Pure point/Continuous decomposition of translation-bounded measures and diffraction
aa r X i v : . [ m a t h . D S ] M a r PURE POINT/CONTINUOUS DECOMPOSITION OFTRANSLATION-BOUNDED MEASURES AND DIFFRACTION
JEAN-BAPTISTE AUJOGUE
Abstract.
In this work we consider translation-bounded measures over a locally com-pact Abelian group G , with particular interest for their so-called diffraction. Given sucha measure Λ, its diffraction p γ is another measure on the Pontryagin dual p G , whose de-composition into the sum p γ “ p γ p ` p γ c of its atomic and continuous parts is central indiffraction theory. The problem we address here is whether the above decompositionof p γ lifts to Λ itself, that is to say, whether there exists a decomposition Λ “ Λ p ` Λ c ,where Λ p and Λ c are translation-bounded measures having diffraction p γ p and p γ c re-spectively. Our main result here is the almost sure existence, in a sense to be madeprecise, of such a decomposition. It will also be proved that a certain uniqueness prop-erty holds for the above decomposition. Next we will be interested in the situationwhere translation-bounded measures are weighted Meyer sets. In this context, it willbe shown that the decomposition, whether it exists, also consists of weighted Meyersets. We complete this work by discussing a natural generalization of the consideredproblem. Outline
In this work we consider certain measures over a locally compact Abelian (LCA) group G , for which we aim to establish a general result on structures that goes beyond the onescalled purely diffractive . The motivation of this work comes from solid state physics,where materials with atomic-like kinematic diffraction have focused a lot of attentionduring the last 30 years since the discovery by Shechtman et al. [43] of physical materialsnowadays called Quasicrystal , a work for which Schechtman was awarded the Nobel pricein 2011.Mathematically, one describes a material by a translation-bounded measure
Λ overthe group G (thought as the ambient space, and usually taken as R d , d “
1, 2 or 3).Very often a simpler picture is displayed by a weighted Dirac comb over a point sethaving specific geometric properties (lattice, Delone, FLC, Meyer property and so on),supposed to model the position and nature of the constituents of the alloy. However thisis not required for a proper definition of diffraction so we shall not assume this, at leastnot at the beginning. Although when dealing with some translation-bounded measureone thinks of an alloy as made of finitely many atoms, the structures one considers areinfinite-sized idealizations. Now the mathematical diffraction theory as proposed by Hofin [21], see also the recent review article [9], assigns to a general translation-boundedmeasure Λ on a LCA group G another measure γ on G , given as a self-convolutionproduct of Λ passed to the infinite volume limit along a certain sequence p A k q k of sets Keywords:
Translation-bounded measures, Diffraction, Weighted Meyer sets, Quasicrystals.Work supported by the FONDECYT Post-doctoral grant n. 3150535. (a Van Hove sequence) γ : “ lim n Ñ8 | A k | Λ | A k ˚ Ć Λ | A k Such limit, whenever it exists, and it always will after possibly extracting some sub-sequence, is called the autocorrelation of Λ (with respect to the averaging sequence p A k q k ), and is translation-bounded on G . Its Fourier transform p γ is called the diffractionmeasure (or simply diffraction) of Λ, and is a positive translation-bounded measure onthe Pontryagin dual p G . It is this latter which is thought as the outcome of a diffractionexperiment set on an alloy modeled by Λ, see the discussion in [21]. As any complexmeasure over p G , the diffraction measure splits as the sum p γ “ p γ p ` p γ c of two mutually singular summands, namely a purely atomic (or pure point ) part p γ p and a continuous part p γ c . The summand p γ p is called the Bragg spectrum of the un-derlying translation-bounded measure Λ, and any ω P p G in its support, that is, with p γ pt ω uq ą
0, is called a
Bragg peak for Λ. The measure Λ is then called purely diffractive ,or said to have pure point spectrum, whenever p γ is a pure point measure, that is, if itscontinuous part p γ c vanishes. Purely diffractive measures are, at least in the setting ofweighted point sets, rather well understood, see the general characterizations made in[27, 20, 5, 4]. It encompasses the (uniform Dirac combs over) lattices and regular modelsets [6, 42, 37], and is agremented by many explicit examples [12, 13, 6], see again thereview article [9]. The situation of mixed diffraction spectrum is much less understood,although some explicit computations exist in this case [34, 10].In this work we will be specially interested in the situation of mixed diffraction spec-trum. Namely, if Λ is a translation-bounded measure with non-pure point diffraction say p γ , we wish to know whether Λ still admits a natural ”purely diffractive part” Λ p whosediffraction would exactly be the pure point part p γ p , and such that Λ ´ Λ p would havediffraction the non-vanishing continuous part p γ c . That is to say, we raise the followingquestion: Q. Given a translation-bounded measure Λ on G having diffraction p γ “ p γ p ` p γ c , doesthere exist a decomposition of the form Λ “ Λ p ` Λ c , with Λ p and Λ c being translation-bounded measures having diffraction p γ p and p γ c respectively? As we will see, it is possible to give a rigorous answer to this question when one doesnot consider a translation-bounded measure Λ as a deterministic object, but rather a random one. Physically this means that the precise structure of the underlying alloy isunknown, although one knows the value of frequency of appearance of any possible localconfiguration. On the mathematical side this means that we do not consider a singletranslation-bounded measure but the whole collection M p G q of such measures, endowedwith a translation-invariant (ergodic) probability distribution m on it. In other wordswe shall deal here with an ergodic system of translation-bounded measures p X, G , m q , X Ă M p G q being the support of m . In the setting of (uniform Dirac combs over) pointsets one speaks of (ergodic) stationary point process [20, 7, 15], see also the more abstractapproach [28]. This approach by dynamical systems presents the great advantage toconnect the diffraction of a random translation-bounded measure modeled by a dynamical URE POINT/CONTINUOUS DECOMPOSITION OF MEASURES AND DIFFRACTION 3 system p X, G , m q , which is well-defined and unique, with the dynamical spectrum of p X, G , m q via the so-called Dworkin argument, see [16, 4], also Paragraph 3.4 in themain text. Overview on the content . Our main result (see Section 4) is that, for an ergodic randomtranslation-bounded measure the decomposition almost surely exists, and is provided bytwo other ergodic random translation-bounded measures which are naturally obtainedfrom the former. In the vocabulary of dynamical systems, this states as follows (Theorem4.1 and Theorem 4.6 in the main text):
Theorem 1.
Let p X, G , m q be an ergodic system of translation-bounded measures withdiffraction p γ “ p γ p ` p γ c . (i) There exist two ergodic systems of translation-bounded measures: ‚ p X p , G , m p q having diffraction p γ p , ‚ p X c , G , m c q having diffraction p γ c ,which are Borel factors of p X, G , m q under two Borel factor maps X Q Λ ✤ / / Λ p P X p X Q Λ ✤ / / Λ c P X c such that the equality Λ “ Λ p ` Λ c occurs for m -almost every Λ P X . (ii) The pair of systems of translation-bounded measures satisfying part (i) is unique,as well as the involved pair of Borel factor maps up to almost everywhere equality.
In this work we are particularly interested in the case where translation-boundedmeasures are weighted Meyer sets of G . These are weighted Dirac combs supported onsets of very special kind, the so-called Meyer sets. In this setting, the natural question is,given an ergodic system of weighted Meyer sets, to know whether translation-boundedmeasures in the spaces X p and X c resulting from Theorem 1 also are weighted Meyersets. As we will see, this turns out to be correct (see Section 5).In fact, much more is true: A particular property of weighted Meyer sets is thatthey are always supported on some point set of special feature, namely a model set [35, 42, 37, 2, 22]. Such point sets emerge if one not only consider the ”ambient” space G but a product H ˆ G of it with some locally compact Abelian group H , inside whichone consider (whenever it exists) a lattice Σ. Then denoting z } and z K the projections ofan element z P H ˆ G along G and H respectively, if one select a compact topologicallyregular subset W of H then one gives rise to a point set, called here a ”closed” modelset, of the form ∆ : “ ! z } P G : z P Σ , z K P W ) Closed model sets are always Meyer sets, and for a partial converse any weightedMeyer set is supported on a point set of this form [35, 37]. Considering this we have thefollowing (see Theorem 5.10):
Theorem 2.
Let p X, G , m q be an ergodic system of weighted Meyer sets. Then the fol-lowing property holds m -almost everywhere: Whenever Λ is supported on a closed model JEAN-BAPTISTE AUJOGUE set, then so are its summands Λ p and Λ c . This result in particular asserts that once X consists of weighted Meyer sets then sodoes X p and X c . It is in particular interesting (and surprising) to see the systematicemergence of weighted Meyer sets, namely the ones forming X c as long as this latter isnon-trivial, having (for almost all of them) purely continuous diffraction. Only few isknown about this latter type of weighted Meyer sets, although some study have beenperformed (on weighted lattices) for instance in [34].The remaining part (Section 6) of this work is about a generalization of our motivatingquestion. Namely, in the decomposition p γ “ p γ p ` p γ c stated earlier the continuous part p γ c always splits into two mutually singular components p γ c “ p γ ac ` p γ sc where p γ ac is its absolutely continuous part with respect to a fixed Haar measure on p G ,and p γ sc the remaining part, its singular continuous part. It is then natural to ask whethera translation-bounded measure Λ with diffraction p γ admits a decomposition of the formΛ “ Λ p ` Λ ac ` Λ sc , with Λ p , Λ ac and Λ sc being translation-bounded measures havingdiffraction p γ p , p γ ac and p γ sc respectively. As before, we shall consider this question on adynamical point of view and, given an ergodic system of translation bounded measures p X, G , m q , look for dynamical systems of translation-bounded measures p X ac , G , m ac q and p X sc , G , m sc q such that a generalized form of Theorem 1 holds.We could not prove or disprove the existence of such systems. Instead, we propose herea reformulation of this problem: Our observation here is that the two Borel factors welook for have, whenever they exist, diffraction being absolutely continuous with respectto p γ , hence of the form f. p γ for functions f on p G belonging in this case to L p p G , p γ q . Thenwhat we would get is a criterion on the existence of a Borel factor of p X, G , m q , whosediffraction is of the form f. p γ where the density function f is prescribed. Our aim is toprovide here such a criterion, which involves certain operators acting on L p X, m q calledhere admissible (see Definition 6.1), and that states as follows (see Theorem 6.6): Theorem 3.
Let p X, G , m q be a dynamical system of translation-bounded measureswith diffraction p γ . There is a bijective correspondence between: (a) Borel factor maps over some dynamical system of translation-bounded measureshaving diffraction p γ ! p γ , with Radon-Nikodym derivative d p γ d p γ P L p p G , p γ q , (b) admissible operators Q on L p X, m q .Given a Borel subset P Ď p G , one has d p γ d p γ “ I P if and only if | Q | “ E p P q P Θ , with E p P q the spectral projector associated to P and P Θ an appropriate projector on L p X, m q . Therefore, existence of our desired Borel factors rests on the existence of admissible op-erators on L p X, m q whose absolute value is given by an appropriate spectral projectors,something still tedious to prove due to the technicality of our admissibility condition. Itis worth precising that this correspondence does not encompasses many interesting fac-tors, as in some cases the knowledge of the diffraction of all factors allows to completelydetermine the dynamical spectrum, see the discussion in [33, 8]. URE POINT/CONTINUOUS DECOMPOSITION OF MEASURES AND DIFFRACTION 5 Generalities on dynamical systems
We set here some notions, notations and results about abstract dynamical systems. Allalong this article G denotes a second countable locally compact Abelian (LCA) group,which is then also σ -compact and metrizable, with group operation noted additively, andwith fixed Haar measure whose valuation on a Borel set A is noted | A | . The integral withrespect to the Haar measure will everywhere be noted ş G ... dt . Its Pontryagin dual p G isalso a second countable LCA group, whose group operation will be noted multiplicatively.2.1. Dynamical systems and Borel factors . By a dynamical system we mean a pair p X, G q consisting of a compact metrizablespace X together with a continuous G -action (noted on the right) X ˆ G Q p x, t q ✤ / / x.t P X Whenever X is equipped with a G -invariant (resp. ergodic) probability measure m then the resulting triple p X, G , m q will be called a measured (resp. ergodic) dynamicalsystem. The topological support Supp p m q of m is the smallest compact subset of X whose complementary set has measure zero, and p X, G , m q is said to have full supportwhenever Supp p m q “ X .If p X, G q is a dynamical system and x P X is chosen then we let X x , the hull of x , tobe the closure of the G -orbit of x in X , giving rise to another dynamical system p X x , G q with restricted G -action. A dynamical system p X, G q is called minimal if X x “ X forany x P X . The following proposition is proved in [18], Chapter 9, Proposition 2 . . Proposition 2.1.
In any ergodic system p X, G , m q the set of point x P X such that X x “ Supp p m q is Borel of full measure in X . Let us now consider two measured dynamical systems p X, G , m q and p X , G , m q . ABorel map π : X ÝÑ X is called a Borel G -map if for any fixed t P G the set of points x P X such that π p x.t q “ π p x q .t has full measure in X . It is called a Borel factor mapif in addition m is the push-forward of m by π , and in this case the system p X , G , m q is called a Borel factor of p X, G , m q . Whenever π is a Borel G -map (resp. a factormap) then any Borel function coinciding with π on a Borel set of full measure is again a G -map (resp. a factor map). By [18], Chapter 9, Proposition 2 . . G -map π : X ÝÑ X is almost everywhere equal to a Borel G -map ˜ π : X ÝÑ X admittinga G -stable Borel subset ˜ X of full measure in X such that ˜ π p x.t q “ ˜ π p x q .t hold for any x P ˜ X and any t P G . In particular one can deduce that a Borel factor of an ergodicdynamical system is also ergodic.Let p X, G , m q be an ergodic dynamical system, and p X , G , m q some Borel factor withBorel factor map π : X ÝÑ X . Denote L p X q the space of bounded complex-valuedBorel functions on X . Then by the Disintegration Theorem the measure m disintegratesover the factor X into a set of measures supported on fibers for the factor map π :Precisely, there exists a Borel map X Q x ✤ / / µ x P P rob p X q which satisfies the following two statements: JEAN-BAPTISTE AUJOGUE p D q The push-forward measure π p µ x q “ δ x for m -almost every x P X , p D q For each f P L p X q one has ż X f dm “ ż x P X „ż X f dµ x dm p x q .Moreover any two such maps coincide m -almost everywhere on X . The affirmationthat a disintegration is a Borel map means that it is a Borel map when P rob p X q isendowed with its vague topology, which exactly means that for each f P L p X q themapping associating to x P X the value ş X f dµ x is a Borel map. This also ensure thatthe right term of the equality in p D q is well defined. As a direct consequence of thealmost everywhere uniqueness, a disintegration is always a G -map in the following sense:As G acts on X continuously it also acts on the space M b p X q of bounded complex Radonmeasures of X via the formula µ.t : “ µ p . p´ t qq , and leave stable the space P rob p X q ofprobability measures on X . Then the almost everywhere uniqueness ensures that forany t P G one has µ x.t “ p µ x q .t for m -almost every x P X . Since any change of thedisintegration map on a set of measure 0 in X does not violate p D q and p D q one canchoose the disintegration to also satisfy p D q One has µ x.t “ p µ x q .t for any t P G and any x in a full Borel subset ˜ X .2.2. Unitary representation and dynamical spectrum . A measured dynamical system p X, G , m q gives naturally rise to a strongly continuousunitary representation of G on the separable Hilbert space L p X, m q , where on a function h P L p X, m q the unitary operator associated with t P G reads U t p h q “ h p . p´ t qq . FromStone’s theorem ([31], Section 36D, or Volume 2 of [17], Section 10.2 therein) there is anassociated projection-valued measure on the Borel sets of p G , E : Borel sets of p G ÝÑ Projectors of L p X, m q that is to say, a map such that E p P q is an orthogonal projector on L p X, m q for eachBorel subset P Ď p G , with E pHq the trivial operator and E p p G q the identity operator, andwhich is σ -additive on the Borel sets of p G . This projection-valued measure is uniquelycharacterized according to the following property: For any two h, k P L p X, m q the scalarproduct x h, E p . q k y sets a bounded complex measure on p G , such that one has x h, U t k y “ ż ω P p G ω p t q d x h, E p ω q k y (1)In other words x h, E p . q k y is the unique bounded complex measure on p G whose Fouriertransform is the continuous bounded function on G mapping t to x h, U t k y . A projectorof the form E p P q is called a spectral projector . One can deduces from equality (1) thata bounded operator commuting with any operators U t also commutes with any spec-tral projectors. In particular the spectral projectors themselves commute with any U t .As a result, the range of any spectral projector is a closed G -stable subspace of L p X, m q .For each ω P p G we simply denote E p ω q the spectral projector associated with the single-ton t ω u . As a consequence of (1), the range of E p ω q exactly consists of the eigenfunctions of the system p X, G , m q with eigenvalue ω , that is to say, functions h P L p X, m q suchthat U t p h q “ ω p t q h as L -functions for any t P G . A ω P p G is called an eigenvalue of thesystem p X, G , m q if it admits a non-trivial eigenfunction, that is, if the spectral projector E p ω q is non-trivial. A measure dynamical system p X, G , m q is said to have pure point URE POINT/CONTINUOUS DECOMPOSITION OF MEASURES AND DIFFRACTION 7 dynamical spectrum if the eigenvalues span L p X, m q , or equivalently, if the associatedprojection-valued measure E is purely atomic.2.3. The maximal Kronecker Borel factor of an ergodic dynamical system . When the measured dynamical system p X, G , m q is ergodic each eigenvalue ω has aunique eigenfunction up to a multiplicative constant, that is, the range of E p ω q is 1-dimensional in L p X, m q , and moreover any such eigenfunction is almost everywhereconstant in modulus and thus belongs to L p X, m q . Therefore the product of an eigen-function associated with ω with another associated with ω is well-defined, and is aneigenfunction associated with ω.ω . Hence the set of eigenvalues of an ergodic dynamicalsystem p X, G , m q always forms a (countable) subgroup E of p G .Consider now any subgroup E of p G , endowed with the discrete topology for which itis a LCA group. As discrete Abelian group, it admits a Pontryagin dual T E which is acompact Abelian group, on which G acts by rotation : The natural continuous injectionmorphism of E in p G leads by Pontryagin dualization a continuous group morphism i : G / / T E with dense range, which in turns define a G -action (set on the right) by z.t : “ z. i p t q .The resulting dynamical system p T E , G q is then a minimal Kronecker system , and thenormalized Haar measure on T E is the unique invariant measure with respect to thisaction, providing so the ergodic dynamical system p T E , G , m Haar q . Proposition 2.2.
Let p X, G , m q be an ergodic dynamical system and let E Ă p G itseigenvalue group. Then p T E , G , m Haar q is a Borel factor of p X, G , m q . For p X, G , m q an ergodic dynamical system with eigenvalue group E , the Borel factor p T E , G , m Haar q associated with E is called its maximal Kronecker Borel factor []. Theterm ”maximal” refers to the fact that any Kronecker Borel factor of p X, G , m q factorsthrough p T E , G , m Haar q . The celebrated Halmos-Von Neumann Theorem asserts that anergodic dynamical system has pure point dynamical spectrum if and only if it is Borelconjugated with its maximal Kronecker Borel factor.Given an ergodic dynamical system p X, G , m q with eigenvalue group E , any Borelfactor map π onto its maximal Kronecker Borel factor p T E , G , m Haar q induces a G -commuting isometric embedding of Hilbert spaces L p T E , m Haar q Q h ✤ / / h ˝ π P L p X, m q It maps a continuous character on T E , which corresponds to some ω P E by Pontryaginduality, to a normalized eigenfunction of the system whose eigenvalue is ω . As a byprod-uct, its range is precisely the closed G -stable Hilbert subspace H p of L p X, m q spannedby the eigenfunctions of the system. The orthogonal projection onto this subspace is thespectral projector E p E q associated to the countable subset E Ă p G , and can be linked withthe disintegration theorem applied to the factor map π over T E (regardless the choice of π , which is essentially unique): JEAN-BAPTISTE AUJOGUE
Proposition 2.3. E p E qp f q is equal, for any f P L p X, m q , to the L -class of x ✤ / / ż X f dµ π p x q Translation-bounded measures
Basics on Fourier analysis . Here we mainly follow [4, 42]. The C -vector space of compactly supported complexvalued continuous functions on G will be denoted C c p G q , and is topologized as follows:For each compact subset K Ă G let C K p G q be the subspace of functions supported on K equipped with the suppremum norm } . } , for which it is a Banach space. Then let p K N q N be a nested sequence of compact regular sets in G whose interiors cover G : Itgives C c p G q “ Y N C K N p G q and defines an inductive limit topology T lim on C c p G q whichis the weakest making the natural inclusions p C K N p G q , } . } q (cid:31) (cid:127) / / p C c p G q , T lim q continuous. This topology does not depend on the choice of nested sequence p K N q N ,and the resulting space p C c p G q , T lim q is a separable topological vector space. The space C c p G q is closed under the operations φ ´ : “ φ p´ . q and r φ : “ φ p´ . q , and under convolutionproduct φ ˚ ψ p t q : “ ş G φ p s q ψ p t ´ s q ds . One has the usual Fourier transform ([41]) C c p G q Q φ ✤ / / p φ P C p p G q with p φ p ω q : “ ż G φ p t q ω p t q dt The space C p p G q of continuous complex valued functions on p G null at infinity is closedunder pointwise product, and one has the relations z p φ ´ q “ p p φ q ´ , pr φ “ p φ and z φ ˚ ψ “ p φ. p ψ . We now turn to complex Borel measures on G . The dual space C c p G q ˚ of C c p G q isby construction of the topology T lim the space of linear functionals on C c p G q which arebounded on each subspace p C K N p G q , } . } q (with a constant depending on K N ), and isprecisely given, according to the Riesz-Markov representation Theorem ([39], TheoremIV. 18 therein), by the space M p G q of (possibly unbounded) complex Borel measures on G , where a functional in C c p G q ˚ is given by the integral against a measure in M p G q . Thevague topology T vague on M p G q is the weak- ˚ topology on C c p G q ˚ after identification,which is the weakest topology making the functionsΛ ✤ / / N φ p Λ q : “ ż G φ ´ d Λ “ ż G φ p´ s q d Λ p s q (2)continuous on M p G q for each φ P C c p G q . Hence defined the space p M p G q , T vague q is alocally convex topological vector space. It carries a natural G -action defined for t P G byΛ ✤ / / Λ ˚ δ t , where the convolution product of two convolable measures is determinedon φ P C c p G q by ş G φ d Λ ˚ Λ : “ ş G ş G φ p s ` t q d Λ p s q d Λ p t q . There is a natural involutionon Λ P M p G q set by r Λ p φ q “ Λ p r φ q for φ P C c p G q .3.2. Dynamical systems of translation-bounded measures . We denote below p K, M q to be a pair with K Ă G compact of non-empty interior and M ě
0. A complex Borel measure Λ P M p G q is called p K, M q -translation-bounded if | Λ | p K ` t q ď M for each t P G URE POINT/CONTINUOUS DECOMPOSITION OF MEASURES AND DIFFRACTION 9
Here, | Λ | is the absolute value, or total variation of Λ, which is the least positivecomplex Borel measure on G such that | Λ p B q| ď | Λ | p B q for any Borel B . A complexBorel measure in M p G q is called translation-bounded if it is p K, M q -translation-boundedfor some p K, M q . The collection of p K, M q -translation-bounded measures on G will bedenoted M p K,M q p G q , and that of translation-bounded measures M p G q . It is clear that M p G q , when endowed with the vague topology, is a topological vector subspace of M p G q that remains stable under the natural G -action. Proposition 3.1. [4]
Each set M p K,M q p G q is a metrizable compact convex subset of M p G q , on which the natural G -action is well defined and continuous. Compactness and metrizability is proved in [4], Theorem 2, and continuity of the G -action in Proposition 2 therein. The convexity statement is ensured by the inequality | α Λ ` p ´ α q Λ | ď α | Λ | ` p ´ α q | Λ | holding for any two measures Λ , Λ and α P r , s . Definition 3.2.
A measured dynamical system of translation-bounded measures is ameasured dynamical system p X, G , m q , with X a compact subset of some M p K,M q p G q and the G -action given by X ˆ G Q p Λ , t q ✤ / / Λ ˚ δ t P X . A way to generate dynamical systems of translation-bounded measures is to start witha particular measure Λ in some M p K,M q p G q , and consider the closure X Λ of its G -orbitin M p K,M q p G q which we shall call its hull . Once a G -invariant probability measure m on X Λ is chosen one ends up with a system of the desired form. To a translation-boundedmeasure can be associated as in [4] its rubber local isomorphism class , or RLI-class, tobe the collection of all translation-bounded measures locally isomorphic with Λ,RLI p Λ q : “ t Λ P M p G q | X Λ “ X Λ u (3)It is not hard to show that the RLI-class of a translation-bounded measure Λ is a Borel G -stable subset of M p G q (it is even a G δ ), and moreover one clearly has RLI p Λ q Ď X Λ .Given a dynamical system of translation-bounded measures p X, G , m q , it is clear thatwhenever Λ belongs to X then the entire class RLI p Λ q also belongs to X , so that X always partitions into disjoint RLI-classes. Obviously p X, G q is minimal if and only if X contains a single RLI-class. If p X, G , m q is ergodic, one knows by Proposition 2.1 thatalmost all Λ P X have the support of m as hull. These elements form, as it is easy toremark, a single RLI-class in X , yielding: Proposition 3.3.
An ergodic system of translation-bounded measures p X, G , m q admits asingle RLI -class of full m -measure in X , formed of the Λ P X satisfying X Λ “ Supp p m q . This will be of main importance in Section 5. Any dynamical system of translation-bounded measures p X, G , m q has the natural set of continuous functions on X given bythe restriction of functions (2) on the compact subset X , C c p G q Q φ ✤ / / N φ “ N Xφ P C p X q , N φ p Λ q “ ż G φ ´ d Λ(4)This map is C -linear and moreover satisfies N φ p . ˚ δ t q “ N φ ˚ δ t whenever t P G . Intensities . Let M p K,M q p G q be the compact and convex subset of p K, M q -translation-boundedmeasures in M p G q . The following statement is a particular case of a general result onvector-valued integration theory, see [41], Theorem 3.27 therein: Theorem 3.4.
Suppose that X is a compact subset of M p K,M q p G q . Then any boundedcomplex measure µ P M b p X q defines a translation-bounded measure i p µ q P M p G q by ż G φ d i p µ q “ ż X ż G φ d Λ dµ p Λ q If moreover µ is a probability measure then i p µ q belongs to M p K,M q p G q . Given a compact subset X of some M p K,M q p G q and µ a complex measure on X , the re-sulting translation-bounded measure i p µ q is sometimes called the integral, or barycenter,or also first moment of µ on X . However we shall call it here intensity . This terminol-ogy comes from point-set theory, where one studies Delone sets of G (see for instance[4]): Given a compact space X of Delone sets of G (with common radius of uniformdiscreteness), the intensity I p µ q of a bounded measure µ on X is the translation-boundedmeasure obtained ([19, 20]) on any Borel set A Ď G by I p µ qp A q “ ż X S X A q dµ p S q A compact space X of Delone sets can be identified with a compact subset X of some M p K,M q p G q by replacing a point set S by the Dirac comb δ S : “ ř t P S δ t (see [4], Section4 and Theorem 4 therein). Hence a measure µ on X can alternatively be viewed as livingon X , and its intensity as given just above is equal to the translation-bounded measureyield by Theorem 3.4.Recall that if X is a compact G -stable subset of M p K,M q p G q then G acts on the space M b p X q of bounded complex Radon measures of X via the formula µ.t : “ µ p . ˚ δ ´ t q . Thefollowing is then straightforward to prove: Proposition 3.5.
Let X be a compact G -stable subset of M p K,M q p G q , with M b p X q itsspace of bounded complex measures. Then the map p M b p X q , T vague q Q µ ✤ / / i p µ q P p M p G q , T vague q is continuous and G -commuting, that is, i p µ p . ˚ δ ´ t qq “ i p µ q ˚ δ t for any t P G . Mathematical diffraction and connection with dynamical spectrum . We provide here a very concise survey on diffraction, and refer the reader to [21, 42, 4,5] for more details, comments and proofs. The diffraction of a translation-bounded mea-sure Λ is defined by first considering a Van Hove sequence [42] of subsets p A n q n P N , formwhich one considers the following vague limit in M p G q of truncated auto-correlations γ Λ : “ lim n Ñ8 | A n | Λ | A n ˚ Ć Λ | A n (5)The above sequence may not converges in M p G q but it always will once one extractan appropriate sub-sequence (one shows by mimicking the proof of Proposition 2.2 of [21] URE POINT/CONTINUOUS DECOMPOSITION OF MEASURES AND DIFFRACTION 11 and using Lemma 1.1 in [42] that such sequence belongs to a certain M p K,M q p G q , and theclaim comes from compacity and metrizability of this latter set). Thus by consideringthis sub-sequence in (5) one can get rid of this difficulty. Moreover again up to extractionone can assume that the Van Hove sequence is tempered ([30], Proposition 1.4). The limit γ Λ is the auto-correlation measure of Λ P M p G q (with respect to the Van Hove sequence p A n q n P N ), and is translation-bounded. Such measure is moreover positive-definite , henceFourier transformable ([47], Theorem 4.7): There exists a unique Borel measure p γ Λ onthe Pontryagin dual p G , positive and translation-bounded, such that at any φ P C c p G q ż p G | p φ ´ | d p γ Λ “ ż G φ ˚ r φ dγ Λ (6)Whenever it exists, we call p γ Λ the diffraction measure of Λ P M p G q (with respect tothe sequence p A n q n P N ). Within a given dynamical system of translation-bounded mea-sures p X, G , m q different measures will have, in principle, different diffraction measures.However when m is ergodic there is a unique typical resulting diffraction, that is, a mea-sure p γ on p G such that p γ “ p γ Λ for m -almost every Λ P X and along any tempered VanHove sequence (see [4], Theorem 5). In fact, it is possible to get rid of the ergodicityassumption and provide a certain definition for the diffraction measure of a measureddynamical system of translation-bounded measures p X, G , m q , in a way that supplies the m -almost sure diffraction in the case of ergodicity: Theorem 3.6. [4]
Let p X, G , m q be a measured dynamical system of translation-boundedmeasures. Then there exists a unique positive measure p γ on p G satisfying the equality foreach φ, ψ P C c p G q ż p G p φ p ψ d p γ “ ż X N φ N ψ dm (7) When p X, G , m q is ergodic the measure p γ is its m -almost sure diffraction measure. The proof is given by combining Theorem 5(b) and Lemma 7 of [4] together withthe formula defining the diffraction (6). The formula (7) defining the diffraction of ameasured dynamical system of translation-bounded measures extends to the followingformula, as shown in Proposition 7 in [4], for any Borel subset P Ď p G ż P p φ p ψ d p γ “ ż X E p P qp N φ q E p P qp N ψ q dm (8)where E p P q stands for the spectral projector associated with P on L p X, m q . Thetheorem above provides an efficient tool to deal with the diffraction measure: As it wassuggested in [16, 42, 4] and latter on explicitly formulated in [15, 28, 33], the equalityset in the theorem above is nothing but an isometric embeddingΘ : L p p G , p γ q (cid:31) (cid:127) / / L p X, m q (9)where on the dense subspace of L p p G , p γ q formed of the Fourier transforms p φ of com-pactly supported functions φ P C c p G q the mapping Θ writes Θ p p φ q : “ N φ . This is theso-called diffraction-to-dynamic map of p X, G , m q , called after [28]. Its range H Θ isthe closed subspace of L p X, m q spanned by continuous functions on X of the form N φ with φ P C c p G q , which is in general not be the whole space L p X, m q . This subspace is G -stable and thus its orthogonal projection, which we denote P Θ , commutes with the G -representation, and consequently also with any spectral projector. Now using the map-ping Θ one can check that formula (8) admits the following equivalent form: Denotingfor f P L p p G , p γ q the associated multiplication operator on L p p G , p γ q by M f , one has forany Borel subset P Ď p G Θ ˝ M I P ˝ Θ ´ “ E p P q P Θ (10)An alternative way to observe this is to note that both M I P and E p P q P Θ are projec-tors, which are in fact spectral projectors for appropriates G -representations on L p p G , p γ q and H Θ (see Section 3 of [28] for more about these representations). These represen-tations are intertwined by the diffraction to dynamic map, hence naturally yielding theintertwining formula (10) for the associated spectral projectors. The fundamental con-nection between the dynamical and diffraction spectra of a measured dynamical systemof translation-bounded measures states as follows, where point p i q straightly follows fromformula (10) above whereas point p ii q is shown in [4], Theorems 6, 7 and 9 therein: Theorem 3.7. [4]
Let p X, G , m q be a measured dynamical system of translation-boundedmeasures, with diffraction measure p γ and projection-valued measure E . Then:(i) p γ is absolutely continuous with respect to E . In particular the set S of Bragg peaks,the atoms of p γ , belongs to the eigenvalue group E , the atoms of E .(ii) p γ is a pure point measure if and only if E is a pure point measure. In this casethe set of Bragg peaks S algebraically generates the eigenvalue group E . The decomposition of translation-bounded measures
Existence of the decomposition . In this section we give the proof of our main result:
Theorem 4.1.
Let p X, G , m q be an ergodic system of translation-bounded measures withdiffraction p γ “ p γ p ` p γ c . There exist two ergodic systems of translation-bounded measures: ‚ p X p , G , m p q having diffraction p γ p , ‚ p X c , G , m c q having diffraction p γ c ,which are Borel factors of p X, G , m q under two Borel factor maps X Q Λ ✤ / / Λ p P X p X Q Λ ✤ / / Λ c P X c such that the equality Λ “ Λ p ` Λ c occurs for m -almost every Λ P X ..Proof. Let us begin with the construction of the two desired dynamical systems. First,denoting E the eigenvalue group of p X, G , m q one has by Proposition 2.2 a Borel factormap form X to T E . Composing this factor map with the disintegration of the measure m over T E gives us a Borel G -map µ p : X Q Λ ✤ / / µ pΛ : “ µ π p Λ q P M b p X q URE POINT/CONTINUOUS DECOMPOSITION OF MEASURES AND DIFFRACTION 13 having values in the subset of probability measures on X . Now the Borel factor mapform X to T E dualizes in a G -commuting isometric embedding of L p T E q in L p X, m q with range the closed Hilbert subspace H p spanned by eigenfunctions of the system p X, G , m q . The orthogonal projector onto H p is the spectral projector P p : “ E p E q associ-ated with the Borel subset E Ă p G , and for each f P C p X q the L -class of Λ ÞÝÑ ş X f dµ pΛ is by Proposition 2.3 equal to P p p f q . On the other hand one has another Borel map,which is straightforwardly shown to be a G -map, µ c : X Q Λ ✤ / / µ cΛ : “ δ Λ ´ µ pΛ P M b p X q with the property that for each f P C p X q the L -class of Λ ÞÝÑ ş X f dµ cΛ is given by f ´ E p E qp f q “ E p p G z E qp f q “ : P c p f q , the orthogonal projection of f onto the subspace H c orthogonal to H p . It is obvious that both µ p and µ c are valued in the compact G -stablesubset M b p X q of M b p X q of signed Radon measures of total variation less of equal to 2.Denote by Cv p X q the image under the intensity map of Proposition 3.5 of the compact G -stable subset M b p X q : It is then a compact G -stable subset of M p G q . By composingthe previous Borel G -maps with the barycenter map one gets two Borel G -maps π p : X Q Λ / / π p p Λ q : “ i p µ pΛ q P Cv p X q π c : X Q Λ / / π c p Λ q : “ i p µ cΛ q P Cv p X q Pushforwarding the ergodic measure m by there Borel G -maps yield two ergodic prob-ability measures m p and m c supported on Cv p X q , whose support are two compact G -stable subsets of Cv p X q which we denote X p and X c respectively. The subset of Λ P X having images π p p Λ q and π c p Λ q in X p and X c respectively is a full Borel subset, so afterpossibly modifying the maps π p and π c on a set of measure 0 of X we can assume thatthese are valued in X p and X c respectively. What we therefore have is two ergodic sys-tems of translation-bounded measures p X p , G , m p q and p X c , G , m c q as well as two Borelfactor maps X π p / / X p X π c / / X c Our aim in the sequel is then to show that these two Borel factors satisfy the desiredproperties. First we point a remark on the summands p γ p and p γ c of the diffraction p γ : firstthese are the restriction of p γ on the Borel subsets E and p G z E respectively. Moreover,formula (8) writes ż P p φ p ψ d p γ “ ż X E p P qp N φ q E p P qp N ψ q dm with E p P q the spectral projector associated to P on L p X, m q so in the particularcases of P “ E or p G z E this gives, with the notations P p : “ E p E q and P c : “ E p p G z E q , ż p G p φ p ψ d p γ p “ ż X P p p N φ q P p p N ψ q dm ż p G p φ p ψ d p γ c “ ż X P c p N φ q P c p N ψ q dm (11)Let us then show that our two systems have diffraction p γ p and p γ c equal to p γ p and p γ c respectively. For either α “ p or c consider C c p G q Q φ ✤ / / N αφ P C p X α q , N αφ p Λ q : “ ż φ ´ d Λ It comes for each φ P C c p G q that N αφ ˝ π α has L -class equal to P α p N φ q in L p X, m q .Indeed it follows from the series of almost everywhere equalities N αφ ˝ π α p Λ q “ ż G φ ´ dπ α p Λ q “ ż G φ ´ d i p µ α Λ q “ ż X „ż G φ ´ d Λ dµ α Λ p Λ q “ ż X N φ dµ α Λ whose L -class is from what has been said earlier in this proof equal to P α p N φ q in L p X, m q . One therefore has, invoking equality of Theorem 3.6 combined with (11), ż p φ p ψ d p γ α “ ż X α N αφ N αψ dm α “ ż X P α p N φ q P α p N ψ q dm “ ż p φ p ψ d p γ α yielding the desired equalities. It then remains to show the decomposition statement,that is, we need to show that Λ “ π p p Λ q ` π c p Λ q for almost every Λ P X , but this comesfrom the straightforward computation for any φ P C c p G q ż G φ ´ d p π p p Λ q ` π c p Λ qq “ ż G φ ´ dπ p p Λ q ` ż G φ ´ dπ c p Λ q “ N p φ p π p p Λ qq ` N c φ p π c p Λ qq which is in turn equal for m -almost every Λ P X to P p p N φ qp Λ q ` P c p N φ qp Λ q “ N φ p Λ q “ ż G φ ´ d ΛApplying this to a countable dense collection of functions φ P C c p G q one deduces theexistence of a Borel subset of full measure in X such that the equality Λ “ π p p Λ q ` π c p Λ q occurs, yielding the proof. (cid:3) Let us make a few remarks on this result. Here we consider an ergodic system oftranslation-bounded measures p X, G , m q with eigenvalue group E and diffraction p γ . Remark 4.2.
We shall point a remark about the eigenvalue groups of the two systemsyield by Theorem 4.1. The first associated system p X p , G , m p q has by constructionpure point diffraction, and thus by Theorem 3.7 pure point dynamical spectrum, witheigenvalue group E p a subgroup of E . This latter inclusion may however be strict: Indeedthe set of Bragg peaks of p X p , G , m p q is precisely the set S of atoms of p γ , and thereforethe eigenvalue group E p is the subgroup of E algebraically generated by S , which mightnot be the entire group E . On the other hand, The second associated system p X c , G , m c q has by construction no Bragg peak (not even at 0), but it remains unclear to us whetherits eigenvalue group is trivial or not (this latter must however belong to the group E ). Remark 4.3.
The autoccorelation γ of a generic element in X is positive-definite, hencea weakly almost periodic measure on G [1]. As such, its admits a unique decompositioninto the sum of a strongly almost periodic measure γ sap and a null-weakly almost periodic one γ wap , which are both Fourier transformable, and such that y γ sap “ p γ p and z γ wap “ p γ c ,see [1] for definitions and proofs. In our situation the two systems p X p , G , m p q and p X c , G , m c q have diffraction p γ p and p γ c respectively, so it straightly follows that the auto-correlation of a generic element of p X p , G , m p q is exactly the strong almost periodic part γ sap of γ while a generic element of p X c , G , m c q has auto-correlation the null weaklyalmost periodic part γ wap of γ . Remark 4.4.
The translation-bounded measures belonging in p X p , G , m p q are, for al-most all of them, the intensity of a certain probability measure on X . It then a directverification that if X consists of positive translation-bounded measures then so does X p .In case translation-bounded measures in X are only signed then the translation-bounded URE POINT/CONTINUOUS DECOMPOSITION OF MEASURES AND DIFFRACTION 15 measures of both X p and X c are signed as well.In Theorem 4.1 just above the uniqueness part, that is, part (ii) in the statement ofTheorem 1 in the introduction, is absent. We shall prove this in Paragraph 4.3 below.Before we found elegant to turn this uniqueness statement into an operator-theoreticformalism, as it is done in next paragraph, allowing us to set a proof by only invokingstandard arguments from operator theory.4.2. From Borel factors to operators . Let p X, G , m q be a measured dynamical system of translation-bounded measures, withdiffraction to dynamic mapΘ : L p p G , p γ q o o / / H Θ Ď L p X, m q , Θ p p φ q “ N φ for any φ P C c p G q . Recall that P Θ stands for the orthogonal projection onto H Θ . Nowassume we are given a dynamical system of translation-bounded measures p X π , G , m π q which is a Borel factor of p X, G , m q under a Borel factor map π : X ÝÑ X π . Suppose inaddition that its diffraction measure p γ π is absolutely continuous with respect to p γ , withan essentially bounded Radon-Nikodym differential f π : “ d p γ π d p γ P L p p G , p γ q . Let us denote C c p G q Q φ ✤ / / N πφ P C p X π q , N πφ p Λ q : “ ż φ ´ d ΛWe shall here construct a bounded linear operator Q π on L p X, m q , which characterizesthe factor map π , hence the factor system p X π , G , m π q itself, in a unique way: Proposition 4.5.
Let π : X ÝÑ X π with p X π , G , m π q as above. (i) There exists a unique Q π P B p L p X, m qq such that N πφ ˝ π “ Q π p N φ q in the L sense for any φ P C c p G q and subject to the condition that Q π “ Q π P Θ . (ii) For π , π as above then Q π “ Q π if and only if π “ π almost everywhere on X . (iii) The following equality holds, with M f π the multiplication operator by f π on L p p G , p γ q , Θ ˝ M f π ˝ Θ ´ “ Q ˚ π Q π Proof. (i) First the factor map π induces a G -comuting isometry i π : L p X π , m π q Q f ✤ / / f ˝ π P L p X, m q Denote the diffraction to dynamic map of p X π , G , m π q byΘ π : L p p G , p γ π q o o / / H Θ π Ď L p X π , m π q Now p γ π “ f π p γ for some f π P L p p G , p γ q and therefore } f } L p p G , p γπ q ď } f π } ,ess } f } L p p G , p γ q for any f P L p p G , p γ q . This shows that the identity map from L p p G , p γ q to L p p G , p γ π q is well-defined and a bounded linear map, and thus there exists a unique bounded operator r Q π making the diagram commutative L p p G , p γ q id (cid:15) (cid:15) o o Θ / / H Θ r Q π (cid:15) (cid:15) L p p G , p γ π q o o Θ π / / H Θ π Now the operator Q π : “ i π ˝ r Q π ˝ P Θ is a well-defined linear operator of L p X, m q ,which is bounded with operator norm } Q π } op ď } f π } ,ess as it is not difficult to check.Byconstruction one has Q π “ Q π P Θ , and for any φ P C c p G q the almost everywhere equalities Q π p N φ q “ i π ˝ r Q π ˝ Θ p p φ q “ i π p Θ π p p φ qqq “ i π pp N πφ qq “ N πφ ˝ π (12)Uniqueness of the operator Q π having these properties is clear, as it is uniquely definedon the whole subspace H Θ and must vanish on its orthogonal space, giving (i).(ii) The given equivalence statement easily follows, as one can observe that π “ π almost everywhere on X if and only if, for any φ P C c p G q , one has for almost all Λ P X ż φ ´ d π p Λ q “ ż φ ´ d π p Λ q , that is, according to equalities (12), if and only is Q π p N φ q “ Q π p N φ q for any φ P C c p G q .This latter condition holds if and only if Q π and Q π coincide on the closed subspace H Θ , and since by construction one has always Q π “ Q π P Θ , with P Θ the orthogonalprojection onto H Θ , this is equivalent to have Q π “ Q π on all L p X, m q , as desired.(iii) For the last point it suffices to check the equality in scalar products againstfunctions N φ , N ψ for φ, ψ P C c p G q : One has x Θ ˝ M f π ˝ Θ ´ p N φ q , N ψ y m “ x M f π p φ, p ψ y p γ “ ż f π p φ p ψ d p γ “ ż p φ p ψ d p γ π “ x N πφ , N πψ y m π on one hand, equal by point (i) of this proof to x Q π p N φ q , Q π p N ψ qy m “ x Q ˚ π Q π p N φ q , N ψ y m which settles the proof. (cid:3) Uniqueness of decomposition of measures . Using the result of the earlier paragraph we shall now prove uniqueness of the decom-position set in Theorem 4.1:
Theorem 4.6.
Let p X, G , m q be an ergodic system of translation-bounded measures.Then the two ergodic systems provided in Theorem 4.1 exist in a unique way, as well asthe two Borel factor maps up to almost everywhere equality.Proof. Let S be the countable subset of atoms of p γ , with complementary set p G z S . Letthen p X p , G , m p q and p X p , G , m p q be Borel factors of p X, G , m q under two Borel factormaps π p and π c , with respective diffraction p γ p “ I S p γ and p γ p “ I p G z S p γ and such thatΛ “ π p p Λ q ` π c p Λ q hold for almost every Λ P X , as in Theorem 4.1. By Proposition 4.5 URE POINT/CONTINUOUS DECOMPOSITION OF MEASURES AND DIFFRACTION 17 the factor maps π p and π c give rise to a pair of operators Q and Q on L p X, m q . Thenit will be sufficient to show that Q “ E p S q P Θ , Q “ E p p G z S q P Θ (13)Indeed by the equivalence set in Proposition 4.5 this will readily ensure uniqueness ofthe pair of Borel factors, and that of Borel factor maps up to almost everywhere equality.To show equalities (13), first observe that Q ` Q “ P Θ : Indeed this follows from theequalities holding for any φ P C c p G q and almost everyΛ P XQ p N φ qp Λ q ` Q p N φ qp Λ q “ N π p φ p Λ q ` N π c φ p Λ q “ ż φ ´ d π p p Λ q ` ż φ ´ d π c p Λ q “ N φ p Λ q the latest equality being given by the almost everywhere equality Λ “ π p p Λ q ` π c p Λ q .Second, observe that the assumption on diffraction implies by point (iii) of Proposition4.5 that Q ˚ Q and Q ˚ Q are equal to the operators Θ ˝ M I S ˝ Θ ´ and Θ ˝ M I p G z S ˝ Θ ´ respectively. From formula (10) one deduces that the absolute values of Q and Q (see[39] for definition) satisfy | Q | “ | Q | “ Q ˚ Q “ E p S q P Θ whereas | Q | “ | Q | “ Q ˚ Q “ E p p G z S q P Θ . Now the bounded operators Q and Q admit a polar decomposition ([39],Theorem VI.10 therein) of in the form Q “ V | Q | and Q “ V | Q | , where V and V arepartial isometries with initial spaces the ranges of | Q | and | Q | respectively, that is to say,the ranges of the projectors E p S q P Θ and E p p G z S q P Θ . Then to show equalities (13) it onlyremains to show that these partial isometries are the identity on their respective initialspace. Since Q ` Q “ P Θ one has for any h in the initial space of V that, remindingthat E p S q P Θ and E p p G z S q P Θ have orthogonal ranges, V p h q “ V E p S q P Θ p h q “ V E p S q P Θ p h q ` V E p p G z S q P Θ p h q “ p Q ` Q qp h q “ h whereas for any h in the initial space of V that V p h q “ V E p p G z S q P Θ p h q “ V E p S q P Θ p h q ` V E p p G z S q P Θ p h q “ p Q ` Q qp h q “ h as desired. (cid:3) Let us remark that, given dynamical system of translation-bounded measures p X, G , m q with diffraction p γ , if one drops the requirement of forming a decomposition of (almostevery) measures in X then one may exhibit several different Borel factors having diffrac-tion p γ p and p γ c respectively. Indeed given a pair of such Borel factors it suffices to considerfor each another system of translation-bounded measures which is Borel conjugated and homometric , that is, with same diffraction, which even in the purely diffractive situation(in this case any two homometric systems are Borel conjugated) seems very likely toexist in general [46].4.4. Support of positive translation-bounded measures . In this paragraph we shall exclusively deal with positive translation-bounded measures.Here we will be interested, given an ergodic system p X, G , m q of positive translation-bounded measures, by the support of the positive, according to Remark 4.4, translation-bounded measures of the system p X p , G , m p q resulting from Theorem 4.1. At first oneexpect, as a very consequence of the ergodicity property, that the support of these mea-sures should be large. But large in which sense ? For instance, does these measureshave relatively dense support, or at least almost all of them ? Here a subset B Ă G isrelatively dense if there exists a compact set K Ă G with B ` K “ G . This is not atrivial point and it seems judicious to deal with a weaker notion than relative density. Such a weaker notion is set in Proposition 4.7 just below. First select a Van Hovesequence p A k q k P N of G , which always exist in σ -compact LCA groups [42]. We let thensuch a sequence be chosen. Then one defines the density of a subset B Ď G (along theVan Hove sequence p A k q k P N ) is, whenever it exists, given by dens p B q : “ lim k Ñ8 | B X A k || A k | This definition of density may differ from the one used elsewhere in the literature,specially for point sets, whose density always vanish in our sense. The density of asubset B Ď G as defined above is, if it exists, a value comprised between 0 and 1, andoften depends on the choice of Van Hove sequence made at the beginning. Using thisone has: Proposition 4.7.
Let p X, G , m q be a non-trivial ergodic system of positive translation-bounded measures. Then almost any Λ P X has the following property: for any ε ą there is a compact set K Ă G such that dens p Supp p Λ q ` K q is greater than ´ ε .Proof. Consider an increasing sequence of symmetric open sets U n Ă G with compactclosure and whose union is all G , and set Q n to be the subset of Λ P X such that0 P Supp p Λ q ` U n . Then each Q n is Borel in X . For, consider a sequence φ n P C c p G q ofcontinuous functions, with φ n being nowhere vanishing inside U n and entirely vanishingoutside for each n P N (such functions always exist because G is metrizable). Thenfor each Λ P X and s P G there is an equivalence between having s R Supp p Λ q ` U n , Λ p U n ` s q “ ş φ n p . ´ s q d Λ “
0. Indeed equivalence between the two lastconditions is obvious, whereas if Λ p U n ` s q ą Supp p Λ q must cross U n ` s , yielding s P Supp p Λ q ´ U n “ Supp p Λ q ` U n , and in the other direction if there is an s withΛ p U n ` s q “ Supp p Λ q cannot intersect the open set U n ` s and thus s cannotbelong to Supp p Λ q ` U n . As a result, one is allowed to write Q n “ N ´ φ n ps `8rq , whichis open and thus Borel in X .Now the Pointwise Ergodic Theorem (see Theorem 7.1, and [38] for further details),when applied to the Borel indicator function I Q n , yields a Borel subset of full measure X p n q Ď X such that one has for any Λ P X p n q the convergence1 | A k | ż A k I Q n p Λ .s q ds k Ñ 8 / / ż X I Q n dm “ m p Q n q However I Q n p Λ .s q ‰ P Supp p Λ .s q ` U n “ Supp p Λ q ´ s ` U n , that is, s P Supp p Λ q ` U n so that the left terms are nothing but1 | A k | ż A k I Q n p Λ .s q ds “ |p Supp p Λ q ` U n q X A k || A k | so that we arrive by taking limit to the equality, which holds for any Λ P X p n q , dens p Supp p Λ q ` U n q “ m p Q n q Let us show that m p Q n q converges to 1: First, an the sequence U n is increasing in G thesequence Q n is also increasing in X . Therefore m p Q n q converges to m p Ť n Q n q . Now a Λnot belonging in Ť n Q n must satisfy 0 R Supp p Λ q ` U n for each integer n , which is onlypossible for the everywhere trivial measure. Since p X, G , m q was assumed non-trivial itfollows that Ť n Q n has complementary set of measure 0, so is of measure 1 in X , asdesired. URE POINT/CONTINUOUS DECOMPOSITION OF MEASURES AND DIFFRACTION 19
As a result, for any Λ belonging in the Borel susbset of full measure X p8q given bythe countable intersection of the X p n q one has the convergence dens p Supp p Λ q ` U n q “ m p Q n q n Ñ 8 / / (cid:3) This proposition in particular apply for the system p X p , G , m p q associated by Theorem4.1 to any given ergodic system p X, G , m q of positive translation-bounded measures. Nowreturning to relative density, to complete the above Proposition we have the followingresult: Proposition 4.8.
Let p X, G , m q be an ergodic system of positive translation-boundedmeasures. Assume that almost any measure in X have relatively dense support. Thenalmost any measure in X p also have relatively dense support.Proof. Let U Ă G be a symmetric open subset with compact closure. Then the collection X U Ď X of translation-bounded measures having U -relatively dense support is a G -stableBorel subset of X . For, it is obviously G -stable and moreover, from an argument of theproof of Proposition 4.7,Λ P X U ðñ Λ p U ` s q ą s P G (14)Now considering a continuous function φ P C c p G q nowhere vanishing inside U and nulloutside U (such a function exists since G is metrizable), one gets that the conditionΛ p U ` s q ą s P G is equivalent to have ş φ p . ` s q d Λ ą s P G .Now consider a sequence p K k q k P N of compact sets in G whose union covers G . Since ş φ p . ` s q Λ is continuous in the variable s , our condition is in turns equivalent to have inf s P K k ş φ p . ` s q d Λ ě δ k for some δ ą
0, for each k P N . Again by continuity inthe variable s its infimum over s P K k can be set on a countable dense subset of K k ,independent on Λ, showing that F k p Λ q : “ inf s P K k ş φ p . ` s q d Λ is the infimum of acountable collection of continuous functions on X and thus is Borel on X . One concludesthat X U is Borel by observing that X U “ č k P N F ´ k ps , `8rq Now consider an increasing sequence of symmetric open subset U n Ă G with compactclosure and whose union covers G . Since almost any measure in X have relatively densesupport it follows that the increasing countable union of Borel sets X U n has measure1. Therefore some of those must have non zero measure, which we simply denote X U ,and since it is a Borel G -stable subset then ergodicity ensure that m p X U q “
1. Nowthe Disintegration Theorem ensures, when applied to the Borel indicator function I X U ,that the set X U of Λ P X such that X U has µ π p Λ q -measure 1 is also of full measure in X . Let moreover X U be the Borel subset of full measure in X such that the equalityΛ p “ i p µ π p Λ q q holds. Then for any Λ belonging simultaneously to these three sets onehas Λ p p U ` t q “ ż X Λ p U ` t q dµ π p Λ q p Λ q “ ż X U Λ p U ` t q dµ π p Λ q p Λ q which is the integral of a strictly positive function (by (14)) with respect to a probabilitymeasure, and thus must be strictly positive for any t P G . Therefore the image of such measures in X p have U -relatively dense support, so that the measures belonging to X p and having U -relatively dense support is of full m p -measure, as desired. (cid:3) Decomposition of weighted Meyer sets
In this section we focus on a very particular type of translation-bounded measures.Very often, the translation-bounded measures considered in the literature are
Diraccombs , with support being, form the most particular to the most general, a subset of alattice [3, 34], a point set with an extra geometric property such as uniform discreteness,finite local complexity or the Meyer property [12, 5, 14, 45], or ultimately a possiblydense (yet countable) subset [40, 29]. In the remaining part of this work, the translation-bounded measures we will consider are Dirac combs supported on a Meyer set, simplyreferred as weighted Meyer sets . Our main result here is the claim that for an ergodicsystem of weighted Meyer sets p X, G , m q , the two associated systems p X p , G , m p q and p X c , G , m c q also consist of weighted Meyer sets.5.1. Weighted FLC sets . A point set S of G is called uniformly discrete if there is an open set U such that anyof its translates by an element of G contains at most one element of S . If one considersits difference set S ´ S : “ t ´ t : t, t P S ( then this means that 0 in isolated in S ´ S . A uniformly discrete set is called of finitelocal complexity (FLC) if the difference set S ´ S is closed and all its points are isolated.In studying a particular FLC set S one usually consider a whole ensemble of FLC sets X S called its hull , stable under the natural G -action shifting sets point by point (seeSection 2 of [42]). The hull X S of a FLC set S is then a compact space with jointlycontinuous G -action when equipped with a topology, the so-called local topology , forwhich a neighborhood basis at each S P X S is yield by O U,K p S q : “ S P X S : D s P U such that S X K ” p S ´ s q X K ( for 0 P U Ă G open and K compact (see [42] for details and a proof). An importantremark that we will further need is that whenever a set S belongs to the hull X S of someFLC set S then its difference set S ´ S is included in S ´ S .On the other hand a FLC set S obviously defines a translation-bounded measure δ S by setting a Dirac mass at any point of S . It is therefore natural to look at itsassociated dynamical system of translation-bounded measures p X δ S , G q , and as discussedin [4], Section 4 therein, this system is conjugated with p X S , G q under the natural mapassociating a FLC set its corresponding Dirac comb. In the light of this correspondence,a natural generalization of FLC sets is the following: Definition 5.1.
A weighted FLC set is a translation-bounded measure supported on aFLC set.
A weighted FLC set is thus always of the formΛ “ ÿ t P S c t δ t with " ď | c t | ď MS FLC setWeighted lattices [3] are important examples of such measures.
URE POINT/CONTINUOUS DECOMPOSITION OF MEASURES AND DIFFRACTION 21
Proposition 5.2.
If a weighted FLC set Λ is supported on some FLC set S then any Λ P X Λ is supported on some S P X S .Proof. Let Λ P X Λ be chosen. It is a vague limit of a sequence of translates Λ ˚ δ s n for p s n q n P N Ă G , each respective translate having support in S ´ s n . Since X S is compactfor the topology described earlier the sequence S ´ s n must accumulates at some FLCset S , and after possibly extracting one can suppose that S ´ s n also converges to S in X S . Select a decreasing sequence p U k q k of neighborhoods of 0 in G whose intersection isthe origin, and furthermore let p K k q k be an increasing sequence of compact sets whoseinteriors cover G . Thus one is able to extract a subsequence p s n k q k of p s n q n such that foreach k P N one has S X K k ” p S ´ s n k ´ ǫ k qX K k for some ǫ k P U k . Let s k : “ s n k ` ǫ k : Then Λ ˚ δ s k converges vaguely to Λ, each having support in S ´ s k where S X K k ” p S ´ s k qX K k for all integer k P N . Therefore on each compact K k the restriction of Λ ˚ δ s k on K k iseventually supported on the finite set S X K k , and it follows that the restriction of Λon K k is supported on S X K k . Therefore Λ is a weighted Dirac comb supported on S ,as desired. (cid:3) Remark 5.3.
Note that if Λ is a weighted FLC set with support the FLC set S p Λ q , thenthe mapping associating a Λ P X Λ its support is in general neither valued in X S p Λ q norcontinuous between these two spaces. One has in fact that it yields a well-defined andcontinuous map X Λ ÝÑ X S p Λ q if and only if any non-zero coefficient of the Dirac comb Λ has absolute value bounded from below by some positive constant.Let Λ be a weighted FLC set. It will later be convenient to have at our disposal a ”localtopology” description of the topology of the hull X Λ . This is provided by consideringthe collection of subsets of X Λ ˆ X Λ O U,K,ε : “ " p Λ , Λ q P X Λ ˆ X Λ : inf s P U „ sup v P K ˇˇ Λ p v q ´ Λ p v ` s q ˇˇ ă ε * for 0 P U Ă G open, K compact and ε ą
0, which constitutes a basis of a uniformity on X Λ (see [24, Chapter 6] for more on uniformities). Here and all along this section we noteΛ p v q for the Λ-measure of the singleton t v u . This defines a unique topology T loc whichwe shall call the local topology of X Λ , such that a neighborhood basis at any Λ P X Λ isprovided by O U,K,ε p Λ q : “ t Λ P X Λ : p Λ , Λ q P O U,K,ε u for 0 P U Ă G open, K compactand ε ą G is σ -compact and 1 st countable one easilyextracts a countable family of sets O U,K,ε forming a basis of the same uniformity, and by[24, Chapter 6, Theorem 13] the local topology is consequently metrizable.
Proposition 5.4.
Let Λ be a weighted FLC set. Then the vague topology coincides withthe local topology on X Λ . In particular p X Λ , T loc q is compact.Proof. We only need to show the continuity of the identity map from p X Λ , T vague q to p X Λ , T loc q , and the result will follows from compacity of the former topology and Haus-dorff property (which is straightforward to show since elements of X Λ are atomic mea-sures) of the latter. Since both topologies are metrizable it suffices to show that is asequence p Λ n q n converges vaguely to Λ then it converges to Λ in the local topology. Let thus a sequence p Λ n q n converging vaguely to a limit Λ: To show that this sequence alsoconverges to Λ in the local topology is is sufficient to show that from any subsequencecan be extracted another subsequence converging to Λ in the local topology.Denote by S the support of the weighted FLC set Λ . Then let p Λ n k q k be a subsequence,which still converges vaguely to Λ. Repeating the argument of the proof of Proposition ?? one can extract a subsequence p Λ n kl q l (still vaguely converging to Λ), as well a a sequence p S k l q l in X S with limit S P X S , such that: Λ n kl is supported on S l , Λ is supported on S ,and for a given decreasing sequence p U l q l of neighborhoods of 0 in G whose intersectionis the origin and a given increasing sequence p K l q l of compact sets whose interiors cover G one has for each l P N that S X K l ” p S k l ´ ǫ l q X K l for some ǫ l P U l We claim now that the subsubsequence p Λ n kl q l converges to Λ in the local topology:Indeed for any 0 P U Ă G open, K compact and ε ą L suchthat whenever l ě L then S X K ” p S k l ´ s l q X K for some s l P U l Ď U Therefore for l ě L each Λ n kl admits a s l P U such that Λ and Λ n kl ˚ δ s l are all, whenrestricted to the compact set K , supported on the finite set S X K , that is to say, suchthat ˇˇˇ Λ p v q ´ Λ n kl p v ´ s l q ˇˇˇ “ ˇˇˇ Λ p v q ´ Λ n kl ˚ δ s l p v q ˇˇˇ “ ă ε @ v P K z S Hence we will be done once we find an integer L ě L such that for l ě L one alsohas | Λ p p q ´ Λ n kl p p ´ s l q| ă ε at points p P S X K . To do so consider an open set U such that p S X K ´ S X K q X U “ t u , and moreover consider an open set 0 P U Ă G whose closure is strictly included in U : Then one can exhibit a compactly supportedcontinuous function φ identically equal to 1 on U and vanishing outside U , and we thuslet L ě L be such that ´ s l P U and ˇˇˇˇż φ ˚ δ p d Λ ´ ż φ ˚ δ p d Λ n kl ˇˇˇˇ ă ε @ p P S X K and l ě L which exists since p s l q l converges to 0 and p Λ n kl q l converges vaguely to Λ. From ourparticular choice of φ , for such an L one has for any p P S X K that ˇˇˇ Λ p p q ´ Λ n kl p p ´ s l q ˇˇˇ “ ˇˇˇˇż φ ˚ δ p d Λ ´ ż φ ˚ δ p d Λ n kl ˇˇˇˇ ă ε This shows that the subsubsequence p Λ n kl q l converges to Λ in the local topology, whichfinishes the proof. (cid:3) Weighted Meyer sets and Cut & Project representations . A strengthening of the FLC property for point sets is to require two additional facts:First the uniformly discrete set S should be relatively dense, meaning that there exists acompact set K such S ` K covers the whole group G (we call such set a Delone set ), andsecond the difference set S ´ S should itself be uniformly discrete. A uniformly discreteobeying these two additional properties is called a Meyer set . Meyer sets have severaldifferent by equivalent characterizations, see [36] as well as [25, 26] for the case G “ R d and the improvement [45] for the general case. If S is a Meyer set then as an FLC set ithas a hull X S , and it is not hard to show that any S P X S is also a Meyer set. URE POINT/CONTINUOUS DECOMPOSITION OF MEASURES AND DIFFRACTION 23
We now present the widely studied
Cut & Project formalism for point sets, see thedifferent works [42, 22]. Suppose we are given a triple p H, Γ , s H q where H is a LCAgroup, Γ a finitely generated subgroup of G and a group morphism s H : Γ / / H with range s H p Γ q dense in H , and whose graph G p s H q : “ tp s H p t q , t q P H ˆ G : t P Γ u is furthermore a lattice, that is, a discrete and co-compact subgroup of H ˆ G . Such atriple is called a cut & project scheme (CPS for short). The LCA group H of a CPS iscommonly called the internal space (or internal group), the subgroup Γ of G the structuregroup of the CPS, and the morphism s H the *-map of the CPS. In addition, compacttopologically regular subset W of H , that is, a compact set which is the closure of itsinterior ˚ W in H , will be called a window . Definition 5.5.
Let S be a point set of G . We call Cut & Project representation of S any CPS p H, Γ , s H q such that S belongs to Γ , and whose image s H p S q is relativelycompact in H . Not all point sets admit a Cut & Project representation, see Theorem 5.7 below.Whenever a point set S belongs to Γ for some CPS p H, Γ , s H q then it can thus belifted in a subset of a lattice (namely G p s H q ) in H ˆ G , and the condition of relativecompactness for s H p S q means that this lifting stand in a ”not too thick” strip about G (when naturally embedded into H ˆ G ). Given some point set S , We will call a Cut &Project representation p H, Γ , s H q of S irredundant whenever the closure W S of s H p S q in H is irredundant in H , that is, if there is no non-trivial element w of H satisfying W S ` w “ W S . It is always possible to turn a given Cut & Project representationinto an irredundant one by simply modding out a certain compact subgroup of H , see[32, 23] for details and a proof. Thus a point set having a Cut & Project representationautomatically also admits an irredundant Cut & Project representation. Definition 5.6.
A closed model set is a point set obtained from a CPS p H, Γ , s H q and awindow W in H by P H p W q : “ t t P Γ : s H p t q P W u It is a true fact that a closed model set is always a Meyer set. Moreover a closed modelset S always comes with a Cut & project representation, namely that of the CPS usedto construct it, where s H p S q a compact closure W in H . Theorem 5.7.
Let S be a point set of G . The following assertions are equivalent: p i q S is a subset of a Meyer set, p ii q S admits a Cut & Project representation, p iii q S is a subset of a closed model set.If it holds true then the Cut & Project representation can be chosen irredundant.
The hard part of this Theorem is p i q ñ p ii q , which is sufficient to prove when S isitself a Meyer set, and which was proved by Meyer [35, Chapter II, Section 5, Proposition4], later on followed by several works [25, 36, 5, 2]. Assuming this let us then provide ashort proof of the remaining statements: Proof.
Assume p i q : Then S is contained in some Meyer set S , which by [45], Theorem? admits a Cut & Project representation p H, Γ , s H q . It follows that the closure W S of s H p S q is contained in the closure of s H p S q which is compact in H , and therefore p H, Γ , s H q is a Cut & Project representation of S , giving p ii q . Then assuming p ii q onegives rise by considering any compact topologicaly regular subset W of H containing s H p S q to a closed model set P H p W q : “ t t P Γ : s H p t q P W u which contains S , yielding p iii q . As any closed model set is a Meyer set this immediately gives p i q .Finally, following [32], Section 9 therein, given a Cut & Project representation p H, Γ , s H q of S with closure W S of s H p S q in H one mods out the compact subgroup K S of elements w of H such that W S ` w “ W S , which results in a an irredundant representation p H , Γ , s H q of S with internal group H “ H modulo K S , ˚´ map s H “ s H modulo K S and s H p S q having closure the compact subset W S “ W S modulo K S . (cid:3) Definition 5.8.
A weighted Meyer set is a translation-bounded measure supported on aMeyer set.
A weighted Meyer set is hence always of the formΛ “ ÿ t P S c t δ t with " ď | c t | ď MS Meyer setObviously a weighted Meyer set is always a weighted FLC set, and its support admits by definition a Meyer super-set containing it. A weighted Meyer set may fail to have aMeyer set support since it may not be relatively dense. Any subset (even the finite ones)of a Meyer set then yields a Dirac comb which is according to this definition a weightedMeyer set. A weighted Meyer set could have also been called weighted model set , butsince this terminology makes explicitly reference to a Cut & Project scheme representingit we preferred the former appellation.5.3.
The torus parametrization of a Cut & Project representation . In [11] the authors illustrated in a particular case a relation, which is of main impor-tance here and more generally in the whole point sets theory, between Cut & Projectrepresentations of a Meyer set S and its associated dynamical system p X S , G q : Theynamely showed that existence of a Cut & Project representation of S always gives riseto a Kronecker factor of the system p X S , G q (and thus to a group of eigenvalues for p X S , G q ). Such result has later on been stated and proved in its greatest generality bySchlottmann in [42], Section 4 therein.To be more precise, it is shown in [42] that if a repetitive Meyer set S admits anirredundant Cut & Project representation p H, Γ , s H q then there is a (continuous) factormap from X S onto r H ˆ G s G p s H q , the compact Abelian group yield by the quotient of H ˆ G by its subgroup G p s H q and endowed with the G -action ”by rotation”, set for s P G on an element r w, t s by r w, t s .s : “ r w, t ` s s . Here repetitivity of a Meyer set meansminimality of its dynamical system p X S , G q . The factor map from X S onto the Kroneckerfactor issued from a Cut & Project representation of S is called a torus parametrization .We propose here to set this result in a slightly more general situation, namely that ofweighted Meyer sets, by carefully adapting the proof of [42] to our setting. The originalproof has been set with the assumption of repetitivity on the Meyer sets, and we shallnot assume this here. As a result, the torus parametrization yield by a Cut & Projectrepresentation of a weighted Meyer set Λ will no longer be defined on the entire hull X Λ URE POINT/CONTINUOUS DECOMPOSITION OF MEASURES AND DIFFRACTION 25 but rather only on the rubber local isomorphism class RLI( Λ ) (one will notices that repet-itivity of Λ , that is to say, minimality of p X Λ , G q , precisely means that RLI( Λ q “ X Λ ).This will however be sufficient to prove the main result of this section, namely Theorem5.10 in next paragraph (Theorem 4 in the introductory part). Theorem 5.9.
Let Λ be a weighted Meyer set with support S p Λ q having an irredundantCut & Project representation p H, Γ , s H q , with W the closure of s H p S p Λ qq in H . p i q For each Λ P X Λ there exists p w, t q P H ˆ G such that S p Λ q Ď P H p W ` w q ` t , p ii q When Λ P RLI p Λ q the element p w, t q P H ˆ G is unique modulo G p s H q , p iii q The mapping
RLI p Λ q Q Λ ✤ / / r w, t s P r H ˆ G s G p sH q is a continuous G -map.Proof. p i q : Let Λ P X Λ , which we can assume to not be the trivial measure on G , andsupposed to be supported on the structure group Γ. For such Λ denote by W p Λ q theclosure of s H p S p Λ qq in H . Hence with respect to this notation W is nothing but W p Λ q .Then it is not hard to show the equivalence of conditions, for w P H , w P č γ P S p Λ q s H p γ q ´ W ðñ W p Λ q Ď W ` w. (15)Let us show that any Λ P X Λ supported on Γ admits an element w Λ P H where theseequivalent conditions hold: As Λ P X Λ one can find a sequence p t n q n of elements in G such that Λ ˚ δ t n converges to Λ, which according to Proposition 5.4 means that for eachcompact K and ε ą N K,ε such that, after possibly slightly moving each t n , one has for each n ě N K,ε that | Λ p v q ´ Λ ˚ δ t n p v q| ă ε for any v P K . Therefore,whenever p P S p Λ q then Λ ˚ δ t n p p q is eventually non-zero, so that p eventually belongsto the support S p Λ q ` t n of Λ ˚ δ t n . Since Λ is by assumption non-trivial one can pickup some p P S p Λ q . Then the sequence t n eventually lies in p ´ S p Λ q Ă S p Λ q ´ S p Λ q ,which lies in the group Γ since both Λ and Λ are supported on this latter group. Thus s H p t n q eventually makes sense, and since p P S p Λ q ` t n eventually then one eventuallygets s H p p q P W ` s H p t n q . Thus the sequence s H p t n q lies in the compact set s H p p q ´ W eventually, and therefore accumulates at some element w Λ P H . We can suppose afterpossibly extracting a subsequence that s H p t n q converges to w Λ in H . The latter mustsatisfy s H p p q P W ` w Λ , and since this argument is independent upon the choice of p P S p Λ q we deduce that s H p S p Λ qq Ă W ` w Λ , which yields W p Λ q Ă W ` w Λ as desired.Now given any Λ P X Λ one has for any chosen element t P S p Λ q that Λ ˚ δ t is supportedon the structure group Γ, and from we just said there exists a w P H such that (15)holds, yielding a p w, t q P H ˆ G with S p Λ q Ď P p W ` w q ` t , as desired. p ii q : We show that when Λ P RLI p Λ q with support in Γ then the w P H satisfying (15)is unique. To that end we know from the above analysis that the set W p Λ q is contained insome translate W ` w of the compact set W and thus is compact in H . Since Λ P RLI p Λ q we can interchange the roles of Λ and Λ in the previous argument shows that there equallyexists some w P H with W Ď W p Λ q ` w . It follows that W Ď W p Λ q ` w Ď W ` w ` w with W compact, which forces W “ W ` w ` w . From the irredundancy assumption of W one gets w “ ´ w , and this in turns gives that W p Λ q “ W ` w . Such an equality canhold for at most one element w , giving unicity. Now suppose Λ P RLI p Λ q admits p w, t q and p w , t q in H ˆ G for which inclusion set in the first point of the statement hold: S p Λ q Ď P p W ` w q ` t and S p Λ q Ď P p W ` w q ` t Then from what have been just said one has W p Λ ´ t q “ W ` w and W p Λ ´ t q “ W ` w Given any p P S p Λ q one has t ´ t “ p p ´ t q ´ p p ´ t q P P H p W ` w q ´ P H p W ` w q Ď Γ ´ Γ “ ΓTherefore s H p t ´ t q makes sense and is an element of H such that W ` w ` s H p t ´ t q “ W p Λ ´ t q ` s H p t ´ t q “ W p Λ ´ t q “ W ` w which shows by irredundancy that w “ w ` s H p t ´ t q . Thus the element p w, t q P H ˆ G for which inclusion if point p i q of the statement holds is unique modulo G p s H q , as desired. p iii q : The mapping RLI p Λ q Q Λ ✤ / / r w, t s P r H ˆ G s G p s H q is from the conclusion ofpoint p ii q well-defined. Moreover when Λ P RLI p Λ q is such that S p Λ q Ď P p W ` w q ` t then S p Λ ˚ δ s q Ď P p W ` w q ` t ` s for any s P G , so by the unicity part of point p ii q onededuces that the map is a G -map. We show continuity at any given Λ P RLI p Λ q . Let p w, t q P H ˆ G such that S p Λ q Ď P p W ` w q ` t : Given a neighborhood U of 0 in H anda neighborhood U of 0 in G , since w satisfies t w u “ č p P S p Λ q´ t s H p p q ´ W (16)there exists a sufficiently large compact set K such that č p Pp S p Λ q´ t qX K s H p p q ´ W Ď w ` U. Now the set S p Λ q X p K ` t q being finite one has a minimal value M ą | Λ p v q| onthat set. Then we are done if we can show that if Λ P RLI p Λ q is such thatinf s P U „ sup v P K ` t | Λ p v q ´ Λ p v ` s q| ă M P RLI p Λ q belongs to a neighborhood of Λ uniquely defined by Λ, U and U ) then it admits a representative p w , t q P H ˆ G of its class such that p w , t q P p w, t q ` U ˆ U This is in turns true since for such Λ one has an s P U such that, letting t : “ t ` s , p S p Λ q ´ t q X K is included in p S p Λ q ´ t q X K , and thus w P č p Pp S p Λ q´ t qX K s H p p q ´ W Ď č p Pp S p Λ q´ t qX K s H p p q ´ W Ď w ` U where w is the unique element of H in the intersection Ş p Pp S p Λ q´ t q s H p p q ´ W . Wetherefore end up with a pair p w , t q such that, from the selection of w satisfies S p Λ q Ď P p W ` w q ` t and thus is a representative in H ˆ G of the class of Λ in r H ˆ G s G p s H q , such that p w , t q P p w, t q ` U ˆ U . This shows continuity at any Λ P RLI p Λ q , as desired. (cid:3) URE POINT/CONTINUOUS DECOMPOSITION OF MEASURES AND DIFFRACTION 27
The decomposition of weighted Meyer sets . We shall show here our main result of this section, see Theorem 5.10 just below. Wewill need to consider here the Borel G -maps yield by Theorem 4.1. These maps are notuniquely defined but by the uniqueness statement of Theorem 4.6 any two choices agree m -almost everywhere on X . We assume here that such choice is made. Theorem 5.10.
Let p X, G , m q be an ergodic system of weighted Meyer sets. Then thefollowing property holds true m -almost everywhere: Whenever Λ is supported on a closedmodel set, then so are its summands Λ p and Λ c .Proof. Let p X, G , m q be an ergodic system of weighted Meyer sets with eigenvalue group E , and let p X p , G , m p q and p X c , G , m c q be as in Theorem 4.1. Recall from Theorem4.6 that there exists a unique (up to almost everywhere equality) Borel factor map X / / X p , and from the proof of Theorem 4.1 that it is almost everywhere given bythe composition of the three Borel G -maps X π / / T E µ / / M b p X q i / / M p G q for some choice of Borel factor map π as in Proposition 2.2, and µ the disintegrationof m over T E . The following proposition is at the core of our proof: Proposition 5.11.
Almost any Λ P X admits a Borel set F Λ of full µ π p Λ q -measure suchthat, whenever Λ is supported on a closed model set ∆ , then so are any Λ P F Λ .Proof. As p X, G , m q is a ergodic system the set X p q : “ t Λ P X | X Λ “ Supp p m qu is a Borel set of full measure in X , and choosing any Λ P X p q yield X p q “ RLI p Λ q .Let Λ P X p q be now given. By construction of the compact Abelian group T E , eacheigenvalue ω P E has an associated character χ ω on it, and thus the Borel map Φ ω : “ χ ω ˝ π is an eigenfunction with eigenvalue ω on p X, G , m q , forming so a collection of Borel mapsΦ ω on X , with ω P E . Now consider an irredundant Cut & Project representation ofΛ in a CPS p H, Γ , s H q : By part p iii q of Theorem 5.9 there is a continuous G -map p from X p q “ RLI p Λ q to r H ˆ G s G p sH q . An eigenvalue ω of the Kronecker action of G on r H ˆ G s G p sH q corresponds to a continuous character ξ ω on this latter, which lifts in acontinuous functionΦ Λ ,ω : “ ξ ω ˝ p : X p q / / T such that Φ Λ ,ω p Λ q “ , Φ Λ ,ω p Λ .t q “ ω p t q Φ Λ ,ω p Λ q In particular since X p q is of full measure Φ Λ ,ω is an eigenvalue of p X, G , m q (perhapsnot continuously extendable on X ) with eigenfunction ω . Therefore there is one and onlyone function r Φ ω on X p q , continuous and an eigenfunction for ω , such that ‚ r Φ ω “ Φ ω almost everywhere on X p q , ‚ r Φ ω “ c. Φ Λ ,ω everywhere on X p q , for some constant c P T . Such a function r Φ ω does not depend on the used irredundant Cut & Project repre-sentation of Λ because Φ Λ ,ω is continuous on X p q and the G -orbit of Λ is dense in it.Moreover if the same element ω P E has an eigenfunction of the form Φ Λ ,ω for anotherΛ P X p q then again by continuity there is a constant c P T such that Φ Λ ,ω “ c. Φ Λ ,ω everywhere on X p q . This shows that r Φ ω , whenever it exists, is independent on the choiceof Λ P X p q and on the irredundant Cut & Project representation of Λ. Let us denote E the subset of elements in E having an associated function r Φ ω arising in the aboveway. Since the eigenvalue group E is countable then E is countable, and therefore thefollowing Borel set has full mesure in X : X p q : “ ! Λ P X p q | r Φ ω p Λ q “ Φ ω p Λ q @ ω P E ) Now, as µ is a disintegration of m over the Borel factor T E one has by an applicationof the Disintegration Theorem the Borel subset of full measure in XX p q : “ Λ P X | µ π p Λ q is supported on π ´ p π p Λ qq ( and as a result the Borel set X p q : “ X p q X X p q is of full measure in X . Moreover,applying the Disintegration Theorem to the Borel almost everywhere 1 function I X p q one deduces that the set X : “ ! Λ P X p q | µ π p Λ q p X p q q “ ) is a Borel set of full measure in X . Form the very construction of this Borel set, forany Λ P X the Borel set F Λ : “ π ´ p π p Λ qq X X p q has µ π p Λ q -measure equal to 1 and always contains the element Λ. We now claim thatfor any Λ P X the sets F Λ make our statement holding. For, let Λ P X be given, andsuppose it is supported on some closed model set ∆, that is, let p H, Γ , s H q be a CPS and V a compact set in H such that Λ is supported on ∆ : “ P H p V q . Then s H p S p Λ qq makessense and its closure W in H is contained in V , and thus is compact. Modding out theredundancy subgroup R W in H leads to an irredundant Cut & Project representation of Λ in a new CPS p H , Γ , s H q with same structure group, with closure of s H p S p Λ qq in H being a compact set W , and such that P H p W q “ P H p W q Ď P H p V q “ : ∆(17)It is then obviously sufficient to show that S p Λ q Ď P H p W q for any Λ P F Λ (18)Let us show this: From Theorem 5.9 there is a continuous map X Ď X p q “ RLI p Λ q Q Λ ✤ p / / r w, t s P r H ˆ G s G p s H q where p p Λ q “ r w, t s is the unique G p s H q -class such that any representative p w, t q P H ˆ G satisfies S p Λ q Ď P H p W ` w q ` t URE POINT/CONTINUOUS DECOMPOSITION OF MEASURES AND DIFFRACTION 29
Denote E the eigenvalue group of the Kronecker action of G on r H ˆ G s G p s H q . Then E Ď E naturally, and each ω P E yields functions Φ ω , r Φ ω and r Φ Λ ,ω as done before.Now for Λ P F Λ one has Λ , Λ P π ´ p π p Λ qq so satisfy Φ ω p Λ q “ Φ ω p Λ q , and moreover Λ , Λ P X Ď X p q so satisfy as well r Φ ω p Λ q “ r Φ ω p Λ q . It therefore comes that r Φ Λ ,ω p Λ q “ r Φ Λ ,ω p Λ q for any ω P E , that is, Λ and Λ are identified under any function of the form χ ω ˝ p where χ ω is any character on the compact Abelian group r H ˆ G s G p s H q . One must then have p p Λ q “ p p Λ q “ r , s in r H ˆ G s G p sH . This precisely ensures that S p Λ q Ď P H p W q forany Λ P F Λ , yielding (18) and thus the proof of Proposition 5.11. (cid:3) From this we can easily settle the proof of Theorem 5.10: Indeed let Λ P X such thatthe conclusion of Proposition 5.11 holds true for some Borel set F Λ . Without restrictionone can suppose Λ to belong in Borel subset of full measure Λ P X | Λ p “ i p µ π p Λ q q and Λ c “ Λ ´ Λ p ( We shall show that the supports of Λ p and Λ c satisfy S p Λ p q , S p Λ c q Ď ď Λ P F Λ S p Λ q (19)Indeed for any Borel set B of G one has Λ p p B q “ i p µ π p Λ q qp B q , and since F Λ is of full µ π p Λ q -measure one deduces by definition of iΛ p p B q “ ż X Λ p B q dµ π p Λ q p Λ q “ ż F Λ Λ p B q dµ π p Λ q p Λ q which is null when B belongs to the complementary set of right term of (19). Thisshows the desired inclusion for S p Λ p q . Now since Λ c “ Λ ´ Λ p , S p Λ c q is therefore supportedon S p Λ q Y S p Λ p q , clearly included in the union of (19), given the desired inclusion for Λ c .Therefore, whenever our Λ is supported on some closed model set ∆ then combining theinclusions (19) with the statement of Proposition 5.11 one deduces S p Λ p q , S p Λ c q Ď « ď Λ P F Λ S p Λ q ff Ď ∆as desired. (cid:3) Remark 5.12.
From the above result we know that, given an ergodic system p X, G , m q of weighted Meyer sets with auto-correlation γ , then once a generic Λ P X is supported ina closed model set ∆ then so does Λ p and Λ c . Therefore using formula (5) of Paragraph3.4 for the auto-correlation, as well as the conclusion of Remark 4.3, one shows that both γ , its strong almost periodic part (the auto-correlation of a generic choice of Λ p P X p ) andits null-weakly almost periodic part (the auto-correlation of a generic choice of Λ c P X c )are supported on the difference set ∆ ´ ∆. This statement was obtained (among otherthings) in [44]. Remark 5.13.
A lattice Γ of G is always a closed model set, for instance coming fromthe Cut & Project scheme having trivial internal group and Γ itself as lattice in t u ˆ G .As a result if X consists of weighted translates of some lattice Γ then so does X p and X c . Remark 5.14.
Suppose that p X, G , m q is an ergodic system of Dirac combs of Meyersets (thus with constant weight equal to 1). Let Ξ t stands for the subset of Dirac combsin X having t in its support, for any t P G . Then it is not difficult to show, for m -almostevery Λ P X , the formulaΛ p “ ÿ Λ p p t q δ t with Λ p p t q “ µ π p Λ q p Ξ t q Thus, on a probabilistic viewpoint, Λ p can be interpreted as the sum of Dirac combsof elements t P G with each coefficient Λ p p t q being the m -expectation of the point t toappear, conditioned by the event of belonging in the fiber of Λ over T E . In the generalweighted case a similar formula, yet less trivial, can be stated which involves the weightfunction c : X / / C , c p Λ q : “ Λ p q , the measure of the singleton 0. Remark 5.15.
Suppose that p X, G , m q is an ergodic system of weighted Meyer setswith non-negative weights. Then by Remark 4.4 any weighted Meyer set of p X p , G , m p q also has non-negative weights. Suppose moreover that m -almost any measure in X have relatively dense support. Then it follows from Proposition 4.8 that m p -almost anyweighted Meyer set in X p also have relatively dense support.6. Operator methods for the analysis of Borel factors
In this section we will have a second look on the process associating to certain Borelfactors of a measured dynamical system of translation-bounded measures p X, G , m q anoperator on L p X, m q , as done in Paragraph 4.2. Our main result here is the identificationof the class of operators involved in this process, see Theorem 6.6, as well as a relationbetween such operators and the diffraction of the corresponding Borel factors.6.1. Admissible operators . Let p X, G , m q be a measured dynamical system of translation-bounded measures, withdiffraction to dynamic mapΘ : L p p G , p γ q o o / / H Θ Ď L p X, m q , Θ p p φ q “ N φ for any φ P C c p G q . Recall that P Θ stands for the G -commuting orthogonal projectiononto the Hilbert subspace H Θ . Definition 6.1.
A bounded operator Q on the Hilbert space L p X, m q is admissible if itis G -commuting, if Q “ QP Θ on L p X, m q and if for each compact set K Ă G there is aconstant R K ě such that } Q p N φ q} ,ess ď R K } φ } @ φ P C K p G q It is obvious that the identity operator is admissible in the above sense, and that thecollection of admissible operators of the system p X, G , m q forms a linear subspace of thespace of bounded operators on L p X, m q . Remark 6.2.
Any bounded operator on L p X, m q admits a certain weak form of admis-sibility: If Q is any bounded operator then due to the continuity of the map φ ÝÑ N φ one always has, for any compact set K Ă G , a certain constant R K ě } Q p N φ q} ď R K } φ } for all φ P C K p G q . URE POINT/CONTINUOUS DECOMPOSITION OF MEASURES AND DIFFRACTION 31
Suppose that we are given a Borel factor map π : X ÝÑ X π over a measured dynamicalsystem of translation-bounded measures p X π , G , m π q having diffraction p γ π ! p γ withessentially bounded Radon-Nikodym differential f π : “ d p γ π d p γ P L p p G , p γ q . Then Proposition4.5 gives rise to a bounded operator Q π on L p X, m q , which we aim here to show that itis admissible: Proposition 6.3.
The operator Q π P B p L p X, m qq associated by Proposition 4.5 to anyBorel factor map π : X ÝÑ X π as above is admissible.Proof. One has Q π “ Q π P Θ by construction, whereas G -commutativity, that is to say, U t Q π “ Q π U t for any t P G , is straightforward to check on functions of the form N φ for φ P C c p G q and therefore holds on the closure H Θ , hence on the whole space L p X, m q .For each φ P C K p G q supported on a given compact set K Ă G one has a bound holdingfor almost every Λ P X | Q π p N φ qp Λ q| “ ˇˇ N πφ ˝ π p Λ q ˇˇ ď } N πφ } ď R K } φ } where the constant R K only depends on the translation-bound of measures in X π andon the compact set K , settling the proof. (cid:3) From admissible operators to Borel factors . Given a dynamical system p X, G , m q of translation-bounded measures on G , we shallnow arrive at the reciprocal statement of Proposition 4.5, that is, given an admissibleoperator Q we shall reconstruct the underlying Borel factor of translation-bounded mea-sures p X Q , G , m Q q . Anticipating things a bit, for such constructed dynamical system oftranslation-bounded measures we denote as usual C c p G q Q φ ✤ / / N Qφ P C p X Q q , N Qφ p Λ Q q “ ż G φ ´ d Λ Q The result concerning this is as follows:
Proposition 6.4.
Let Q P B p L p X, m qq be admissible. Then there exists a dynamicalsystem p X Q , G , m Q q of translation bounded measures on G and a Borel factor map π Q : X Q Λ ✤ / / Λ Q P X Q such that N Qφ ˝ π Q “ Q p N φ q in the L sense for each φ P C c p G q .Proof. We proceed by decomposing the proof into five steps (i)-(v) . (i) We fix here our notations: We denote U to be an open relatively compact subset of G , and M ą X consists of p U, M q -translation-bounded measures.We set G to be a dense countable subset of G , and let p K N q N be a nested sequenceof compact subsets which are the closure of their interior in G , with first term K containing U in its interior, and whose union is G . Next, as the group G is locallycompact second countable, it is possible to set a countable collection p φ Nn q n P N Ă C K N p G q which is dense in C K N p G q for the uniform norm. In particular the countable collection p φ Nn q n,N P N is dense in C c p G q for the inductive limit topology. Third we denote by V the G -stable Q ` i Q -vector sub-space of C c p G q spanned by p φ Nn ˚ δ t q n,N P N ,t P G . It may bewritten as the countable union of countable sets V “ ď l “ l ÿ j ,j ,j “ λ j ,j ,j φ j j ˚ δ t j | λ j ,j ,j P Q ` i Q , t j P G + and is therefore itself countable. For each N the Q ` i Q -vector sub-space V N : “ V X C K N p G q is in particular countable and dense in C K N p G q for the uniform norm. Tofinish these preparations we make the following: Since Q is assumed admissible one hasfor each N a constant R N ě } Q r N φ s } ,ess ď R N } φ } @ φ P V N Therefore we let, for each integer N and each φ P V N , Q p N φ q to be a chosen repre-sentative Borel function of the L -class Q r N φ s which is uniformly bounded on X by thevalue R N } φ } . In particular Q p N φ qp Λ q makes sense for each φ P V and any Λ P X , andone has for each integer N | Q p N φ qp Λ q| ď R N } φ } @ φ P V N , Λ P X (20) (ii) We here construct our desired collection of measures Λ Q on G , where Λ P X .Given any two functions φ, φ P V and λ P Q ` i Q , the Borel set of points x P X where Q p N λφ ` φ qp Λ q “ λQ p N φ qp Λ q ` Q p N φ qp Λ q (21)is of full measure in X . As a byproduct the countable intersection X p q : “ č φ,φ P V, λ P Q ` i Q t Λ P X | equation (21) holds at Λ u is a Borel subset of full measure in X . Since m is G -invariant on X the countableintersection X p q : “ X t P G X p q ˚ δ ´ t Ă X p q is Borel of full measure in X . Moreover since Q comutes with the unitary operator U t on L p X, m q the set of Λ P X where Q p N φ ˚ δ t qp Λ q “ QU t p N φ qp Λ q is equal to U t Q p N φ qp Λ q “ Q p N φ qp Λ ˚ δ t q (22)holds whatever φ P V and t P G is a Borel set of full measure in X and thus intersects X p q into a G -stable Borel subset of full measure X p q in X . Now for any Λ P X p q themapping Q Λ : φ ✤ / / Q p N φ qp Λ q is well defined on the Q ` i Q -vector space V , and is Q ` i Q -linear. Thanks to (20) Q Λ is uniformly continuous on each Q ` i Q -vector space p V N , } . } q so extends in a continuous Q ` i Q -linear form on each p C K N p G q , } . } q . It thuspasses to the inductive limit in a continuous Q ` i Q -linear form on C c p G q , and it directlycomes that it is in fact C -linear on C c p G q . Thus, by the Riesz Representation Theoremthere exists for each Λ P X p q a complex Borel measure Λ Q P M p G q such that, Q p N φ qp Λ q “ ż G φ ´ d Λ Q @ φ P V, Λ P X p q . (23)Setting π Q : X / / M p G q to be π Q p Λ q : “ Λ Q as just constructed when Λ belongsto the G -stable Borel subset of full measure X p q , and to be for instance the trivialmeasure when Λ P X z X p q , yields a mapping. URE POINT/CONTINUOUS DECOMPOSITION OF MEASURES AND DIFFRACTION 33 (iii)
We show that π Q is valued in the compact set M p U ,R q p G q of p U , R q -translation-bounded measures, where U is any open subset of U with U Ă U in G , and R is theconstant involved in (20) at integer N “
0. Such set U always exists: Indeed if onepicks a point s P U then one results with a compact set t s u disjoint from the closed set G z U , so G being a topological group it is a regular space and thus has an open subset U containing s and disjoint from G z U , as desired. Let us now assume Λ P X p q , and let t P G be given: As any complex measure the measure Λ Q satisfies (see [4], Proposition1 therein for validity of the first equality) | Λ Q |p U ` t q “ sup "ˇˇˇˇż G φ d Λ Q ˇˇˇˇ : φ P C U ` t p G q , } φ } ď * “ sup "ˇˇˇˇż G φ p . ´ t q d Λ Q ˇˇˇˇ : φ P C U p G q , } φ } ď * “ sup "ˇˇˇˇż G φ ˚ δ t d Λ Q ˇˇˇˇ : φ P C U p G q , } φ } ď * ď sup "ˇˇˇˇż G φ ˚ δ t d Λ Q ˇˇˇˇ : φ P C K p G q , } φ } ď * The later being true since we supposed K to contain U from the beginning. This isin turn equal, with V : “ V X C K p G q as in (i) , dense in C K p G q , tosup "ˇˇˇˇż G φ ˚ δ t d Λ Q ˇˇˇˇ : φ P V , } φ } ď * Now as φ P V Ă V and t P G then φ ˚ δ t is again in V , and by (23) this quantity issup t| Q p N φ ˚ t qp Λ q| : φ P V , } φ } ď u which by (20) is bounded by R . Therefore | Λ Q |p U ` t q ď R holds for t P G . Nowif an s P G is given then from the choice of U one can in fact find a t P G suchthat U ` s Ă U ` t : Indeed one easily prove (by contradiction) that there is an openneighborhood B of 0 in G such that U ` B Ă U , so by density of G one can pick up a t P G with s ´ t P B , yielding U ` s “ U ` p s ´ t q ` t Ă U ` B ` t Ă U ` t, as desired. This yields that | Λ Q |p U ` s q ď | Λ Q |p U ` t q ď R for any s P G whenever Λbelongs to X p q , that is, any such Λ Q is p U , R q -translation-bounded. By constructionΛ Q is trivial when Λ P X z X p q so is also p U , R q -translation-bounded in that case. Onehas therefore as desired π Q : X Q Λ ✤ / / Λ Q P M p U ,R q p G q (iv) We show here that this map is a Borel G -map. To show that it is Borel, since M p G q carries the vague topology it suffices to check that the map Λ ÞÝÑ ş G φ d p Λ Q q isBorel for each φ taken in the dense subspace V of C c p G q . But this map is precisely theBorel map Q p N φ q on the Borel subset X p q and the constant nul map on X z X p q , so theresult follows. Now we infer thatΛ Q ˚ δ t “ p Λ ˚ δ t q Q @ Λ P X p q , t P G (24) Here p Λ ˚ δ t q Q makes sense since X p q is G -stable. This is due to the very constructionof X p q since for such measures one has (22) holding, yielding at each φ P V ż G φ ´ d p Λ Q ˚ δ t q “ ż G φ ´ ˚ δ t d p Λ Q q “ Q p N φ ˚ δ t qp Λ q “ Q p N φ qp Λ ˚ δ t q “ ż G φ ´ d pp Λ ˚ δ t q Q q so that by density of V in C c p G q the equality Λ Q ˚ δ t “ p Λ ˚ δ t q Q occurs on the wholespace C c p G q , whenever x P X p q and t P G . Now to see that π Q is a G -map observe thatif, given t P G , one constructs another mapping π Q as we just did, but with some G containing t playing the role of G , some G -stable countable Q ` i Q -vector subspace V Ă C c p G q containing our space V together with Borel representatives Q p N φ q for each φ P V (which possibly differ from Q p N φ q when φ P V Ď V ) then we also end up witha Borel map π Q : X Q Λ ÞÝÑ Λ Q P M p U ,R q p G q together with a Borel subset of fullmeasure X p q in X such that, combining (23) and (24) in this case, Q p N φ q p Λ q “ ż G φ ´ d Λ Q and Λ Q ˚ δ t “ p Λ ˚ δ t q Q @ φ P V , Λ P X p q , t P G Now the countable collection V being contained in V , and since the Borel maps Q p N φ q and Q p N φ q are representatives of a same L -class so that they coincide almosteverywhere on X , one have for each Λ in the Borel set of full measure X p q X X p q X č φ P V Q p N φ q “ Q p N φ q ( equalities of Λ Q with Λ Q on V , and thus on the whole space C c p G q . This shows that π Q “ π Q almost everywhere on X , and since from its construction one has π Q p Λ q ˚ δ t “ π Q p Λ ˚ δ t q for almost every Λ one deduces that π Q p Λ q ˚ δ t “ π Q p Λ ˚ δ t q for almost everyΛ P X , for any t P G , as desired. (v) We now set properly the dynamical system p X Q , G , m Q q and the stated Borel G -map π Q , and prove validity of formula of the statement. Let m Q be the push-forwardprobability measure of m through π Q , which is supported on the compact G -stable set M p U ,R q p G q and ergodic. Its support X Q is then a compact G -stable set of p U , R q -translation-bounded measures, yielding our desired system p X Q , G , m Q q . It now sufficesto modify π Q , if needed, on a set of null measure in order to have a Borel G -map properlyvalued in X Q , while having a Borel subset of full measure X where π Q as modified stillsatisfies form (23) π Q p Λ q “ Λ Q with Q p N φ qp Λ q “ ż G φ ´ d Λ Q @ φ P V, Λ P X This in particular ensure the equality N Qφ ˝ π Q “ Q p N φ q in the L sense for each φ P V .Since the linear maps C c p G q Q φ ✤ / / N Qφ ˝ π Q P L p X, m q C c p G q Q φ ✤ / / Q p N φ q P L p X, m q are continuous and coincide on the dense collection V as we just saw, they must agreeeverywhere, concluding the proof. (cid:3) The Borel map π Q as set in the above Proposition is uniquely defined up to almosteverywhere equality due to the formula N Qφ ˝ π Q “ Q p N φ q required to hold for each URE POINT/CONTINUOUS DECOMPOSITION OF MEASURES AND DIFFRACTION 35 φ P C c p G q . The constructed Borel factor of translation-bounded measures p X Q , G , m Q q is however uniquely defined (if one requires m Q to have full support).6.3. Diffraction properties . We check here the absolute continuity property for the diffraction p γ Q arising from thesystem p X Q , G , m Q q previously constructed from an admissible operator Q : Proposition 6.5.
Any admissible operator Q P B p L p X, m qq has its associated system p X Q , G , m Q q having diffraction of the form p γ Q “ f Q . p γ for some f Q P L p p G , p γ q .Proof. One has the following set of equalities for any two φ, ψ P C c p G q , the first ensuredby Theorem 3.6 and the second coming from the equalities N Qφ ˝ π Q “ Q p N φ q and N Qψ ˝ π Q “ Q p N ψ q , ż p φ p ψ d p γ Q “ ż X Q N Qφ N Qψ dm Q “ ż X Q p N φ q Q p N ψ q dm (25)It directly follows from this equality that for each φ P C c p G q} p φ } L p p G , p γQ q “ } Q p N φ q} L p X,m q ď } Q } op } N φ } L p X,m q “ } Q } op } p φ } L p p G , p γ q where } Q } op is the operator norm of Q as a bounded operator on L p X, m q . Thereforethe identity map L p p G , p γ q / / L p p G , p γ Q q is well-defined and uniformly continuous on the subspace formed of functions p φ with φ P C c p G q , which is dense in L p p G , p γ q since p γ is translation-bounded. Thus the identitymap is well-defined and of operator norm less or equal to } Q } op on all L p p G , p γ q . Thismeans that } f } L p p G , p γQ q ď } Q } op } f } L p p G , p γ q for any f P L p p G , p γ q , and in particular one hasfor each f ě L p p G , p γ q the inequality0 ď ż f d p γ Q ď } Q } op ż f d p γ (26)Applying this to the indicator function of compact sets of the locally compact Abeliangroup p G shows that p γ Q is absolutely continuous with respect to p γ , and thus admits aRadon-Nikodym differential f Q : “ d p γ Q d p γ in L loc p p G , p γ q , which by an application of (26) ispositive, essentially bounded by } Q } op , yielding the proof. (cid:3) The correspondence between Borel factors and admissible operators . Let as usual p X, G , m q be a measured dynamical system of translation-bounded mea-sures with diffraction to dynamic mapΘ : L p p G , p γ q o o / / H Θ Ď L p X, m q , Θ p p φ q “ N φ for any φ P C c p G q . Then a combined formulation of Propositions 4.5, 6.4 and 6.5 isthe following correspondence Theorem: Theorem 6.6.
Let p X, G , m q be a dynamical system of translation-bounded measureswith diffraction p γ . There is a bijective correspondence between: (a) Borel factor maps π over some dynamical system of translation-bounded measureshaving diffraction p γ π ! p γ , with Radon-Nikodym derivative f π : “ d p γ π d p γ P L p p G , p γ q , (b) admissible operators Q on L p X, m q .Given such a Borel factor map π with admissible operator Q π , the multiplication operator M f π by f π on L p p G , p γ q satisfies Θ ˝ M f π ˝ Θ ´ “ Q ˚ π Q π . In particular, given a Borel subset P Ď p G then f π “ I P if and only is | Q π | “ E p P q P Θ . Here, | Q | denotes the absolute value of the operator Q , that is, the unique positiveoperator whose square is Q ˚ Q , see for instance [39, Section VI.4] for definition andexistence. Remark 6.7.
If we are given a Borel factor map π with associated dynamical systemof translation-bounded measures having diffraction f π . p γ , f π P L p p G , p γ q , and letting Q π its associated admissible operator, then it is not difficult to show the equality of norms } Q π } op “ } f π } L p p G , p γ q Indeed the operator norm is a C ˚ -norm so gives } Q π } op “ } Q ˚ π Q π } op , equal to theoperator norm of the pullback of Q ˚ π Q π under the isometry Θ, in turn equal to theoperator norm of the multiplication operator by f π on L p p G , p γ q according to Theorem6.6, which is nothing but } f π } L p p G , p γ q . Remark 6.8.
As it is easy to note the collection of admissible operators of a given dy-namical system of translation-bounded measures p X, G , m q is a vector space of boundedoperators on L p X, m q . This has the following meaning when one considers the associ-ated Borel factor maps: Given two admissible operators Q, Q and a complex number λ one can consider the mapping π Q ` λπ Q form X to M p G q defined in the obviousway, which is a Borel G -map valued in a certain compact space of translation-boundedmeasures. In fact, what one has is the equality (holding almost everywhere) π λQ “ λπ Q π Q ` Q “ π Q ` π Q There is a similar yet less trivial result about diffraction measures, namely p γ λQ “ | λ | p γ Q p γ Q ` Q “ p γ Q ` p γ Q ` ω where ω is a positive measure of the form g. p γ such that the multiplication operator M g is equal to the pullback of Q ˚ Q ` Q Q on L p p G , p γ q under the isometry Θ. Inparticular, whenever two admissible projectors P and P are orthogonal, meaning that P P “ P P “ L p X, m q , then the sum P ` P is again an admissible projector andthere exist therefore dynamical systems of translation-bounded measures associated with P , P and P ` P respectively. It is then not hard to show that one almost surely has π P ` P “ π P ` π P p γ P ` P “ p γ P ` p γ P URE POINT/CONTINUOUS DECOMPOSITION OF MEASURES AND DIFFRACTION 37
This generalizes to any finite sums of pairwise orthogonal admissible projectors on L p X, m q .6.5. Examples of admissible operators: Convolution operators . We shall see here that, given a dynamical system of translation-bounded measures p X, G , m q , one can define a whole class of admissible projectors, each associated withsome bounded measure σ P M b p G q . Given such σ define on L p X, m q the convolutionoperator Q σ by the operator-valued integral Q σ : “ ż G U t dσ p t q Such operator is well-defined, bounded on L p X, m q and commutes with the unitaryrepresentation of G . Recall that the convolution product between a bounded measure σ and a compactly supported continuous function φ is a continuous function belongingto L p G q . From [1], Theorem 1.2 therein, one knows that the convolution Λ ˚ σ ofany translation-bounded measure Λ with any bounded measure σ is well-defined, andis again a translation-bounded measure. Also recall that any σ P M b p G q is Fouriertransformable, with Fourier transform given by a density p σ (its usual Fourier-Stieltjestransform) belonging to the space C bu p p G q of bounded uniformly continuous functions on p G ([41], Paragraph 1.3.3 therein). Proposition 6.9.
The operator Q σ is admissible, with dynamical system of translation-bounded measures p X σ , G , m σ q and continuous G -map of the form X Q Λ ✤ / / Λ σ : “ Λ ˚ σ P X σ Moreover p X σ , G , m σ q has diffraction | p σ | p γ with | p σ | P C bu p p G q .Proof. Let us show that Q σ p N φ q “ N σ ˚ φ at each φ P C c p G q . Let φ P C K p G q for somecompact K Ă G : One has at any Λ P XQ σ p N φ qp Λ q “ ż G U t p N φ qp Λ q dσ p t q “ ż G N φ p Λ ˚ δ t q dσ p t q in turns equal to ż G ż G φ p´ s q d p Λ ˚ δ t qp s q dσ p t q “ ż G ż G φ p´ s ´ t q d Λ p s q dσ p t q “ ż G ż G φ p´ s ´ t q dσ p t q d Λ p s q which is precisely equal to N σ ˚ φ p Λ q . This in particular gives, with K σ and K σ appro-priate constants only depending on σ , X and on the domain K , } Q σ p N φ q} ,ess “ } N σ ˚ φ } ,ess ď K σ } σ ˚ φ } ď K σ } φ } showing admissibility. Now for (almost every) Λ P X the measure Λ σ is uniquelydefined according to ż G φ ´ d Λ σ “ N Q σ φ p Λ σ q “ Q σ p N φ qp Λ q “ N σ ˚ φ p Λ q “ ż G φ ´ d p σ ˚ Λ q after computation, so π σ and p X σ , G , m σ q are of the stated form. The diffraction p γ σ of p X σ , G , m σ q satisfies for any pair φ, ψ P C c p G q ż p G p φ p ψ d p γ σ “ ż X Q σ p N φ q Q σ p N ψ q dm “ ż X N σ ˚ φ N σ ˚ ψ dm “ ż p G z σ ˚ φ z σ ˚ ψ d p γ “ ż p G p φ p ψ | p σ | d p γ showing that p γ σ “ | p σ | p γ , as desired. (cid:3) Appendix: Existence of a Maximal Kronecker Borel factor
We wish here, as we could not find any reference for this statement which seems tobe classical though, to provide a proof of Proposition 2.2, that is, to show that for anyergodic dynamical system p X, G , m q with eigenvalue group E Ă p G the Kronecker system p T E , G , m Haar q is a Borel factor of p X, G , m q . Proof of Proposition 2.2.
By ergodicity each eigenfunction Φ ω P L p X, m q , with ω P E ,is unique up to a multiplicative constant and can moreover be chosen of absolute valuealmost everywhere constant equal to 1 on X . Also, as discussed in Paragraph 2.1 eachΦ ω can also be chosen such that the equality Φ ω p x.t q “ ω p t q Φ ω p x q occurs everywhereon a G -stable Borel set ˜ X and for any t P G . Now the celebrated Pointwise ErgodicTheorem (see for instance [38, Theorem 2.14]) states as: Theorem 7.1. (Pointwise Ergodic Theorem) Let p X, G , m q be an ergodic dynamicalsystem. Then for any tempered Van Hove sequence p A n q n P N in G and any Borel m -integrable function f on X , one has for m -almost every x P X ż X f dm “ lim n Ñ8 | A n | ż A n f p x.t q dt Using this theorem we can prove:
Lemma 7.2.
The eigenfunctions Φ ω can be chosen to satisfy Φ ω . Φ ω “ Φ ω.ω m-almost everywhere @ ω, ω P E . Proof.
Consider a family p Φ ω q ω P E of eigenfunctions of modulus constant equal to 1 on X ,and such that the equality Φ ω p x.t q “ ω p t q Φ ω p x q occurs on a G -stable Borel set ˜ X andfor any t P G . Since this collection of functions is countable there exists, by ergodicity of m , a Borel subset of full measure on X such that | Φ ω | “ t| λ Φ ω . Φ ω ´ λ Φ ω.ω | : ω P E , λ , λ P C u Indeed one can find such a Borel subset where the conclusion of the Pointwise ErgodicTheorem holds when λ , λ are taken in a countable dense subset of C , and one easilyshows that on such set the conclusion of the Pointwise Ergodic Theorem also holds forany parameters λ , λ P C . Now pick up such an element x and set Φ ω : “ Φ ω p x q Φ ω :Then Φ ω . Φ ω “ Φ ω.ω along the orbit of x as it is straightforward to observe, and itfollows from the very choice of x that ż X | Φ ω . Φ ω ´ Φ ω.ω | dm “ lim n Ñ8 | A n | ż A n | Φ ω . Φ ω ´ Φ ω.ω |p x .t q dt “ URE POINT/CONTINUOUS DECOMPOSITION OF MEASURES AND DIFFRACTION 39 which shows that Φ ω . Φ ω “ Φ ω.ω almost everywhere on X . Thus one has a new family p Φ ω q ω P E of eigenfunctions, still of modulus constant equal to 1 on X and such that theequality Φ ω p x.t q “ ω p t q Φ ω p x q occurs on a G -stable Borel set ˜ X and for any t P G , andobeying the stated equality. (cid:3) We continue our proof of Proposition 2.2: Considering a family p Φ ω q ω P E of eigenfunc-tions as chosen and moreover satisfying the statement of Lemma 7.2, one ends up witha G -invariant Borel subset X : “ č ω,ω P E t Φ ω . Φ ω “ Φ ω.ω u of full m -measure in X . For each x in this Borel set, define π p x q to be the unique characteron E defined by ω ✤ / / Φ ω p x q , and to be the identity element elsewhere on X . Thisdefines a map π : X / / T E obeying the desired properties: Indeed it is a Borel mapwhich is G -equivariant in the sense that for any t P G one has π p x.t q “ π p x q . i p t q for m -almost every x P X . From this latter property it comes that the probability measure m push forwarded on T E is translation invariant by the dense subgroup i p G q and thus is theHaar measure on the compact Abelian group T E , completing the proof of the statement. Acknowledgments.
The author is grateful to Daniel Lenz for his careful reading ofthe manuscript.
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