0.1 Delone dynamical systems: ergodic features and applications 1

0.1 Delone dynamical systems: ergodic features andapplications

Steffen Klassert, Daniel Lenz and Peter Stollmann

Fakultat fur MathematikTechnische Universitat ChemnitzD-09107 Chemnitz, Germany

0.1.1 Introduction

The effect of disorder (i.e. deviation from periodicity) insolid state models is of fundamentalimportance. It has stimulated an enormous effort that has led to quite a number of resultssince the late 1950’s. From mathematical point of view, there is still a lot to be done untilbasic questions can be considered as rigorously settled. Quasicrystals provide an interestingand very challenging type of disorder.

In our research project we contribute to the rigorous study of the underlying mathematicalmodels.

The characteristic properties of quasicrystals suggest a form of aperiodicity which is veryclose to periodicity. Thus, the relevant models exhibit a behaviour between order and disorderwhich is very close to the ordered case. To describe this phenomenon the term long rangeaperiodic order has been introduced. This leads to a number of conjectures concerning thespectral properties of the hamiltonians of quasicrystals.

But let us set the stage and review how order as well as disorder can be cast in a commonmathematical framework. The main idea is to pass from singlemodels to whole familieswhose parametrization encodes the internal symmetries.

Let us start by discussing a periodic solid. It can naturallybe represented by a periodichamiltonianHper onL2(Rd ). Here,L2(Rd ) is the set of all square-integrable functions on thed-dimensional euclidean spaceRd . Typically,Hper = �+Vper with a periodic potentialVperthat is invariant under shifts from a periodicity lattice�. Thus, periodicity ofVper is reflectedin invariance properties of the function~H : Rd ! S(L2(Rd)); t 7! ��+ Vper(�+ t);whereVper(� + t) is the functionVper translated. Here,S(L2(Rd )) denotes the self ad-joint operators onL2(Rd). By invariance, the mapping~H above reduces to a mappingH : Rd=� ! S(L2(Rd )), if one factors out the periodicity lattice. Although this manip-ulation to go fromHper to a whole family may seem to be quite innocent, there are deepconsequences on the level of spectral theory. In fact, this altered point of view is at the basisof the insight that the associated operator will exhibit purely absolutely continuous spectrum.This latter mathematical statement can be recast by saying that there are no localized but onlyextended states. Thus, we have seen that expressing order, periodicity namely, in terms ofpassing to a family of operators can be fruitful.

For the strongly disordered case this is in fact the startingpoint of the whole study. Thereone adopts the point of view that disorder is better not modeled by a single deterministic

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hamiltonian. Instead one consideres a collection (family)of possible hamiltonians along withthe probabilities with which these realization are supposed to occur. Typical examples are theAnderson models, in which case the hamiltonian readsH(!) = ��+ V! on `2(Zd);where = IZd, I � R some interval,V!(k) = !(k) for k 2 Zd, ! = (!(k))k2Zd 2 andthe! is picked with a product probabilityP. Now, we face a situation in which the functionH is much more complicated than in the first case. The differenthamiltoniansH(!1) andH(!2) need no longer be translates of each other nor are they similar, in general. Therefore,we really face a nontrivial operator function. Nevertheless, every! 2 can be thought of asone specific realisation of a disordered model, whose statistical properties are entirely encodedin the probability measureP.

The first step in our program was to identify the right parameter spaces. In order todecide what consequences order - disorder aspect is going tohave, the properties of this spaceare to be studied. In the periodic case, we found a compact parameter space and a smoothoperator valued function.

In the Anderson model, independence is the fundamental property and leads to rigorousproofs of localization in certain energy regions (see [1, 2,3]).

Both cases have one important aspect in common: ergodicity with respect to the naturaltranslations. Thus we are seeking, in fact, an ergodic dynamical system encoding the proper-ties of a quasicrystal.

A natural first approach is to consider the set! � Rd of sites occupied by atoms/ionsof a quasicrystal. This set! should at least share the properties of aDeloneset. Moreover,together with!, also its translates! + t =: Tt! have the same right to be considered. Therelevant notion here is that of Delone sets andDelone dynamical systems(DDS). We discussthis point in Section 0.1.2 below. Most of the basic issues here had already been studied eitherin the Delone dynamical or in the more or less equivalent tiling dynamical framework, at leastunder the assumption of finite local complexity (FLC, see below), [4, 5]. However, it turnedout that in the non-FLC case important questions concerningthe topology had not been settledcompletely before. Our contribution in [6] was to define a suitable topology called the naturaltopology on the set of closed subsets ofRd that has the desired properties when restrictedto Delone sets. It is worthwhile to point out that our topology works without assumptions offinite local complexity. The third section is devoted to the study of ergodic and combinatoricalfeatures of Delone dynamical systems. After what we said above it is not too astonishing thatsuitable parameter spaces are Delone dynamical systems(; T ). By this we simply mean thata set of Delone sets is given such that is invariant under translationsT = (Tt)t2Rd andclosed in the natural topology. By the results from Section 0.1.2 this will imply that is acompact set.

Now there are two different levels of combinatorical properties that are intimately linked.On the one hand, every Delone set! 2 has its intrinsic complexity. In many interestingcases, only finitely many different (up to translation) patterns occur in balls of finite radii. Thissituation is described by the notion of finite local complextiy (FLC). It is now natural to askfor frequencies of patterns, whenever such frequencies exists, and so on. On the other handthere are the ergodic properties of the DDS. It turns out thatthe latter are often determined by

0.1 Delone dynamical systems: ergodic features and applications 3

combinatorical properties of the individual Delone sets! that build the DDS(; T ). More-over, these ergodic properties can be phrased in terms of thevalidity of certain forms of theergodic theorem. Another aspect concerns the consequencesfor the naturally associated op-erator algebras as well as for the individual operators (rather operator families) that constitutethese algebras.

This leads us to Section 0.1.4 in which the relevant operators are introduced. Lookingback to what we said above, it still remains to specify the operator functionH on . Herewe meet a fundamental difference when compared to the cases above. The natural tight bind-ing approach leads to a family(`2(!))!2 of different Hilbert spaces. The natural candidatefrom the point of view of parametrising the order - disorder aspect present in a DDS is nowa family (H(!))!2 in whichH(!) acts in`2(!) for each! 2 . The natural requirementthat this family should respect translations leads to the notion of covariant operator families.We define an associated algebraN (; T; �) given any invariant measure� on (; T ). Usingnoncommutative integration theory, one can prove that these algebras are von Neumann alge-bras. We discuss almost sure constancy of spectral featuresof selfadjoint families of randomoperators. Here, almost sure means that these features holdfor all ! 2 not belonging toa certain set of�-measure zero. AC?- subalgebra is the algebra generated by finite rangeoperators. Here we rely on topological properties to define asuitabe bundle over. Someimportant properties concern almost sure constancy of the spectrum. The existence of the in-tegrated density of states (IDOS) for operators of finite range is discussed in Section 0.1.5. Itis the strong ergodic theorem that allows one to realize thatthe IDOS is given by a uniformlimit. This has a striking application: it has been known since quite some time that the nearestneighbor laplacian of the Penrose tiling allows for eigenfunctions of finite support. Of coursethis leads to a jump in the integrated density of states. Using the uniform convergence, we cansee that finitely supported eigenfunctions are the only possible means to get a discontinuity ofthe IDOS. Moreover, we proved a Shubin’s trace formula that relates the integrated density ofstates with a trace on the corresponding von Neumann algebra.

0.1.2 Delone sets and Delone dynamical systems

We start with defining a suitable mathematical model to describe quasicristals. We use thelanguage of Delone sets to describe these kind of models. Roughly speaking, a Delone setis a discrete point set inRd whose points are not too close and not too far apart.

More precisely, ledd � 1 be fixed. All Delone sets, patterns etc. will be subsets ofRd .The Euclidean norm onRd will be denoted byk � k. Forr 2 R+ andp 2 Rd , we letB(p; r)be the closed ball inRd centered atp with radiusr.

A subset! of Rd is called Delone set if there existr(!) andR(!) > 0 such that2r(!) �kx� yk wheneverx; y 2 ! with x 6= y, andB(x;R(!)) \ ! 6= ; for all x 2 Rd .Note that this definition includes sets with global translation symmetries. In this point of

view a perfect crystal is a special kind of a quasicrystal. Inthe general case we have only localsymmetries.

Therefore we have to deal with local structures of Delone sets and the restrictions of!to bounded subsets ofRd are of particular interest. In order to treat these restrictions, weintroduce the following definition.

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Definition 0.1.1 (a) A pair (�; Q) consisting of a bounded subsetQ of Rd and� � Q finiteis calledpattern. The setQ is called thesupport of the pattern.(b) A pattern(�; Q) is called aball patternif Q = B(x; r) with x 2 � for suitablex 2 Rdandr 2 (0;1).

The pattern(�1; Q1) is contained in the pattern(�2; Q2) written as(�1; Q1) � (�2; Q2)if Q1 � Q2 and�1 = Q1 \ �2. Diameter, volume etc. of a pattern are defined to bethe diameter, volume etc of its support. For patternsX1 = (�1; Q1) andX2 = (�2; Q2),we define℄X1X2, the number of occurences ofX1 in X2, to be the number of elements inft 2 Rd : �1 + t � �2; Q1 + t � Q2g. Note the relation of ball patterns withs-patches asconsidered in [7]. We will identify patterns wich are equal up to translations. The notions ofdiameter, volume etc. can easily be carried over to pattern classes. The class of a patternPwill be denoted by[P ℄.

Every Delone set! gives rise to a set of pattern classes,P(!) = f[Q ^ !℄ : Q �Rd bounded and measurableg, and to a set of ball pattern classesPB(!) = f[B(p; r) ^ !℄ :p 2 !; r 2 R+g. Here we setQ ^ ! = (! \ Q;Q). We define the radiuss = s(P ) of anarbitrary ball patternP to be the radius of the underlying ball. Fors 2 (0;1), we denote byPsB(!) the set of ball patterns with radiuss. A Delone set is said to be offinite typeor of finitelocal complexityif for every radiuss > 0 the setPsB(!) is finite.

Examples� The simplest example of a Delone set is the latticeZd in Rd .� The set of vertices of the Penrose tiling is a Delone set.

Next we introduce a suitable topology on the setF(Rd ) of closed subsets ofRd . Themethod we are going to outline now has most definitely been pointed out to us by someoneelse. Unfortunately, we were not able to find out by whom.

We use the stereographic projection to identify pointsx 2 Rd [ f1g in the one-point-compactification ofRd with the corresponding points~x 2 Sd. Clearly, the latter denotes thed-dimensional unit sphereSd = f� 2 Rd+1 : k�k = 1g. NowSd carries the euclidean metric�. Since the unit sphere is compact and complete, the Hausdorff metric �H makes the setK(Sd) of compact subsets of it into a complete and compact metric space.

ForF 2 F(Rd) write ~F for the corresponding subset ofSd and define�(F;G) := �H( ^F [ f1g; ^G [ f1g) for F;G 2 F(Rd):Although this constitutes a slight abuse of notation it makes sense since^F [ f1g; ^G [ f1gare compact inSd providedF;G are closed inRd .

We have the following result:

Theorem 1 F(Rd ) endowed with the natural topology�nat is compact.

Note that, interestingly, no additional properties are needed for compactness. Of course,this result immediately gives compactness of certain subsets of the set of all Delone sets, e.g.compactness of the union overR of the(r; R)-sets for any fixed value ofr.

0.1 Delone dynamical systems: ergodic features and applications 5

Let us note in passing that the metric� proposed in [5] as well as the metric in [8] do notsatisfy the triangle inequality. See, however, [9], in which a metric on the set of Delone sets isconstructed. A discussion in a more general framework can befound in [4], where the authorconstructs a topology on the set of closed discrete subsets of a locally compact�-compactspace. In the case ofRd this topology coincides with the restriction of the above given naturaltopology.

For a quite different approach to the topology on the set of Delone sets we refer to [10],where Delone sets are identified with the sum of delta measures sitting at the points of theDelone set. Then one has the w�-topology on the set of measures at ones disposal, providinggood compactness properties. The approach presented here has the advantage that a topologyis induced on the set of closed sets.

Next, we define Delone dynamical systems, following the study of repetitive Delone setsin [11] and single out some important properties:

Definition 0.1.2 (a)Let be a set of Delone sets. The pair(; T ) is called aDelone dynam-ical system(DDS) if is invariant under the shiftT and closed in the natural topology.(b) A DDS(; T ) is said to be offinite type(DDSF) if[!2P sB(!) is finite for everys > 0.(c) Let 0 < r;R < 1 be given. A DDS(; T ) is said to be an(r; R)-systemif every! 2 is an(r; R)-set.(d) The setP() of pattern classes associated to a DDS is defined byP() = [!2P(!).Remark 0.1.3 (a) Whenever(; T ) is a Delone dynamical system, there exists anR > 0with BR(x) \ ! 6= ; for every! 2 and everyx 2 Rd . This follows easily as is closedand invariant under the action ofT .(b) Every DDSF is an(r; R)-system for suitable0 < r;R <1.(c) For a thorough study of Delone sets of finite local complexity we refer the reader to [7] (d)Let ! be an(r; R)-set and let! be the closure offTt! : t 2 Rdg in F(Rd ) with respect tothe natural topology. Then(! ; T ) is an(r; R)-system.

For a DDSF, there is a simple way to describe convergence in the natural topology. Thisis shown in the following lemma taken from [6].

Lemma 0.1.4 If (; T ) is a DDSF then a sequence(!n) converges to! in the natural topol-ogy if and only if there exists a sequence(tn) converging to0 such that for everyL > 0 thereis ann0 2 N with (!n + tn) \ B(0; L) = ! \ B(0; L) for n � n0.0.1.3 ergodic Ergodic and combinatorial features of DDS

We start with a result that characterizes unique ergodicityof a Delone dynamical system interms of the validity of a Banach space valued ergodic theorem. First, let us record the fol-lowing notions of ergodic theory along with an equivalent “combinatorial” characterizationavailable for Delone dynamical systems:(; T ) is callednon-periodicif Tt! 6= whenever! 2 andt 2 Rd with t 6= 0. Is is calledminimal if every orbit is dense. This is equivalentto P() = P(!) for every! 2 . This latter property is calledlocal isomorphism propertyin the tiling framework [12]. It is also referred to as repetitivity. Namely, it is equivalent tothere existing anR(P ) > 0 for everyP 2 P() such thatB(p;R(P ))^! contains a copy ofP for everyp 2 Rd and every! 2 . Note also that every minimal DDS is an(r; R)-system.

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We are interested in ergodic averages. More precisely, we will take means of suitablefunctions along suitable sequences of patterns and patternclasses. These functions and se-quences will be introduced next. Here and in the sequel we will use the following notation:ForQ � Rdands > 0 we defineQs � fx 2 Q : dist(x; �Q) � sg; Qs � fx 2 Rd : dist(x;Q) � sg;where, of course, dist denotes the usual distance and�Q is the boundary ofQ. Moreover, wedenote the Lebesgue measure of a measurable subsetQ � Rd by jQj. Then, a sequence(Qn)of subsets inRd is called a van Hove sequence if the sequence(jQnj�1jQsn nQn;sj) tends tozero for everys 2 (0;1). Similarly, a sequence(Pn) of pattern classes, (i.e.Pn = [(�n; Qn)℄with suitableQn;�n) is called a van Hove sequence ifQn is a van Hove sequence. (This isobviously well defined.) We can now discuss unique ergodicity. A dynamical system(; T )is calleduniquely ergodicif it admits only oneT -invariant measure (up to normalization). Fora Delone dynamical system, this is equivalent to the fact that for every pattern classP thefrequencyf(P ) � limn!1 jQnj�1℄P (! ^Qn); (1)

exists uniformly in! 2 for every van Hove sequence(Qn). This equivalence was shownin Theorem 1.6 in [6] (see [9] as well). It goes back to [12], Theorem 3.3, in the tiling setting.

For patternsXi = (�i; Qi), i = 1; : : : ; k, andX = (�; Q), we writeX = �ki=1Xi if� = [�i, Q = [Qi and theQi are disjoint up to their boundaries with the obvious extensionto pattern classes.

Definition 0.1.5 Let be a DDS andB be a vector space with seminormk � k). A functionF : P() :�! B is called almost additive (with respect tok � k) if there exists a functionb : P() �! (0;1) with limn!1 jPnj�1b(Pn) = 0 for every van Hove sequence(Pn) anda constantD > 0 such that

(A1) kF (�ki=1Pi)�Pki=1 F (Pi)k �Pki=1 b(Pi),(A2) kF (P )k � DjP j+ b(P ).(A3) b(P1) � b(P ) + b(P2) wheneverP = P1 � P2.

Now, our first result reads as follows.

Theorem 2 For a minimal, aperiodic DDSF(; T ) the following are equivalent:(i) (; T ) is uniquely ergodic.(ii) The limit limn!1jPnj�1F (Pn) exists for every van Hove sequence(Pn) and every almostadditiveF on (; T ) with values in a Banach space.

Remark 0.1.6 (a) The proof uses methods of Geerse/Hof [13] and ideas from Priebe [14]. Infact, Geerse/Hof established a similar result for a tiling associated to a primitive substitution.(b) In the one dimensional case related results have been shown by one of the authors in [15].

0.1 Delone dynamical systems: ergodic features and applications 7

The proof of the theorem makes use of completeness of the Banach space in crucial man-ner. However, it does not use the nondegeneracy of the norm. Thus, we get the followingcorollary (of its proof).

Corollary 0.1.7 Let (; T ) be aperiodic and strictly ergodic. Let the vector spaceB becomplete with respect to the topology induced by the seminormsk � k�, � 2 I. If F : P �! Bis almost additive with respect to everyk � k�. � 2 I , thenlimn!1 jPnj�1F (Pn) exists forevery van Hove sequence(Pn) in P().

The theorem and the corollary underline the strong averaging features of strictly ergodicDDS. Of course, quite some work has been devoted to studying local conditions ensuringunique ergodicity and minimality of a DDS. Two such conditions arelinear repetitiveityanddense repetitivity. These conditions have recently been discussed and shown toimply strictergodicity by Lagarias and Pleasants in [11] (for one-dimensional linearly repetitive systemssee the work of Durand [16] as well). A DDSF is said to be linearly repetitive if there existsa constantC such that, forr > 0 given, every ball pattern of radiusCr contains every ballpattern of radiusCr. A DDSF is said to be densely repetitive if there exist a constantD suchtthat every ball pattern of radiusr is contained in every ball pattern of radiusCN(r)1=d, whereN(r) denotes the number of different ball patterns of radiusr. As shown in [11] aperiodiclinearly repetitive systems and aperiodic densely repetitive systems are in some sense closestto periodic systems among the non-periodic systems. The twoconcepts are related. In fact,as conjectured by Lagarias/Pleasants in [11] and recently shown in [17] by one of the authorsthe following holds.

Theorem 3 Every aperiodic linearly repetitive DDS is densely repetitive.

As far as uniform ergodic theorems go, linearly repetitive systems are rather special. Namely,they allow for a uniform subadditive ergodic theorem. As shown in [18, 19] the followingholds. LetC(2) be the set of all cubes inRd whose sidelengthsli(C), i = 1; : : : ; d satisfy1=2 � li(C)=lj(C) � 2 for all i; j 2 1; : : : ; 2.

Theorem 4 Let (; T ) be a linearly repetititive DDS. LetF : P() �! R be subadditive(i.e. F (A�B) � F (A) + F (B), wheneverA andB are disjoint up to their boundary), thenlimn!1 F (Pn)jPnjexists for every van Hove sequence(Pn) with supports ofPn contained inC(2).

Remark (1) The theorem gives a new proof of the uniquely ergodicity of linearly repetitivesystems.(2) The theorem holds also for functions which are subadditive up to a boundary term [18].

In d = 1 it is actually possible to characterize the subshifts for which the averages for ar-bitrary subadditive functions exist. They are those satisfying linear repetitivity on the average.More precisely, the following holds as shown by one of the authors in [15].

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Theorem 5 Let (; T ) be a minimal subshift over the finite alphabetA. Then, the followingare equivalent:(i) limjvj!1 F (v)=jvj exists for arbitrary subadditiveF on the associated set of finite words.

(ii) There exists a constantC > 0 with lim inf jxj!1 ℄v(x)jxj jvj � C for arbitrary v in theassociated set of words. Here,℄v(x) denotes the number of copies ofv in x and jxj is thelength ofx.

Generalizations of this one-dimensional result to DDS in arbitrary dimensions for an ap-plication to diffraction theory, are currently under investigation (see [20]). There also applica-tions to computations of the eigenvalues of the DDS are given.

Let us close this section by emphasizing that the theorems onuniform existence of av-erages of subadditive functions have proven rather useful in recent works. They allow toprove Cantor spectrum of measure zero for large classes of one-dimensional quasicrystalSchrodinger operators [21, 22]. Moreover, they can also beused to study uniform existenceof certain averages in lattice gas theory.

0.1.4 The associated algebras and operators

In this Section we introduce aC�-algebra that had already been encountered in a differentform in [10, 23]. Our presentation here is geared towards using the elements of theC�-algebraas tight binding hamiltonians in a quantum mechanical description of disordered solids. Werelate certain spectral properties of the members of such operator families to ergodic featuresof the underlying dynamical system. Moreover, we show that the eigenvalue counting func-tions of these operators are convergent. The limit, known asthe integrated density of states,is an object of fundamental importance from the solid state physics point of view. Apart fromproving its existence, we also relate it to the canonical trace on the von Neumann algebraN (; T; �) in case that the Delone dynamical system(; T ) uniquely ergodic. Results of thisgenre are known as Shubin’s trace formula due to the celebrated results from [24].

Moreover, we review how noncommutative integration theoryis used to construct the vonNeumann algebraN (; T; �) of observables that reflects certain ergodic properties of theunderlying DDS as well.

The results as well as the proofs can be found in [25]. First westudy a C�-subalgebra ofN (; T; �) that contains those operators that might be used as hamiltonians for quasicrystals.We define X � X := f(p; !; q) 2 Rd � � Rd : p; q 2 !g;

which is a closed subspace ofRd � � Rd for any DDS.

Definition 0.1.8 A kernel of finite rangeis a functionk 2 C(X � X ) that satisfies thefollowing properties:

(i) k is bounded.

(ii) k has finite range, i.e., there existsRk such thatk(p; !; q) = 0, wheneverjp� qj � Rk.

(iii) k is invariant, i.e., k(p+ t; ! + t; q + t) = k(p; !; q);for (p; !; q) 2 X � X andt 2 Rd .

0.1 Delone dynamical systems: ergodic features and applications 9

The set of these kernels is denoted byKfin(; T ).We record a few quite elementary observations in the following.

Remarks 0.1.9 (1) For any kernelk 2 Kfin(; T ) denote by�!k := K! the operatorK! 2 B(`2(!)), induced by(K!ÆqjÆp) := k(p; !; q) for p; q 2 !:Clearly, the familyK := �k, K = (K!)!2, is bounded in the product (equipped withthe supremum norm)�!2B(`2(!)).

(2) Pointwise sum, the convolution (matrix) product(a � b)(p; !; q) :=Xx2! a(p; !; x)b(x; !; q)and the involutionk�(p; !; q) := k(q; !; p) makeAfin(; T ) into a�-algebra.

(3) The mapping� : Kfin(; T ) ! �!2B(`2(!)) defined in (1) is a faithful�-representation. We denoteAfin(; T ) := �(Kfin(; T )) and call it theoperators offinite range.

(4) We denote the completion ofAfin(; T )with respect to the normkAk := sup!2 kA!kbyA(; T ).

(5) The mapping�! : Afin(; T )! B(`2(!));K 7! K! with K! as in (1) is a representa-tion that extends by continuity to a representation ofA(; T ) that we denote by the samesymbol.

We get the following result that relates ergodicity properties of(; T ), spectral properties ofthe operator families fromA(; T ) and properties of the representations�!.

Theorem 6 The following conditions on a DDS(; T ) are equivalent:

(i) (; T ) is minimal.

(ii) For any selfadjointA 2 A(; T ) the spectrum�(A!) is independent of! 2 .

(iii) �! is faithful for every! 2 .

Now we introduce algebras of observables that extend the C�-algebra considered above.

Definition 0.1.10 Let(; T ) be an(r; R)-system and let� be an invariant measure on. De-note byV1 the set of allf : X �! C which are measurable and satisfyf(!; �) 2 `2(X!; �!)for every! 2 .

A family (A!)!2 of bounded operatorsA! : `2(!; �!) �! `2(!; �!) is called mea-surable if! 7! hf(!); (A!g)(!)i! is measurablefor all f; g 2 V1. It is calledboundedifthe norms of theA! are uniformly bounded. It is calledcovariantif it satisfies the covariancecondition H!+t = UtH!U�t ; ! 2 ; t 2 Rd ;

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whereUt : `2(!) �! `2(! + t) is the unitary operator induced by translation. Now, we candefine N (; T; �) := fA = (A!)!2jA covariant, measurable and boundedg= �;where� means that we identify families which agree� almost everywhere.

As is clear from the definition, the elements ofN (; T; �) are classes of families of oper-ators. However, we will not distinguish too pedantically between classes and their representa-tives in the sequel.

Remark 0.1.11 It is possible to defineN (; T; �) by requiring seemingly weaker condi-tions. Namely, one can consider families(A!) that are essentially bounded and satisfy thecovariance condition almost everywhere. However, by standard procedures (see [26, 27]), it ispossible to show that each of these families agrees almost everywhere with a family satisfyingthe stronger conditions discussed above.

Obviously,N (; T; �) depends on the measure class of� only. Hence, for uniquelyergodic(; T ), N (; T; �) =: N (; T ) gives a canonical algebra. This case has been con-sidered in [28, 6].

Apparently,N (; T; �) is an involutive algebra under the obvious operations. To see thatit has a predual, i.e., that it is a weak-*-algebra is not obvious. We prove this in [25] byidentifyingN (; T; �) with the algebra of random operators for a suitable random Hilbertspace, as will be outlined now. For details we refer to [6, 25]where we use the followingconcepts from Connes non-commutative integration theory [26]: We introduced� a suitablegroupoidG(; T ),� a transversal measure� = �� for a given invariant measure� on (; T )� and a�-random Hilbert spaceH = (H!)!2leading to the von Neumann algebraN (; T; �) := End�(H)of random operators, all in the terminology of [26]. When everything is put together, thisgives:

Theorem 7 Let (; T ) be an(r; R)-system and let� be an invariant measure on. ThenN (; T; �) is a weak-*-algebra. More precisely,N (; T; �) = End�(H);where� = �� andH = (`2(X! ; �!))!2 are defined as above.

We can use the measurable structure to identifyL2(X ;m), wherem = R �!�(!) withR � `2(X! ; �!) d�(!). This gives the faithful representation� : N (; T; �) �! B(L2(X ;m)); �(A)f((!; x)) = (A!f!)((!; x))and the following immediate consequence.

0.1 Delone dynamical systems: ergodic features and applications 11

Corollary 0.1.12 �(N (; T; �)) � B(L2(X ;m)) is a von Neumann algebra.

Next we want to identify conditions under which�(N (; T; �)) is a factor. Recall that aDelone set! is said to benon-periodicif ! + t = ! implies thatt = 0.

Theorem 8 Let (; T ) be an(r; R)-system and let� be an ergodic invariant measure on.If ! is non-periodic for�-a.e.! 2 thenN (; T; �) is a factor.

Remark 0.1.13 Since� is ergodic, the assumption of non-periodicity in the Theorem can bereplaced by assuming that there is a set of positive measure consisting of non-periodic!.

The following theorem is a consequence of [29]. It deals withalmost sure constancy ofspectral features of random operators.

Theorem 9 Let (; T ) be an(r; R)-system and� be T -invariant. Let� be ergodic and(A!) 2 N (; T; �) be selfadjoint. Then there exist sets�;�a ;�s ;�pp;�ess � R and asubsete of of full measure such that� = �(A!) and��(A!) = �� for � = a ; s ; pp; essand�dis (A!) = ; for every! 2 e.

0.1.5 The integrated density of states and Shubins trace formula

We now define a particular trace onN (; T; �). To this end, choose a nonnegativeu 2C (Rd) withRRd u(x)dx = 1. LetMu be the operator of multiplication byu. It can be shown

that � : N (; T; �) �! C ; �(A) = Z tr(A!Mu) d�(!)does not depend on the choice ofu as long as the integral is one [6, 25]. Important features of� are given in the following lemma.

Lemma 0.1.14Let (; T ) be an(r; R)-system and� be T -invariant. Then the map� :N (; T; �) �! C is continuous, faithful, nonegative onN (; T; �)+ and satisfies�(A) =�(U�AU) for every unitaryU 2 N (; T; �) and arbitraryA 2 N (; T; �), i.e.,� is a trace.

Having defined� , we can now associate a canonical measure�A to every selfadjointA 2N (; T; �).Definition 0.1.15 For A 2 N (; T; �) selfadjoint, andB � R Borel measurable, we set�A(B) � �(�B(A)), where�B is the characteristic function ofB.

For the next result we refer to [29].

Lemma 0.1.16Let (; T ) be an(r; R)-system and� beT -invariant. LetA 2 N (; T; �)selfadjoint be given. Then�A is a spectral measure forA. In particular, the support of�Aagrees with the spectrum� ofA and the equality�A(F ) = �(F (A)) holds for every boundedmeasurableF onR.

We now come to relate the abstract trace� defined above with the mean trace per unitvolume. The latter object is quite often considered by physicists and bears the nameintegrateddensity of states. Its proper definition rests on ergodicity.

12

Proposition 0.1.17Assume that(; T ) is a uniquely ergodic(r; R)-system with invariantprobability measure� andA 2 A(; T ). Then, for any van Hove sequence(Qn) it followsthat limn2N 1jQnj tr(A!jQn) = �(A)for all ! 2 .

Clearly,A!jQ denotes the restriction ofA! to the finite dimensional subspace`2(! \ Q),wheneverQ � Rd is bounded.

The following result finally establishes an identity that one might call an abstract Shubin’strace formula. It says that the abstractly defined trace� is determined by the integrated densityof states. The latter is the limit of the following eigenvalue counting measures. Let, forselfadjointA 2 A(; T ) andQ � Rd :h�[A!; Q℄; 'i := 1jQj tr('(A! jQ)); ' 2 C(R):Its distribution function is denoted byn[A!; Q℄, i.e. n[A!; Q℄(E) gives the number of eigen-values per volume belowE (counting multiplicities).

Theorem 10 Let (; T ) be a uniquely ergodic(r; R)-system and� its invariant probabilitymeasure. Then for selfadjointA 2 A(; T ) and any van Hove sequence(Qn) we get that forevery' 2 C(R) and every!h�[A! ; Qn℄; 'i ! �('(A)) asn!1:Consequently, the measures�Qn! converge weakly to the measure�A defined above byh�A; 'i := �('(A)).

The above statement has many precursors: [30, 31, 32, 2, 24] in the context of almostperiodic, random or almost random operators on`2(Zd) or L2(Rd ). It generalizes results byKellendonk [23] on tilings associated with primitive substitutions. Its proof relies on ideasfrom [30, 31, 32, 23] and [33]. Nevertheless, it is new in the present context.

The primary object from the physicists point of view is the finite volume limit:N [A℄(E) := limn!1n[A!; Qn℄(E)known as the integrated density of states. It has a striking relevance as the number of energylevels belowE per unit volume, once its existence and independence of! are settled.

The last theorem provides the mathematically rigorous version. Namely, the distributionfunctionNA(E) := �A(�1; E℄ of �A is the right choice. It gives a limit of finite volumecounting measures since �[A! ; Qn℄! �A weakly asn!1:Therefore, the desired independence of! is also clear. Moreover, by standard arguments weget that the distribution functions of the finite volume counting functions converge toNA atpoints of continuity of the latter.

0.1 Delone dynamical systems: ergodic features and applications 13

A better convergence result can be deduced from the strong ergodic result Theorem 2presented in the Section 0.1.3. To do so we proceed as follows.

The spectral counting functions appearing as approximantsof the integrated density ofstates are obviously elements of the vector spaceD consisting of all bounded right continuousfunctionsf : R �! R equipped for whichlimx!�1 f(x) = 0 and limx!1 f(x) exists.Equipped withe the supremum normkfk1 � supx2R jf(x)j this vector space is a Banachspace. It turns out that the spectral counting function is essentially an almost additive function.More precisely the following holds [34].

Theorem 11 Let(; T ) be a DDS. LetA be an operator of finite range. ThenFA : P() �!D, defined byFA(P ) � n(A!; QRA) for P = [(! ^ Q)℄ is a well defined almost additivefunction.

Remark 0.1.18 The theorem seems to be new even in the one-dimensional case.(There, ofcourse, it is very easy to prove.)

Now, our results on convergence of the integrated density ofstates reads as follows.

Theorem 12 Let (; T ) be an aperiodic strictly ergodic DDSF. LetA be a selfadjoint oper-ator of finite range and(Qn) be an arbitrary van Hove sequence. Then the measures�A!Qnconverge in distribution to the measure�A and this convergence is uniform in! 2 .

As a consequence of this stronger convergence we can explainthe occurrence of discon-tinuities of the integrated density of states. Namely such discontinuities are always due tofinitely supported eigenfunctions, i.e., to very strongly localized states. This has been workedout in [35].

Theorem 13 Let (; T ) be a strictly ergodic DDSF. LetA be a selfajoint random operatorof finite range. ThenE is a point of discontinuity of�A if and only if there exists a locallysupported eigenfunction ofA! toE for one (all)! 2 .

It is rather straightforward to see that locally supported eigenfunctions lead to a discon-tinuity of the IDS. Moreover one can show easily that every DDS can be changed in such away that the original and the new system are “more or less the same” and such that the newsystem exhibits locally supported eigenfunctions. The idea is to replace points in the originalsystem by a “small” graph that allows an eigenfunction. The equivalence of the two systemscan be phrased in the notion ofmutually locally derivable(MLD), going back to [36]. For athorough discussion, see [35]. Here we only give the result and a picture of the small graphthat one can choose to implement. Here the little squares mark the vertices of the graph andthe straight lines are the edges of the graph. Next to each vertex we indicate the value of alocally supported eigenfunction of the nearest neighbor laplacian.

Theorem 14 Let (; T ) be a arbitrary DDSF. Then there exists a DDSF(b; T ) and a ran-dom operator of finite range(Abw) such that(; T ) and(b; T ) are mutually locally derivableand(Abw) has locally supported eigenfunctions with eigenvalueE for every! 2 b. More-over,(Abw) can be chosen to be the nearest neighbor laplacian of suitable graph.

14

0 0

−1 −1

1

1

0 0

Figure 1: The finite graphGfin.

The starting point is a small graphGfin = (Vfin; Efin) and an eigenfunctionufin of theassociated nearest neighbor laplacian. For definiteness sake consider Figure 1. The values ofufin are indicated near the corresponding vertices. Here the eigenvalue isE = 0.

It is clear that whatever edges reach out of the four corners in a larger graph extendingGfin, the extension ofufin by 0 to the larger vertex set will still constitute an eigenfunctionof the laplacian on the large graph. It is now easy to implement this picture into a given DDSF.

For those who prefer tiling examples we now indicate how to view the construction abovein this framework. Take a tiling dynamical system (see [6, 5]) and replace one given tileT bya suitable homeomorphic image ofT b indicated in Figure 2 by the full lines. We also indicatedthe next neighbor relations, showing that the resulting graph is justGfin above.

0.1.6 Conclusion

To sum up, using Delone sets one can define natural tight binding hamiltonians for quasicrys-tals. These hamiltonians have many aspects in common with the hamiltonians of disorderedsystems. They can be considered as families indexed by the points of a Delone dynamicalsystem.

However, the complex geometry of Delone sets can lead to effects like discontinuities ofthe integrated density of states. Moreover, these models exhibit a striking uniformity as faras their ergodic properties are concerned. Thus, despite all analogies, there are importantdifferences to disordered systems.

ergodic Ergodic properties of the Delone dynamical system in question are of central im-

0.1 Delone dynamical systems: ergodic features and applications 15

Figure 2: The tiling ofGfin.

portance in different aspects as well and they can be deducedto questions concerning thecomplexity of the individual Delone sets that belong to the DDS. Certain spectral informa-tion concerning the hamiltonian for a quasicrystal can be deduced from ergodic information.Moreover, the hamiltonians can be regarded as elements of certain operator algebras and thelatter reflect ergodic features of the underlying DDS.

Acknowledgements:Our research has benefited from stimulating discussions with var-ious people. We would particularly like to thank Michael Baake, Anne Boutet de Monvel,David Damanik, Uwe Grimm, Johannes Kellendonk, Jeff Lagarias, Yoram Last, Peter Pleas-ants, Boris Solomyak and Gunter Stolz.

We would also like to take the opportunity to gratefully acknowledge financial supportfrom the DFG within the Schwerpunktprogramm 1031.

Index

almost additive, 6, 7, 13

Delone dynamical system, 2densely repetitve, 7discontinuity, 3, 13dynamical system, 2, 3, 5, 6, 8, 14, 19

eigenfunction, 3, 13, 14ergodic, 2, 3, 5–8, 10–15

factor, 1, 11finite range, 3, 8, 9, 13

graph, 13, 14

hamiltonian, 1, 2, 8, 14, 15

integrated density of states, 3, 8, 11–14

linear repetitiv, 7

minimal, 5, 6, 8, 9mutually locally derivable, 13

Shubin’s trace formula, 3, 8, 12spectrum, 1, 3, 8, 9, 11subadditive, 7, 8

trace, 3, 8, 11, 12

von Neumann algebra, 3, 8, 10, 11

Bibliography

[1] R. Carmona and J. Lacroix, Birkhauser, Boston, (1990).[2] L. Pastur and A. Figotin, Springer-Verlag, Berlin, (1992).[3] P. Stollmann, Progress in Math. Phys., vol 20, Birkhauser, Boston (2001).[4] M. Schlottmann, Directions in Mathematical Quasicrystals, eds M. Baake and R.V.

Moody, CRM Monograph series, AMS, Providence RI (2000), 143-159.[5] B. Solomyak, Quasicrystals and Discrete Geometry, ed. J. Patera, Fields Institute

Monographs, vol. 10, AMS, Providence, RI (1998), 265-275.[6] D. Lenz, P. Stollmann, Proc. OAMP, Constanta 2001, to appear, eprint:

arXiv math-ph/0202142.[7] J. C. Lagarias, Discrete and Computational Geometry, toappear.[8] J. C. Lagarias, Discrete and Computational Geometry, toappear.[9] J.-Y. Lee, R.V. Moody and B. Solomyak, available from

http://www.math.washington.edu/ solomyak/personal.html.[10] J. Bellissard, D. J. L. Hermann, and M. ZARROUATI, Directions in mathematical

quasicrystals, CRM Monogr. Ser., 13, Amer. Math. Soc., Provicence, RI, (2000),207–258.

[11] J. C. Lagarias, P. A. B. Pleasants, Ergod. Th. & Dynam. Sys., to appear.[12] B. Solomyak, Ergodic Theory Dynam. Systems, 17, (1997), 695-738.[13] C.P.M. Geerse and A. Hof, Rev. Math. Phys., 3, (1991), 163-221.[14] N. Priebe, Geometriae Dedicata, 79 (2000), 239-265.[15] D. Lenz, Ergodic theory & s, 22 (2002), 245–255.[16] F. Durand, Ergod. Th. & Dynam. Sys. 20 (2000), 1061-1078.[17] D. Lenz, Discrete and Computational Geometry, to appear.[18] D. Lenz, PhD Thesis, Frankfurt, Logos Verlag, Berlin, (2000).[19] D. Damanik, D. Lenz, Discrete and Computational Geometry, 26 (2001), 411-428.[20] S. Klassert, D. Lenz, in preparation.[21] D. Lenz, preprint (2002).[22] D. Lenz, Commun. Math. Phys. 227 (2002), 119-130.[23] J. Kellendonk, Rev. Math. Phys., 7, (1995), 1133-1180.[24] M. Shubin, Russian Math. Surveys, vol 34, (1979).[25] D. Lenz, P. Stollmann,eprint: arXiv math-ph/0210031.[26] A. Connes, Algebres d’operateurs (Sem., Les Plans-sur-Bex, 1978), 19-143, LNM,

vol. 725, Springer, Berlin, (1979).

20 Bibliography

[27] D. Lenz, Math. Phys. Anal. Geom., 2, (1999), 197-220.[28] D. Lenz, P. Stollmann, C.R. Acad. Sci. Paris, Ser. I, 334(2002), 1131-1136.[29] D. Lenz, N. Peyerimhof and I. Veselic,eprint: arXiv math-

ph/0203026.[30] J. Avron and B. Simon, Duke Math. J., 50, (1982), 369-391.[31] J. Bellissard and R. Lima and D. Testard, Mathematics+ physics. Vol. 1, World Sci.

Publishing, Singapore, (1995), 1-64.[32] J. Bellissard, Statistical mechanics and field theory:mathematical aspects (Gronin-

gen, 1985), Lecture Notes in Physics, Vol. 257, Berlin, 1986, 99-156.[33] A. Hof, J. Statist. Phys., 72, (1993) 1353–1374.[34] D. Lenz, P. Stollmann, In preparation.[35] S. Klassert, D. Lenz, P. Stollmann,eprint: arXiv math-ph/0208027.[36] M. Baake, M. Schlottmann and P.D. Jarvis, J. Phys. A, 24,(1991) 4637-4654.