Baake, Michael (ed.); Damanik, David (ed.); Kellendonk, Johannes (ed.); Lenz, Daniel (ed.) Spectral structures and topological methods in mathematical quasicrystals. Abstracts from the workshop held October 1–7, 2017. (English) Zbl 1409.00064 Oberwolfach Rep. 14, No. 4, 2781-2845 (2017). Summary: The mathematical theory of aperiodic order grew out of various predecessors in discrete geometry, harmonic analysis and mathematical physics, and developed rapidly after the discovery of real world quasicrystals in 1982 by Shechtman. Many mathematical disciplines have contributed to the development of this field. In this meeting, the goal was to bring leading researchers from several of them together to exchange the state of affairs, with special focus on spectral aspects, dynamics and topology. MSC: 00B05 Collections of abstracts of lectures 00B25 Proceedings of conferences of miscellaneous specific interest 52C23 Quasicrystals and aperiodic tilings in discrete geometry 11K70 Harmonic analysis and almost periodicity in probabilistic number theory 28D15 General groups of measure-preserving transformations 37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010) 54H20 Topological dynamics (MSC2010) 60B05 Probability measures on topological spaces 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 52-06 Proceedings, conferences, collections, etc. pertaining to convex and discrete geometry 11-06 Proceedings, conferences, collections, etc. pertaining to number theory Software:CHomP; GAP; Z PDFBibTeX XMLCite \textit{M. Baake} (ed.) et al., Oberwolfach Rep. 14, No. 4, 2781--2845 (2017; Zbl 1409.00064) Full Text: DOI References: [1] S Akiyama, J. Caalim, K. Imai, H. Kaneko, On corona limits: periodic case, preprint, arXiv:1707.02373. · Zbl 1416.52013 [2] S. Akiyama, K. Imai, The corona limit of Penrose tilings is a regular decagon, in Cellular Automata and Discrete Complex Systems, eds. M. Cook, T. Neary, Lecture Notes in Computer Science, vol. 9664, Springer, Cham (2016), pp. 35-48. Topological properties of some quasi-periodic tilings — From structure to spectrum Eric Akkermans (joint work with Yaroslav Don, Eli Levy, Dor Gitelman) The problem addressed is motivated by studies relevant to physical properties of some one-dimensional quasi-periodic tilings and quasicrystals. The meaning of structural and spectral properties is defined below. For the case of periodic tilings (crystals), these two types of properties are related. This constitutes the basis of the Bloch theorem (whose d = 1 version is sometimes referred to as Floquet theory). For quasi-periodic tilings, no such relation between structural and spectral data exists as yet. Our purpose is to present some preliminary results which may prove relevant towards such a relation. We consider a two-letter alphabet {a, b}. An aperiodic tiling can be obtained from different building rules. The first we consider is the substitution rule defined by its action σ on a word w = l1l2. . . lkby the concatenation σ(w) = σ(l1)σ(l2) . . . σ(lk). An occurrence primitive matrix, M =α βγ δso that σ(a) = aαbβand σ(b) = aγbδ, is associated to σ. It allows to define a sequence of numbers FNfrom the recurrence FN+1= tFN− pFN−1where t = TrM , p = detM and 2788Oberwolfach Report 46/2017 F0,1= 0, 1. The largest eigenvalue λ1of M is larger than 1 (Perron-Frobenius) and we consider substitutions whose second eigenvalue |λ2| < 1 (Pisot property). For the Fibonacci substitution M√f= (1 11 0) which we shall use as a generic example, λ1= τ =1+25and FNare the Fibonacci numbers. Another building rule, the Cut & Project method (hereafter C&P), is equivalent to defining a characteristic function, (1)χ(n, φ) = signcos 2π n λ−11+ φ− cos π λ−11. The phason parameter φ ∈ [0, 2π] is an extra gauge degree of freedom which fixes the origin of a given finite word. We consider first the more restrictive case of quasi-periodic tilings which can be described either by a substitution or by the characteristic function (C&P). Endowed with this description of a quasi-periodic tiling, we consider a distribution of identical atoms placed at the vertices xPkbetween consecutive letters. This defines the atomic density ρ(x) =kδ(x − xk). The distances δk= xk+1− xkdiffer depending on the letters a and b. The diffraction spectrum associated to this atomic density is obtained from the structure factor S(ξ) = |g(ξ)|2where we have defined the Fourier transform g(ξ) =Pkeiξxk. For C&P tilings, the diffraction spectrum consists of a dense set of Bragg peaks. We now consider words of finite size FNfor large N . Using (1), we obtain an expression of the atomic density Fourier transform (shifted by a non-relevant constant term), FXN−1 (2)g(ξ, φ) =ω−ξnχ(n, φ) n=0 2iπ with ω ≡ eFN. For C&P tilings, the corresponding structure factor S(ξ, φ) = |g(ξ, φ)|2is φ-independent. This result expresses that the positions of the discrete diffraction spectrum is independent of the choice of the origin. To prove this result we consider s0(n) = χ(n, 0) and apply the translation operator T [s0(n)] = s0(n+ 1). We then define the FN×FNmatrix Σ0whose matrix element Σ0(n, l) = Tl[s0(n)] and, more generally, the set of matrices Σr(n, l) = Tm(l,r)[s0(n)]. We then have Σ1(n, l) = χ(n, φl) with m(l, 1) ≡ lFN−1−1(mod FN). There, φ takes the discrete set of values φl=F2πl. This proves the announced result.1 N By contrast, the phase Θ(ξ, l) ≡ arg g(ξ, φ) = arg ωm(l,1)ξdepends on the phason φ. For each discrete diffraction peak ξq= qFN−1obtained from the structure factor S(ξq) = |g(ξq, φ)|2, the winding number associated to Θ(ξq, l) is Wξq= q. This indicates that topological features of C&P quasi-periodic tilings are encoded in the phase of the Fourier transform of the finite size atomic density. We then consider Schr¨odinger operators defined on C&P tilings as defined previously. Different approaches have been taken, e.g. tight-binding discrete Hamiltonians [3, 4, 5]. Here, since we are interested in properties of finite size tilings, we propose to obtain spectral properties such as density of states or counting function 1 A more thorough study of the group structure of the set of FNmatrices Σrwill be presented in Ref. [8]. Mathematical Quasicrystals2789 (integrated density of states) from the scattering operator. To that purpose, we consider embedding a tiling of finite size FNbetween two semi-infinite identical and periodic tilings built out of either the letter a or b with appropriate boundary conditions. These semi-infinite leads support incoming and outgoing plane wave solutions (see [1] for details). In this setup, the unitary scattering operator which relates incoming to outgoing waves is a 2 × 2 matrix, −→rt i (3)oR=t←−riRL≡ SFN(φ)iiRL where the transmission and reflection complex amplitudes are given by t ≡|t|ei θt, −→r ≡rei−→θand ←−r ≡rei←−θwith the arrow convention indicating incoming waves from left or right. SFN(φ) is diagonalizable under the form diag(eiΦ1(k,φ), eiΦ2(k,φ)). We define δ(k) ≡(Φ1+Φ2)/2 known as the total phase shift. It allows to obtain a simple and useful relation for the density of states ρ(k), known as the Krein-Schwinger formula, namely 1∂1dδ(k) 2πIm∂kln det S(k) =πdk, where ρ0(k) is the density of states of the free system, i.e. without the scattering structure [1]. Using the unitarity condition, −→r∗t + ←−r t∗= 0, we obtain the additional expressions det S = e2iδ= −t/t∗= ←−r /−→r∗and δ(k) = θt(k) + π/2 = 1−→←− 2θ +θ. The notations −→r and ←−r represent the two possible transmission channels which are identical except for the phases of the reflected amplitudes. Therefore, δ(k) may be expressed using either the transmitted phase shift or the sum of the two possible reflected phase shifts. The total phase shift allows to characterize the zero measure Cantor set spectrum of Schr¨odinger operators defined on C&P quasi-periodic tilings (e.g., gap labelling theorem). In addition to numerous theoretical and numerical studies [3, 4, 5], this Cantor spectrum has also been observed experimentally [6]. It has been shown that δ(k) is independent of the phason φ just like the structure factor S(ξ) = |g(ξ, φ)|2previously discussed. A second scattering phase Λ(k, φ) ≡ (Φ2− Φ1)/2, complementary to δ(k) is available from the diagonal form of SFN(φ). It carries additional information regarding the structure, unavailable through δ(k). A useful rewriting of this second phase is Θcav(k, φ) = 2δ(k) + α(φ) with α ≡←−θ −−→θ which conveniently disentangles the k and φ dependencies [2, 7]. The winding properties of the phase Θcav(k, φ) for values of k in spectral gaps are identical to those obtained for the phase Θ(ξ, l) ≡ arg g(ξ, φ) in the diffraction spectrum of the finite size atomic density [8]. These identical behaviours have been observed experimentally [9, 10]. This relation between topological features in the diffraction and Schr¨odinger spectra constitutes a step towards a Bloch theorem for C&P and for certain substitution families of quasi-periodic tilings. 2790Oberwolfach Report 46/2017 References [3] E. Akkermans, G.V. Dunne, E. Levy, Wave propagation in one dimension: Methods and applications to complex and fractal structures, in Optics of Aperiodic Structures: Fundamentals and Device Applications, ed. L. dal Negro, Pan Stanford Publishing, Singapore (2013), pp. 407-450. [4] E. Levy, A. Barak, A. Fisher, E. Akkermans, Topological properties of Fibonacci quasicrystals: A scattering analysis of Chern numbers, preprint, arXiv:1509.04028 (2015). [5] J. Bellissard, A. Bovier, J.-M. Ghez, Spectral properties of a tight binding Hamiltonian with period doubling potential, Commun. Math. Phys. 135 (1991), 379-399. · Zbl 0726.58038 [6] J. Bellissard, B. Iochum, E. Scoppola, D. Testard, Spectral properties of one-dimensional quasicrystals, Commun. Math. Phys. 125 (1989), 527-543. · Zbl 0825.58010 [7] D. Damanik, M. Embree, A. Gorodetski, Spectral properties of Schr¨odinger operators arising in the study of quasicrystals, in Mathematics of Aperiodic Order eds. J. Kellendonk, D. Lenz, J. Savinien, Progress in Mathematics, vol. 309, Birkh¨auser, Basel (2015), pp. 307-370. · Zbl 1378.81031 [8] D. Tanese, E. Gurevich, F. Baboux, T. Jacqmin, A. Lemaˆıtre, E. Galopin, I. Sagnes, A. Amo, J. Bloch, E. Akkermans, Fractal energy spectrum of a polariton gas in a Fibonacci quasiperiodic potential, Phys. Rev. Lett. 112, 146404 (2014). [9] E. Levy, E. Akkermans, Topological boundary states in 1D: An effective Fabry-Perot model, Eur. Phys. J. Special Topics 226 (2017), 1563-1582. [10] Y. Don, E. Levy, D. Gitelman, E. Akkermans, unpublished results (2017). [11] F. Baboux, E. Levy, A. Lemaˆıtre, C. G´omez, E. Galopin, L. Le Gratiet, I. Sagnes, A. Amo, J. Bloch, E. Akkermans, Measuring topological invariants from generalized edge states in polaritonic quasicrystals, Phys. Rev. B 95 (2017), 161114(R). [12] A. Dareau, E. Levy, M. Bosch Aguilera, R. Bouganne, E. Akkermans, F. Gerbier J. Beugnon, Direct measurement of Chern numbers in the diffraction pattern of a Fibonacci chain, to appear in Phys. Rev. Lett. (2017), arXiv:1607.00901. Spectral stability of Schr¨odinger operators in the Hausdorff metric Siegfried Beckus (joint work with Jean Bellissard, Horia Cornean, Giuseppe de Nittis, Felix Pogorzelski) In this talk, recent developments and results are discussed that are based on various collaborations [1, 2, 3, 4, 5]. Within this research project, we seek to connect dynamical and spectral properties of self-adjoint operators. In the centre of our considerations are Schr¨odinger operators arising in quantum mechanical models of non-periodic solids. Up to now, the approach by transfer matrices and trace maps led to amazing results showing that new phenomena appear in physical systems. Unfortunately, this techniques do not extend to higher dimensional systems except if the operators decompose to one-dimensional systems [7, 8]. Inspired by the techniques developed so far, an appropriate approximation theory would be helpful for numerical and analytic results. Such a theory is established in [1, 2, 3, 4, 5]. During the talk we discuss this approach while a special focus is put on the quantitative estimates achieved in [3]. For simplicity of the talk, we restrict ourselves to the simpler case of symbolic dynamical systems over Zdwhile most of the results hold in much larger generality. Mathematical Quasicrystals2791 Given a self-adjoint bounded operator H, its spectrum σ(H) is a compact subset of R. The space K(R) of compact subsets of R is naturally equipped with the Hausdorff metric dHinduced by the Euclidean metric. In [2], a family of selfadjoint bounded operators (Ht)t∈Tis studied indexed by a topological (metric) space T . There the (H¨older-)continuity of the map t 7→ σ(Ht) is characterized by the (H¨older-)continuity of suitable norms of the operators. This approach is used to show the convergence of the spectra if the underlying dynamical systems converge which is described next. For a finite set A, we restrict our considerations to the symbolic dynamical system (AZd, Zd). Specifically, the configuration space AZd:=Qn∈ZdA is equipped with the product topology. Furthermore, Zdacts continuously by translation, i.e., αn(w) := w(· − n) for n ∈ Zdand w ∈ AZd. A family of Schr¨odinger operators Hw: ℓ2(Zd) → ℓ2(Zd), w ∈ AZd, is defined by ! X Hw:=Mqh,wUh+ U−hMqh,w+ Mp,w h∈R where R ⊆ Zdis finite and for f : AZd→ C, Uhu(n) := u(n − h) ,Mf,wu(n) := f α−n(w)u(n) ,u ∈ ℓ2(Zd) . If R generates Zd, the sum over R is the Laplacian on the Cayley graph of Zdwith generators R. The corresponding involved multiplication operators are interpreted as weights on the edges. The multiplication operator Mp,wby the real valued function p ◦ α−•(w) : Zd→ R is called the potential term. The multiplication operators by qh◦ α−•(w) : Zd→ C and p ◦ α−•(w) : Zd→ R should reflect the local structure of the configuration w at the corresponding position in Zd. In light of this, the functions qh: AZd→ C and p: AZd→ R are assumed to be continuous. Motivated by the elaborations for one-dimensional systems, periodic approximations are the most promising so far in order to deal with Schr¨odinger operators Hwwhere w represents a quasicrystal. This is based on the fact that the spectral properties can be analyzed by the Floquet-Bloch theory and, from experience, periodic approximations admit the best convergence rates. Hence, we address the question which notion of convergence of periodic systems (wn) to w implies the convergence of suitably many spectral properties of the associated Schr¨odinger operators. The elaborations [1, 2, 3, 4, 5] show that the Chabauty-Fell topology [6, 9] on dynamical subsystems encodes several spectral properties. More precisely, the compact metrizable space of dynamical subsystems Ω ⊆ AZdclosed, invariant⊆ K AZd equipped with the Chabauty-Fell topology is studied. A metric on I is given by the Hausdorff metric dAHinduced by the metric d : AZd× AZd→ [0, 1] ,d(w, w′) :=1 supr ∈ N0: w|Br= w′|Br+ 1 on AZdwhere Br⊆ Rdis the closed ball with radius r centred at 0. 2792Oberwolfach Report 46/2017 In [1, 4], a special focus is put on the convergence of the spectra of the Schr¨odinger operators. More precisely, the following equivalence is shown σ(Hwn) → σ(Hw) in K(R) for all R ⊆ Zd Orb(wn) → Orb(w) in I⇐⇒ finite and all continuous p and q where Orb(w) :={αn(w) : n ∈ Zd} is the orbit closure in the product topology. The hard direction “⇒” relies on the construction of a continuous field of C∗algebras by gluing together the dynamical systems. The fruitful outcome of this project is based on the embedding of the dynamics in the Chabauty-Fell topology defined on I. This strategy works out in the generality of topological groupoids as shown in [4]. Hence, an analogous result holds for R infinite with suitable decay assumptions on the off-diagonal terms, general dynamical systems (X, G) and Schr¨odinger operators associated with Delone sets. The only restriction is a suitable amenable assumption on the underlying structure. Thus, our approach opens the possibility to handle very interesting examples such as the Penrose tiling. As discussed before, we focus on the existence and construction of periodic approximations for quasicrystals. The existence and construction of periodic approximations is solved for 1-dimensional systems [1, 4]. In higher dimensions, local symmetries of substitutional systems lead to a specific construction of periodic approximations covering known results such as for the Fibonacci sequence but also for higher dimensional systems like the Table tiling [1]. Furthermore, first elaborations [5] show that cut-and-project sets in the Euclidean setting admit suitable periodic approximations for measured quantities. In a recent project [5], we analyze the behaviour of measures associated with Delone dynamical systems in the Chabauty-Fell topology within I. This approach is valid for Delone dynamical systems in general locally compact second countable Hausdorff groups. It turns out that the measures converge if the underlying Delone dynamical systems converge in I and the limit object is uniquely ergodic. More precisely, the density of states measure and the autocorrelation measure is considered. As application, cut-and-project sets are studied by approximating the corresponding lattice and the window function by continuous window functions. It is worth mentioning that small changes on the lattice provide periodic approximations in the Euclidean setting which makes this approach very interesting. Continuous window functions need to be considered as the cutting process is highly non-continuous. It is left for further studies if the approximation of the window function also leads to the convergence of the related spectra. As discussed before, a C∗-algebraic approach is used in [1, 4] to show the convergence of the spectra. In the recent work [3], a different proof is provided without this machinery for Schr¨odinger operators on lattices in Rdadmitting strongly pattern equivariant potentials qhand p. More precisely, the existence of a constant C > 0 (depending on the dimension) is proven such that α dHσ(Hw), σ(Hw′)≤ C · kHkα· dAHOrb(w), Orb(w′),w, w′∈ AZd. Here kHkαdenotes the Schur α-norm where α ∈ [0, 1] is chosen such that kHkαis finite. The Schur α-norm measures the polynomial decay of the off-diagonal terms Mathematical Quasicrystals2793 of the Schr¨odinger operator. Thus, kHk1is finite whenever R is finite implying the Lipschitz continuity of the spectra. The proof is based on a suitable localization of the operator via a quadratic partition of Rd. With this at hand, the main task is to estimate the commutators with such a localization which is discussed during the talk. References [13] S. Beckus, Spectral approximation of aperiodic Schr¨odinger operators, PhD thesis, FriedrichSchiller Universit¨at Jena (2016). [14] S. Beckus, J. Bellissard, Continuity of the spectrum of a field of self-adjoint operators, Ann. Henri Poincar´e 17 (2016), 3425-3442. · Zbl 1354.81018 [15] S. Beckus, J. Bellissard, H. Cornean, Spectral stability of Schr¨odinger operators in the Hausdorff metric, in preparation (2017). [16] S. Beckus, J. Bellissard, G. de Nittis, Spectral continuity for aperiodic quantum systems I. General theory, preprint, arXiv:1709.00975 (2017). · Zbl 1406.81023 [17] S. Beckus, F. Pogorzelski, Delone dynamical systems and spectral convergence, in preparation (2017). · Zbl 1443.37007 [18] C. Chabauty, Limite d’ensembles et g´eom´etrie des nombres, Bull. Soc. Math. France 78 (1950), 143-151. · Zbl 0039.04101 [19] D. Damanik, A. Gorodetski, Sums of regular Cantor sets of large dimension and the square Fibonacci Hamiltonian, preprint, arXiv:1601.01639 (2016). · Zbl 1344.47021 [20] D. Damanik, A. Gorodetski, B. Solomyak, Absolutely continuous convolutions of singular measures and an application to the square Fibonacci Hamiltonian, Duke Math. J. 164 (2015), 1603-1640. · Zbl 1358.37117 [21] J.M.G. Fell, A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc. 13 (1962), 472-476. Orbit equivalence, dimension groups and eigenvalues Mar´ıa Isabel Cortez, Fabien Durand (joint work with Samuel Petite) In a series of papers [3, 1, 2, 6, 5] with X. Bressaud, A. Frank, B. Host, A. Maass and S. Petite we studied continuous and non-continuous eigenvalues of minimal Cantor systems. Among other questions, we were looking for (computable) necessary and sufficient conditions for eigenvalues to be continuous or non-continuous. This was achieved for the continuous eigenvalues in [5]. With S. Petite we investigated the restrictions induced on the groups of eigenvalues that can be realized within a given strong orbit equivalence class. For a minimal Cantor system (X, T ), let E(X, T ) be the set of real numbers α such that λ = exp(2iπα) is a continuous eigenvalue (that is, having a continuous eigenfunction f : f ◦ T = λf). We know from [9] that strong orbit equivalent minimal Cantor systems share the same subgroup of continuous eigenvalues that are roots of unity. It is no longer true for the orbit equivalence as shown again in [9]. Indeed, Ormes proved that in a prescribed orbit equivalence class it is possible to realize any countable subgroup of the circle as a group of measurable eigenvalues. 2794Oberwolfach Report 46/2017 It happens that a first restriction has been shown in [8]: the additive group of eigenvalues, E(X, T ), of a minimal Cantor system (X, T ), is a subgroup of the intersection of all the images of the dimension group by its traces. DynamicallyR speaking, it is a subgroup of I(X, T ) = ∩µ∈M(X,T )f dµ|f ∈ C(X, Z), where M(X, T ) is the set of T -invariant probability measures of (X, T ) and C(X, Z) is the set of continuous functions from X to Z. A different proof of this observation can be found in [3] but it was not pointed out. In [4] we obtained the following additional constraint. Theorem. Suppose that (X, T ) is a minimal Cantor system such that the infinitesimal subgroup of the dimension group K0(X, T ) is trivial. Then the quotient group I(X, T )/E(X, T ) is torsion free. In [7] the hypotheses were relaxed removing the one on the infinitesimal subgroup. To illustrate this result, take K0(X, T ) = Z + αZ = I(X, T ), with α irrational. This is the case for a Sturmian subshift. Then within the strong orbit equivalence class of (X, T ) the only groups of continuous eigenvalues that can be realized are Z, which will provide topologically weakly mixing minimal Cantor systems, and Z+ αZ. Moreover, both can be realized, in the first case using results in [9] and in the second case it is realized by a Sturmian subshift. Apart from such particular examples no general realization results have been obtained so far. References [22] X. Bressaud, F. Durand, A. Maass, Necessary and sufficient conditions to be an eigenvalue for linearly recurrent dynamical Cantor systems, J. London Math. Soc. 72 (2005), 799-816. · Zbl 1095.54016 [23] X. Bressaud, F. Durand, A. Maass, On the eigenvalues of finite rank Bratteli-Vershik dynamical systems, Ergodic Th. & Dynam. Syst. 30 (2010), 639-664. · Zbl 1204.37008 [24] M.I. Cortez, F. Durand, B. Host, A. Maass, Continuous and measurable eigenfunctions of linearly recurrent dynamical Cantor systems, J. London Math. Soc. 67 (2003), 790-804. · Zbl 1045.54011 [25] M.I. Cortez, F. Durand, S. Petite, Eigenvalues and strong orbit equivalence, Ergodic Th. & Dynam. Syst. 36 (2016), 2419-2440. · Zbl 1375.37008 [26] F. Durand, A. Frank, A. Maass, Eigenvalues of minimal Cantor systems, preprint, arXiv:1504.00067 (2015). · Zbl 1416.54018 [27] F. Durand, A. Frank, A. Maass, Eigenvalues of Toeplitz minimal systems of finite topological rank, Ergodic Th. & Dynam. Syst. 35 (2015), 2499-2528. · Zbl 1356.37013 [28] T. Giordano, D. Handelman, M. Hosseini, Orbit equivalence of Cantor minimal systems and their continuous spectra, preprint, arXiv:1606.03824 (2016). · Zbl 1394.37015 [29] B. Itz´a-Ortiz, Eigenvalues, K-theory and minimal flows, Canad. J. Math. 59 (2007), 596– 613. · Zbl 1132.37007 [30] N.S. Ormes, Strong orbit realization for minimal homeomorphisms, J. Anal. Math. 71 (1997), 103-133. Mathematical Quasicrystals2795 Spectral calculations for two-dimensional quasicrystals Mark Embree (joint work with David Damanik, Jake Fillman, Anton Gorodetski, May Mei, Charles Puelz) Mathematical models of one-dimensional quasicrystals, most notably the Fibonacci Hamiltonian, have reached an advanced state of refinement [2]. In contrast, our understanding of two dimensional quasicrystal models remains at a nascent stage. This talk described some analytical and computational results related to twodimensional models. Fast spectral computation for 1d periodic models The simplest two-dimensional models are constructed by combining one-dimensional quasiperiodic models on a square lattice. The spectrum of such a square model then equals the set sum of the corresponding one-dimensional spectra. While these one-dimensional spectra are Cantor sets, the same need not be true for the square model. We presented numerical calculations from [1] that estimate the structure of this spectrum as a function of the coupling constant (i.e., the weight of the potential) for the Fibonacci, Thue-Morse, and period doubling models. In numerous circumstances, periodic approximations of one-dimensional quasiperiodic potentials lead to covers (upper bounds) on the spectrum of the quasiperiodic model; long periods yield more accurate estimates. Floquet-Bloch theory shows that the spectrum of a one-dimensional model of period p comprises the union of p real intervals that are traced out by the eigenvalues of a parameterized p × p matrix Jp(θ); for p = 7, v11e−iθ 1v21 1v31 Jp(θ) =1v41, 1v51 1v61 eiθ1v7 and the spectrum of the one dimensional model is ∪θ∈[0,π]σ(Jp(θ)). (The vjvalues specify the potential; unspecified entries are zero.) The ends of these intervals are given by σ(Jp(0)) and σ(Jp(π)), so one can determine the spectrum of a periodic approximation by computing all the eigenvalues of two p × p symmetric matrices. When p is large, the corner entries e±iθcause the conventional QR eigenvalue algorithm to use O(p2) storage and O(p3) computation time. We describe a simple trick from [6] that relabels the p sites in the periodic potential using a breadth-first ordering, effectively performing an orthogonal similarity transformation (gauge 2796Oberwolfach Report 46/2017 transformation) to obtain the pentadiagonal matrix (illustrated for p = 7) v1e−iθ1 eiθv701 10v201 P Jp(θ)P∗=10v601. 10v301 10v50 10v4 This transformed matrix has fixed bandwidth independent of p, so standard algorithms from numerical linear algebra deliver all the eigenvalues of this matrix with O(p) storage and O(p2) computation. This improvement becomes particularly crucial because large p values often lead to numerically inaccurate eigenvalues (we are seeking good covers of Cantor sets, after all), necessitating the use of expensive extended precision arithmetic. We illustrated this algorithm with results from [6] showing the accuracy of double and quadruple precision computations, along with estimates of the Hausdorff dimension of the Fibonacci model and gap scaling for the Thue-Morse potential. Gap openings in 2d periodic models In the period-p models described above, one can always construct a potential, arbitrarily small in norm, that has a spectrum with p − 1 distinct gaps. Is the same true of (p, q)-periodic models on a square lattice? We described a recent result from [3] that shows that this is not the case, extending earlier work of Kr¨uger [5]. Specifically, if both p and q are even, an arbitrarily small (p, q)-periodic model can open a gap at E = 0; with this exception, arbitrarily small (p, q)-periodic potentials cannot open any gaps. The proof of this fact follows from eigenvalue perturbation theory for symmetric matrices. Eigenvalues of the graph Laplacian for the Penrose tiling Arguably the most intriguing two-dimensional quasicrystal model comes from the Laplacian on a graph generated from the Penrose tiling. We closed the talk by presenting numerical results (with Fillman and Mei) based on Robinson’s stone inflation rule using triangular tiles. We illustrated modes with local support occurring at E = 2 and E = 4 (as observed by Kohmoto and Sutherland [4]), which lead to a jump in the integrated density of states at those energies; we showed other intriguing modal structures for this model that merit further investigation. References [31] D. Damanik, M. Embree, A. Gorodetski, Spectral properties of Schr¨odinger operators arising in the study of quasicrystals, in Mathematics of Aperiodic Order eds. J. Kellendonk, D. Lenz, J. Savinien, Progress in Mathematics, vol. 309, Birkh¨auser, Basel (2015), pp. 307-370. · Zbl 1378.81031 [32] D. Damanik, A. Gorodetski, W. Yessen, The Fibonacci Hamiltonian, Invent. Math. 206 (2016), 629-692. Mathematical Quasicrystals2797 · Zbl 1359.81108 [33] M. Embree, J. Fillman, Spectra of discrete two-dimensional periodic Schr¨odinger operators with small potentials, to appear in J. Spectral Th., arXiv:1701.00863. · Zbl 1436.35095 [34] M. Kohmoto, B. Sutherland, Electronic and vibrational modes on a Penrose lattice: Localized states and band structure, Phys. Rev. B 34 (1986), 3849-3853. [35] H. Kr¨uger, Periodic and limit-periodic discrete Schr¨odinger operators, preprint, arXiv:1108.1584. [36] C. Puelz, M. Embree, J. Fillman, Spectral approximation for quasiperiodic Jacobi operators, Integr. Eq. Oper. Th. 82 (2015), 533-554. Spectral properties of continuum quasicrystal models Jake Fillman (joint work with David Damanik, Mark Embree, Anton Gorodetski, May Mei, Yuki Takahashi, William Yessen) We consider continuum Schr¨odinger operators acting in L2(R) via LVu = −u′′+ V u, where the potential V : R → R models a quasicrystal; for an archetypal example, let X (1)Vω(x) =(1 − ωn)f0(x − n) + ωnf1(x − n), n∈Z where f0, f1∈ L2[0, 1) and ω denotes the Fibonacci sequence √ (2)ωn= χ[1−α)(nα mod 1),n ∈ Z, α =5 − 1. 2 These operators are interesting because their spectra are globally zero-measure Cantor sets [4]; see also [12] for the case of measure-valued potentials. In fact, this holds true for any aperiodic potential of the type (1), as long as the sequence ω is generated by a minimal subshift satisfying Boshernitzan’s criterion (cf. [1, 2, 3]). It is then of interest to assess more delicate fractal properties, such as the Hausdorff dimension. For general potentials, this is currently out of reach, but these questions can be studied in the Fibonacci case using the trace map and tools from hyperbolic dynamics. In particular, for Vωand ω as in Eqs. (1)–(2), one has liminfdimlocHσ(LVω); E= 1; K→∞E∈σ(LVω)∩[K,∞) that is, the local Hausdorff dimension of the spectrum tends to one in the highenergy region [4, 7]. This holds for any choice of f0and f1and hence this property is independent of the shape of the bump functions one uses to pattern the Fibonacci potential. Turning to higher dimensions, one may study separable Schr¨odinger operators, i.e., operators of the form L(2)Vu 1,V2(x, y) = −∇2u(x, y) + V1(x)u(x, y) + V2(y)u(x, y), 2798Oberwolfach Report 46/2017 where the potential is the sum of two pieces: one piece depends only on x, and the other depends only on y. The spectra of such operators are amenable to analysis, as they are simply the Minkowski sum of the 1D spectra, that is, σ L(2)V= σ(L 1,V2V1) + σ(LV2) =a1+ a2: aj∈ σ(LVj). However, even these ostensibly simple models touch on deep questions in geometric measure theory. In particular, the Minkowski sum of two zero-measure Cantor sets can be a Cantor set, an interval, a finite union of intervals, or something even more exotic. There are parameter ranges for which σ L(2)λcontains 1Vω,λ2Vω both intervals and Cantor sets [8], where Vωis defined by Eqs. (1)–(2). However, there is reason to suspect more is true. Motivated in part by the results of [5] for separable discrete models built around the Fibonacci sequence, we pose the following question: Question. If Vωis defined by Eqs. (1)– (2), is it true that σ L(2)λcontains 1Vω,λ2Vω a half-line for every choice of λ1, λ2> 0? This question is also motivated by and connected with the Bethe-Sommerfeld conjecture for periodic Schr¨odinger operators, which inspired substantial contributions from many authors, including (but certainly not limited to) [10, 11, 14, 15, 16, 17, 18], and culminating in the paper of Parnovskii [13]. On that note, we conclude with a discussion of the discrete Bethe-Sommerfeld conjecture, proved in dimension 2 by Embree-Fillman [6] and in higher dimensions by Han-Jitomirskaya [37] . The spectrum of a 2D periodic discrete Schr¨odinger operator with a sufficiently small potential consists of either one or two intervals, and is guaranteed to be a single interval as long as at least one period is odd. Using the simple ℓ∞ perturbation theory, this immediately implies that a large class of limit-periodic Schr¨odinger operators in Z2have spectra with one or two connected components. References [38] M. Boshernitzan, A unique ergodicity of minimal symbolic flows with linear block growth, J. Analyse Math. 44 (1984/85), 77-96. · Zbl 0602.28008 [39] M. Boshernitzan, A condition for minimal interval exchange maps to be uniquely ergodic, Duke Math. J. 52 (1985), 723-752. · Zbl 0602.28009 [40] M. Boshernitzan, A condition for unique ergodicity of minimal symbolic flows, Ergodic Th. & Dynam. Syst. 12 (1992), 425-428. · Zbl 0756.58030 [41] D. Damanik, J. Fillman, A. Gorodetski, Continuum Schr¨odinger operators associated with aperiodic subshifts, Ann. Henri Poincar´e 15 (2014), 1123-1144. · Zbl 1292.81052 [42] D. Damanik, A. Gorodetski, Spectral and quantum dynamical properties of the weakly coupled Fibonacci Hamiltonian, Commun. Math. Phys. 305 (2011), 221-277. · Zbl 1232.81016 [43] M. Embree, J. Fillman, Spectra of discrete two-dimensional periodic Schr¨odinger operators with small potentials, J. Spectr. Theor., in press. · Zbl 1436.35095 [44] J. Fillman, M. Mei, Spectral properties of continuum Fibonacci Schr¨odinger operators, Ann. Henri Poincar´e, in press. [45] J. Fillman, Y. Takahashi, W. Yessen, Mixed spectral regimes for square Fibonacci Hamiltonians, J. Fract. Geom. 3 (2016), 377-405. · Zbl 1370.47033 [46] R. Han, S. Jitomirskaya, Discrete Bethe-Sommerfeld conjecture, preprint, arxiv:1707.03482. Mathematical Quasicrystals2799 · Zbl 1403.35190 [47] B. Helffer, A. Mohamed, Asymptotics of the density of states for the Schr¨odinger operator with periodic electric potential, Duke Math. J. 92 (1998), 1-60. · Zbl 0951.35104 [48] Y.E. Karpeshina, Perturbation Theory for the Schr¨odinger Operator with a Periodic Potential, Lecture Notes in Mathematics, vol. 1663, Springer, Berlin (1997). [49] D. Lenz, C. Seifert, P. Stollmann, Zero measure Cantor spectra for continuum onedimensional quasicrystals, J. Diff. Eq. 256 (2014), 1905-1926. · Zbl 1351.47030 [50] L. Parnovski, Bethe-Sommerfeld conjecture, Ann. Henri Poincar´e 9 (2008), 457-508. · Zbl 1201.81054 [51] V.N. Popov, M. Skriganov, A remark on the spectral structure of the two dimensional Schr¨odinger operator with a periodic potential, Zap. Nauchn. Sem. 109 (1981), 131-133 (in Russian). · Zbl 0492.47024 [52] M. Skriganov, Proof of the Bethe-Sommerfeld conjecture in dimension two, Soviet Math. Dokl. 20 (1979), 89-90. · Zbl 0417.35063 [53] M. Skriganov, Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Proc. Steklov Math. Inst. 171 (1984), 3-122. [54] M. Skriganov, The spectrum band structure of the three-dimensional Schr¨odinger operator with periodic potential, Invent. Math. 80 (1985), 107-121. · Zbl 0578.47003 [55] O.A. Veliev, Spectrum of multidimensional periodic operators, Teor. Funktsi˘ı Funktsional. Anal. i Prilozhen 49 (1988), 17-34 (in Russian). Weak forms of equicontinuity Felipe Garc´ıa-Ramos In this report we will discuss the relationship between the regularity of minimal topological dynamical systems, in terms of forms of equicontinuity, and the regularity of topological and measure theoretic factors to equicontinuous sytems. This will provide a hierarchy of non-chaotic dynamical systems. We say (X, T ) is topological dynamical system (TDS), if X is a compact metric space (with metric d) and T : X → X is continuous. We say (X, T, µ) is a measure preserving topological dynamical system (MP-TDS), if (X, T ) is a TDS and µ is a T -invariant probability measure. We say (X, T ) and (X′, T′) are conjugate if there exists a continuous bijective function f : X → X′such that T′◦ f = T ◦ f. We say (X, T, µ) and (X′, T′, µ′) are isomorphic if there exists a bi-measurable invertible function f : X → X′such that f and f−1push the measures and T′◦ f = T ◦ f. There are some non-symmetric notions of similarity that use both the topology and the measure. We say (X, T, µ) is an isomorphic extension of (X′, T′, µ′) if they are isomorphic but we also require f to be continuous and surjective. We say (X, T, µ) is a regular isomorphic extension of (X′, T′, µ′) if there exists a surjective continuous function f : X → X′such that T′◦ f = T ◦ f and ′′ µx∈ X′: {f−1(x′)} is a singleton= 1. Note that for a pair of MP-TDSs, conjugacy ⇒ regular isomorphic extension ⇒ isomorphic extension ⇒ isomorphism. A TDS is equicontinuous if the family {Ti}i∈Nis equicontinuous or, equivalently, if for every ε > 0 there exists δ > 0 such that for every open set with diam(U ) ≤ ε then diam(TiU ) ≤ δ for all i ∈ N (where diam(U) denotes the diameter of the set). 2800Oberwolfach Report 46/2017 Classical equicontinuity is a very strong property and it is not very useful for studying subshifts or Delone systems. Every equicontinuous subshift or Delone system is periodic [2]. A weaker form of equicontinuity was introduced by Fomin [4]. Definition 1. We say a TDS is mean equicontinuous if for every ε > 0 there exists δ > 0 such that if d(x, y) ≤ δ then XN 1 lim supd(Tix, Tiy) ≤ ε. N→∞N i=1 Every equicontinuous TDS is mean equicontinuous. Given the average nature of the definition one might expect that this topological property is strongly related to ergodic properties. Oxtoby showed that every minimal mean equicontinuous system is uniquely ergodic [14, 1]. It was conjectured that this measure must have discrete spectrum [15] (see definition below). This question was answered independently in [13] and in [5]. Theorem 1 ([3, 13]). A minimal TDS (X, T ) is mean equicontinuous if and only if there exists an equicontinuous TDS (X′, T′) such that (X, T, µ) is an isomorphic extension of (X′, T′, µ′) (where µ and µ′are the respective invariant measures). We say a TDS is BD-mean equicontinuous if for every ε > 0 there exists δ > 0 such that for every open set with diam(U ) ≤ δ 1XN limsupdiam(TiU ) ≤ ε. N−M→∞N − M i=M+1 Theorem 2 ([5, 6]). A minimal TDS (X, T ) is BD-mean equicontinuous if and only if there exists an equicontinuous TDS (X′, T′) such that (X, T, µ) is a regular isomorphic extension of (X′, T′, µ′) (where µ and µ′are the respective invariant measures). It is not hard to show that a TDS is BD-mean equicontinuous if and only if for every ε > 0 and x ∈ X there exists δ > 0 such that BDi ∈ N : diam(TiBδ(x)) > ε< ε, where Bδ(x) is the δ-neighbourhood of x, and BD denotes the upper Banach density. Another form of order is zero topological sequence entropy, also known as null systems. For definition and properties see [12, 10]. Theorem 3 ([5, 6]). Every minimal null TDS is BD-mean equicontinuous. Definition 2. Let (X, T ) be a TDS and µ an invariant Borel probability measure. We say (X, T ) is µ-mean equicontinuous if for every τ > 0 there exists a compact set M ⊂ X with µ(M) ≥ 1 − τ, such that for every ε > 0 there exists δ > 0 such Mathematical Quasicrystals2801 that whenever x, y ∈ M and d(x, y) ≤ δ then XN 1 N→∞Nd(Tix, Tiy) ≤ ε. i=1 Halmos and von Neumann showed that an ergodic dynamical system has discrete spectrum if and only if it is isomorphic to a minimal equicontinuous TDS (respective to its invariant measure) [11]. Theorem 4. Let (X, T ) be a TDS and µ an invariant ergodic probability measure. Then (X, T, µ) is isomorphic to a minimal equicontinuous TDS if and only if (X, T ) is µ-mean equicontinuous. Note that uniquely ergodic topologically weak mixing systems may have discrete spectrum. These systems are never mean equicontinuous [7]. Let (X, T ) be a TDS µ a Borel probability measure and f ∈ L2(X, µ). We say (X, T ) is µ-f -mean equicontinuous if for every τ > 0 there exists a compact set M ⊂ X with µ(M) ≥ 1 − τ such that for every ε > 0 there exists δ > 0 such that whenever x, y ∈ M and d(x, y) ≤ δ then XN 1 f(Ti N→∞Nx) − f(Tiy) ≤ ε. i=1 Theorem 5 ([8]). Let (X, T ) be a TDS, µ an invariant ergodic probability measure and f ∈ L2(X, µ). Then f is almost periodic if and only if (X, T ) is µ-f -mean equicontinuous. We have the following hierarchy for minimal uniquely ergodic systems (each implication is strict): equicontinuous ⇒ null ⇒ BD-mean equicontinuous ⇒ mean equicontinuous ⇒ µ-mean equicontinuous ⇒ µ-f-mean equicontinuous (for some but not every f ∈ L2). References [56] J. Auslander, Mean-L-stable systems, Illinois J. Math. 3 (1959), 566-579. · Zbl 0097.11102 [57] M. Baake, D. Lenz, R.V. Moody, Characterization of model sets by dynamical systems, Ergodic Th. & Dynam. Syst. 27 (2007), 341-382. · Zbl 1114.82022 [58] T. Downarowicz, E. Glasner, Isomorphic extensions and applications, Topol. Methods Nonlinear Anal. 48 (2015), 321-338. · Zbl 1362.37026 [59] S. Fomin, On dynamical systems with a purely point spectrum, Dokl. Akad. Nauk SSSR 77 (1951), 29-32. [60] F. Garc´ıa-Ramos, Weak forms of topological and measure theoretical equicontinuity: relationships with discrete spectrum and sequence entropy, Ergodic Th. & Dynam, Syst. 37 (2017), 1211-1237. · Zbl 1366.37019 [61] F. Garc´ıa-Ramos, E. Glasner, X. Ye, Regular isomorphic extensions and diam-mean equicontinuity, in progress. [62] F. Garc´ıa-Ramos, J. Li, R. Zhang, When is a dynamical system mean sensitive? Ergodic Th. & Dynam. Syst., in press. [63] F. Garc´ıa-Ramos, B. Marcus, Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems, preprint, arXiv:1509.05246. 2802Oberwolfach Report 46/2017 [64] E. Glasner, The structure of tame minimal dynamical systems, Ergodic Th. & Dynam. Syst. 27(2007), 1819-1837. · Zbl 1127.37011 [65] T. Goodman, Topological sequence entropy, Proc. London Math. Soc. 29 (1974), 331-350. · Zbl 0293.54043 [66] P. Halmos, J. von Neumann, Operator methods in classical mechanics, II, Ann. Math. 43 (1942), 332-350. · Zbl 0063.01888 [67] W. Huang, S. Li, S. Shao, X. Ye, Null systems and sequence entropy pairs, Ergodic Th. & Dynam. Syst. 23 (2003), 1505-1523. · Zbl 1134.37308 [68] J. Li, S. Tu, X. Ye, Mean equicontinuity and mean sensitivity, Ergodic Th. & Dynam. Syst. 35(2015), 2587-2612. · Zbl 1356.37016 [69] J.C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc. 58 (1952), 116-136. · Zbl 0046.11504 [70] B. Scarpellini, Stability properties of flows with pure point spectrum, J. London Math. Soc. (2) 26 (1982), 451-464. Stealthy hyperuniform processes Subhro Ghosh (joint work with Joel L. Lebowitz) In recent years, a special class of hyperuniform particle systems, known as stealthy hyperuniform (henceforth abbreviated as SH) systems, have attracted considerable attention [14, 15, 16, 17, 18]. These systems are characterized by the structure function S(k) vanishing in a neighbourhood of k = 0. The quantity S(k) is also referred to as the Bartlett spectrum. A natural generalization of SH point processes is to consider point processes, or random fields, with a gap in the spectrum on an open set U which may not include the origin. We shall denote these processes as Generalized Stealthy (henceforth abbreviated as GS) processes. The nomenclature “stealthy”, as well as the physical interest in SH particle systems, stems from the fact that such systems are optically transparent (invisible) for wave vectors k in the gap U . Numerical and experimental investigations have been carried out regarding how to construct SH particle systems. These systems cannot be equilibrium systems, with tempered potentials, at finite temperatures. They may, however, be ground states of such systems, e.g. the periodic (disordered?) zero temperature states of classical systems, or they can be generated as non-equilibrium states. SH systems are an extension of hyperuniform (superhomogeneous) particle systems. Hyperuniform systems, which have been studied extensively both in the physics and the mathematics literature, have reduced fluctuations: the variance of the particle number in a domain D in Rdor Zdgrows slower than the volume of D. The significance of hyperuniform materials, and in particular SH systems, lies in the fact that they embody properties of both crystalline and disordered or random systems (see [9, 10] and the references therein). For translation invariant systems, an equivalent characterization of hyperuniformity can be obtained by looking at their structure functions. Hyperuniformity then boils down to the vanishing of the structure function S(k) at k = 0. SH systems, therefore, involve a specific manner in which this vanishing of the structure function takes place. Mathematical Quasicrystals2803 In the article [18], Zhang, Stillinger and Torquato provide numerical evidence in support of some remarkable conjectural properties of stealthy hyperuniform processes, in particular that the hole sizes for stealthy hyperuniform processes are uniformly bounded. In [10], we carry out a rigorous mathematical analysis of stealthy hyperuniform processes, and establish the veracity of this conjecture. In particular, we prove that Theorem 1. Let Ξ be a stealthy hyperuniform point process. Let B(x; r) be the ball with centre x and radius r. Then there exists a positive number r0such that P|Ξ ∩ B(x; r0)| = 0= 0. Further, the quantity r0, can be chosen to be Cb−1, where b is the radius of the maximal ball (centred at the origin) that is contained in the gap of the structure function S, and C is a universal constant. We also establish an anti-concentration property for particle numbers of stealthy hyperuniform processes: Theorem 2. Let Ξ be a stealthy hyperuniform point process on Rdor Zdwith one point intensity ρ and b the radius of the largest ball around the origin (in the wave space) on which the structure function of Ξ vanishes. There exists numbers C, c > 0 (independent of all parameters of Ξ) such that, the number of points of Ξ in any given d-dimensional cube of side-length Cb−1is a.s. bounded above by cρb−d. The fact that holes in SH processes cannot be bigger than a deterministic size is suggestive of a high degree of crystalline behaviour in these processes. In our work, we go further, and establish a remarkable maximal rigidity property of these ensembles. We can, in fact, do this in the setting of GS processes. For a point process (more generally, a random field or a random measure) Ξ on Rdand a bounded domain D ⊂ Rd, statistic Ψ defined on Ξ restricted to D is said to be rigid if Ψ is completely determined by (that is, a deterministic function of) the process Ξ restricted to D∁. To put things in perspective, a point process having rigidity is in notable contrast to the Poisson process, where the process inside and outside of D are statistically independent. Rigidity phenomena for particle systems have been investigated quite intensively in the last few years, and have been shown to appear in many natural models which are, nonetheless, far removed from being crystalline. Key examples include the Ginibre ensemble, Gaussian zeros, the Dyson log gas, Coulomb systems and various determinantal processes related to random matrix theory (see, e.g., [11, 6, 7, 8, 5, 12, 9]). In [10], we show that GS random measures on Rdor Zdexhibit maximal rigidity: namely, for any domain D ⊂ Rd, the random measure [Ξ]|D∁determines completely the measure [Ξ]|D(that is, the latter is a deterministic measurable function of the former). Stated in formal terms, we prove: Theorem 3. Let Ξ be a generalized stealthy random measure on Rdor Zd. Then for any bounded domain D, the random measure [Ξ]|Dis almost surely determined by (i.e., is a measurable function of ) the random measure [Ξ]|D∁. 2804Oberwolfach Report 46/2017 We further show that, to have maximal rigidity in the sense discussed above, it suffices that the structure function vanishes faster than any polynomial at some point in the wave space. In the 1D discrete setting (i.e. Z-valued processes on Z), this can also be seen as a consequence of a recent theorem of Borichev, Sodin and Weiss [4]; in higher dimensions or in the continuum, such a phenomenon seems novel. The question of inference about a stochastic process from its diffraction spectrum has a long history in diffraction theory, and we believe the results in the present article would be of interest to that body of literature. We refer the reader who is interested in further exploration of this direction to [1, 2, 3]. Note that the central Bragg peak ρ2δk=0is not included in S(k). References [71] M. Baake, M. Birkner, R.V. Moody, Diffraction of stochastic point sets: Explicitly computable examples, Commun. Math. Phys. 293 (2010), 611-660. · Zbl 1197.82053 [72] M. Baake, U. Grimm, Aperiodic Order. Vol. 1: A Mathematical Invitation, Cambridge University Press, Cambridge (2013). · Zbl 1295.37001 [73] M. Baake, H. K¨osters, R.V. Moody, Diffraction theory of point processes: Systems with clumping and repulsion, J. Stat. Phys. 159 (2015), 915-936. · Zbl 1320.60114 [74] A. Borichev, M. Sodin, B. Weiss, Spectra of stationary processes on Z, preprint, arXiv:1701.03407 (2017). · Zbl 1405.60045 [75] A.I. Bufetov, Rigidity of determinantal point processes with the Airy, the Bessel and the Gamma kernel, Bull. Math. Sci. 6 (2016), 163-172. · Zbl 1335.60075 [76] S. Ghosh, Determinantal processes and completeness of random exponentials: the critical case, Probab. Theory Rel. 163 (2015), 643-665. · Zbl 1334.60083 [77] S. Ghosh, Palm measures and rigidity phenomena in point processes, Electron. Commun. Probab. 21 (2016), 85. · Zbl 1354.60055 [78] S. Ghosh, M. Krishnapur, Rigidity hierarchy in random point fields: random polynomials and determinantal processes, preprint, arXiv:1510.08814 (2015). · Zbl 1489.60084 [79] S. Ghosh, J. Lebowitz, Number rigidity in superhomogeneous random point fields, J. Stat. Phys. 166, 1016-1027. · Zbl 1362.60047 [80] S. Ghosh, J. Lebowitz, Generalized stealthy hyperuniform processes: maximal rigidity and the bounded holes conjecture, preprint, arXiv:1707.04328 (2017). · Zbl 1401.60096 [81] S. Ghosh, Y. Peres, Rigidity and tolerance in point processes: Gaussian zeros and Ginibre eigenvalues, Duke Math. J. 166 (2017), 1789-1858. · Zbl 1405.60067 [82] H. Osada, T. Shirai, Absolute continuity and singularity of Palm measures of the Ginibre point process, Probab. Theory Rel. 165 (2016), 725-770. · Zbl 1344.60042 [83] S. Torquato, F. Stillinger, Local density fluctuations, hyperuniformity, and order metrics, Phys. Rev. E 68 (2003), 041113. [84] S. Torquato, G. Zhang, F.H. Stillinger, Ensemble theory for stealthy hyperuniform disordered ground states, Phys. Rev. X 5 (2015), 021020. [85] G. Zhang, F.H. Stillinger, S. Torquato, Ground states of stealthy hyperuniform potentials: I. Entropically favored configurations, Phys. Rev. E 92 (2015), 022119. [86] G. Zhang, F.H. Stillinger, S. Torquato, Ground states of stealthy hyperuniform potentials: II. Stacked-slider phases, Phys. Rev. E 92 (2015), 022120. [87] G. Zhang, F.H. Stillinger, S. Torquato, Transport, geometrical, and topological properties of stealthy disordered hyperuniform two-phase systems, J. Chem. Phys. 145 (2016), 244109. [88] G. Zhang, F.H. Stillinger, S. Torquato, Can exotic disordered “stealthy” particle configurations tolerate arbitrarily large holes? Preprint, arXiv:1705.04415 (2017). Mathematical Quasicrystals2805 The structure of tame minimal dynamical systems for general groups Eli Glasner A dynamical version of the Bourgain-Fremlin-Talagrand dichotomy [1] shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of βN, or it is a “tame” topological space whose topology is determined by the convergence of sequences. In the latter case the dynamical system is called tame [8, 2]. WAP (weakly almost periodic) as well as HNS (hereditarily non-sensitive) systems are tame, and among the typical examples of tame systems one can find many cut and project systems like the classical Sturmian and some Toeplitz flows, [5]. Minimal tame dynamical systems (X, G) with an Abelian acting group G were studied by several authors, and it was shown that such systems are almost automorphic and uniquely ergodic, and that the canonical continuous map from X onto its largest Kronecker factor is a measure theoretical isomorphism, [6, 7, 3]. What happens when the acting group is not assumed to be commutative, or even not amenable? Here, one discover completely new phenomena and a wealth of new examples. The most prominent among them are boundaries of Gromov hyperbolic groups and linear actions on spheres and projective spaces. In a recent work [4] I use the structure theory of minimal dynamical systems to show that, for a general group G, a tame, metric, minimal dynamical system (X, G) has the following structure: X˜ooθ∗X∗ ⑦ η⑦⑦⑦ ⑦ι ⑦ ⑦⑦ XπZπ∗ σ YooY∗ θ Here (i) ˜X is a metric minimal and tame system (ii) η is a strongly proximal extension, (iii) Y is a strongly proximal system, (iv) π is a point distal and RIM extension with unique section, (v) θ, θ∗and ι are almost one-to-one extensions, and (vi) σ is an isometric extension. When the map π is also open this diagram reduces to X˜ ⑦ η⑦⑦ ⑦ι ⑦ ⑦ ⑦ ⑦ XZπ σ Y 2806Oberwolfach Report 46/2017 In general the presence of the strongly proximal extension η is unavoidable. If the system (X, G) admits an invariant measure µ then Y is trivial and X = ˜X is an almost automorphic system; i.e. X→ Z, where ι is an almost one-to-oneι extension and Z is equicontinuous. Moreover, µ is unique and ι is a measure theoretical isomorphism ι : (X, µ, G) → (Z, λ, G), with λ the Haar measure on Z. Thus, this is always the case when G is amenable. References [89] J. Bourgain, D.H. Fremlin, M. Talagrand, Pointwise compact sets of Baire-measurable functions, Amer. J. Math. 100 (1978), 845-886. · Zbl 0413.54016 [90] E. Glasner, On tame dynamical systems, Colloq. Math. 105 (2006), 283-295. · Zbl 1117.54046 [91] E. Glasner, The structure of tame minimal dynamical systems, Ergodic Th. & Dynam. Syst. 27(2007), 1819-1837. · Zbl 1127.37011 [92] E. Glasner, The structure of tame minimal dynamical systems for general groups. Invent. Math., in press, doi:10.1007/s00222-017-0747-z. · Zbl 1384.54021 [93] E. Glasner, M. Megrelishvili, Hereditarily non-sensitive dynamical systems and linear representations, Colloq. Math. 104 (2006), 223-283. · Zbl 1094.54020 [94] W. Huang, Tame systems and scrambled pairs under an Abelian group action, Ergodic Th. & Dynam. Syst. 26 (2006), 1549-1567. · Zbl 1122.37009 [95] D. Kerr, H. Li, Independence in topological and C∗-dynamics, Math. Ann. 338 (2007), 869-926. · Zbl 1131.46046 [96] A. K¨ohler, Enveloping semigroups for flows, Proc. Roy. Irish Acad. 95A (1995), 179-191. Aperiodic Schr¨odinger operators Anton Gorodetski Most of the questions on spectral properties of higher dimensional aperiodic operators (such as Laplacian on Penrose tilings) are completely open; the only known results are related to existence of a well defined density of states measure and some of its properties [10, 11, 12]. On a one dimensional lattice reasonable models of quasicrystals are substitution sequences (such as Fibonacci, Thue-Morse, period doubling), and Sturmian sequences. In all these cases the spectrum is known to be a Cantor set of zero measure for all non-zero values of the coupling constant [1, 2, 3, 14]. At the same time other characteristics of the spectrum can be very different. For example, gap sizes asymptotics for small coupling is known for Fibonacci Hamiltonian [6], Thue-Morse [1], and period doubling potentials [2], and turn out to be model-dependent. As another example, exact large coupling asymptotics of the Hausdorff dimension of the spectrum of Fibonacci Hamiltonian are known [5] — it tends to zero as an inverse of a logarithm of the coupling; at the same time the Hausdorff dimension of the spectrum in the case of Thue-Morse potential is uniformly bounded away from zero [13]. The Fibonacci Hamiltonian is the most heavily studied since it belongs to both classes — operators with Sturmian potentials, and those with potential given by a substitution sequence. The Trace Map approach allowed to provide a very detailed and almost complete description of the properties of the spectrum, density of states measure, transport properties etc., see [9] and references therein. Mathematical Quasicrystals2807 One of the ways to use the obtained one-dimensional results to gain some intuition on the spectral properties of the higher dimensional aperiodic operators is via separable potentials. In this case the potential on two (or higher) dimensional lattice is given by the sum of two potentials, each of them depends only on one of the coordinates. The spectrum of the discrete Schr¨odinger operator in this case turns out to be the Minkowski sum of the spectra of the corresponding one dimensional operators, and the density of states measure is given by convolution of the corresponding density of states measures. In the case of the Fibonacci Hamiltonian the spectrum is a dynamically defined Cantor set [9]. Questions on the structure of sums of dynamically defined Cantor sets appeared before in dynamical systems and number theory. Applying some of the existing methods and using the known results on the spectrum of the Fibonacci Hamiltonian one can show that the spectrum of the square Fibonacci Hamiltonian is an interval for small values of the coupling constant, and is a Cantor set of zero measure for the large coupling [6]. Moreover, typically the density of states is a.c. with respect to the Lebesgue measure for small couplings [8]. Also, interestingly enough, there is a regime (open set in the space of couplings) where typically the spectrum of the square Fibonacci Hamiltonian has positive Lebesgue measure while the density of states measure is singular [7]. For a detailed recent survey of these and many other results on spectral properties of aperiodic Schr¨odinger operators see [4]. References [97] J. Bellissard, Spectral properties of Schr¨odinger’s operator with a Thue-Morse potential, in Number Theory and Physics, eds. J.M. Luck, P. Moussa, M. Waldschmidt, Springer Proc. Phys., vol. 47, Springer, Berlin (1990), pp. 140-150. [98] J. Bellissard, A. Bovier, J.-M. Ghez, Spectral properties of a tight binding Hamiltonian with period doubling potential, Commun. Math. Phys. 135 (1991), 379-399. · Zbl 0726.58038 [99] J. Bellissard, B. Iochum, E. Scoppola, D. Testard, Spectral properties of one-dimensional quasicrystals, Commun. Math. Phys. 125 (1989), 527-543. · Zbl 0825.58010 [100] D. Damanik, M. Embree, A. Gorodetski, Spectral properties of Schr¨odinger operators arising in the study of quasicrystals, in Mathematics of Aperiodic Order eds. J. Kellendonk, D. Lenz, J. Savinien, Progress in Mathematics, vol. 309, Birkh¨auser, Basel (2015), pp. 307-370. · Zbl 1378.81031 [101] D. Damanik, M. Embree, A. Gorodetski, S. Tcheremchantsev, The fractal dimension of the spectrum of the Fibonacci Hamiltonian, Commun. Math. Phys. 280 (2008), 499-516. · Zbl 1192.81151 [102] D. Damanik, A. Gorodetski, Spectral and quantum dynamical properties of the weakly coupled Fibonacci Hamiltonian, Commun. Math. Phys. 305 (2011), 221-277. · Zbl 1232.81016 [103] D. Damanik, A. Gorodetski, Spectral transitions for the square Fibonacci Hamiltonian, to appear in J. Spectr. Theory. · Zbl 1503.47059 [104] D. Damanik, A. Gorodetski, B. Solomyak, Absolutely continuous convolutions of singular measures and an application to the square Fibonacci Hamiltonian, Duke Math. J. 164 (2015), 1603-1640. · Zbl 1358.37117 [105] D. Damanik, A. Gorodetski, W. Yessen, The Fibonacci Hamiltonian, Invent. Math. 206 (2016), 629-692. · Zbl 1359.81108 [106] A. Hof, Some remarks on discrete aperiodic Schr¨odinger operators, J. Stat. Phys. 72 (1993), 1353-1374. · Zbl 1101.39301 [107] A. Hof, A remark on Schr¨odinger operators on aperiodic tilings, J. Stat. Phys. 81 (1995), 851-855. 2808Oberwolfach Report 46/2017 · Zbl 1081.82521 [108] D. Lenz, P. Stollmann, An ergodic theorem for Delone dynamical systems and existence of the integrated density of states, J. Anal. Math. 97 (2005), 1-24. · Zbl 1131.37015 [109] Q. Liu, Y. Qu, On the Hausdorff dimension of the spectrum of the Thue-Morse Hamiltonian, Commun. Math. Phys. 338 (2015), 867-891. · Zbl 1333.47026 [110] A. S¨ut˝o, The spectrum of a quasiperiodic Schr¨odinger operator, Commun. Math. Phys. 111 (1987), 409-415. Dynamical encodings of patterns in cut and project sets Alan Haynes (joint work with Antoine Julien, Henna Koivusalo, Jens Marklof, Lorenzo Sadun, James Walton) The purpose of this talk was to demonstrate how problems about patterns in cut and project sets can be reformulated in terms of questions about higher rank linear actions on tori. One goal of the talk was to emphasize the strong connections between these types of questions and problems in Diophantine approximation. First we first reviewed classical results of Morse and Hedlund about Sturmian sequences [7, 8], in which patterns of a given size correspond to regions in the circle determined by an irrational rotation α (the slope defining the Sturmian sequence). This point of view leads quickly to detailed knowledge about three quantities associated to patterns in Sturmian sequences: complexity, frequencies, and the repetitivity function. Understanding the complexity is a geometric problem, which corresponds precisely to counting the number of connected components of the circle, with a finite sub-orbit of 0 under the rotation by α removed. Questions about frequencies of patterns are answered by understanding volumes of connected components, which is a simple example of a gaps problem in Diophantine approximation. Questions about repetitivity of patterns are answered by a detailed analysis of the continued fraction expansion of α. Sturmian sequences are examples of one-dimensional cut and project sets obtained by projecting from a two-dimensional total space. For more general cut and project sets, with windows satisfying appropriate regularity conditions (which was a standing assumption in our talk), we can ask analogous questions about complexity, frequencies, and repetitivity of patterns. It is not difficult to see that, as in the case of Sturmian sequences, for k to d cut and project sets there is a correspondence between patterns of a given size and connected components of the window, after a sub-orbit of the boundary under the natural Zd-action determined by the physical space is removed. This idea was used in [6] to satisfactorily understand complexity of patterns in cut and project sets, as well as how the growth of the complexity function is related to the cohomology of associated topological spaces. Problems about the number of distinct frequencies of patterns of a given size in cut and project sets are related to higher dimensional gaps problems in Diophantine approximation. Such problems are in general much more difficult than their one-dimensional counterparts. In [2] we proved that, for any k and d, there Mathematical Quasicrystals2809 is a full Hausdorff dimension set of cut and project sets for which, for any r ≥ 1, the number of distinct frequencies of patterns of size r remains bounded. We also showed that, for almost every k to d cut and project set (with respect to Lebesgue measure), for any ǫ > 0, and for all sufficiently large r, the number of distinct frequencies of patterns of size r is bounded above by (log r)(1+ǫ)(d+1)(k−d). Furthermore, recent work on higher dimensional Steinhaus problems [5] now also implies a previously elusive result, that for almost every cut and project set, the number of distinct frequencies of patterns of size r is not bounded. Finally, problems about repetitivity of patterns in cut and project sets involve a careful study of the volumes and shapes of connected components of the regions in the corresponding dynamical encodings. This type of study has recently been undertaken in [3], where we gave an explicit characterization of the collection of all linearly repetitive cut and project sets with cubical windows. Further work on this problem, including a development of the connections with Diophantine approximation, discrepancy theory, and the Littlewood conjecture, can be found in [1] and [4]. References [111] A. Haynes, A. Julien, H. Koivusalo, J. Walton, Statistics of patterns in typical cut and project sets, to appear in Ergodic Th. & Dynam. Syst., arXiv:1702.04041. · Zbl 1448.37010 [112] A. Haynes, H. Koivusalo, L. Sadun, J. Walton, Gaps problems and frequencies of patches in cut and project sets, Math. Proc. Cambridge Philos. Soc. 161 (2016), 65-85. · Zbl 1371.11110 [113] A. Haynes, H. Koivusalo, J. Walton, Linear repetitivity and subadditive ergodic theorems for cut and project sets, to appear in Nonlinearity, arXiv:1503.04091. · Zbl 1384.52018 [114] A. Haynes, H. Koivusalo, J. Walton, Perfectly ordered quasicrystals and the Littlewood conjecture, to appear in Trans. Amer. Math. Soc., arXiv:1506.05649. · Zbl 1394.11058 [115] A. Haynes, J. Marklof, Higher dimensional Steinhaus and Slater problems via homogeneous dynamics, submitted, arXiv:1707.04094. · Zbl 1475.11130 [116] A. Julien, Complexity and cohomology for cut-and-projection tilings, Ergodic Th. & Dynam. Syst. 30 (2010), 489-523. · Zbl 1185.37031 [117] M. Morse, G. A. Hedlund, Symbolic Dynamics, Amer. J. Math. 60 (1938), 815-866. · JFM 64.0798.04 [118] M. Morse, G. A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math. 62 (1940), 1-42. Quasicrystals beyond amenable groups Tobias Hartnick, Felix Pogorzelski (joint work with Michael Bj¨orklund) In this talk, we sketched the diffraction theory of model sets in homogeneous spaces associated with Gelfand pairs (commutative spaces), as developed in [3, 4]. 1. Model sets in homogeneous spaces By a cut-and-project scheme we shall mean a triple (G, H, Γ), where G and H are locally compact second countable (lcsc) groups and Γ < G × H is a lattice which projects injectively to G and densely to H. Given such a triple (G, H, Γ) and a 2810Oberwolfach Report 46/2017 compact subset W ⊂ H with non-empty interior, we define the associated model set P0= P0(G, H, Γ, W ) as P0:= prG(G × W ) ∩ Γ, where prGdenotes the projection onto the factor G of G×H. If K < G is a compact subgroup, we also refer to the image P of P0in the homogeneous space X = G/K as a model set. For G and H Abelian and K trivial, this is the classical cutand-project construction as used by Meyer and others. In the talk, we discussed non-classical examples in Riemannian symmetric spaces, Bruhat-Tits buildings and nilmanifolds, including both arithmetic and non-arithmetic examples. In the sequel, we reserve the letter P0to denote a model set in an lcsc group G; we will always assume that the corresponding window W satisfies the following regularity conditions, (1)W = Wo,|∂W | = 0,StabH(W ) = {e},∂W ∩ prH(Γ) = ∅. On the other hand, we will (at least initially) not assume that Γ is uniform. We also fix a compact subgroup K < G and denote by P ⊂ X := G/K the image of P0under the canonical projection p : G → X. Both P0and P are Delone sets with respect to natural classes of metrics on G and X, respectively, and both have G-finite local complexity, that is, finitely many patches up to G-translation. Moreover, P0−1P0is uniformly discrete, which can be seen as a long-range order property. 2. The hull of a model set Given a homogeneous space Z of an lcsc group G, we will denote by C(Z) the collection of all closed subsets of Z, considered as a compact metrizable space with respect to the Chabauty-Fell topology, and given a subset Q ⊂ Z, we denote by ΩQ:=g.Q : g ∈ G⊆ C(Z) its hull. Our model sets P0and P then give rise to topological dynamical systems G y ΩP0⊂ C(G) and G y ΩP⊂ C(X), respectively. If the underlying lattice is non-uniform, these hulls contain the empty set as a non-trivial fixpoint. We will thus also consider the punctured hulls Ω×Q:= ΩQ{∅} for Q ∈ {P0, P }. A priori, it is not clear whether Ω×Por Ω×admit any G-invariant probabil0P ity measures. To settle this issue, we establish the following generalization of Schlottmann’s torus periodization map [7]; here, Y := (G×H)/Γ is a parameter space, which generalizes the classic torus parametrization [1, 7]. Theorem 2.1 (Parametrization map, [3]). Let P0⊂ G be a model set as above. (1) There exists a unique surjective Borel G-map β : Ω×P→ Y with closed 0 graph which maps P0to the basepoint (e, e)Γ of Y . If Γ is uniform, then β is continuous. (2) There exists a subset Yns⊂ Y of full Haar measure such that β is oneto-one over Yns. Mathematical Quasicrystals2811 Explicitly, the set Ynsof non-singular parameters is given by Yns:=(g, h)Γ : h−1∂W ∩ πH(Γ) = ∅. From Theorem 2.1, one deduces the following consequences. Corollary 2.2 (Unique ergodicity and minimality of the hull, [3]). Let P0⊂ G and P ⊂ X be model sets as above. (1) The spaces Ω×Pand Ω×each admit a unique G-invariant probability mea0P sure. (2) If Γ is uniform, the dynamical systems G y ΩP0and G y ΩPare minimal. In fact, to establish unique ergodicity of Ω×P, one needs to establish a stronger property of Ω×Pcalled unique stationarity. It then follows that also Ω×is uniquely 0P stationary, hence uniquely ergodic. 3. Autocorrelation of model sets We explained how to associate with our model set P ⊂ X an autocorrelation measure, which is a Radon measure on the double coset space K\G/K, following the general approach of Bartlett from the theory of point processes; compare [5]. The main steps of this construction were as follows: (1) Construct a periodization map X P : Cc(X) → Cc(Ω×P),Pf(Q) =f (x). x∈Q (2) Form the second correlation measure η(2)ν∈ R(X ×X)Gof the G-invariant measure ν on Ω×Pby Z η(2)ν(f ⊗ g) =Pf(Q)Pg(Q) dν(Q)f, g ∈ Cc(X). Ω×P (2) (3) Define the autocorrelation measure ηP∈ R(K\G/K) as the image of ην under the canonical isomorphism R(X × X)G∼=R G \(G/K × G/K) ∼=R(K\G/K). For the so-defined autocorrelation measure, we obtain the following formula. Theorem 3.1 (Autocorrelation formula, [3]). Let P ⊂ X be a model set as above; denote by p : G → X the canonical projection and by PΓ: Cc(G × H) → Cc(Y ) the periodization map along Γ. Then, ηPis uniquely determined by the fact that L2(Y )Kc(K\G/K). Let us compare our definition of the autocorrelation measure to the more classical definition of Hof [6], the latter being a mathematical formulation of the 2812Oberwolfach Report 46/2017 well-known Patterson function. Given a family of subsets Ft⊂ X, let us define a family of Radon measures on K\G/K by 1XX σt(f ) :=f (x−1y)f ∈ Cc(K\G/K). |Ft|x ∈P ∩Fty∈P In the classical Abelian setting, if (Ft) is a van Hove sequence, then it can be shown (see e.g. [2]) that the autocorrelation measure is given by the formula (2)ηP(f ) = limσt(f )f ∈ Cc(K\G/K), t→∞ and the classical argument extends to van Hove sequences in arbitrary amenable groups G. Remarkably, formula (2) holds also in many non-amenable situations, where Følner sequences, let alone van Hove sequences, do not exist. For example, if X is a Riemannian symmetric space, then (2) holds for Riemannian balls Ft. However, the situation is far from simple in general. For example, if X is a tree, then (2) holds along balls of even radius, but in general not along arbitrary balls. Thus, while the dynamical approach to autocorrelation always works in a uniform way, the approach through Hof approximation depends very much on the geometry of the spaces in question. 4. Towards diffraction While autocorrelation can be defined for Delone sets of finite local complexity in arbitrary homogeneous spaces of the form X = G/K, the definition of diffraction requires a Fourier transform on the double coset space K\G/K, hence we need to make additional assumptions on the pair (G, K) from now on. Definition 4.1. We say that (G, K) is a Gelfand pair and that X = G/K is a commutative space if the Hecke (convolution) algebra H(G, K) = Cc(K \G/K) is commutative. This assumption is satisfied in all examples considered above, in particular hyperbolic spaces, Riemannian symmetric spaces, and for nilmanifold pairs. If (G, K) is a Gelfand pair, the Banach algebra L1(K \G/K) is commutative and its Gelfand spectrum can be identified with the space Sb(G, K) of all bounded spherical functions; here, a continuous function ω : G → C is called spherical if the associated measure mω∈ R(G) as given by Z mω(f ) :=f (x)ω(x−1) dxf ∈ Cc(G) G is bi-K-invariant. The spherical Fourier transform of the pair (G, K) is the restriction of the Gelfand transform of L1(K \ G/K) to the subspace S+(G, K) ⊆ Sb(G, K) of positive-definite spherical functions, i.e. for f ∈ H(G, K) and ω ∈ S+(G, K) we define f (ω) := mbω(f ). Mathematical Quasicrystals2813 Similarly, if η is a positive-definite Radon measure on K\G/K, its spherical Fourier transformη is determined by the fact thatb η f∗∗ f=bη | bf |2,f ∈ H(G, K). Since the autocorrelation measure ηPis positive-definite, we may thus define the spherical diffraction measure of P as ηbP∈ R S+(G, K). Theorem 4.2 (Pure point diffraction, [4]). If the lattice Γ is uniform, the spherical diffraction measurebηPis pure point, i.e., there is a countable set S ⊂ S+(G, K) and a function c : S → R>0such that X bηP=c(ω) δω. ω∈S In fact, in the situation of the theorem, we can determine S and the function c explicitly. Indeed, the set S is given by the spherical automorphic spectrum of Γ, that is, by the collection of all ω ∈ S+(G, K) for which the eigenspace L2(Y )Kω:=g ∈ L2(Y )K: ∀ f ∈ H(G, K) : f ∗ g = bf (ω) g is non-zero. The coefficient function c can be computed as the squared L2-norm of a certain integral transform of the characteristic function χW. In the Abelian case, this transform is simply a normalized Fourier transform on H. In the general case, the desired integral transform is obtained as a shadow of the spherical Fourier transform of the pair (G, K) (in the spirit of a Hecke correspondence) and hence is referred to as the shadow transform; see [4] for details. References [119] M. Baake, J. Hermisson, P.A.B. Pleasants, The torus parametrization of quasiperiodic LI classes, J. Phys. A: Math. Gen. 30 (1997), 3029-3056. · Zbl 0919.52015 [120] M. Baake, D. Lenz. Dynamical systems on translation bounded measures: pure point dynamical and diffraction spectra, Ergodic Th. & Dynam. Syst. 24 (2004), 1867-1893. · Zbl 1127.37004 [121] M. Bj¨orklund, T. Hartnick, F. Pogorzelski. Aperiodic order and spherical diffraction, I: autocorrelation of model sets, to appear in Proc. London Math. Soc. (2017), arXiv:1602.08928. [122] M. Bj¨orklund, T. Hartnick, F. Pogorzelski. Aperiodic order and spherical diffraction, II: The shadow transform and the diffraction formula, preprint, arXiv:1704.00302. · Zbl 1485.43005 [123] D.J. Daley, D. Vere-Jones, An Introduction to the Theory of Point Processes. Vol. I: Elementary Theory and Methods. Springer, New York (1988). · Zbl 0657.60069 [124] A. Hof, On diffraction by aperiodic structures, Commun. Math. Phys. 169 (1995), 25-43. · Zbl 0821.60099 [125] M. Schlottmann, Generalized model sets and dynamical systems, in Directions in Mathematical Quasicrystals, eds. M. Baake, R.V. Moody, CRM Monograph Series, vol. 13, AMS, Providence, RI (2000), pp. 143-159. 2814Oberwolfach Report 46/2017 Topological invariants for tilings John Hunton This talk gave a brief — and personal — overview of some of the main themes in the recent and current study of aperiodic tilings by methods from topology. It was clearly not possible to cover everything, and similarly it is not possible to give a comprehensive bibliography in the space available here, even for the subjects touched upon. The interested reader should explore the topics further through the selected papers mentioned below and the further work they cite. We restrict ourselves mainly to tilings of d-dimensional Euclidean space which are repetitive, aperiodic and of translationally finite local complexity (FLC). For such a tiling T ⊂ Rd, the key to the topological approach is the space Ω = ΩT, variously known as the tiling space, or continuous hull of T , the completion of the set of translates of T under the tiling metric. Under the assumptions above Ω naturally carries a minimal action of the translation group Rd, and in many of the most popular classes of tilings, a unique ergodic probability measure. The structure of Ω is fundamental to this work. Most lines of approach start from one or other of the observations that Ω can be (a) described (up to shape equivalence – see later) as an inverse limit of convenient finite CW complexes (approximants), or (b) given (up to homeomorphism) the structure of a fibre bundle over a d-torus with fibre a Cantor set [24]. The space may also be described as the classifying space of the holonomy groupoid associated with Ω. For description (a), there are a number of useful models. For primitive substitution tilings, the first constructions were those of [1, 14]. The desire to produce smaller models for the approximants led to a number of developments, including [2, 3] which implicitly involved working in the shape category, a notion formally explored in [6]. Recent work has explored further the use of minimal homotopy models for the approximants. For general tilings, inverse limit descriptions exist via various models [1, 3, 10], but without specific structure these are principally of theoretical use. Similarly, the Cantor bundle structure is computationally practical only in the case of a tangible description of the holonomy action of Zdon the Cantor fibre; this can be given explicitly in the case of cut and project tilings [9]. Various results have been established exploring the relationships between the spaces ΩTand ΩSand the possible relationships of the underlying tilings T and S. Notable work in this thread includes [8] on deformations of tilings, and most recently [12] characterising homeomorphisms of tiling spaces. Topological invariants for tilings typically study ΩT, with or without additional structure, through the application of methods from algebraic topology. Typical applications to date have included characterization results, identification of geometric properties of T , issues related to questions about pure point diffraction (for example work related to the Pisot substitution conjecture, see [16, Ch. 2] for an overview), labelling of gaps in the spectrum of the Schr¨odinger operator associated to T [4, 5, 13], and results on the complexity of T , [11]. Mathematical Quasicrystals2815 What algebro-topological tools should be employed? Homotopy groups are rich but hard to compute. For tiling spaces, the relevant variant of these are the shape groups πsh∗(−). The case of d = 1 was studied in [6] where the fundamental shape group πsh1(Ω) was shown to collect information relevant to embedding onedimensional tiling spaces in surfaces: the non-Abelian nature of π1registered aspects not picked up by commutative invariants such as cohomology or K-theory. This is taken further in recent work of G¨ahler who uses the representation variety of πsh1(Ω) (more readily computable than πsh1(Ω) itself) in his classification of certain classes of one-dimensional substitutions. Cohomology is a long standing tool used for tiling spaces, but there are several variants in common use; we mention just three. ˇCech cohomology was the first, and perhaps most natural choice from its behaviour on inverse limits (in which it differs from singular or simplicial cohomology). The models [1, 2] for substitutions mentioned above make this is a computable and well understood invariant for such tilings, at least in low dimensions [23]. Recent work has explored more general situations, such as mixed substitutions [19, 21]. Cohomology gives some clear characterizations: for example, H∗(Ω; Q) is finite rank for an FLC substitution, but infinite for a generic cut and project tiling; the first cohomology H1(ΩT, Rd) counts degrees of freedom for deformations of T , and so on. Pattern equivariant cohomology [15, 22] has proved a useful alternative approach, yielding the same algebraic invariant as the ˇCech theory, but in a way that elements can be realized in terms of geometric patches of T . A homological variant [25] shows that tiling spaces satisfy a Poincar´e duality property analogous to that of manifolds, and has offered computational advantage, for example in the study of spaces remembering the symmetries of T [26]. The third variant can be thought of as the cohomology of the tiling groupoid, but in the case of an explicit Cantor bundle structure over a d-torus Td, this is equivalent to the group cohomology of Zd= π1(Td) with coefficients the continuous Z-valued functions on the fibre. This too has its strengths, especially in the case of an explicit description of the bundle, such as for many of the cut and project tilings. See [16, Ch. 4] for a general introduction. Similar methods become natural to apply when studying tilings with rotations, as explored in recent work of the author with Walton. Cohomology may be enriched with various additional structures, producing finer invariants. Included here are the Ruelle-Sullivan map of [18], the ordered cohomology of [20] and the homology core of [7]. The reader should consult those papers for statements of the advantages gained. Aperiodic tilings are a fruitful source of examples for non-commutative geometry. Several C∗-algebras AThave been constructed to model ΩTand its paraphenalia, and their K-groups reflect the space and Rdaction; in the case of a unique ergodic measure, there is also a trace map K∗(AT) → R. See [17] for a discussion. Connes’ Thom isomorphism identifies K∗(AT) with the topological K-theory K∗(Ω), and an Atiyah-Hirzebruch spectral sequence gives a method of 2816Oberwolfach Report 46/2017 calculating K∗(Ω) from the ˇCech cohomology H∗(Ω). Through these the noncommutative invariants can frequently be computed. A key object of study here has been the image of the tracial state, which is related to Bellissard’s gap labelling [4, 5, 13]. References [126] J. Anderson, I. Putnam, Topological invariants for substitution tilings and their C*-algebras, Ergodic Th. & Dynam. Syst. 18 (1998), 509-537. · Zbl 1053.46520 [127] M. Barge, B. Diamond, Cohomology in one-dimensional substitution tiling spaces, Proc. Amer. Math. Soc. 136 (2008), 2183-2191. · Zbl 1139.37002 [128] M. Barge, B. Diamond, J. Hunton, L. Sadun, Cohomology of substitution tiling spaces, Ergodic Th. & Dynam. Syst. 30 (2010), 1607-1627. · Zbl 1225.37021 [129] J. Bellissard, R. Benedetti, J.-M. Gambaudo, Spaces of tilings finite telescopic approximations and gap-labelling, Commun. Math. Phys. 261 (2006), 1-41. · Zbl 1116.46063 [130] M. Benameur, H. Oyono-Oyono, Calcul du label des gaps pour les quasi-cristaux, C. R. Acad. Sci. Paris, Ser. I 334 (2002), 667-670. · Zbl 0996.19006 [131] A. Clark, J. Hunton, Tiling spaces, codimension one attractors and shape, New York J. Math. 18 (2012), 765-796. · Zbl 1263.37057 [132] A. Clark, J. Hunton, The homology core of matchbox manifolds and invariant measures, to appear in Trans. Amer. Math. Soc, doi:10.1090/tran/7398. · Zbl 1412.37040 [133] A. Clark, L. Sadun, When shape matters: deformations of tiling spaces, Ergodic Th. & Dynam. Syst. 26 (2006), 69-86. · Zbl 1085.37011 [134] A. Forrest, J. Hunton, J. Kellendonk, Topological invariants for projection method patterns, Memoirs Amer. Math. Soc. 758 (2002). · Zbl 1011.52008 [135] F. G¨ahler, Lectures given at workshops Applications of Topology to Physics and Biology, Max-Planck-Institut f¨ur Physik komplexer Systeme, Dresden, June 2002, & Aperiodic Order, Dynamical Systems, Operator Algebras and Topology, Victoria, BC, August 2002. [136] A. Julien, Complexity and cohomology for cut-and-projection tilings, Ergodic Th. & Dynam. Syst. 30 (2010), 489-523. · Zbl 1185.37031 [137] A. Julien, L. Sadun, Tiling deformations, cohomology, and orbit equivalence of tiling spaces, preprint, arXiv:1506.02694. · Zbl 1417.37085 [138] J. Kaminker, I. Putnam, A proof of the gap labeling conjecture, Mich. Math. J. 51 (2003), 537-546. · Zbl 1054.46047 [139] J. Kellendonk, The local structure of tilings and their integer group of coinvariants, Commun. Math. Phys. 187 (1997), 115-157. · Zbl 0887.52013 [140] J. Kellendonk, Pattern-equivariant functions & cohomology, J. Phys. A: Math. Gen. 36 (2003), 5765-5772. · Zbl 1055.53029 [141] J. Kellendonk, D. Lenz, J. Savinien (eds.), Mathematics of Aperiodic Order, Birk¨auser, Basel (2015). [142] J. Kellendonk, I. Putnam, Tilings, C∗-algebras and K-theory, in Directions in Mathematical Quasicrystals, eds. M. Baake, R.V. Moody, CRM Monograph Series, vol. 13, AMS, Providence, RI (2000), pp. 177-206. · Zbl 0972.52015 [143] J. Kellendonk, I. Putnam, The Ruelle-Sullivan map for actions of Rd, Math. Ann. 334 (2006), 693-711. · Zbl 1098.37004 [144] G. Maloney, D. Rust, Beyond primitivity for one-dimensional substitution subshifts and tiling spaces, to appear in Ergodic Th. & Dynam. Syst., doi:10.1017/etds.2016.58. · Zbl 1396.37023 [145] N. Ormes, C. Radin, L. Sadun, A homeomorphism invariant for substitution tiling spaces, Geom. Dedicata 90 (2002), 153-182. · Zbl 0997.37006 [146] D. Rust, An uncountable set of tiling spaces with distinct cohomology, Topol. Appl. 205 (2016), 58-81. · Zbl 1356.37023 [147] L. Sadun, Pattern-equivariant cohomology with integer coefficients, Ergodic Th. & Dynam. Syst. 27 (2007), 1991-1998. Mathematical Quasicrystals2817 · Zbl 1127.37006 [148] L. Sadun, Topology of Tiling Spaces, University Lecture Series, vol. 46, AMS, Providence, RI (2008). · Zbl 1166.52001 [149] L. Sadun, R.F. Williams, Tiling spaces are Cantor set bundles, Ergodic Th. & Dynam. Syst. 23(2003), 307-316. · Zbl 1038.37014 [150] J. Walton, Pattern-equivariant homology, Algebr. Geom. Topol. 17 (2017), 1323-1373. · Zbl 1373.52027 [151] J. Walton, Cohomology of rotational tiling spaces, to appear in Bull. London Math. Soc., doi:10.1112/blms.12098. Irregular model sets Tobias J¨ager (joint work with Michael Baake, Gabriel Fuhrmann, Daniel Lenz, Christian Oertel) Cut and project schemes. A cut and project scheme (CPS) is a triple (G, H, L) consisting of two locally compact Abelian groups G and H and a co-compact discrete subgroup L ⊆ G×H (called lattice) that projects injectively to G and densely to H. Given a compact set W ⊆ that satisfies int(W ) = W and is called window in this context, a CPS defines a model set or cut and project set by Λ(W ) = πG(L ∩ (G × W )) , where πG: G × H → G denotes the projection to the first coordinate. Under the above assumptions, the resulting model set Λ(W ) is always Delone (relatively dense and uniformly discrete) [10]. CPS were introduced by Meyer in 1972 [9] and have emerged as one of the main constructions to obtain aperiodic structures. In particular, paradigmatic examples such as the Fibonacci quasicrystal or the Penrose tiling can be represented as model sets. Hull dynamics and torus parametrization.Given a suitable topology on the space of Delone sets, a model set Λ(W ) defines a topological dynamical system, which is given by the action of G on the dynamical hull Ω(Λ(W )) = ({Λ(W ) − t | t ∈ Γ}) . An important fact for the analysis of this system is the existence of a torus parametrization (see [13, 2]), that is, a flow morphism β : Ω(Λ(W )) → T = (G × H)/L from the action on the hull, (Ω(Λ(W )), G), to the canonical G-action on the ‘torus’ G given by G × T → T ,(t, (g, h) + L) 7→ (g + t, h) + L . Thereby, the map β is uniquely defined by the condition β(Γ) = (g, h) + L⇔Λ(int(W ) + h) − g ⊆ Γ ⊆ Λ(W + h) − g . Regular model sets. One case which is quite well-understood is that of regular model sets, by which we mean model sets Λ(W ) for which |∂W | = 0, where | . | denotes the Haar measure on H. The reason is the fact that in this situation the flow morphism β is almost surely 1-1, that is, β−1((g, h) + L)) is a singleton for µ-almost every (g, h) + L ∈ T with respect to the Haar measure µ on T. This 2818Oberwolfach Report 46/2017 further entails that the system (Ω(Λ(W )), G) is uniquely ergodic and isomorphic to its factor (T, G) and has pure point dynamical spectrum and zero topological entropy [13, 2]. Moreover, it can be shown that the diffraction spectrum of Λ(W ) (which we will not define in detail here) is pure point as well [8]. The latter gives a motivation to consider regular model sets as appropriate models for quasicrystals. Irregular model sets. In contrast to this, a situation that is much less understood is that of irregular model sets, that is, of windows with |∂W | > 0. In this case, one expects that ‘typically‘ the dynamics of Ω(Λ(W )) should be more complex, and a number of questions have been raised in the literature in this direction. In particular, we want to point out the following two problems, which are attributed to Moody (see [11, 12]) and Schlottmann [13], respectively. • Does |∂W | > 0 imply positive topological entropy (see [11, 12])? • Does |∂W | > 0 imply unique ergodicity [13]? In order to address these questions, we consider two different settings. Toeplitz flows.The first is that of so-called Toeplitz flows. A sequence ξ = (ξn)n∈Z∈ Σ = {0, 1}Zis called a Toeplitz sequence if it is aperiodic1and for all n ∈ Z there exists a period p ∈ N such that ξn+kp= ξnfor all k ∈ Z. In other words, every symbol in a Toeplitz sequence is repeated periodically, but the period depends on the position n of the symbol. If we let Per(ξ, p) = {n ∈ Z | ξSn+kp= ξnfor all k ∈ Z}, then ξ is Toeplitz if and only ifp∈NPer(ξ, p) = Z. For any Toeplitz sequence ξ, one can choose a period structure (pSℓ)ℓ∈Nof integers such that pQℓdivides pℓ+1andℓ∈NPer(ξ, pℓ) = Z. Let qℓ= pℓ+1/pℓand denote by Ω =∞ℓ=1Z/qℓZthe corresponding odometer with minimal group rotation R. Then there exists a flow morphism π from subshift given by the orbit closure of ξ to the odometer (Ω, R). Note that different period structures for a given Toeplitz sequence always define the same odometers up to isomorphism (see [3]). An important distinction between two basic types of Toeplitz flows is the following. For any p ∈ N, we denote by D(ξ, p) = #(Per(ξ, p) ∩ [0, p − 1])/p the density of the p-periodic positions. If limℓ→∞D(ξ, pℓ) = 1, then ξ is called a regular Toeplitz sequence. Otherwise limℓ→∞D(ξ, pℓ) < 1 and the Toeplitz sequence ξ is called irregular. In the regular case, the above flow morphism π is ν-almost surely one-to-one, where ν is the Haar measure on Ω. Hence, similar to the situation for model sets, regular Toeplitz flows are uniquely ergodic and isomorphic to the corresponding odometer and consequently have zero topological entropy and purely discrete dynamical spectrum. These analogies are no coincidence. Theorem 1 ([1]). If ξ is a Toeplitz sequence and Ω is the corresponding odometer, then the point set Λξ= {n ∈ Z | ξn= 1} can be represented as a model set with CPS (Z, Ω, L), where L =(n, Rn(0)) | n ∈ Zand the window W satisfies |∂W | = 1 − limℓ→∞D(ξ, pℓ). 1 Here ξ ∈ Σ is called aperiodic if σp(ξ) 6= ξ for all p ∈ N, where σ : Σ → Σ denotes the left shift map. Mathematical Quasicrystals2819 Hence, any Toeplitz system can be interpreted and used as an example in the context of model sets, and the notion of regularity coincides in both settings. As there exist examples of irregular Toeplitz flows which are uniquely ergodic and have zero entropy, this allows to give a negative answer to the above questions by Moody and Schlottmann (and a variety of further questions in the same direction). Irregular model sets in Euclidean CPS. The Toeplitz examples allow to answer the above questions in the general setting, where arbitrary locally compact Abelian groups are allowed in the CPS. However, this still leaves the possibility that stronger restrictions exist in the Euclidean setting, where both G and H are Euclidean spaces. In this situation, the following result guarantees that a positive measure of the window boundary “typically” leads to positive entropy (where “typical” is understood in a probabilistic sense). Theorem 2 ([7]). Suppose (RN, R, L) is a Euclidean CPS, C ⊆ R is a Cantor set of positive measure, (Gn)n∈Nis a numbering of the gaps of C, ω ∈ {0, 1}Nand [ W (ω) = C ∪Gn. n∈N:ωn=1 Then for P-almost every ω we have int(W ) = W ,|∂W | = C and (Ω(Λ(W )), RN) has positive entropy, where P refers to an arbitrary Bernoulli measure on {0, 1}N. However, even in the Euclidean case there exist exceptions. Theorem 3 ([4]). Given any CPS (R, R, L), there exists a window W with |∂W | > 0 such that (Ω(Λ(W )), R) is uniquely ergodic and has zero topological entropy. CPS and symbolic dynamics. Finally, we want to close with an announcement of a result that has been obtained during and shortly after the week in Oberwolfach and was inspired by discussions with Eli Glasner and Felipe Garc´ıaRamos during this time. It can therefore be considered a direct outcome of the workshop. We say a minimal subshift Σ ⊆ {0, 1}Zis almost automorphic if it has a maximal equicontinuous factor (Ω, ρ) for which the corresponding factor map π is almost one-to-one (there exists a point with unique preimage). Fact 4. Any minimal almost automorphic subshift (Σ, Z) is equivalent (up to conjugacy) to the system (Ω(Λ(W )), (Z)) obtained from the CPS (Z, Ω, L) with lattice L = {(n, ρn(ω0) | n ∈ Z}, where • ω0∈ Ω has unique preimage under the factor map π; • W = π([1]), where [1] = {ξ ∈ {0, 1}Z| ξ0= 1}. Moreover, the window W satisfies the topological regularity condition int(W ) = W . Analogous to the above situation, (Σ, Z) is called regular if π is ν-almost surely one-to-one, where ν denotes the unique invariant probability measure on Ω, and irregular otherwise. As the examples discussed above already indicate, a subshift may be uniquely ergodic and have zero entropy even if it is regular. This prompts 2820Oberwolfach Report 46/2017 the obvious question whether irregularity has any dynamical consequences at all. Here, the CPS formalism can be used as a tool to obtain a positive answer. We say the subshift (Σ, Z) has an infinite free set, if there exists an infinite set S ⊆ Z such that for any a ∈ {0, 1}Sthere exists ξ ∈ Σ such that ξs= asfor all s ∈ S.2 Theorem 5. If a minimal almost automorphic subshift is irregular, then it has an infinite free set. In particular, it has positive topological sequence entropy. The advantage of the CPS formalism in this context is the fact that it translates this dynamical problem into a purely topological questions concerning the structure of the window, which is easier to address. An analogous statement can be obtained for arbitrary irregular model sets with more general groups G and H. An important consequence concerns the notion of tame systems. (See [5, 6] for a definition and discussion of this notion.) Due to work of Glasner and Megrelishvili [152] , it is known that tame subshifts do not allow infinite free sets. Hence, we obtain Corollary 6. For minimal almost automorphic subshifts, tame implies regular. The analogous result for Toeplitz flows is due to Downarowicz. An extension to more general model sets will be the subject of future research. References [153] M. Baake, T. J¨ager, D. Lenz, Toeplitz flows and model sets, Bull. London Math. Soc. 48 (2016), 691-698. · Zbl 1362.37030 [154] M. Baake, D. Lenz, R.V. Moody, Characterization of model sets by dynamical systems, Ergodic Th. & Dynam. Syst. (2007), 341-382. · Zbl 1114.82022 [155] T. Downarowicz, Survey of odometers and Toeplitz flows, Contemp. Math. 385 (2005), 7-37. · Zbl 1096.37002 [156] G. Fuhrmann, T. J¨ager, C. Oertel, Irregular model sets, in preparation. [157] E. Glasner, The structure of tame minimal dynamical systems, Ergodic Th. Dynam. Syst. 27(2007), 1819-1837. · Zbl 1127.37011 [158] E. Glasner, M. Megrelishvili, Eventual nonsensitivity and tame dynamical systems, preprint, arXiv:1405.2588. · Zbl 1407.37011 [159] T. J¨ager, D. Lenz, C. Oertel, Model sets with positive entropy in Euclidean cut and project schemes, preprint, arXiv:1605.01167. [160] J.-Y. Lee, R.V. Moody, B. Solomyak, Pure point dynamical and diffraction spectra, Ann. Henri Poincar´e 5 (2002), 1003-1018. · Zbl 1025.37004 [161] Y. Meyer, Algebraic Number Theory and Harmonic Analysis, North Holland, Amsterdam (1972). [162] R.V. Moody, Meyer sets and their duals, in The Mathematics of Long-Range Aperiodic Order, ed. R.V. Moody,, NATO ASI Series C 489, Kluwer, Dordrecht (1997), pp. 403-441. · Zbl 0880.43008 [163] P.A.B. Pleasants, C. Huck, Entropy and diffraction of the k-free points in n-dimensional lattices, Discrete Comput. Geom. 50 (2013), 39-68. · Zbl 1307.37010 [164] P.A.B. Pleasants, Entropy of the visible points and kth-power-free numbers, unpublished manuscript (2006). [165] M. Schlottmann, Generalized model sets and dynamical systems, in Directions in Mathematical Quasicrystals, eds. M. Baake, R.V. Moody, CRM Monograph Series, vol. 13, AMS, Providence, RI (2000), pp. 143-159. 2 Note that if S has positive asymptotic density, then this implies positive entropy for (Σ, Z), but this is not required. Mathematical Quasicrystals2821 Homeomorphisms between tiling spaces Antoine Julien (joint work with Lorenzo Sadun) A common method for studying aperiodic tilings is to study a topological space associated with a given tiling, rather than the tiling itself. This approach is fruitful because properties of the space reflect properties of the tiling. This talk, based on the results of [3], addresses the following two questions: • Whenever two spaces are equivalent what can be said about the tilings? • What remains of a tiling space when one forgets about the tiling? The answer to the first question depends of course of what is meant by equivalent. Two tiling spaces are equivalent whenever there is a map between them which preserves some structure. How much structure is given to a space by the virtue of being a tiling space is the answer to the second question. Given a tiling T of Rd, its space is Ω :={T − x : x ∈ Rd}, where the closure in the set of all tilings is taken for an appropriate topology. It is an Rd-dynamical system. We assume that T is aperiodic, repetitive, with finite local complexity — FLC (meaning T has finitely many patterns of size r, for all r). This implies that Ω is compact, minimal and has no periodic orbit. The tiling space in itself can be given several structures: (1) it is a topological space; (2) with an orbit structure (equivalence relation inherited by the Rd-action); (3) it is a dynamical system (a specific parametrization of the orbits by Rd); (4) with a certain transverse “rigid” structure. Let us specify point (4). It is known that a tiling space with finite local complexity can be given an atlas of charts in which neighbourhoods are all homeomorphic to B(0, r) × X where B(0, r) is an Euclidean ball of Rdand X is a Cantor set. The image in Ω of a set of the form {0} × X is called a vertical transversal. The translate of a vertical transversal is still a vertical transversal (or a finite union of such sets), and the property of being a (finite union of) vertical transversals does not depend on the chart. Spaces having such a transverse structure are sometimes called tileable laminations in the literature. We can now categorize maps between tiling spaces according to which structure they preserve. We will always require our maps to be continuous (or homeomorphisms if invertible). Because of the local structure of FLC tiling spaces, such a map always sends path-connected component to path-connected component hence orbit to orbit. The weakest notion of equivalence we therefore consider is orbitequivalence, i.e., a homeomorphism sending orbit to orbit. Additional structure can be preserved: • maps preserving (3) are the topological conjugacies; • maps preserving (3) and (4) are the mutual local derivations (MLD), as introduced in [1]; 2822Oberwolfach Report 46/2017 • maps preserving (4) are called local maps. The archetypes of local maps between tiling spaces are given by shape changes (or tiling deformations), see [2]. For example: consider the Fibonacci sequence given by the substitution a 7→ ab; b 7→ a. We can consider two tiling spaces based on it: one in which the a and b-tiles have respective length φ and 1 (with φ the golden ratio); one with constant tile length c. It is known that these two spaces are conjugate for a good choice of c: they are the same dynamical system. However, their canonical transverse structures are different, and it is not preserved by the conjugacy. In essence, our work establishes that any homeomorphism between FLC tiling spaces is within bounded distance of (and isotopic to) a homeomorphism which preserves the transverse structure. In the results below, all spaces are aperiodic, minimal, FLC tiling spaces. Theorem 1. Let h : Ω → Ω′be a homeomorphism between tiling spaces. Assume Ω is uniquely ergodic. Then there exists α : Ω → Ω′a homeomorphism, which is local in the sense above, such that h = α ◦ τs, where τs(T ) = T − s(T ) for some continuous (hence bounded) function s : Ω → Rd. While any homeomorphism between two tiling spaces is isotopic to a homeomorphism which preserves (4), there is also a topological invariant measuring how this map changes the parametrization of the orbits by the action. Theorem 2. A continuous map h : Ω → Ω′defines a cohomology class [h] ∈ Hˇ1(Ω; Rd). Furthermore, whenever h1, h2are homeomorphisms hi: Ω → Ωisuch that [h1] = [h2], then there exists an MLD map between Ω1and Ω2. If Ω is uniquely ergodic, one defines Cµ: ˇH1(Ω; Rd) → Md(R), a matrix-valued map (see [4]). A non-singular class in ˇH1is a class having a non-singular image under Cµ. Given a homeomorphism h, the matrix Cµ[h] describes how h maps orbits to orbits at large scales. Theorem 3. Given a homeomorphism h : Ω → Ω′(with Ω uniquely ergodic), Cµ[h] is a non-singular matrix. Conversely, for any non-singular [α] ∈ ˇH1(Ω; Rd), there exists an FLC tiling space Ωαand a homeomorphism hα: Ω → Ωαsuch that [hα] = [α]. References [166] M. Baake, M. Schlottmann, P. Jarvis, Quasiperiodic tilings with tenfold symmetry and equivalence with respect to local derivability, J. Phys. A: Math. Gen. 24 (1991), 4637-4654. · Zbl 0755.52006 [167] A. Clark, L. Sadun, When shape matters: deformations of tiling spaces, Ergodic Th. & Dynam. Syst. 26 (2006), 69-86. · Zbl 1085.37011 [168] A. Julien, L. Sadun, Tiling deformations, cohomology, and orbit equivalence of tiling spaces, preprint, arXiv:1506.02694. · Zbl 1417.37085 [169] J. Kellendonk, I. Putnam, The Ruelle-Sullivan map for actions of Rn, Math. Ann. 334 (2006), 693-711. Mathematical Quasicrystals2823 Dynamics of B-free sets Mariusz Lema´nczyk Given an infinite B ⊂ N {1}, the set FB:= {n ∈ Z : no b ∈ B divides n} is called the set of B-free numbers. Although such sets need not possess asymptotic density, they always have logarithmic density (Davenport-Erd¨os theorem from 1936). Prominent examples of B-free sets are: the set of square-free numbers, the set of deficient numbers or even the set of prime numbers itself (consider B = {pq : p, q are primes}). It is not hard to see that A ⊂ Z is B-free, i.e. A = FB (for some B) if and only if A is closed under taking divisors. By setting η to be the characteristic function of FBand treating it as a point in the shift space {0, 1}Z, we obtain the subshift (Xη, S), where S stands for the left shift. B-free subshifts constitute an important class of examples of dynamical systems arising in the cut-Q and-project scheme. Indeed, set G = Z for the physical space, H =b∈BZ/bZ for the external space (we assume for simplicity that B is coprime but this is not essential), L = {(n, n) : n ∈ Z} for the lattice in G × H (here, n = (n, n, . . .)) and W = {h = (hb) ∈ H : hb6= 0 for all b ∈ B} for the window. However, some other natural subshifts appear in this context, for example, (XB, S) the subshift of B-admissible sequences (those sequence whose support taken mod an arbitrary b ∈ B misses at least one residue class mod b). It is not hard to see that Xη⊂ XB and the latter set is hereditary. This yields Xη⊂ eXη⊂ XB, where eXηstands for the hereditary closure of Xη. The talk, based mainly on two recent papers [2] and [4], focuses on dynamical properties of subshifts given by B-free sets. Several theorems classifying characteristic properties of such subshifts are usually given as an equivalence between dynamical, topological and arithmetical viewpoints. For example, the proximality (dynamics) of (Xη, S) is equivalent to the fact that B contains an infinite coprime subset (arithmetic) which in turn is equivalent to Int(W ) = ∅ (topology). In this spirit we go through proximality, minimality (which turns out to be closely related to the theory of Toeplitz systems), Behrend property and, the most surprising part concerning the tautness property of B. Tautness is a classical, purely arithmetical property telling us that, for each b ∈ B, the logarithmic density of FB{b [170] M. Baake, C. Huck, N. Strungaru, On weak model sets of extremal density, Indag. Math. 28(2017), 3-31, arXiv:1512.07129. · Zbl 1364.82011 [171] A. Bartnicka, S. Kasjan, J. Ku laga-Przymus, M. Lema´nczyk, B-free sets and dynamics, to appear in Trans. Amer. Math. Soc., arXiv:1509.08010. [172] V. Bergelson, J. Ku laga-Przymus, M. Lema´nczyk, F. Richter, Rational sets, polynomial recurrence and symbolic dynamics, to appear in Ergodic Th. & Dynam. Syst., arXiv:1611.08352. [173] S. Kasjan, G. Keller, M. Lema´nczyk, Dynamics of B-free sets: a view through the window, to appear in Intern. Math. Res. Notices, arXiv:1702.02375. [174] G. Keller, C. Richard, Dynamics on the graph of the torus parametrisation, to appear in Ergodic Th. & Dynam. Syst., arXiv:1511.06137. Fourier quasicrystals and Poisson summation formulas Nir Lev (joint work with Alexander Olevskii) By a Fourier quasicrystal one often means an (infinite) pure point measure µ on Rdwhose Fourier transform is also a pure point measure. The classical example of such a measure is the sum of unit masses over a lattice, and the spectrum is the dual lattice. The subject has received a new peak of interest after the experimental discovery in the middle of the 80’s of non-periodic atomic structures with diffraction patterns consisting of spots. The “cut-and-project” construction, introduced by Y. Meyer in the beginning of the 70’s, may serve as a good model for this phenomenon. It provides many examples of measures with uniformly discrete support and dense countable spectrum. On the other hand, we proved with A. Olevskii [1, 2] that if both the support and the spectrum of a measure on R are uniformly discrete sets, then the measure has a periodic structure. A similar result was proved for positive measures on Rd. In our paper [3] with A. Olevskii we establish in a strong sense the sharpness of the uniform discreteness requirement in this result. Namely, we proved there the existence of a measure µ on R whose support and spectrum are both discrete closed sets, but such that the support contains only finitely many elements of any arithmetic progression. The latter result thus reveals the existence of “nonclassical” Poisson summation formulas. Mathematical Quasicrystals2825 In the crystallography community, it seems to be commonly agreed that the support of the measure µ should be a uniformly discrete set. So it is a natural problem, to what extent can the spectrum of a non-periodic quasicrystal be discrete, assuming that the support is uniformly discrete? In our paper [4] with A. Olevskii we address this problem, and consider quasicrystals with non-symmetric discreteness assumptions on the support and the spectrum. We obtain several results which show that, under various conditions, if the spectrum is a discrete closed set, then in fact it must be uniformly discrete. These results thus reduce the situation to the setting in [2], which in turn allows us to conclude that the measure has a periodic structure. On the other hand, we present an example of a non-periodic quasicrystal such that the spectrum S is a nowhere dense countable set. Finally, we extend our results to the more general situation, where the Fourier transform of the measure µ has both a pure point component and a continuous one. References [175] N. Lev, A. Olevskii, Measures with uniformly discrete support and spectrum, C. R. Math. Acad. Sci. Paris 351 (2013), 599-603. · Zbl 1293.28001 [176] N. Lev, A. Olevskii, Quasicrystals and Poisson’s summation formula, Invent. Math. 200 (2015), 585-606. · Zbl 1402.28002 [177] N. Lev, A. Olevskii, Quasicrystals with discrete support and spectrum, Rev. Mat. Iberoam 32(2016), 1341-1352. · Zbl 1366.42011 [178] N. Lev, A. Olevskii, Fourier quasicrystals and discreteness of the diffraction spectrum, Adv. Math. 315 (2017), 1-26. On the union of spectra for all Sturm potentials Qinghui Liu (joint work with Bassam Fayad, Yanhui Qu) 1. Introduction Taking V > 0, irrational α ∈ (0, 1) and θ ∈ [0, 1), the Schr¨odinger operator with Sturm potential HV,α,θacting on l2(Z) is defined by, for any (φ(n))n∈Z∈ l2(Z), HV,α,θφ(n) = φ(n + 1) + φ(n − 1) + vnφ(n), where vn= V χ[1−α,1[({nα + θ}) and χ[1−α,1[is the characteristic function, and V is called coupling, α is called frequency, θ is called phase. Since the spectrum σ(HV,α,θ) is independent of θ, we take θ = 0 and denote the operator by HV,α. We study the union of spectra of the Schr¨odinger operator with Sturm potential of fixed coupling and all frequencies, i.e, for any V > 0, the set [ SV=σ(HV,α). α∈Qc∩(0,1) Theorem 1. [1] L(σ(HV,α)) = 0. 2826Oberwolfach Report 46/2017 In this paper, they show that σ(HV,α) ⊂ [−2, 2] ∪ [−2 + V, 2 + V ] = [−2, 2] + {0, V } := Γ. Notice that [−2, 2] = {2 cos tπ : 0 ≤ t ≤ 1}, we have Theorem 2 (Fayad, Liu, Qu, preprint). For V > 0, there exists Θ ⊂ Γ at most countable such that SV= Γ\Θ, where Θ ⊂ {2 cos tπ : 0 ≤ t ≤ 1, rational} + {0, V } Θ ⊃ {±2, 0, ±2 + V, V } = {2 cos tπ : t = 0, 1/2, 1} + {0, V }, V > 4 Θ ∩ {2 cos tπ : ε < t < 1 − ε} + {0, V }is a finite set. 2. Transfer matrix and trace polynomial Define the transfer matrices by Tn(E) =E − vn−1 10 and by T1→n(E) = Tn(E)Tn−1(E) · · · T1(E). For α ∈ [0, 1]\Q, let α = [0; a1, a2, · · · ] be the continued fraction expansion. For any k ≥ 0, let pk/qk= [0; a1, a2, · · · , ak], which satisfies, p−1= 1,p0= 0,pk+1= ak+1pk+ pk−1, k ≥ 0, q−1= 0,q0= 1,qk+1= ak+1qk+ qk−1, k ≥ 0. For k ≥ 0, define Mk(E) := T1→q(E) k xk(E) := tr Mk(E) σk:=E ∈ R : |xk(E)| ≤ 2. Note that xk(E) is a polynomial with degree qk. Theorem 3. [1] One has Mk+1(E) = Mk−1(E)Mkak+1(E), ∀k ≥ 0, σ(HV,α) =(σk−1∪ σk), k≥0 where 010(E) =E1−10, and σk:=E ∈ R : tr Mk(E) ≤ 2for k = 0, −1. Note that x−1(E) := tr M−1(E) ≡ 2, and x0(E) := tr M0(E) = E. Mathematical Quasicrystals2827 3. Sketch of2 cos tπ : t ∈ [0, 1]\Q⊂ SV We choose akstep by step. The idea in the proof comes from [4, 3]. Lemma 1. For E ∈ R, k ≥ 0, if xk−1(E) < 2, xk(E) < 2, then there exists ak+1such that xk+1(E) < 2. Corollary 1. For E ∈ R, if there exists k ≥ 0 so that |xk−1(E)| < 2, |xk(E)| < 2, then E ∈ SV. Proposition 1. If t ∈ [0, 1] be irrational, then there exists a1> 0 so that x0(2 cos tπ) < 2, x1(2 cos tπ) < 2, i.e., 2 cos tπ ∈ SV. 4. If V > 4, then 2 6∈ SV Lemma 2. [1] For any V > 0, α irrational and E ∈ R, (xk(E))k≥−1grow exponentially if and only if there exists k ≥ 0 such that xk−1(E) ≤ 2, xk(E) > 2, xk+1(E) > 2. Lemma 3. [2] For any V > 0, α irrational, δ ≥ 0 and E ∈ C, (xk(E))k≥−1grow exponentially if and only if there exists k ≥ 0 such that xk−1(E) ≤ 2 + δ, xk(E) > 2 + δ, tr Mk−1Mk(E) > 2 + δ. We can modify these results by Lemma 4. Take any V > 0, α irrational, and E ∈ C. If there exists k ≥ 0 such that xk−1(E) ≤ 2, xk(E) ≥ 2, tr Mk−1Mk(E) > 2, k≥−1grow exponentially. Since x−1(2) = 2, x0(2) = 2, tr M−1M0(2) = 2 − V , we have 2 6∈ SV. References [179] J. Bellissard, B. Iochum, E. Scoppola, D. Testard, Spectral properties of one-dimensional quasicrystals, Commun. Math. Phys. 125 (1989), 527-543. · Zbl 0825.58010 [180] D. Damanik, A. Gorodetski, Q.H. Liu, Y.H. Qu, Transport exponents of Sturmian Hamiltonians, J. Funct. Anal. 269 (2015), 1404-1440. · Zbl 1323.82038 [181] Q.H. Liu, Z.Y. Wen, Hausdorff dimension of spectrum of one-dimensional Schr¨odinger operator with Sturmian potentials, Potential Anal. 20 (2004), 33-59. · Zbl 1049.81023 [182] L. Raymond, A constructive gap labelling theorem for the discrete Schr¨odinger operator on a quasiperiodic chain, preprint (1997). 2828Oberwolfach Report 46/2017 On continuous and measure-theoretical eigenvalues of minimal Cantor systems and applications Alejandro Maass (joint work with Fabien Durand, Alexander Frank) The study of eigenvalues of topological dynamical systems, either from a measuretheoretical or a topological perspective, is a fundamental topic in ergodic theory. Particularly interesting and rich has been the study of eigenvalues and weakly mixing properties of classical systems like interval exchange transformations or other systems arising from translations on surfaces. From the symbolic dynamics point of view most of these systems have representations as minimal Cantor systems of finite topological rank, i.e., there is a symbolic extension that can be represented by a Bratteli-Vershik system such that the number of Kakutani-Rohlin towers per level is globally bounded. To characterize eigenvalues of the original systems it is enough to consider this class of Cantor systems. With these examples in mind and extensions to the study of tiling systems, our main motivation is to provide general necessary and sufficient conditions for a complex number to be the eigenvalue, either continuous or measure-theoretical, of a minimal Cantor system of finite topological rank and when possible to get the same kind of results for any minimal Cantor system. Some results for different subclasses of minimal Cantor systems of finite topological rank have been produced since the pioneering work of Dekking [5] and Host [183] . There, it was stated that measurable eigenvalues of primitive substitution dynamical systems are always associated to continuous eigenfunctions. Later, necessary and sufficient conditions to characterize continuous and measurable eigenvalues of linearly recurrent minimal Cantor systems were provided in [3] and [1]. These conditions are very effective and rely on the combinatorial data carried by their Bratteli-Vershik representations. Even if linearly recurrent systems are natural from the symbolic dynamics point of view (see [6, 7]), this class is “small”, meaning that in many classical cases, like interval exchange transformations, only a few maps have a symbolic representation of this kind. In fact, most of them are of finite topological rank and not linearly recurrent. There are few general results concerning eigenvalues of minimal Cantor systems of finite topological rank. Some preliminary results are given in [2] and a detailed study of eigenvalues of Toeplitz systems of finite topological rank is given in [8]. After reviewing the results described above we provide novel necessary and sufficient conditions that a complex number should satisfy to be a measurable eigenvalue of a minimal Cantor system of finite topological rank (we follow [9]). In addition, we give a necessary and sufficient condition for a complex number to be a continuous eigenvalue of a minimal Cantor system, that is, we succeeded in dropping the finite rank hypothesis. In its conception, the conditions are very similar to those proposed for linearly recurrent systems. They are given in the form of the convergence of some series or special sequences and only depend on the combinatorial data provided by the Bratteli-Vershik representations. The main Mathematical Quasicrystals2829 difference here is that we need to include in an algebraic way the information of the local orders carried by these representations. We illustrate the use of the conditions giving examples and applications. First we prove that our conditions extend the results in [8] to characterize eigenvalues of finite rank Toeplitz minimal systems. Then, a first application relates the notions of continuous eigenvalues and strong orbit equivalence. We use our necessary and sufficient condition in the continuous case to prove that, by doing controlled modifications of the local orders of a Bratteli-Vershik system, one can alter the group of continuous eigenvalues. In particular, starting from a minimal Cantor system without roots of unity as continuous eigenvalues we produce a strong orbit equivalent system that is topologically weakly mixing and which shares the Kronecker factor with the original system for any ergodic measure. In [12] a similar example is developed in the context of tiling systems. In a second example, the conditions to be measurable eigenvalues and previous application are used to construct a topologically weakly mixing minimal Cantor system of rank two admitting all rational numbers as measure theoretical eigenvalues, showing that topological rank is not an obstruction to have non continuous rational eigenvalues as in the Toeplitz case. Finally, inspired by questions in [4] and [10], we use our main theorems to produce an expansive minimal Cantor system whose group of continuous eigenvalues coincides with the intersection of the images of the so-called group of traces. References [184] X. Bressaud, F. Durand, A. Maass, Necessary and sufficient conditions to be an eigenvalue for linearly recurrent dynamical Cantor systems, J. London Math. Soc. 72 (2005), 799-816. · Zbl 1095.54016 [185] X. Bressaud, F. Durand, A. Maass, On the eigenvalues of finite rank Bratteli-Vershik dynamical systems, Ergodic Th. & Dynam. Syst. 30 (2010), 639-664. · Zbl 1204.37008 [186] M.I. Cortez, F. Durand, B. Host, A. Maass, Continuous and measurable eigenfunctions of linearly recurrent dynamical Cantor systems, J. London Math. Soc. 67 (2003), 790-804. · Zbl 1045.54011 [187] M.I. Cortez, F. Durand, S. Petite, Eigenvalues and strong orbit equivalence, Ergodic Th. & Dynam. Syst. 36 (2016), 2419-2440. · Zbl 1375.37008 [188] F.M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrscheinlichkeitsth. verw. Gebiete 41 (1978), 221-239. · Zbl 0348.54034 [189] F. Durand, Linearly recurrent subshifts have a finite number of non-periodic subshift factors, Ergodic Th. & Dynam. Syst. 20 (2000), 1061-1078. · Zbl 0965.37013 [190] F. Durand, Corrigendum and addendum to ‘Linearly recurrent subshifts have a finite number of non-periodic subshift factors’, Ergodic Th. & Dynam. Syst. 23 (2003), 663-669. [191] F. Durand, A. Frank, A. Maass, Eigenvalues of Toeplitz minimal systems of finite topological rank, Ergodic Th. & Dynam. Syst. 35 (2015), 2499-2528. · Zbl 1356.37013 [192] F. Durand, A. Frank, A. Maass, Eigenvalues of minimal Cantor systems, preprint, arXiv:1504.00067v2 (2017). · Zbl 1416.54018 [193] T. Giordano, D. Handelman, M. Hosseini, Orbit equivalence of Cantor minimal systems and their continuous spectra, preprint, arXiv:1606.03824 (2016). · Zbl 1394.37015 [194] B. Host, Valeurs propres des syst‘emes dynamiques d´efinis par des substitutions de longueur variable, Ergodic Th. & Dynam. Syst. 6 (1986), 529-540. · Zbl 0625.28011 [195] N.P. Frank, L. Sadun, Fusion: a general framework for hierarchical tilings of Rd, Geom. Dedicata 171 (2014), 149-186. 2830Oberwolfach Report 46/2017 Spectral analysis of primitive inflation rules Neil Ma˜nibo (joint work with Michael Baake, Michael Coons, Nathalie P. Frank, Franz G¨ahler, Uwe Grimm and E. Arthur Robinson Jr.) The spectral analysis of inflation rules is explained for a characteristic class of examples in one dimension. We determine the spectral type of the diffraction measurebγ for the one-parameter family of binary substitutions given by (1)̺m:0 7→ 01m,1 7→ 0 with corresponding inflation factor λ = λm, where m > 2 (we exclude m = 1 because this is the well-known Fibonacci inflation). We achieve this by using the geometric realization as an inflation tiling with prototiles (intervals) of natural length [196] and by examining exact renormalization equations for the corresponding pair correlations νij(z); see [3], as well as Eq. (6) below for a generalization to arbitrary dimension. These relations extend to a (measure-valued) renormalization equationP for the measure vector Υ, whose components are given by Υij=z∈Λνij(z) δz. These components determine the autocorrelation γ for general weights ui∈ C via a simple quadratic form [2]. All measures Υijare Fourier transformable. Via Fourier transform, one obtains a measure-valued renormalization equation for bΥ; compare [3, 2, 6]. This new equation, which holds for each of the three spectral types (pp, sc and ac) separately, involves the Fourier matrices B(k), where Bij(k) = dδT(k) with Tijbeing the set of positions of tiles of type i in level-1 ij supertiles of type j. In particular, for the absolutely continuous components, when described by a vector h of Radon-Nikodym densities, this implies an iterative equation for a.e. k ∈ R, (2)h(λk) = λ B−1(k) ⊗ B−1(k)h(k). This iteration can be reduced to an equation of lower dimension [2, 4], namely √ (3)v(λk) =λ B−1(k) v(k), where hij(k) = vi(k) vj(k). Exponential growth of kv(k)k implies an exponential growth of the norm of h and hence contradicts the translation-boundedness of the corresponding measure bΥacif v(k) 6= 0 for a subset of positive measure [2]. To rule out the existence of a non-trivial component bΥac, it thus suffices to show that, for any chosen 0 6= v(k) ∈ C2and a.e. k ∈ R, kv(k)k grows exponentially under the iteration (3). One way to analyse this is to obtain bounds for the Lyapunov exponents of the associated cocycle B(n)(k) = B(k)B(λk) · · · B(λn−1k); see [14] for background. This was done rigorously for m = 3 in [2] and extended to the entire family in [6]. These substitutions give rise to 2-dimensional cocyles, which ensures that there can be at most two distinct exponents, denoted by χminand χmax. Whenever λ is not an integer, i.e., for cases other than m = ℓ(ℓ + 1) with ℓ ∈ N, the existence of these exponents and Lyapunov regularity for a.e. k are not guaranteed (Oseledec’s Mathematical Quasicrystals2831 theorem does not necessarily apply here [2]). Nevertheless, it can be shown that the exponents (then defined via a lim sup) add up to log(λm) for all m via some extension of Sobol’s theorem to almost periodic functions [7]. A useful sufficient criterion for the positivity of all exponents, i.e., the positivity of the smallest exponent, is given by (4)log(λm) > M log kB(k)k2F, where M denotes the mean of a function and k.kFis the Frobenius norm. It can be shown [4] that this mean is recoverable as the logarithmic Mahler measure m(qm) of a polynomial qm∈ Z[x] given by (5)qm(x) = 2xm−1+ (1 + x + x2+ . . . + xm−1)2. Furthermore, one finds that this family of Mahler measures is bounded. In particular, it is dominated by log(λm) for all m > 18; see [6] for a proof. For m < 18, the smallest exponent has a bound that depends on the mean of a quasiperiodic functionN1Mlog kB(N )(k)k2Fwith two incommensurate frequencies, which cannot be expressed as a one-dimensional Mahler measure. However, this mean is computable as a finite integral over T2, and hence an appropriate N can be chosen so that this quantity is surpassed by log(λ). From this, we conclude the desired positivity by invoking a one-sided inequality due to some version of the subadditive ergodic theorem; see [2, 12]. This confirms the absence of bΥacfor all m, of whichbγac= 0 is an immediate consequence. For all ̺mwith non-Pisot inflation multiplier, this means that the diffraction is singularly continuous (except for the Bragg peak at k = 0, which corresponds to the constant eigenfunction of the inflation dynamical system). In contrast, the systems with an integer inflation multiplier (m = ℓ(ℓ + 1) and λm= ℓ + 1 for some ℓ ∈ N) are MLD to constant-length substitutions with a coincidence at the first column, and hence are automatically pure point due to Dekking’s classic result [11]. Oseledec’s multiplicative ergodic theorem can be applied to this class, from which one can obtain a closed form of χminthat is related to the (logarithmic) Mahler measure of a {−1, 0, 1}-polynomial in one variable [6, 13, 1]. This minimal exponent can be shown to be strictly positive, which provides an independent argument of why the diffraction is singular. Some general results on the absence ofbγacin the one-dimensional case via this method have already been written down; in particular, it has been shown for all binary aperiodic constant-length substitution in [13], for which the exponents are bounded appropriately by considering relevant polynomials of height 1; compare [8, 9]. General constant-length substitutions on n letters follow a similar scheme, and positivity can be proved for some general families (bijective Abelian, some families with coincidences [4]). We comment briefly that the renormalization scheme in [3] also holds for higher-dimensional analogues (primitive stone inflations of finite local complexity with a suitably chosen reference point in each prototile [5]). 2832Oberwolfach Report 46/2017 The renormalization equations then read 1XXX |det(As)|νkℓA−1s(z + u − v), k,ℓu∈Tikv∈Tjℓ where z, u, v ∈ Rd, while Asis the linear map that expands the system to one that is MLD with the original one via the stone inflation in question [5, 4]. The indices run over a set of labels for the finite prototile set. An application to the higher-dimensional case of block substitutions boils down to finding appropriate bounds for Mahler measures of polynomials in more than one variable [10, 1]. References [197] M. Baake, M. Coons, N. Ma˜nibo, Binary constant-length substitutions and Mahler measures of Borwein polynomials, preprint, submitted; arXiv:1711.02492. · Zbl 1461.11143 [198] M. Baake, N.P. Frank, U. Grimm, E.A. Robinson, Geometric properties of a binary non-Pisot inflation and absence of absolutely continuous diffraction, preprint, submitted; arXiv:1706.03976. · Zbl 1419.37017 [199] M. Baake, F. G¨ahler, Pair correlations of aperiodic inflation rules via renormalisation: Some interesting examples, Topol. Appl. 205 (2016), 4-27; arXiv:1511.00885. · Zbl 1359.37025 [200] M. Baake, F. G¨ahler, N. Ma˜nibo, Renormalisation of pair correlation measures for primitive inflation rules and absence of absolutely continuous diffraction, in preparation. [201] M. Baake, U. Grimm, Aperiodic Order. Vol. 1: A Mathematical Invitation, Cambridge University Press, Cambridge (2013). · Zbl 1295.37001 [202] M. Baake, U. Grimm, N. Ma˜nibo, Spectral analysis of a family of binary inflation rules, preprint, submitted; arXiv:1709.09083. [203] M. Baake, A. Haynes, D. Lenz, Averaging almost periodic functions along exponential sequences, in Aperiodic Order. Vol. 2:Crystallography and Almost Periodicity, M. Baake, U. Grimm (eds.), Cambridge University Press, Cambridge (2017), pp. 343-362; arXiv:1704.08120. · Zbl 1435.42004 [204] P. Borwein, S. Choi, J. Jankauskas, Extremal Mahler measures and Lsnorms of polynomials related to Barker sequences, Proc. Amer. Math. Soc. 141 (2013), 2653-2663. · Zbl 1272.11041 [205] P. Borwein, M. Mossinghoff, Barker sequences and flat polynomials, in Number Theory and Polynomials, J. McKee, C. Smyth (eds.), Cambridge University Press, New York (2008), pp. 71-88. · Zbl 1266.11051 [206] D. Boyd, M. Mossinghoff, Small limit points of Mahler’s measure, Experim. Math. 15 (2005), 403-414. · Zbl 1152.11343 [207] F.M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrscheinlichkeitsth. verw. Geb. 41 (1978), 221-239. · Zbl 0348.54034 [208] A.-H. Fan, B. Saussol, J. Schmeling, Products of non-stationary random matrices and multiperiodic equations of several scaling factors, Pacific J. Math. 214 (2004), 31-54; arXiv:math.DS/0210347. · Zbl 1061.37034 [209] N. Ma˜nibo, Lyapunov exponents for binary substitutions of constant length, J. Math. Phys., in press; arXiv:1706.00451. [210] M. Viana, Lectures on Lyapunov Exponents, Cambridge University Press, Cambridge (2013). Mathematical Quasicrystals2833 Quasicrystalline structures and quantum walks Darren C. Ong (joint work with David Damanik, Jake Fillman) Consider first the following classical random walk problem. A walker is travelling on the integers, and flips a coin. If the coin lands heads, the walker moves to the left, and if the coin lands tail, the walker moves right. Furthermore, imagine that that weighted coins are distributed at each integer, and whenever the walker is at that integer we use the coin placed there. We consider now a “quantum mechanical” version of this problem. In this model, the walker (which we imagine as a quantum particle) possesses a spin (either ↑ or ↓) as well as an integer location; moreover, the walker may be in a superposition of pure states, rather than being purely localized at a particular site with a definite spin. Instead of a weighted coin at each location, we have a unitary operator (which we call the quantum coin) at each location that interacts with the particle differently depending on its spin. This model has attracted a lot of interest in mathematics, computer science, and physics. Please refer to [6, 7] for some recent surveys on the subject. We now introduce a second object, the CMV operator. These are operators on ℓ2(N) or ℓ2(Z) that can be viewed as a unitary analogue to the Jacobi operator. See [4, 5] for a survey on the spectral theory of the CMV operator. The CMV operator on ℓ2(Z) looks like . ... ... ... .. α0ρ−1−α0α−1α1ρ0ρ1ρ0 E =ρ0ρ−1−ρ0α−1−α1α0−ρ1α0 α2ρ1−α2α1α3ρ2ρ3ρ2. ρ2ρ1−ρ2α1−α3α2−ρ3α2 . ... ... ... .. Here, the αpnare a sequence of complex numbers in the open unit disk, and ρn= 1 − |αn|2. Cantero, Gr¨unbaum, Moral and Velazquez discovered in [1] that CMV operators can be used to understand quantum walks. Furthermore, in [3] the authors discover a connection between the spectral properties of a CMV operator and the spreading rate of a walker in a quantum walk. In our paper [2], we discover upper and lower bounds on quantum walk spreading that depend on the growth rates of the transfer matrices of the corresponding CMV operator. As an application, this enables us to understand quantum walk problems where the coin distributions are given by an aperiodically ordered sequence. To be more precise, consider two types of coins, each weighted differently. We arrange these coins on the integers using a Fibonacci binary string, such that the 2834Oberwolfach Report 46/2017 0’s correspond to the first type of coin and the 1’s correspond to the second. Using the upper and lower bounds we developed, we can calculate that it is possible to obtain anomalous transport this way: that is, the quantum walker leaves the origin, but at sub-ballistic speeds. To our knowledge, this was the first example of anomalous transport in quantum walks when the coins do not vary in time. References [211] M.-J. Cantero, A. Gr¨unbaum, L. Moral, L. Vel´azquez, Matrix-valued Szeg˝o polynomials and quantum random walks, Commun. Pure Appl. Math. 63 (2010), 464-507. [212] D. Damanik, J. Fillman, D.C. Ong, Spreading estimates for quantum walks on the integer lattice via power-law bounds on transfer matrices, J. Math. Pures Appl. 105 (2016), 293– 341. · Zbl 1332.81066 [213] D. Damanik, J. Fillman, R. Vance, Dynamics of unitary operators, J. Fractal Geom. 1 (2014), 391-425. · Zbl 1321.47076 [214] B. Simon, Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory, Colloquium Publications, vol. 54 part 1, AMS, Providence, RI (2005). · Zbl 1082.42020 [215] B. Simon, Orthogonal Polynomials on the Unit Circle. Part 2. Spectral Theory, Colloquium Publications, vol. 54 part 2, AMS, Providence, RI (2005). · Zbl 1082.42021 [216] S.E. Venegas-Andraca, Quantum walks: a comprehensive review, Quantum Inf. Process. 11 (2012), 1015-1106. · Zbl 1283.81040 [217] S.E. Venegas-Andraca, Quantum Walks for Computer Scientists, Synthesis Lectures on Quantum Computing, vol. 1, Morgan & Claypool, San Rafael, CA (2008). Automorphism and extended symmetry groups of shifts Samuel Petite, Reem Yassawi A d-dimensional shift is a closed set X of sequences on a finite alphabet, indexed by Zd, and invariant by the d shift maps σ1, . . . σddefined by the canonical basis for Zd. Shifts form a rich class of dynamical systems. An automorphism (or symmetry) of a shift is a homeomorphism Φ : X → X that commutes with the shift maps σ1, . . . σd. The Curtis-Hedlund-Lyndon theorem tells us that such an automorphism is defined by a sliding block map; i.e., there exists a local rule, defined on a set of finite configurations, such that the image of a point x at index m= m1, . . . mdis defined by applying the local rule table to a neighbourhood of σ1m1. . . σmddx . The set of automorphisms Aut(X) of a shift, endowed with the composition operation, form a group, which, other than being countable, is in general hard to describe. We will present in this talk a survey of recent results, where we give a finer description of the automorphism groups of certain small onedimensional shifts. In particular, if the complexity function of a one-dimensional minimal shift is linear, we show that Aut(X)/hσi is finite. We also give conditions on an automorphism Φ, in terms of its radius, so that it has finite order, and deduce that if such an automorphism Φ exists, then the automorphism group does not contain a group with an exponentially distorted element. Staying in the zerocomplexity case, we show that for a minimal shift whose complexity function is o(n5), any finitely generated, torsion-free subgroup of Aut(X) is virtually Abelian. Mathematical Quasicrystals2835 We also consider extended symmetry groups of shifts. While the automorphism group can be recast as the centralizer of the group hσi generated by the shift action in the group H(X) of homeomorphisms on X, the extended symmetry group R(X) is defined to be the normalizer of hσi in H(X). If (X, σ) is a minimal onedimensional shift, then work of Putnam, Giordano and Skau tells us that R(X)/hσi is the group of outer autormphisms of the topological full group of a shift, whose commutator subgroup has recently been shown by Juschenko and Monod to have interesting algebraic properties. We also define and discuss extended symmetry groups in higher dimension. We illuminate these concepts by computing the extended symmetry groups of two celebrated and qualitatively different shifts: the chair shift and the Ledrappier shift. This is based on recent and active works [1, 2, 3, 4, 5, 6, 7, 8]. References [218] M. Baake, J.A.G. Roberts, R. Yassawi, Reversing and extended symmetries of shift spaces, Discr. Cont. Dynam. Syst. A 38 (2018), 835-866; arXiv:1611.05756. · Zbl 1377.37027 [219] E. Coven, A. Quas, R. Yassawi, Computing automorphism groups of shifts using atypical equivalence classes, Discr. Anal. (2016), no. 3, 28 pp; arXiv:1505.02482. · Zbl 1378.54035 [220] V. Cyr, B. Kra, The automorphism group of a shift of linear growth: beyond transitivity, Forum Math. Sigma 3 (2015), e5, 27 pp; arXiv:1411.0180. · Zbl 1321.37010 [221] V. Cyr, B. Kra, The automorphism group of a shift of subquadratic growth, Proc. Amer. Math. Soc. 144 (2016), 613-621; arXiv:1403.0238. · Zbl 1365.37019 [222] V. Cyr, B. Kra, The automorphism group of a minimal shift of stretched exponential growth, J. Mod. Dyn. 10 (2016), 483-495; arXiv:1509.08493. · Zbl 1402.37013 [223] V. Cyr, J. Franks, B. Kra, S. Petite, Distortion and the automorphism group of a shift, preprint, arXiv:1611.05913. · Zbl 1407.37022 [224] S. Donoso, F. Durand, A. Maass, S. Petite, On automorphism groups of low complexity subshifts, Ergodic Th. & Dynam. Syst. 36 (2016), 64-95; arXiv:1501.00510. · Zbl 1354.37024 [225] S. Donoso, F. Durand, A. Maass, S. Petite, On automorphism groups of Toeplitz subshifts, Discr. Anal. (2017), no. 11, 19 pp; arXiv:1701.00999. Topological boundary spectrum in physical systems Emil Prodan Let P be a Delone set in Rd, Ω be its discrete hull (transversal) and HΩ= {Hω}ω∈Ω a family of covariant Hamiltonians on CN⊗ ℓ2(P). We denote by spec(HΩ) = ∪ω∈Ωspec(Hω). A spectral gap G is defined as a connected component of R spec(HΩ). A mobility gap is a connected region ∆ of the real axis where the direct conductivity tensor d−1Pdi=1σii(EF, T = 0) vanishes, whenever EF∈ ∆. We use the terminology (mobility) gapped Hamiltonians for a pair (HΩ, G) with G a non-empty spectral (mobility) gap. Given a covariant bulk family HΩ, one can define a family of Hamiltonians HbΩ= { ˆHω}ω∈Ωwith a boundary, defined on CN⊗ ℓ2(P ∩ Rd+), Rd+= {x = (x1, . . . , xd) ∈ Rd, xd≥ 0}. More precisely, bHΩ= bHΩ(D)+ eHΩ, where bHΩ(D)is obtained from HΩby imposing the Dirichlet boundary condition and eHΩis a family of covariant Hamiltonians w.r.t. translations parallel to the boundary that 2836Oberwolfach Report 46/2017 are localized near the boundary (hence eHΩcan be seen as defining the boundary condition). As it turns out, spec(HΩ) ⊆ spec( bHΩ) and one defines the boundary spectrum as specb( bHΩ) = spec( bHΩ) spec(HΩ). For such data, the physics community propose the following programs. Bulk. (1) Classify the gapped Hamiltonians w.r.t. the equivalence relation defined by (HΩ, G) ∼ (HΩ′, G′) iff there exists a homotopy of gapped Hamiltonians (HΩ(t), G(t))t∈[0,1]such that (HΩ(0), G(0)) = (HΩ, G) and (HΩ(1), G(1)) = (HΩ′, G′). (2) Same as above but with spectral gap replaced by mobility gap. Boundary. (1) Find all (HΩ, G) such that specb( bHΩ) ∩ G = G, regardless of the boundary condition. This type of boundary spectrum is called topological. (2) Same as above but with additional requirement that specb( bHΩ) is not Anderson localized. Bulk+Boundary. (1) Find a relations between the bulk and boundary programs. Whenever such relations exist, they go by the name of bulk-boundary principle. The most ambitious programs are Bulk #2, Boundary #2 and, of course, establishing the bulk-boundary principle which relates the two. In the context of the electronic degrees of freedom in disordered crystals, the physics community put forward a conjecture which comes in the form of a classification table of all possible topological phases displaying delocalized topological boundary spectrum [13, 7, 12]. The conjecture survived a large number of numerical tests (too many to be mentioned here). For the phases classified by the Z group, a proof of the conjecture can be found in the monograph [11]. It is based on the index theorems discovered in [9, 10], which extended the previous pioneering works [1, 6] on the integer quantum Hall effect. For the topological phases classified by Z2, progress with program Bulk #2 has been made in [4], while the program Bulk #1 has been carried out in [14, 2, 5]. A unifying bulk-boundary principle for the programs Bulk #1 and Boundary #1 has been carried in [3]. Work on the conjecture contained in the classification table continues these days. Recently, the interest of the physics community is rapidly shifting from crystals to meta-materials and from the electronic degrees of freedom to the electromagnetic and acoustic degrees of freedom. The search for topological photonic and acoustic crystals is vigorously underway, on both theoretical and experimental fronts. Since meta-materials give great control over the structure of the materials, there is a strong movement among the physics community to move away from the Mathematical Quasicrystals2837 simple disordered crystalline patterns and experiment with more interesting patterns. Following this effort, myself and collaborators [8] introduced the concept of dynamically generated patterns, defined as below. Proposition. Let (Ω, Zd, τ ) be a topological dynamical system. Denote the set of generators of Zd, taken with both signs, by Gd. Assume the existence of the continuous maps: (1)Fe: Ω → Rα,e∈ Gd,α ∈ N+, obeying the consistency relations: (2)F−e= −Fe◦ τe,Fe′− Fe′◦ τ−e= Fe− Fe◦ τ−e′. Then, for each ω ∈ Ω, the algorithm: (3)p0= 0,pn+e= pn+ (Fe◦ τn+e)(ω),n∈ Zd,e∈ Gd, generates a point pattern P = {pn}n∈Zdwhose points are indexed by Zd. Under precise conditions, the discrete hull of these patterns coincides with Ω. While these patterns are algorithmically simple, the patterns themselves can be very complex and, through various limits, one can explore patterns that are not from this category. Hence, we think they are very interesting. For many examples, the bulk-boundary principle for programs Bulk #1 and Boundary #1 can be carried out completely using tools from K-theory. As a result, we obtained a large number of new classes of systems displaying topological boundary spectrum. The goals of my talk given for the workshop “Spectral Structures and Topological Methods in Mathematical Quasicrystals”, organized at Oberwolfach Institute, were: 1) communicate the research programs described in the first part of this note; 2) exemplify the success stories with physical examples; 3) communicate the interest of the physics community in patterns that go beyond disordered crystals; 4) introduce the dynamically-generated patterns; 5) establish the bulk-boundary principle for equivariant Hamiltonians defined over such patterns; 6) exemplify the physical consequences using laboratory results and numerical simulations. References [226] J. Bellissard, A. van Elst, H. Schulz-Baldes, The non-commutative geometry of the quantum Hall-effect, J. Math. Phys. 35 (1994), 5373-5451. · Zbl 0824.46086 [227] C. Bourne, A.L. Carey, A. Rennie, A noncommutative framework for topological insulators, Rev. Math. Phys. 28 (2015), 1650004. · Zbl 1364.81269 [228] C. Bourne, J. Kellendonk, A. Rennie, The K-theoretic bulk-edge correspondence for topological insulators, to appear in Ann. Henri Poincar´e. · Zbl 1372.82023 [229] J. Grossmann, H. Schulz-Baldes: Index pairings in presence of symmetries with applications to topological insulators, Commun. Math. Phys. 343 (2016), 477-513. · Zbl 1348.82083 [230] J. Kellendonk, On the C∗-algebraic approach to topological phases for insulators, Ann. Henri Poincar´e 18 (2017), 2251-2300. · Zbl 1382.82045 [231] J. Kellendonk, T. Richter, H. Schulz-Baldes, Edge current channels and Chern numbers in the integer quantum Hall effect, Rev. Math. Phys. 14 (2002), 87-119. · Zbl 1037.81106 [232] A. Kitaev, Periodic table for topological insulators and superconductors, in Adv. Theor. Phys.: Landau Memorial Conference, vol. 1134, eds. V. Lebedev, M. Feigel’man, AIP (2009), pp. 22-30. 2838Oberwolfach Report 46/2017 · Zbl 1180.82221 [233] M. Herman, E. Prodan, Y. Shmalo, The K-theoretic bulk-boundary principle for dynamically patterned resonators, in preparation. · Zbl 1471.37004 [234] E. Prodan, B. Leung, J. Bellissard, The non-commutative n-th Chern number (n ≥ 1), J. Phys. A: Math. Theor. 46 (2013), 485202. · Zbl 1290.82016 [235] E. Prodan, H. Schulz-Baldes, Non-commutative odd Chern numbers and topological phases of disordered chiral systems, J. Func. Anal. 271 (2016), 1150-1176. · Zbl 1344.82055 [236] E. Prodan, H. Schulz-Baldes, Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics, Springer, Berlin (2016). · Zbl 1342.82002 [237] S. Ryu, A.P. Schnyder, A. Furusaki, A.W. Ludwig. Topological insulators and superconductors: tenfold way and dimensional hierarchy, New J. Phys. 12 (2010), 065010. [238] A.P. Schnyder, S. Ryu, A. Furusaki, A.W.W. Ludwig, Classification of topological insulators and superconductors in three spatial dimensions, Phys. Rev. B 78 (2008), 195125. · Zbl 1180.82228 [239] G.C. Thiang, On the K-theoretic classification of topological phases of matter, Ann. Henri Poincar´e 17 (2016), 757-794. The “mixed” spectral nature of the Thue-Morse Hamiltonian Yanhui Qu (joint work with Qinghui Liu, Xiao Yao) We find a subset Σ of the spectrum of Thue-Morse Hamiltonian Hλ, such that for any E ∈ Σ, the following properties hold: (i) The related trace orbit {tn(E) : n ≥ 1} is unbounded; (ii) The norms of the transfer matrices grow as √√ ec1γn≤Tn(E) ≤ ec2γn, where 0 < c1< c2are two absolute constants, γ > 0 is a constant only depending on E; (iii) There exists a subordinate solution ψ of Hλψ = Eψ, such that |ψn| is polynomially bounded; |ψ±22n| decreases as e−2nγ; (ψ22n+1+1, ψ22n+1) tends to (±1, 1). We call such a ψ a pseudo-localized state. It is known that there exists a dense subset ˜Σ of the spectrum such that, for any E ∈ ˜Σ and any solution φ of Hλφ = Eφ, φ is an extended state [1, 2]. Since the extended states and pseudo-localized states co-exist, we may say that the Thue-Morse Hamiltonian exhibits “mixed” spectral nature [3]. References [240] F. Axel, J. Peyriere, Spectrum and extended states in a harmonic chain with controlled disorder: effects of the Thue-Morse symmetry, J. Stat. Phys. 57 (1989), 1013-1047. · Zbl 0724.11011 [241] D. Damanik, S. Tcheremchantsev, Power-law bounds on transfer matrices and quantum dynamics in one dimension, Commun. Math. Phys. 236 (2003), 513-534. · Zbl 1033.81032 [242] Q.-H. Liu, Y.-H. Qu, X. Yao, Unbounded trace orbits of Thue-Morse Hamiltonian, J. Stat. Phys. 166 (2017), 1509-1557. Mathematical Quasicrystals2839 Almost periodic measures and diffraction Nicolae Strungaru (joint work with Robert V. Moody) Almost periodicity plays an important role in the study of mathematical diffraction. Given a point set Λ ⊂ Rd, representing the positions of atoms in an idealized solid, Hof [7] defined the diffractionbγ of Λ as the Fourier transform of the positive and positive definite measure γ, the autocorrelation (or 2-point correlation) measure of Λ; see [2] for a general exposition. Each positive definite measure γ (or, more generally, each weakly almost periodic measure γ) admits a (unique) Eberlein decomposition γ = γs+ γ0 into a strongly almost periodic measure γsand a null-weakly almost periodic measure γ0(see the review [8] or [4, 5, 6] for details). As proved by Eberlein for finite measures [5] and by Gil de Lamadrid-Argabright for twice Fourier transformable measures [6], the Eberlein decomposition of the autocorrelation γ is Fourier dual to the Lebesgue decomposition of the diffraction measurebγ. Recently, we proved that this result more generally holds for translation bounded, Fourier transformable measures as follows. Theorem 1 ([8]). Let γ be a translation bounded, Fourier transformable measure. Then, γ is weakly almost periodic, γsand γ0are Fourier transformable, and γbs= (bγ)pp,γb0= (bγ)c. An immediate consequence of this result is that strong almost periodicity and pure point diffraction are Fourier dual concepts: • a Fourier transformable measure µ is strongly almost periodic if and only ifµ is pure point [8];b • a Fourier transformable measure µ is pure point if and only if bµ is strongly almost periodic [6]. These results allow us to study the pure point spectrum (bγ)ppand the continuous spectrum (bγ)cof Λ, which are measures in the Fourier dual space cRd≃ Rd, by studying instead the measures γsand γ0, respectively, in the real space Rd. This approach has led to many general results about the diffraction of Meyer sets (see [9, 10] for example). To gain further insight into the absolutely continuous and the singular continuous spectrum, we would like to extend the Eberlein decomposition to another decomposition step, γ0= γ0a+ γ0s, which is Fourier dual to the spectral decomposition (bγ)c= (bγ)ac+ (bγ)sc. While the general question about the existence of this decomposition is still open, recent progress has been made in the case of positive definite measures with Meyer set support as follows. 2840Oberwolfach Report 46/2017 Theorem 2. [11] Let γ be a translation bounded, positive definite measure that is supported inside a Meyer set. Then, there exist three positive definite measures γs, γ0s, γ0asupported inside Meyer sets such that γ = γs+ γ0s+ γ0aand γbs= (bγ)pp,γc0s= (bγ)sc,γc0a= (bγ)ac. If Λ is a Meyer set, and γ its autocorrelation, it follows that each of the measures (bγ)pp, (bγ)ac, (bγ)scis strongly almost periodic. In particular, each of these measures is either trivial or has a relatively dense support. Theorem 2 holds if Rdis replaced by an arbitrary metrizable locally compact Abelian group G. It follows more generally that, if ω is any translation bounded measure with Meyer set support in G, and γ is any autocorrelation of ω, then each of the measures (bγ)pp, (bγ)ac, (bγ)scis either trivial or has a relatively dense support. References [243] L.N. Argabright, J. Gil de Lamadrid, Fourier analysis of unbounded measures on locally compact Abelian groups, Memoirs Amer. Math. Soc. no. 145, AMS, Providence, RI (1974). · Zbl 0294.43002 [244] M. Baake, U. Grimm, Aperiodic Order. Vol. 1: A Mathematical Invitation, Cambridge University Press, Cambridge (2013). · Zbl 1295.37001 [245] M. Baake, U. Grimm, Aperiodic Order. Vol. 2: Crystallography and Almost Periodicity, Cambridge University Press, Cambridge (2017). · Zbl 1415.52001 [246] W.F. Eberlein, Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc. 67 (1949), 217-240. · Zbl 0034.06404 [247] W.F. Eberlein, The point spectrum of weakly almost periodic functions, Michigan Math. J. 3(1955/56), 137-139. · Zbl 0073.30702 [248] J. Gil de Lamadrid, L.N. Argabright, Almost periodic measures, Memoirs Amer. Math. Soc. 85, no. 428, AMS, Providence, RI (1990). · Zbl 0719.43006 [249] A. Hof, On diffraction by aperiodic structures, Commun. Math. Phys. 169 (1995), 25-43. · Zbl 0821.60099 [250] R.V. Moody, N. Strungaru, Almost periodic measures and their Fourier transforms, in [3], pp. 173-270. · Zbl 1421.42004 [251] N. Strungaru, Almost periodic measures and long-range order in Meyer sets, Discr. Comput. Geom. 33 (2005), 483-505. · Zbl 1062.43008 [252] N. Strungaru, Almost periodic measures and Meyer sets, in [3], pp. 271-342. · Zbl 1062.43008 [253] N. Strungaru This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.