiment, we have tried to deal with the problem of interoperability in the domain of VANETs and have been able to show that this problem can be tackled to a certain degree. The results of this experiment demonstrate that the use of ontology combined with SWRL and SQWRL can help perform a matching between any two packets. This forms the basis of comparing any two concepts, which is the starting point for handling interoperability between them. If we try to expand this idea in a much broader context where different networked systems are trying to interoperate with one another, we would need to create an ontology for every such system in order to capture the meaning of the concepts present within the domain. Thus, the deployment of ontologies creates yet another challenge which is the differences arising among the different ontologies of a same general domain, making their manipulation even more complex and difficult.

To deal with such different application ontologies, a new type of ontology is surfacing called the Reference ontology [START_REF] Wang | MFI4Onto: Towards Ontology Registration on the Semantic Web[END_REF], which aims at providing links between heterogeneous ontologies. However, the authors in paper [START_REF] Wang | MFI4Onto: Towards Ontology Registration on the Semantic Web[END_REF] argue that if ontologies expand a particular reference ontology in a coherent way, then matching their different concepts can be made easier. Providing an initial matching between distinct ontologies of a general domain through a reference ontology is indispensable. If we are moving towards the inception of an emergent middleware to tackle dynamic interoperability, then the reference ontology can provide a benchmark to compare related ontologies and hence facilitate matching the different concepts through the application of SWRL and SQWRL-like rules.

Related Work

Universal interoperability is a long-standing objective of distributed systems research. The traditional approach to resolve interoperability problems is to agree on a standard, i.e., everyone uses the same protocols and interface description languages; CORBA [START_REF]The common object request broker: Architecture and specification Version 2.0[END_REF], DCOM [START_REF] Booth | W3C Working Group Note[END_REF], and Web Services are good examples of this approach. For situations where systems can agree to a common standard, the approaches are highly effective. However, for both long-lived and universal interoperability these solutions have demonstrably failed, indeed, future attempts at such global standards are destined to fail too. Such one size fits all standards and middleware platforms cannot cope with the extreme heterogeneity of distributed systems, e.g., from sensor applications through to large-scale Internet applications; and a single communication paradigm, e.g. RPC, cannot meet all application requirements. Moreover, new distributed systems and applications emerge fast, while standards development is a slow, incremental process; it is likely that new technologies will appear that will make a pre-existing standard obsolete. Finally, new standards do not typically embrace an existing legacy standard, which leads to immediate interoperability problems.

Software bridges have been proposed to enable communication between different middleware environments. The bridge acts as a one-to-one mapping between domains; it will take messages from a client in one format and then marshal this to the format of the server middleware; the response is then mapped to the original message format. Many bridging solutions have been produced between established commercial platforms. The OMG created the DCOM/CORBA Inter-working specification [START_REF]COM/CORBA Interworking Specification Part A & B[END_REF]. OrbixCOMet [15] is an implementation of the DCOM-CORBA bridge, while SOAP2CORBA [START_REF] Brueckne | SOAP2CORBA[END_REF] bridges SOAP and CORBA middleware. Further, Model Driven Architecture advocates the generation of such bridges to underpin deployed interoperable solutions. However, developing bridges is a resource intensive, timeconsuming task, which for universal interoperability would be required for every protocol pair; further a future protocol requires a mapping to every existing protocol. Finally, software bridges must normally be deployed and available in the network; for many environments (particularly resource-constrained) this is not possible.

Intermediary-based solutions take the ideas of software bridges further; rather than a one-to-one mapping, the protocol or data is translated to an intermediary representation at the source and then translated to the legacy format at the destination (and vice versa for a response). Enterprise Service Buses (ESB), INDISS [START_REF] Bromberg | INDISS: Interoperable Discovery System for Networked Services[END_REF], uMiddle [START_REF] Nakazawa | A Bridging Framework for Universal Interoperability in Pervasive Systems[END_REF] and SeDIM [START_REF] Flores | An Adaptive Middleware to Overcome Service Discovery Heterogeneity in Mobile Ad Hoc Environments[END_REF] are examples that follow this philosophy. However, this approach suffers from the greatest common divisor problem, i.e., between two protocols the intermediary is where their behaviour matches, they cannot interoperate beyond this defined subset. As the number of protocols grows this common divisor then becomes smaller such that only limited interoperability is possible.

Substitution solutions (e.g., ReMMoC [START_REF] Grace | A Reflective Framework for Discovery and Interaction in Heterogeneous Mobile Environments[END_REF] and WSIF [START_REF] Duftler | Web Services Invocation Framework (WSIF)[END_REF]) embrace the philosophy of speaking the peer's language. That is, they substitute the communication middleware to be the same as the peer or server they wish to use. A local abstraction maps the behaviour onto the substituted middleware. Like for software bridges this is particularly resource consuming; every potential (and future) middleware must be developed such that it can be substituted. Further, it is generally limited to client-side interoperability with heterogeneous server.

Semantic Middleware solutions e.g. S-ARIADNE [START_REF] Ben Mokhtar | Efficient Semantic Service Discovery in Pervasive Computing Environments[END_REF] employs efficient, semantic, service-oriented solutions to achieve interoperability; this is utilized at the discovery and matching stage to ensure that only services that semantically match attempt to interoperate with one another; hence, concerning itself with only the application data and function differences, not the heterogeneity of middleware (indeed a common middleware platform is required).

There is a distinct disconnection between the mainstream middleware work and the work on semantic interoperability. Our solution embraces and integrates the ideas of both (as far as we are aware it is the first to employ ontologies to resolve communication protocol interoperability); because of this, we argue that it is better placed to achieve long-lived, universal interoperability. By employing ontologies to classify, match and map communication protocols, we have the ability to automatically generate a bridge between two legacy communication protocols. The nature of the solution means that the problems of standards, exhaustive resource requirements, and minimal matches can be overcome.

Conclusions

This paper puts forward the approach of using ontologies to handle the semantic differences that arise between different systems, in order to enable them to interoperate. We have elicited three major steps required for performing interoperability, which are: discovery/learning, matching and synthesis of mapping. We have also elicited the role ontology plays in handling these steps and we advocate that the ontology has a crosscutting role in the steps involved during the interoperability process. All three processes are intertwined and each step provides the necessary input to perform the next step. For instance, the result from the matching process provides a sound notion on the type of mapping that needs to be performed. We have been able to validate the discovery/learning and matching phases through a case study on VANETs, however, the synthesis of mapping between two different systems remains part of our future work.

Fig 2 .

 2 Fig 2. Dynamic Interoperability Framework

Fig 6 .Fig. 7 .

 67 Fig 6. Inferred Vehicular Ontology

Fig. 8 .

 8 Fig. 8. Inferred Vehicular Ontology with the generated Test cases

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Furthermore, we also intend to extend in the future our experiment on VANETs through the investigation of user-defined SWRL built-ins in order to compare the data types of the fields in a message to achieve richer interoperability between protocols through the handling of data heterogeneity. We also plan to explore a wider range of middleware protocols including traditional technologies where bridging has been attempted e.g., RPC protocols, message-based platforms and service discovery.