THE TOPOLOGY of TILING SPACES
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1 Seoul National University March20, THE TOPOLOGY of Sponsoring TILING SPACES This material is based upon work supported by the National Science Foundation Grant No. DMS Jean BELLISSARD Georgia Institute of Technology, Atlanta School of Mathematics & School of Physics jeanbel@math.gatech.edu Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
2 Seoul National University March20, Contributors R. Benedetti, U. Pisa, Italy J.-M. Gambaudo, U. Nice, France D.J.L. Herrmann, U. Tübingen, Germany J. Kellendonk, U. Lyon I, France J. Savinien, U. Metz, France M. Zarrouati, U. Toulouse, France
3 Seoul National University March20, Main References J. Bellissard, The Gap Labeling Theorems for Schrödinger s Operators, in From Number Theory to Physics, pp , Les Houches March 89, Springer, J.M. Luck, P. Moussa & M. Waldschmidt Eds., (1993). J. Kellendonk, The local structure of tilings and their integer group of coinvariants, Comm. Math. Phys., 187, (1997), J. E. Anderson, I. Putnam, Topological invariants for substitution tilings and their associated C -algebras, Ergodic Theory Dynam. Systems, 18, (1998), A. H. Forrest, J. Hunton, The cohomology and K-theory of commuting homeomorphisms of the Cantor set, Ergodic Theory Dynam. Systems, 19, (1999), J. C. Lagarias, Geometric models for quasicrystals I & II, Discrete Comput. Geom., 21, (1999), &
4 Seoul National University March20, J. Bellissard, D. Herrmann, M. Zarrouati, Hull of Aperiodic Solids and Gap Labeling Theorems, In Directions in Mathematical Quasicrystals, CRM Monograph Series, Volume 13, (2000), , M.B. Baake & R.V. Moody Eds., AMS Providence. L. Sadun, R. F. Williams, Tiling spaces are Cantor set fiber bundles, Ergodic Theory Dynam. Systems, 23, (2003), J. Bellissard, R. Benedetti, J. M. Gambaudo, Spaces of Tilings, Finite Telescopic Approximations, Comm. Math. Phys., 261, (2006), J. Bellissard, J. Savinien, A Spectral Sequence for the K-theory of Tiling Spaces, arxiv: , submitted to Ergod. Th. Dyn. Syst., 2007.
5 Seoul National University March20, Content 1. Tilings, Tilings The Hull as a Dynamical System 3. Branched Oriented Flat Riemannian Manifolds 4. Cohomology and K-Theory 5. Conclusion
6 Seoul National University March20, I - Tilings, Tilings,...
7 Seoul National University March20, A triangle tiling -
8 Seoul National University March20, Dominos on a triangular lattice -
9 Seoul National University March20, Building the chair tiling -
10 Seoul National University March20, The chair tiling -
11 Seoul National University March20, The Penrose tiling -
12 Seoul National University March20, Kites and Darts -
13 Seoul National University March20, Rhombi in Penrose s tiling -
14 Seoul National University March20, Inflation rules in Penrose s tiling -
15 Seoul National University March20, The Penrose tiling -
16 Seoul National University March20, The octagonal tiling -
17 Seoul National University March20, Octagonal tiling: inflation rules -
18 Seoul National University March20, Another octagonal tiling -
19 Seoul National University March20, Another octagonal tiling -
20 Seoul National University March20, Building the Pinwheel Tiling -
21 Seoul National University March20, The Pinwheel Tiling -
22 Seoul National University March20, Aperiodic Materials 1. Periodic Crystals in d-dimensions: translation and crystal symmetries. Translation group T Z d. 2. Periodic Crystals in a Uniform Magnetic Field; magnetic oscillations, Shubnikov-de Haas, de Haas-van Alfen. The magnetic field breaks the translation invariance to give some quasiperiodicity.
23 Seoul National University March20, Quasicrystals: no translation symmetry, but icosahedral symmetry. Ex.: (a) Al 62.5 Cu 25 Fe 12.5 ; (b) Al 70 Pd 22 Mn 8 ; (c) Al 70 Pd 22 Re 8 ; 4. Disordered Media: random atomic positions (a) Normal metals (with defects or impurities); (b) Alloys (c) Doped semiconductors (Si, AsGa,...);
24 Seoul National University March20, The icosahedral quasicrystal AlPdMn -
25 Seoul National University March20, The icosahedral quasicrystal HoMgZn-
26 Seoul National University March20, II - The Hull as a Dynamical System
27 Seoul National University March20, Point Sets A subset L R d may be: 1. Discrete. 2. Uniformly discrete: r > 0 s.t. each ball of radius r contains at most one point of L. 3. Relatively dense: R > 0 s.t. each ball of radius R contains at least one points of L. 4. A Delone set: L is uniformly discrete and relatively dense. 5. Finite Local Complexity (FLC): L L is discrete and closed. 6. Meyer set: L and L L are Delone.
28 Seoul National University March20, Point Sets and Point Measures M(R d ) is the set of Radon measures on R d namely the dual space to C c (R d ) (continuous functions with compact support), endowed with the weak topology. For L a uniformly discrete point set in R d : ν := ν L = y L δ(x y) M(R d ).
29 Seoul National University March20, Point Sets and Tilings Given a tiling with finitely many tiles (modulo translations), a Delone set is obtained by defining a point in the interior of each (translation equivalence class of) tile. Conversely, given a Delone set, a tiling is built through the Voronoi cells V(x) = {a R d ; a x < a y, yl \ {x}} 1. V(x) is an open convex polyhedron containing B(x; r) and contained into B(x; R). 2. Two Voronoi cells touch face-to-face. 3. If L is FLC, then the Voronoi tiling has finitely many tiles modulo translations.
30 Seoul National University March20, Building a Voronoi cell-
31 Seoul National University March20, A Delone set and its Voronoi Tiling-
32 Seoul National University March20, The Hull A point measure is µ M(R d ) such that µ(b) N for any ball B R d. Its support is 1. Discrete. 2. r-uniformly discrete: iff B ball of radius r, µ(b) R-Relatively dense: iff for each ball B of radius R, µ(b) 1. R d acts on M(R d ) by translation.
33 Seoul National University March20, Theorem 1 The set of r-uniformly discrete point measures is compact and R d -invariant. Its subset of R-relatively dense measures is compact and R d -invariant. Definition 1 Given L a uniformly discrete subset of R d, the Hull of L is the closure in M(R d ) of the R d -orbit of ν L. Hence the Hull is a compact metrizable space on which R d acts by homeomorphisms.
34 Seoul National University March20, Properties of the Hull If L R d is r-uniformly discrete with Hull Ω then using compactness 1. each point ω Ω is an r-uniformly discrete point measure with support L ω. 2. if L is (r, R)-Delone, so are all L ω s. 3. if, in addition, L is FLC, so are all the L ω s. Moreover then L L = L ω L ω ω Ω. Definition 2 The transversal of the Hull Ω of a uniformly discrete set is the set of ω Ω such that 0 L ω. Theorem 2 If L is FLC, then its transversal is completely discontinuous.
35 Seoul National University March20, Local Isomorphism Classes and Tiling Space A patch is a finite subset of L of the form p = (L x) B(0, r 1 ) x L, r 1 0 Given L a repetitive, FLC, Delone set let W be its set of finite patches: it is called the the L-dictionary. A Delone set (or a Tiling) L is locally isomorphic to L if it has the same dictionary. The Tiling Space of L is the set of Local Isomorphism Classes of L. Theorem 3 The Tiling Space of L coincides with its Hull.
36 Seoul National University March20, Minimality L is repetitive if for any finite patch p there is R > 0 such that each ball of radius R contains an ɛ-approximant of a translated of p. Theorem 4 R d acts minimaly on Ω if and only if L is repetitive.
37 Seoul National University March20, Examples 1. Crystals : Ω = R d /T T d with the quotient action of R d on itself. (Here T is the translation group leaving the lattice invariant. T is isomorphic to Z D.) The transversal is a finite set (number of point per unit cell). 2. Impurities in Si : let L be the lattices sites for Si atoms (it is a Bravais lattice). Let A be a finite set (alphabet) indexing the types of impurities. The transversal is X = A Zd with Z d -action given by shifts. The Hull Ω is the mapping torus of X.
38 Seoul National University March20, The Hull of a Periodic Lattice -
39 Seoul National University March20, Quasicrystals Use the cut-and-project construction: R d E π R n π E R n d Here 1. L is a lattice in R n, L π L π W, 2. the window W is a compact polytope. 3. L is the quasilattice in E defined as L = {π (m) E ; m L, π (m) W}
40 Seoul National University March20,
41 Seoul National University March20, The transversal of the Octagonal Tiling is completely disconnected -
42 Seoul National University March20, III - The Gap Labeling Theorem J. Bellissard, R. Benedetti, J.-. Gambaudo, Commun. Math. Phys., 261, (2006), J. Kaminker, I. Putnam, Michigan Math. J., 51, (2003), M. Benameur, H. Oyono-Oyono, C. R. Math. Acad. Sci. Paris, 334, (2002),
43 Seoul National University March20, Schrödinger s Operator Ignoring electrons-electrons interactions, the one-electron Hamiltonian is given by H ω = 2 2m + y L ω v( y) Its integrated density of states (IDS) is defined by 1 N(E) = lim Λ R d Λ # { eigenvalues of H ω Λ E } For any R d -invariant probability measure P on Ω the limit exists a.e. and is independent of ω. It defines a nondecreasing function of E constant on the spectral gaps of H ω. It is asymptotic at large E s to the IDS of the free Hamiltonian.
44 Seoul National University March20, An example of IDS -
45 Seoul National University March20, Phonons, Vibrational Modes Atom vibrations in the harmonic approximation are solution of M d2 u (ω,x) dt 2 = y L ω ;y x K ω (x, y) ( u (ω,x) u (ω,y) ) M= atomic mass, u (ω,x) displacement vector of the atom located at x L ω, K ω (x, y) is the matrix of spring constants. The density of vibrational modes (IDVM) is the IDS of K 1/2 ω.
46 Seoul National University March20, Gap Labels Theorem 5 The value of the IDS or of the IDVM on gaps is a linear combination of the occurrence probabilities of finite patches with integer coefficients. The proof goes through the group of K-theory of the hull. The result is model independent. The abstract result goes back to 1982 (J.B). In 1D, proved in 1993 (JB). Recent proof in any dimension for aperiodic, repetitive, aperiodic tilings by Kaminker-Putnam, Benameur & Oyono-Oyono, JB- Bendetti-Gambaudo in 2001.
47 Seoul National University March20, IV - Branched Oriented Flat Riemannian Manifolds
48 Seoul National University March20, Laminations and Boxes A lamination is a foliated manifold with C -structure along the leaves but only finite C 0 -structure transversally. The Hull of a Delone set is a lamination with R d -orbits as leaves. If L is a FLC, repetitive, Delone set, with Hull Ω a box is the homeomorphic image of φ : (ω, x) F U t x ω Ω if F is a clopen subset of the transversal, U R d is open and t denotes the R d -action on Ω.
49 Seoul National University March20, A quasi-partition is a family (B i ) n i=1 of boxes such that i B i = Ω and B i B j =. Theorem 6 The Hull of a FLC, repetitive, Delone set admits a finite quasi-partition. It is always possible to choose these boxes as φ(f U) with U a d-rectangle.
50 Seoul National University March20, Branched Oriented Flat Manifolds Flattening a box decomposition along the transverse direction leads to a Branched Oriented Flat manifold. Such manifolds can be built from the tiling itself as follows Step 1: 1. X is the disjoint union of all prototiles; 2. glue prototiles T 1 and T 2 along a face F 1 T 1 and F 2 T 2 if F 2 is a translated of F 1 and if there are x 1, x 2 R d such that x i + T i are tiles of T with (x 1 + T 1 ) (x 2 + T 2 ) = x 1 + F 1 = x 2 + F 2 ; 3. after identification of faces, X becomes a branched oriented flat manifold (BOF) B 0.
51 Seoul National University March20, Branching -
52 Seoul National University March20, Vertex branching for the octagonal tiling -
53 Seoul National University March20, Step 2: 1. Having defined the patch p n for n 0, let L n be the subset of L of points centered at a translated of p n. By repetitivity this is a FLC repetitive Delone set too. Its prototiles are tiled by tiles of L and define a finite family P n of patches. 2. Each patch in T P n will be collared by the patches of P n 1 touching it from outside along its frontier. Call such a patch modulo translation a a collared patch and P c n their set. 3. Proceed then as in Step 1 by replacing prototiles by collared patches to get the BOF-manifold B n. 4. Then choose p n+1 to be the collared patch in P c n containing p n.
54 Seoul National University March20, A collared patch -
55 Seoul National University March20, Step 3: 1. Define a BOF-submersion f n : B n+1 B n by identifying patches of order n in B n+1 with the prototiles of B n. Note that D f n = Call Ω the projective limit of the sequence fn+1 B n+1 f n B n f n 1 3. X 1, X d are the commuting constant vector fields on B n generating local translations and giving rise to a R d action t on Ω. Theorem 7 The dynamical system (Ω, R d, t) = lim (B n, f n ) obtained as inverse limit of branched oriented flat manifolds, is conjugate to the Hull of the Delone set of the tiling T by an homemorphism.
56 Seoul National University March20, V - Cohomology and K-Theory
57 Seoul National University March20, Čech Cohomology of the Hull Let U be an open covering of the Hull. If U U, F (U) is the space of integer valued locally constant function on U. For n N, the n-chains are the element of C n (U), namely the free abelian group generated by the elements of F (U 0 U n ) when the U i varies in U. A differential is defined by d : C n (U) C n+1 (U) n+1 n d f ( U i ) = ( 1) j f ( U i ) i=0 j=0 i:i j
58 Seoul National University March20, This defines a complex with cohomology Ȟ n (U, Z). cohomology group of the Hull Ω is defined as The Čech Ȟ n (Ω, Z) = lim U Ȟ n (U, Z) with ordering given by refinement on the set of open covers. Thanks to properties of the cohomology, if fn is the map induced by f n on the cohomology Ȟ n (Ω, Z) = lim n (Ȟn (B n, Z), fn )
59 Seoul National University March20, Examples J. E. Anderson, I. Putnam, Ergodic Theory Dynam. Systems, 18, (1998), L. Sadun, Topology of Tiling Spaces. AMS (2008)
60 Seoul National University March20, Examples J. E. Anderson, I. Putnam, Ergodic Theory Dynam. Systems, 18, (1998), L. Sadun, Topology of Tiling Spaces. AMS (2008) Fibonacci: divides R into intervals a, b of length 1, σ = ( 5 1)/2 according to the substitution rule a ab, b a. Then H 0 = Z, H 1 = Z 2
61 Seoul National University March20, The Anderson-Putnam complex for the Fibonacci tiling -
62 Seoul National University March20, Examples J. E. Anderson, I. Putnam, Ergodic Theory Dynam. Systems, 18, (1998), L. Sadun, Topology of Tiling Spaces. AMS (2008) Fibonacci: divides R into intervals a, b of length 1, σ = ( 5 1)/2 according to the substitution rule a ab, b a. Then H 0 = Z, H 1 = Z 2
63 Seoul National University March20, Examples J. E. Anderson, I. Putnam, Ergodic Theory Dynam. Systems, 18, (1998), L. Sadun, Topology of Tiling Spaces. AMS (2008) Fibonacci: divides R into intervals a, b of length 1, σ = ( 5 1)/2 according to the substitution rule a ab, b a. Then H 0 = Z, H 1 = Z 2 Thue-Morse: substitution a ab, b ba H 0 = Z, H 1 = Z[1/2] Z
64 Seoul National University March20, Examples J. E. Anderson, I. Putnam, Ergodic Theory Dynam. Systems, 18, (1998), L. Sadun, Topology of Tiling Spaces. AMS (2008) Fibonacci: divides R into intervals a, b of length 1, σ = ( 5 1)/2 according to the substitution rule a ab, b a. Then H 0 = Z, H 1 = Z 2 Thue-Morse: substitution a ab, b ba H 0 = Z, H 1 = Z[1/2] Z Penrose 2D: H 0 = Z, H 1 = Z 5, H 2 = Z 8
65 Seoul National University March20, Examples J. E. Anderson, I. Putnam, Ergodic Theory Dynam. Systems, 18, (1998), L. Sadun, Topology of Tiling Spaces. AMS (2008) Fibonacci: divides R into intervals a, b of length 1, σ = ( 5 1)/2 according to the substitution rule a ab, b a. Then H 0 = Z, H 1 = Z 2 Thue-Morse: substitution a ab, b ba H 0 = Z, H 1 = Z[1/2] Z Penrose 2D: H 0 = Z, H 1 = Z 5, H 2 = Z 8 Chair tiling: H 0 = Z, H 1 = Z[1/2] Z[1/2], H 2 = Z[1/4] Z[1/2] Z[1/2]
66 Seoul National University March20, Other Cohomologies Longitudinal Cohomology (Connes, Moore-Schochet) Pattern-equivariant cohomology (Kellendonk-Putnam, Sadun) PV-cohomology (Savinien-Bellissard) In maximal degree the Čech Homology does exists. It contains a natural positive cone isomorphic to the set of positive R d -invariant measures on the Hull (Bellissard-Benedetti-Gambaudo).
67 Seoul National University March20, Cohomology and K-theory The main topological property of the Hull (or tiling psace) is summarized in the following Theorem 8 (i) The various cohomologies, Čech, longitudinal, patternequivariant and PV, are isomorphic. (ii) There is a spectral sequence converging to the K-group of the Hull with page 2 given by the cohomology of the Hull. (iii) In dimension d 3 the K-group coincides with the cohomology.
68 Seoul National University March20, Conclusion 1. Tilings can be equivalently be represented by Delone sets or point measures. 2. The Hull allows to give tilings the structure of a dynamical system with a transversal. 3. This dynamical system can be seen as a lamination or, equivalently, as the inverse limit of Branched Oriented Flat Riemannian Manifolds.
69 Seoul National University March20, The Čech cohomology is equivalent to the longitudinal one, obtained by inverse limit, to the pattern-equivariant one or to the Pimsner cohomology are equivalent Cohomology of the Hull. The K-group of the Hull can be computed through a spectral sequence with the cohomology in page In maximum degree, the Homology gives the family of invariant measures and the Gap Labelling Theorem.
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