• No results found

Cut-and-Project Schemes for Pisot Family Substitution Tilings

N/A
N/A
Protected

Academic year: 2020

Share "Cut-and-Project Schemes for Pisot Family Substitution Tilings"

Copied!
10
0
0

Loading.... (view fulltext now)

Full text

(1)

Cut-and-Project Schemes for Pisot Family

Substitution Tilings

Jeong-Yup Lee1,*ID, Shigeki Akiyama2and Yasushi Nagai3

1 Department of Mathematics Education, Catholic Kwandong University, Gangneung, Gangwon 210-701,

Korea and KIAS, 85 Hoegiro, Dongdaemun-gu, Seoul, 02455, Korea; jylee@cku.ac.kr

2 Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, Japan (zip:305-8571);

akiyama@math.tsukuba.ac.jp

3 Montanuniversität, Department Mathematik und Informationstechnologie, Lehrstuhl für Mathematik und

Statistik, Franz Josef Strasse 18, A-8700 Leoben, Austria; yasushi.nagai@unileoben.ac.at

* Correspondence: jylee@cku.ac.kr; Tel.: +82-10-7726-8606 Version September 19, 2018 submitted to Preprints

Abstract: We consider Pisot family substitution tilings inRdwhose dynamical spectrum is pure 1

point. There are two cut-and-project schemes(CPS) which arise naturally: one from the Pisot family

2

property and the other from the pure point spectrum respectively. The first CPS has an internal

3

spaceRmfor some integerm∈Ndefined from the Pisot family property, and the second CPS has 4

an internal spaceHwhich is an abstract space defined from the property of the pure point spectrum.

5

However it is not known how these two CPS’s are related. Here we provide a sufficient condition

6

to make a connection between the two CPS’s. In the case of Pisot unimodular substitution tiling in

7

R, the two CPS’s turn out to be same due to [5, Remark 18.5]. 8

Keywords:Pisot substitution tilings; pure point spectrum; regular model set; algebraic coincidence

9

0. Introduction

10

There has been a lot of study on Pisot substitution sequences or Pisot family substitution tilings

11

onRdwhich characterizes the property of pure point spectrum (see [2] and therein). There are two 12

natural cut-and-project schemes(CPS) arising in this study. One CPS is constructed with an Euclidean

13

internal space using the Pisot property [5]. We extend the idea of constructing the CPS from the Pisot

14

property to the Pisot family property. The other CPS is made by constructing an abstract internal

15

space from the property of pure point spectrum [4]. These two CPS’s got developed independently

16

from different aims of study. It is not known yet if these two CPS’s have any relation to each other.

17

Here we would like to provide how these two CPS’s are related showing that two internal spaces are

18

basically isomorphic to each other in Theorem3.1.

19

In this paper we are mainly interested in primitive substitution tilings inRdwith pure point 20

spectrum. It is shown in [17] that primitive substitution tilings with pure point spectrum always

21

have finite local complexity (FLC). So it is not necessary to make an assumption of FLC in the

22

consideration of pure point spectrum.

23

In section1, we visit the basic definitions of the terms that we use. In section2, we construct a

24

natural CPS which arises from the property of Pisot family substitution. In section3, we recall the

25

other CPS constructed from the property of pure point spectrum. We show in Theorem3.1that the

26

two CPS’s are closely related showing that there is an isomorphism between two internal spaces of

27

the CPS’s under certain model set condition. In section4, we raise a few questions for later study.

28

(2)

1. Preliminary

29

1.1. Tilings

30

We begin with a set of types (or colours){1, . . . ,κ}, which we fix once and for all. AtileinRd 31

is defined as a pairT= (A,i)whereA=supp(T)(the support ofT) is a compact set inRd, which 32

is the closure of its interior, andi=l(T)∈ {1, . . . ,κ}is the type ofT. We letg+T= (g+A,i)for

33

g∈Rd. We say that a setPof tiles is apatchif the number of tiles inPis finite and the tiles ofPhave 34

mutually disjoint interiors. A tiling ofRdis a setT of tiles such thatRd=S{supp(T):T∈ T }and 35

distinct tiles have disjoint interiors. Given a tilingT, a finite set of tiles ofT is calledT-patch. We

36

always assume that any twoT-tiles with the same colour are translationally equivalent (hence there

37

are finitely manyT-tiles up to translations).

38

We will make use of the following notation:

39

F+r:={x ∈Rd: dist(x,F)≤r} and F−r :={x ∈F: dist(x,∂F)≥r}. (1.1)

Avan Hove sequenceforRdis a sequenceF ={Fn}n≥1of bounded measurable subsets ofRd 40

satisfying

41

lim

n→∞Vol((∂Fn)

+r)/Vol(Fn) =0, for allr>0. (1.2)

1.2. Deloneκ-sets

42

Recall that a Delone set is a relatively dense and uniformly discrete subset inRd. We say that 43

Λ= (Λi)i≤κis aDeloneκ-setinRdif eachΛiis Delone and supp(Λ):= Sκ

i=1Λi ⊂Rdis Delone.

44

A Deloneκ-setΛ= (Λi)i≤κ is calledrepresentable(by tiles) if there exist tilesTi= (Ai,i),i≤κ, 45

so that{x+Ti:x∈Λi,i≤κ}is a tiling ofRd, that is,Rd=Si≤κ S

x∈Λi(x+Ai), and the sets in this

46

union have disjoint interiors.

47

1.3. Substitutions

48

Definition1. LetA={T1, . . . ,Tκ}be a finite set of tiles inRdsuch thatTi= (Ai,i); we will call them prototiles. Denote byPAthe set of patches made of tiles each of which is a translate of one ofTi’s. We say thatω:A → PAis atile-substitution(or simplysubstitution) with expansive mapφif there exist finite setsDijRdfori,j

κ, such that

ω(Tj) ={u+Ti: u∈ Dij, i=1, . . . ,κ} (1.3)

with

49

φAj =

κ [

i=1

(Dij+Ai) forj≤κ. (1.4)

Here all sets in the right-hand side must have disjoint interiors; it is possible for some of theDijto be

50

empty. Thesubstitutionκ×κmatrixSis defined byS(i,j) =#Dij. IfSm>0 for somem∈N, we say 51

that the substitution tilingT isprimitive.

(3)

A set of algebraic integersΘ={θ1,· · ·,θr}is aPisot familyif for any 1≤ j≤r, every Galois

53

conjugateγofθj, with|γ| ≥1, is contained inΘ. Forr=1, withθ1real and|θ1|>1, this reduces to

54

|θ1|being a real Pisot number, and forr=2, withθ1non-real and|θ1| >1, toθ1being a complex

55

Pisot number. We say thatT is aPisot substitution tilingif the expansive mapφ : Rd → Rd is a 56

Pisot expansive factorλ, and aPisot family substitution tilingif the eigenvalues of the expansive map

57

φ: Rd→Rdform a Pisot family. 58

1.4. Cut and project scheme

59

Definition2. Acut and project scheme(CPS) consists of a collection of spaces and mappings as follows;

60

Rd ←−π1 Rd×H −→π2 H S

e

L

(1.5)

whereRdis a real Euclidean space,His a locally compact Abelian group,π1andπ2are the canonical

61

projections,eL⊂Rd×His a lattice, i.e. a discrete subgroup for which the quotient group(Rd×H)/eL 62

is compact,π1|eLis injective, andπ2(eL)is dense inH.

63

For a subsetV⊂H, we denoteΛ(V):={π1(x)∈Rd:x∈eL,π2(x)∈V}. 64

Amodel setinRdis a subsetΛofRdfor which, up to translation,Λ(W◦) ⊂Γ ⊂Λ(W),Wis 65

compact inH,W =W◦ 6=. The model setΛisregularif the boundary∂W =W\WofWis of

66

(Haar) measure 0. We say thatΛ= (Λi)i≤κis amodelκ-set(resp.regular modelκ-set) if eachΛiis a 67

model set (resp. regular model set) with respect to the same CPS.

68

Without loss of generality, we assume thatHis generated by the windowsWi’s, whereΛi = Λ(Wi)for alli≤κ. WhenHsatisfies the following

{t∈ H:t+Wi=Wifor alli≤κ}={0}, (1.6)

we say that the windowsWi’s haveirredundancy.

69

1.5. Pure point spectrum

70

LetXT be the collection of all primitive substitution tilings inRdeach of whose clusters is a translate of a T-patch. We give a usual metric δon tilings in such a way that two tilings are close if there is a large agreement on a big area with small shift (see [15,24,26]). Then XT = {−h+T :h∈Rd}where the closure is taken in the topology induced by the metricδ. We have a natural action ofRdon the dynamical hullXT ofT by translations and get a topological dynamical system(XT,Rd). Let(XT,µ,Rd)be a measure preserving dynamical system with a unique ergodic measureµ. We consider the associated group of unitary operators{Tx}x∈Rd onL2(XT,µ):

Txg(T0) =g(−x+T0).

Everyg∈L2(XT,µ)defines a function onRdbyx7→ hTxg,gi. This function is positive definite on 71

Rd, so its Fourier transform is a positive measureσgonRdcalled thespectral measurecorresponding 72

tog. The dynamical system(XT,µ,Rd)is said to havepure point spectrumifσgis pure point for every

(4)

g∈L2(XT,µ). We also say thatT has pure point spectrum if the dynamical system(XT,µ,Rd)has 74

pure point spectrum.

75

2. Cut-and-project scheme for Pisot family substitution tilings

76

LetT be a primitive substitution tiling onRdwith expansion mapφ. There is a standard way to

77

choose distinguished points in the tiles of primitive substitution tiling so that they form aφ-invariant

78

Delone set. They are calledcontrol points. A tilingT is called a fixed point of the substitutionωif

79

ω(T) =T.

80

Definition3 ([23,28]). LetT be a fixed point of a primitive substitution with expansion mapφ. For each T-tile T, fix a tileγT in the patchω(T); chooseγT with the same relative position for all tiles of the same type. This defines a mapγ:T → T called the tile map. Then define the control point for a tile T∈ T by

{c(T)}= ∞ \

n=0

φ−n(γnT).

The control points have the following properties:

81

(a) T0=T+c(T0)−c(T), for any tilesT,T0of the same type;

82

(b) φ(c(T)) =c(γT), forT∈ T.

83

Control points are also fixed for tiles of any tilingS ∈ XT: they have the same relative position as inT-tiles. Note that the choice of control points is non-unique, but there are only finitely many possibilities, determined by the choice of the tile map. Let

C:=C(T) ={c(T):T∈ T }

be a set of control points of the tilingT inRd. Let

Ξ:=Ξ(T) =

κ [

i=1

(Ci− Ci)

whereCiis the set of control points of tiles of typei. Equivalently,Ξis the set of translation vectors

84

between twoT-tiles of the same type.

85

Let us assume that φ is diagonalizable over C and the eigenvalues of φ are algebraically

86

conjugate with multiplicity one. For a complex eigenvalueλofφ, the 2×2 diagonal block

"

λ 0

0 λ

#

87

is similar to a real 2×2 matrix

88

"

a −b

b a

#

=S−1

"

λ 0

0 λ

#

S, (2.1)

whereλ = a+ib,a,b∈ R, andS = √12

"

1 i 1 −i

#

. So we can assume, by appropriate choice of

89

basis, thatφis diagonal with the diagonal entries equal toλcorresponding to real eigenvalues, and

90

diagonal 2×2 blocks of the form

"

aj −bj bj aj

#

corresponding to complex eigenvaluesaj+ibj.

(5)

Without loss of generality, we can assume thatφis a diagonal matrix.

92

We recall the following theorem. The theorem is not in the form as shown here, but one can

93

readily note that from the proof of [16, Thm.4.1].

94

Theorem 2.1. [16, Thm.4.1] LetT be a primitive substitution tiling onRdwith expansion mapφ. Assume thatT has FLC,φis diagonalizable, and all the eigenvalues ofφare algebraically conjugate with multiplicity one. Then there exists an isomorphismσ:Rd→Rdsuch that

σφ=φσ and k·Z[φ]ασ(C(T))⊂Z[φ]α,

whereα= (1, 1, . . . , 1)∈Rdand k∈Z. 95

Let us assume now thatT has FLC,φis diagonalizable, the eigenvalues ofφare all algebraically conjugate with multiplicity one, and there exists at least one other algebraic conjugate different from eigenvalues ofφ. Suppose thatφhasenumber of real eigenvalues, and f number of 2×2 blocks of the form of complex eigenvalues whered=e+2f. Let all the algebraic conjugates of eigenvalues of φbe real numbersλ1, . . . ,λsand complex numbersλs+1,λs+1, . . . ,λs+t,λs+t. Letm:=s+2tand writeλs+t+i =λs+ifori=1, . . .tfor the convenience. Let us consider a spaceK, where

K:=Rs−e×Ct−f wRm−d.

Let us consider a following map

96

Ψ:Z[φ]ξ→K, (2.2)

P(φ)ξ7→(P(λe+1), . . . ,P(λs),P(λs+f+1), . . . ,P(λs+t)), (2.3)

whereP(x)is a polynomial overZ. Let us construct a new cut and project scheme : 97

Rd ←−π1 Rd×K −→π2 K ∪

L ←− eL −→ Ψ(L)

x ←−[ (x,Ψ(x)) 7−→ Ψ(x),

(2.4)

whereπ1andπ2are canonical projections,L=hCiii≤κ andeL={(x,Ψ(x)):x∈ L}. It is clear to see 98

thatπ1|eLis injective. We now show thatπ2(eL)is dense inKandeLis a lattice inR d×

K. 99

Lemma 4. eL is a lattice inRd×K. 100

Proof. Since Z[φ]α is a free Z-module of rank m and m×m matrix A = (λj−i 1)i,j∈{1,...,m} is

101

non-degenerate by the Vandermonde determinant, the natural embedding combining all conjugates;

102

f :Z[φ]α→Rs×Ct 'Rd×Kgives a lattice f(Z[φ]α)inRd×K. ConsequentlyeLis isomorphic 103

to a freeZ-submodule of f(Z[φ]α)due to the theory of elementary divisors. From Theorem2.1,

104

(6)

claim is shown. The case with complex conjugates can be shown in a similar manner, taking care of

106

embeddingsCtoR2. 107

Lemma 5. Ψ(L) =π2(eL)is dense inK. 108

Proof. As in the proof of Lemma4, we showed thateLis a sub-lattice of f(Z[φ]α), it suffices to prove

thatΨ(Z[φ])is dense inK. We prove the totally real case, i.e.,λi∈Rfor alli. By [27, Theorem 24], Ψ(Z[φ])is dense if

m

i=d+1

xiλij−1∈Z (j=1, . . . ,m)

impliesxi=0 fori=d+1, . . . ,m. The condition is equivalent to

ξA∈Zm

withξ = (xi) = (0, . . . , 0,xd+1, . . . ,xm) ∈ Rmin the terminology of Lemma 4. Multiplying the

109

inverse ofA, we see that entries ofξmust be Galois conjugates. Asξhas at least one zero entry, we

110

obtainξ=0 which showsxi =0 fori=d+1, . . . ,m. In fact, this discussion is using the Pontryagin

111

duality that theΨ :Zm → Rm−dhas a dense image if and only if its dual mapΨb : Rm−d → Tm 112

is injective (see also [18, Chapter II, Section 1], [10] and [1]). The case with complex conjugates is

113

similar.

114

3. Two cut-and-project schemes

115

3.1.φ-topology

116

LetT be a primitive substitution tiling onRdwith expansion mapφ. Define

L:=hCiii≤κ

be the group generated byCi,i≤κ, whereC = (Ci)i≤κis a control point set ofT and

K:={x∈Rd:T +x=T }

be the set of periods ofT. We say thatT admits analgebraic coincidenceif there existM∈Z+and 117

ξ∈ Cifor somei≤κsuch thatξ+QMΞ(T)⊂ Ci. It is known in [11] thatT has pure point spectrum

118

if and only ifT admits an algebraic coincidence.

119

Under the assumption thatT admits an algebraic coincidence, we introduce a topology onLand

120

find a completionHof the topological groupLsuch that the image ofLis a dense subgroup ofH. This

121

enables us to construct a cut and project scheme (CPS) such that each point setCi,i≤κ, arises from

122

the CPS. From the following lemma, we know that the system{α+φnΞ(T) +K:n∈Z+,α∈L}

123

satisfies the topological properties for the groupLto be a topological group ([19], [7] and [20]).

124

Lemma 6. [11, Lemma 4.1] LetT be a primitive substitution tiling with an expansive mapφ. Suppose that

125

T admits an algebraic coincidence. Then the system{φnΞ(T) +K:n

Z+}serves as a neighbourhood 126

base for0∈ L of the topology on L relative to which L becomes a topological group.

(7)

We call the topology on Lwith the neighbourhood base{α+φnΞ(T) +K :n ∈ Z+,α ∈ L}

128

φ-topology. LetLφbe the spaceLwithφ-topology. 129

LetL0 = L/K. From [7, III. §3.4, §3.5] and Lemma6, we know that there exists a complete

130

Hausdorff topological group ofL0, which we denote by H, for whichL0is isomorphic to a dense

131

subgroup of the complete groupH(see [4] and [14]). Furthermore there is a uniformly continuous

132

mappingψ:L→Hwhich is the composition of the canonical injection ofL0intoHand the canonical

133

homomorphism ofLontoL0for whichψ(L)is dense inHand the mappingψfromLontoψ(L)is

134

an open map, the latter with the induced topology of the completionH. One can directly considerH

135

as the Hausdorff completion ofLvanishingK.

136

3.2. Pe-topology 137

We introduce another topology on L which becomes equivalent to φ-topology under the

138

assumption of algebraic coincidence.

139

Let{Fn}n∈Z+ be a van Hove sequence and letT

0,T00be two tilings in

Rd, whereΛ0 = (Λ0i)i≤κ

andΛ00= (Λ00i)i≤κare representable Deloneκ-sets of the tilingsT

0,T00. We define

ρ(T0,T00):= lim n→∞sup

κ

i=1]((Λ0i4Λ00i)∩Fn)

Vol(Fn) . (3.1)

Here4is the symmetric difference operator. LetPe ={x ∈ L: ρ(x+T,T)<e}for eache >0. 140

IfT admits an algebraic coincidence, then, for anye>0,Peis relatively dense [3,4,15,22]. In this 141

case the system{Pe:e>0}serves as a neighbourhood base for 0∈Lof the topology onLrelative 142

to whichLbecomes a topological group. We namePe-topology for this topology onLand denote

143

the spaceLwithPe-topology byLP(see [3,4] forPe-topology under the name of autocorrelation

144

topology).

145

Proposition 7. [11, Prop. 4.6, 4.7] LetT be a primitive substitution tiling. Suppose thatT admits an

146

algebraic coincidence, then the mappingι:x 7→x from Lφonto LPis topologically isomorphic. 147

Remark 8. From Prop.7, LP is topologically isomorphic to Lφ. Thus the completion of LP is

148

topologically isomorphic to the completion Hof Lφ. We will identify the former withH. Thus 149

ϕ:= ψ·ι−1: LP→ His uniformly continuous,ϕ(LP)is dense inH, and the mapping ϕfromLP

150

ontoϕ(LP)is an open map, the latter with the induced topology of the completionH. Therefore we

151

can consider the CPS (1.5) with an internal spaceHwhich is a completion ofLP. Note that sinceT is

152

repetitive,T

e>0Pe=KandK={0}inLφ. 153

We observe thatLPandΨ(L)are all topologically isomorphic when the control point setCis a

154

regular modelκ-set in CPS(2.4).

155

Theorem 3.1. LetT be a primitive Pisot family substitution tiling inRdwith an expansive mapφ. Suppose 156

thatφis diagonalizable, all the eigenvalues ofφare algebraic conjugates with multiplicity one, and there

157

exists at least one algebraic conjugateλof eigenvalues ofφfor which|λ|<1. IfCis a regular modelκ-set

158

in CPS(2.4), then the internal space H which is the completion of Lφwithφ-topology is isomorphic to the 159

(8)

PROOF. Sinceφis an expansive map and satisfies the Pisot family condition, we first note that there

161

is no algebraic conjugateγof eigenvalues ofφwith|γ|=1.

162

We will show that if fort∈L,Ψ(t)is close to 0 inK, thenρ(t+T,T)is close to 0 inH. Since

163

each point setCiis a regular model set by the assumption whereC= (Ci)i≤κ andCi =Λ(Wi)in the

164

CPS (2.4), fort∈L

165

ρ(t+T,T) = lim n→∞sup

κ

i=1](((t+Ci)4 Ci)∩An) Vol(An)

=

κ

i=1

lim n→∞

](((t+Ci)4 Ci)∩An) Vol(An)

=

κ

i=1

(θ(Wi\(Ψ(t) +Wi)) +θ(Wi\(−Ψ(t) +Wi))), (3.2)

whereθis a Haar measure inK(see [21, Thm. 1]). 166

Note that

θ(Wi\(s+Wi)) =θ(Wi)−1Wi∗g1Wi(s)

is uniformly continuous ins ∈ K (see [25, Subsec. 1.1.6]). So ifΨ(t)converges to 0 inK, then 167

ρ(t+T,T)converges to 0 inR. 168

On the other hand, suppose that{tn}is a sequence such thatρ(tn+T,T)→0 asn→∞. Then for eachi≤κ

{θ(Wi\(Ψ(tn) +Wi))}n→0 asn→∞.

Note that for large enoughn,Wi∩(Ψ(tn) +Wi) 6= ∅and soΨ(tn) ∈Wi−Wifor alli≤ κ. Since Wi−Wi is compact,{Ψ(tn)}n has a converging subsequence{Ψ(tnk)}k. For any such sequence definet0∗:=limk→∞Ψ(tnk). Then

θ(Wi\(t0∗+Wi)) =0

and soθ(Wi◦\(t0∗+Wi)) =0 for eachi≤κ. ThusWi◦⊂t0∗+Wiand this impliesWi⊂t0∗+Wi.

169

On the other hand, limk→∞−Ψ(tnk) = −t0

and

θ(Wi◦\(−t0∗+Wi)) = 0. SoWi ⊂ −t0∗+Wi.

170

HenceWi ⊂ t0∗+Wi ⊂ t0∗−t0∗+Wi andWi = t0∗+Wi. This equality is for eachi ≤ κ. Since

171

Kis isomorphic toRm−d, each model setWihas irredundancy. Thust0∗ = 0. So all converging

172

subsequences{Ψ(tnk)}kconverge to 0 and{Ψ(tn)}n →0 asn→∞.

173

This establishes the equivalence of the two topologies. By [7, Prop 5, III §3.3], there exists an

174

isomorphism betweenHontoK. 175

The above theorem shows that the internal spaceHconstructed fromLφwithφ-topology is 176

isomorphic to Euclidean spaceK(i.e.Rm−d). 177

It is known in [5,9], [2, Thm. 3.6] that unimodular irreducible Pisot substitution tilings inRwith 178

pure point spectrum give rise to regular model sets. We give a precise statement below.

179

Theorem 3.2. [5, Remark 18.5] LetT be a primitive substitution tiling inRwith expansion factorβbeing a

180

unimodular irreducible Pisot number. ThenT has pure point spectrum if and only if for any1≤i≤κ, each

181

Ciis a regular model set in CPS (2.4).

(9)

Corollary 9. LetT be a primitive Pisot substitution tiling inRwith an expansion factorβ. Assume that 183

there exists at least one algebraic conjugateλofβfor which|λ| < 1. IfT has pure point spectrum, then

184

the internal space H which is the completion of Lφwithφ-topology can be realized by Euclidean spaceRm−1 185

where m is the degree of the characteristic polynomial ofβ.

186

PROOF. By Theorem3.2, it is known that for a primitive Pisot substitution tiling inR, ifT has pure 187

point spectrum, thenCis a regular modelκ-set in CPS(2.4).

188

4. Further study

189

We are left with the following questions extending Theorem3.1.

190

Question 10. Can we replace the assumption of regular modelκ-set by pure point spectrum? Another words,

191

for a primitive Pisot family substitution tilingT inRdwith an expansion mapφ, does the pure point spectrum 192

ofT imply thatCis a regular modelκ-set with an Euclidean internal space.

193

Question 11. Can the theorem be still extended into the case that the multiplicity of eigenvalues ofφis not

194

one?

195

Author Contributions:Writing – original draft, Jeong-Yup Lee, Shigeki Akiyama and Yasushi Nagai

196

Acknowledgments: The first author would like to acknowledge the support by Local University Excellent

197

Researcher Supporting Project through the Ministry of Education of the Republic of Korea and National Research

198

Foundation of Korea (NRF) (2017078374). She is also grateful to the Korea Institute for Advanced Study (KIAS),

199

where part of this work was done, during her sabbatical year in KIAS. The second author is partially supported

200

by JSPS grants (17K05159, 17H02849, BBD30028). The third author was supported by the project I3346 of the

201

Japan Society for the Promotion of Science (JSPS) and the Austrian Science Fund (FWF).

202

References

203

1. Akiyama, S. Self affine tiling and Pisot numeration system. Number theory and its applications (Kyoto,

204

1997),Dev. Math.1999,2, 7—17,

205

2. Akiyama, S.; Barge, M.; Berthé, V.; Lee, J.-Y.; Siegel, A. On the Pisot substitution conjecture. inMathematics 206

of aperiodic order, eds. J. Kellendonk, D. Lenz, J. Savinien,; Birkhäuser/Springer, Basel, Swiss, 2015; Volume

207

309, pp. 33–72.

208

3. M. Baake and U. Grimm,Aperiodic order, Vol. 2. Crystallography and almost periodicity. With a foreword by 209

Jeffrey C. Lagarias.Encyclopedia of Mathematics and its Applications, 166. Cambridge University Press,

210

Cambridge, 2017.

211

4. Baake, M.; Moody, R. V. Weighted Dirac combs with pure point diffraction,J. Reine Angew. Math.2004,573,

212

61–94.

213

5. Barge, M.; Kwapisz, J. Geometric theory of unimodular Pisot substitutions,Amer. J. Math.2006,128no. 5,

214

1219–1282.

215

6. Berthé, V.; Siegel, A. Tilings associated with beta-numeration and substitutions,Integers2005,5, 46.

216

7. Bourbaki, NElements of mathematics: General Topology, Springer- Verlag, Berlin, 1998; Chapters 5–10.

217

8. Borevich, A. I; Shafarevich,I.R.Number Theory; Academic Press, New York-London, 1966.

218

9. Ito, S.; Rao, H. Atomic surfaces, tilings and coincidences I. Irreducible case,Israel J. Math.,2006,153,

219

129–156.

220

10. Iizuka, S; Akama, Y.; Akazawa, Y. Asymmetries of cut-and-project sets and related tilings,Interdiscip. 221

Inform. Sci.2009,15, no. 1, 99—113.

222

11. Lee, J.-Y. Substitution Delone sets with pure point spectrum are inter-model sets,Journal of Geometry and 223

Physics2007,57, 2263–2285.

(10)

12. Lagarias, J.C.; Wang, Y. Substitution Delone sets,Discrete Comput. Geom.2003,29, 175–209.

225

13. Kenyon, R.; Solomyak, B. On the characterization of expansion maps for self-affine tilings,Discrete Comput. 226

Geom.2010,43, no. 3, 577–593.

227

14. Lee, J.-Y.; Moody, R.V. Characterization of model multi-colour sets,Ann. Henri Poincaré2006,7, 125–143.

228

15. Lee,J.-Y.; Moody, R.V.; Solomyak, B. Consequences of pure point diffraction spectra for multiset substitution

229

systems,Discrete Comp. Geom.2003,29, 525–560.

230

16. Lee, J.-Y.; Solomyak, B. Pisot family self-affine tilings, discrete spectrum, and the Meyer property,Discrete 231

Contin. Dyn. Syst.,2012,32, no. 3, 935–959.

232

17. Lee, J.-Y.; Solomyak, B. On substitution tilings without finite local complexity, Preprint.

233

18. Meyer, Y.Algebraic Numbers and Harmonic Analysis, North-Holland Publishing Co., Amsterdam, 1972.

234

19. Havin, V.P.; Nikolski, N.K.Commutative Harmonic Analysis II, Encyclopaedia of Mathematical Sciences,

235

Springer-Verlag, Berlin, 1998.

236

20. Hewitt, E.; Ross, K.Abstract Harmonic Analysis, Vol. I, Springer-Verlag, 1963.

237

21. Moody, R.V. Uniform Distribution in Model sets,Can. Math. Bulletin2002,45, no. 1, 123–130.

238

22. Moody, R.V.; Strungaru, N. Point sets and dynamical systems in the autocorrelation topology,Can. Math. 239

Bulletin2004,47, 82-–99.

240

23. Praggastis, B. Numeration systems and Markov partitions from self similar tilings,Trans. Amer. Math. Soc. 241

1999,351, 3315–3349.

242

24. Radin, C.; Wolff, M. Space tilings and local isomorphism,Geometriae Dedicata1992,42, 355–360.

243

25. Rudin, W.Fourier Analysis on Groups, Wiley, New York, 1962; reprint, 1990.

244

26. Schlottmann, M. Generalized model sets and dynamical systems, in: Directions in Mathematical 245

Quasicrystals,; Baake, M. and Moody, R.V., CRM Monograph series, AMS, Providence RI, 2000, pp. 143—159.

246

27. C. L. Siegel,Lectures on the Geometry of Numbers, Springer-Verlag, Berlin, 1989.

247

28. W. Thurston,Groups, Tilings, and Finite State Automata, AMS lecture notes, 1989.

References

Related documents

Questions were grouped into eight sections on the (i) overview of the project (name of the initiative, and institutions involved); (ii) characteristics of the initiative (the number

Our contributions are the following: (1) we design a crowdsourcing task for efficiently labeling speech events in classroom videos, (2) we develop a neural network-based

Colon adenocarcinoma; CRC : Colorectal cancer; DFS: Disease-free survival; DM: Distant metastasis; E2F1: E2F transcription factor 1; EGFR: Epidermal growth factor receptor;

Permission to use, copy, modify, and distribute this software and its documentation for any purpose and without fee is hereby granted, provided that the above copyright notice

Note: The figure shows analysis of the impact of the implied contract doctrine on state union membership rates using equation 3, a weighted OLS difference and differences

Though the distribution of bismuth oxide on titanium dioxide was found to depend strongly on particle size and deposition time used, the composite materials exhibited

Water- gated organic thin film transistors (OTFTs) using the hole transporting semiconducting polymer, poly(2,5-bis(3-hexadecylthiophen-2-yl)thieno[3,2-b]thiophene)