Substitution Tilings and Separated Nets with Similarities to the Integer Lattice

0.17 MB·English

by  Yaar Solomon

Substitution Tilings and Separated Nets with
9
0 Similarities to the Integer Lattice
0
2

n Yaar Solomon
a
J
18.01.2009

8
1

]
G Abstract
M We show that any primitive substitution tiling of R2 createsa sep-
. arated net which is biLipschitz to Z2. Then we show that if H is a
h primitivePisotsubstitutioninRd,foreveryseparatednetY,thatcor-
t
a responds to some tiling τ X , there exists a bijection Φ between
H
m ∈
Y and the integer lattice such that sup Φ(y) y < . As a
y∈Y k − k ∞
[ corollarywe get that we havesuch a Φ for anyseparatednet that cor-

respondstoaPenroseTiling. TheproofsrelyonresultsofLaczkovich,
2
v and Burago and Kleiner.
5
2
2 1 Introduction
5
.
0 Definition. A set Y Rd is called a separated net, or a Delone set, if there
1 ⊆
exist constants R,r > 0 such that every ball of radius R intersects Y and
8
0 every ball of radius r contains at most one point of Y.
:
v
Definition 1.1. Let Y and Y be separated nets. We say that a mapping
i 1 2
X
Φ : Y Y is biLipschitz if there exists a constant C 1 such that for
1 2
r → ≥
every y,y Y we have
a ′ 1
∈
1
y y Φ(y) Φ(y ) C y y ,
′ ′ ′
C · − ≤ − ≤ · −
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
where is some ((cid:13)any) n(cid:13)orm(cid:13)on Rd. Φ is ca(cid:13)lled a b(cid:13)ounded(cid:13)displacement if
k·k
sup Φ(y) y < .
k − k ∞
y Y1
∈
Consider two equivalence relations on the set of all separated nets. In
one, two nets are equivalent if there exists a biLipschitz bijection between
1
them. In the other, the relation holds if there is a bijection which is also a
bounded displacement. Since we are dealing with functions between sepa-
rated nets, it is easy to verify that the second relation refines the first. A
natural question is: Is every separated net in Rd biLipschitz to Zd? This
question was firstposed by Gromov in [Gr93]. Itwas answered negatively in
1998 by McMullen [McM98] and also independently by Burago and Kleiner
[BK98]. Their results imply, in particular, that there are separated nets in
Rd which are not a bounded displacement of Zd, not even after rescaling.
Inthispaperwedealwithseparatednetswhichareobtained fromtilings
of Euclidean spaces. When a tiling of Rd is given, by placing one point in
each tile, and keeping the minimal distance property, one gets a separated
net. Since we are studying the equivalence classes under bounded displace-
ment, the positions of the points in the tiles does not matter. In particular
a tiling of Rd gives rise to a separated net; more precisely, an equivalence
class of nets, in both of the above senses. Our main objective is to prove
the two following theorems:
Theorem 1.2. Any separated net that corresponds to a primitive substitu-
tion tiling of R2 is biLipschitz to Z2.
Definition 1.3. Let H be a primitive substitution in Rd and denote by λ
2
an eigenvalue of A which is second in absolute value (see Definitions 2.1,
H
2.2, 2.5). If λ < 1 we say that H is a Pisot substitution.
2
| |
Theorem 1.4. Let H be aPisot substitution inRd. Then foreverysubstitu-
tion tiling of H there exists a constant β and bounded displacement between
the corresponding separated net Y and β Zd.
·
Here we also answer a question of Burago and Kleiner ([BK02], p.2).
Corollary 1.5. Any separated net that is created from a Penrose Tiling is a
bounded displacement of β Z2, for some β >0 (and in particular biLipschitz
·
to Z2).
TheproofsofTheorem 1.2andof Theorem1.4rely onaresultofBurago
and Kleiner [BK02] and a result of Laczkovich [L92] respectively. Both of
these results deal with the difference between the number of tiles in a large
(bounded) set U and the area of U. We use the Perron Frobenius Theorem
and some dynamical properties of the matrix of the tiling in order to get
good estimates for the number of tiles in large sets of certain kind. Then,
by using properties of substitution tilings, we fill U with such sets, that get
smaller and smaller near the boundary of U, and get an estimate for the
number of tiles in U.
2
Acknowledgements: ThisresearchwassupportedbytheIsraelScience
Foundation. This work is a part of the author’s Master thesis under the
supervision of Barak Weiss whose endless support and guidance are deeply
appreciated. The author also wishes to thank Bruce Kleiner for suggesting
the problem about the Penrose Tiling. After [S08] appeared on the web it
was brought to the author’s attention that there is another paper, [DSS95],
where a sketch of proof for Corollary 1.5 is given.
2 Basic Definitions of Tilings
We use standard definitions of tilings. Similar definitions can be found at
[GS87], [Ra99], [Ro04].
A set S Rd is a tile if it is homeomorphic to a closed d-dimensional
⊆
ball. A tiling of a set U Rd is a countable collection of tiles, with pairwise
⊆
disjoint interiors, such that their union is equal to U. We say that two tiles
aretranslation equivalent ifoneisatranslationoftheother. Representatives
of the equivalence classes are called prototiles. A tiling space, X , is the set
of all tilings of Rd by prototiles from . A tiling P of a boundedTset U Rd
T ⊂
is called a patch. We call the set U the support of P and we denote it by
supp(P). We extend the equivalence relation from the last Definition to
patches and denote by the equivalence class representatives.
∗
T
Substitution Tilings
Let ξ > 1 and let = S ,...,S be a set of d-dimensional prototiles.
1 k
T { }
Definition 2.1. A substitution is a mapping H : ξ 1 such that
− ∗
T → T
for every i we have supp(S ) = supp(H(S )). In other words it is a set of
i i
dissection rulesthatshowsushowtodividetheprototiles tootherprototiles
from with a smaller scale. We extend H to the set of all tiles (in a given
T
tiling), to and to any tiling τ X by applying H separately on every
∗
T ∈ T
tile. The constant ξ is called the inflation constant of H.
Definition 2.2. Let H be a substitution defined on . Consider the fol-
T
lowing set of patches:
= (ξH)m(T ) :m N,i = 1,...,k .
i
P { ∈ }
The substitution tiling space X is the set of all tilings of Rd that for every
H
patch P in them there is a patch P such that P is a sub-patch of P .
′ ′
∈ P
Every tiling τ X is a substitution tiling of H.
H
∈
3
Proposition 2.3. If H is a primitive substitution then X = and for
H
6 ∅
every τ X and for every m N there exists a tiling τ X that
H m H
∈ ∈ ∈
satisfies (ξH)mτ = τ.
m
Proof. See [Ro04].
The construction of substitution tilings is explained with more details in
[Ro04]. We denote by H( m)(τ) a tiling τ that satisfies (ξH)mτ = τ.
− ′ ′
Matrices of Substitution
Definition. A matrix A is called positive, and denoted A > 0, if all its
entries are positive. A is called nonnegative, and denoted A 0, if the
≥
entries of A are nonnegative. A is called primitive if there exists an m N
∈
such that Am > 0.
Definition 2.4. For a substitution H, the representative matrix of H is a
k k matrix B = (b ), where b is the number of prototiles which are
H ij ij
×
translation equivalent to S in ξH(S ). We say that H is primitive if B is
i j H
primitive.
The matrix B can be very large sometimes and does not describe ex-
H
actly what we need here. Consider the following equivalence relation on
prototiles: S S if there exists an isometry O such that S = O(S )
i j i j
∼
and H(S ) = O(H(S )) (this is actually a condition on the representatives,
i j
and it is obvious that it is well defined). We call the representatives of the
equivalence classes basic tiles. By this definition, we can also think of H as
a dissection rule on the basic tiles and extend it to tiles, patches and tilings
as before.
Definition 2.5. Denote by T ,...,T ,(n k) the set of the basic tiles.
1 n
{ } ≤
Define the substitution matrix of H to be an n n matrix, A = (a ),
H ij
×
where a is the number of basic tiles in ξH(T ) which are equivalent to T .
ij j i
Example. Let H be the substitution of the Penrose Tiling. There are 20
different prototiles (rotations and reflections are not allowed):
vHvHvHvHvHvH(cid:21)vH)(cid:21))(cid:21))vH(cid:21))(cid:21))(cid:21))vHvH(cid:21))(cid:21))vH(cid:21))vH(cid:21))(cid:21))vH(cid:21))vH(cid:21))(cid:21))vHvH(cid:21))(cid:21))Hv)(cid:21)vH)(cid:21))(cid:21)Hv)(cid:21)vH)(cid:21))Hv(cid:21)HvHvHvHvHv)(cid:21)Hv)(cid:21))(cid:21))(cid:21)Hv)(cid:21))(cid:21) (cid:21))(cid:21))(cid:21))(cid:21))(cid:21))(cid:21))HvvHHvvHHvvHHvvHHvvHHvvHHv(cid:21))(cid:21))(cid:21)Hv)(cid:21))(cid:21))Hv(cid:21))(cid:21))(cid:21))Hv(cid:21))(cid:21))Hv)(cid:21))(cid:21))(cid:21)Hv)(cid:21))(cid:21))(cid:21)Hv)(cid:21))(cid:21)Hv)(cid:21))(cid:21)vHHvvHHvvHHvvHHvvHHvHvvH)(cid:21))(cid:21))(cid:21))(cid:21))(cid:21))(cid:21)
4
Then B is a 20 20 matrix. On the other hand, there are only two different
H
×
basic tiles, with the following dissection rule:
)
(cid:21))
(cid:21) )
(cid:21) )
(cid:21) )
(cid:21) )
(cid:21) )
vvvvvvvvvv(cid:21)(cid:21)(cid:21)v(cid:21)(cid:21)(cid:21)v(cid:21)(cid:21)HHHHHHHHHHHH (cid:21)v(cid:21)(cid:21)v(cid:21)(cid:21)(cid:21)v(cid:21)(cid:21)v(cid:21)(cid:21)(cid:21)v(cid:21)H(cid:21)vHvHvHvHvHvH)))vH)))))))))
2 1
then A = , a 2 2 matrix.
H 1 1 ×
(cid:20) (cid:21)
Denote by e the i’th element of the standard basis of Rn (or Rk). Then
i
A (e ) is the i’th column of A (and the same for B in Rk). Thus, if
H i H H
e represents one tile of type i, multiplying the vector e by these matrices
i i
gives usavector thatrepresentsthenumberofbasictiles (prototiles) ofeach
kind obtained after applying H on the corresponding tile. By linearity, this
is true for any vector in Rn (or Rk). Denote by π : Rk Rn the quotient
→
map that defines the relation . Then by the definition of , ker(π) is
∼ ∼
B -invariant and the following diagram commutes
H
Rk BH // Rk
π π
(cid:15)(cid:15) (cid:15)(cid:15)
Rn AH // Rn
Hence,itiseasytoverifythattheeigenvaluesofB ,thathasaneigenvector
H
v / ker(π), are also eigenvalues of A .
H
∈
3 Properties of Substitution
We denote by Rn the set of all nonnegative vectors in Rn. For a finite set P
+
we denote by #P the number of elements of P. We also use the notations
µ (), and ford-dimensional Lebesguemeasure, themaxnormand
d · k·k k·k2
the Eucli∞dean norm in Rn respectively.
Definition. Let λ ,...,λ be the eigenvalues of a matrix A. The spectral
1 n
radius of A is ρ(A) = max λ . For a nonnegative primitive matrix A, an
i i
{| |}
eigenvalueλthatsatisfies λ = ρ(A)iscalledaPerron Frobenius eigenvalue.
| |
5
Let H be a primitive substitution with an inflation constant ξ and let
τ X . Denote by T ,...,T the set of d-dimensional basic tiles. We
0 H 1 n
∈ { }
denote by λ the Perron Frobenius eigenvalue of A and let λ be an eigen-
1 H 2
value which is second in absolute value. For every m N we denote τ =
m
∈
H( m)(τ ), these are substitution tilings with basic tiles ξmT ,...,ξmT .
− 0 1 n
{ }
Then for every patch P of τ , Hm(P) is a patch of τ and supp(Hm(P)) =
m 0
supp(P). We denote by t(m) the number of tiles from Hm(P) which are
i
equivalent to T .
i
Our main objective in this section is to prove the following proposition:
Proposition3.1. LetH beaprimitivesubstitution, thenthere areconstants
a ,a and C , that depend only on H, such that for every 0 < ǫ < λ λ
1 2 2 1 2
−| |
and for any τ X there exists an N such that for every m N and a
0 H
∈ ≥
patch P τ we have
m
∈
t(m)(a C δm) #P t(m)(a +C δm)
(1) 1 1− 2 ≤ ≤ 1 1 2
t(m)(a C δm) µ (V) t(m)(a +C δm),
1 2− 2 ≤ d ≤ 1 2 2
where V = supp(P) and
λ +ǫ
2
(2) δ = | | < 1.
λ
1
We start with some notions of matrix theory. See also [BP79], [H07].
Proposition 3.2. Let H be a primitive substitution with an inflation con-
stant ξ and let T ,...,T be the set of d-dimensional basic tiles, then
1 n
{ }
(a) ρ(A ) is an eigenvalue of A , with an algebraic multiplicity one, and
H H
the associate eigenvector is positive.
(b) If v > 0 is an eigenvector of A then v corresponds to ρ(A ).
H H
(c) Denote by (v ,...,v ) a basis of generalized eigenvectors of A with
1 n H
v = 1 for all i, where v is the eigenvector that corresponds to
k ik2 1
ρ(A ). Then sp v ,...,v Rn = 0 .
H { 2 n}∩ + { }
(d) In addition, ξd = ρ(A ), and in particular ρ(A ) >1.
H H
Proof. (a) ThisisthewellknownPerronFrobeniusTheorem,alsosee[Q87],
p.91.
(b) See [BP79] Theorem 2.1.4.
6
(c) This is a well known fact, see [H07], 26.1.4.(d).
s
1
.
(d) Consider the vector v =  ..  where si is the area of Ti. Since the
s
n
 
substitution divides everyTito smaller basic tiles, by using Definition
2.5 one can easily show that ξdv = At v, where At is the transpose
H H
matrix of A . By (b) the proof is complete.
H
Let A 0 be a primitive matrix. We use the notations λ and λ as
1 2
≥
before. We also denote by v the eigenvector of λ with v = 1 and by
1 1 k 1k2
W the generalized eigenspace of the other eigenvalues. For a vector u Rn
∈
we write u= β (u)v +β (u)w where w W with w = 1.
1 1 2 u u ∈ k uk2
Proposition 3.3. For a primitive matrix A 0 there exists a constant
≥
C > 0, that depends only on A, such that for every ǫ > 0 there exists an N
such that for every m N and a vector u Rn we have
≥ ∈ +
Amu
(3) v δm,
β (u)λm − 1 ≤
(cid:13) 1 1 (cid:13)
(cid:13) (cid:13)∞
where δ as in (2). (cid:13) (cid:13)
(cid:13) (cid:13)
Proof. We denote K = v Rn : v = 1 and consider the restrictions of
{ ∈ + k k2 }
β and β to K. By Proposition 3.2 (c) we have β (u) > 0 for every u K.
1 2 1
∈
By the compactness of K we denote
α = min β (u) > 0 and α = max β (u) > 0.
1 1 2 2
u K{ } u K{| |}
∈ ∈
Denote C = α2, then C > 0 and for every u Rn we have:
′ α1 ′ ∈ +
u β ( u )
|ββ2((uu))| = k uk2·(cid:12)β2(kuuk2)(cid:12) ≤ αα2 = C′.
1 k k2·(cid:12)(cid:12) 1 kuk2 (cid:12)(cid:12) 1
Notice that there is a constant a> 0 such that for every m N and w W
∈ ∈
with w = 1 we have Am(w) a( λ +ǫ)m. Then
k k2 k k ≤ | 2|
∞
Amu Am(β (u)v +β (u)w ) β (u)λmv
v = 1 1 2 u − 1 1 1
β (u)λm − 1 β (u)λm
(cid:13) 1 1 (cid:13) (cid:13) 1 1 (cid:13)
(cid:13) (cid:13)∞ (cid:13) (cid:13)∞
(cid:13) Am(β (u)w (cid:13)) (cid:13) β (u) Am(w ) a(λ +ǫ(cid:13))m
=(cid:13) β (2u)λmu(cid:13) =(cid:13)| 2 |β·(ku)λm u k∞ ≤ C′· | 2λ|m (cid:13) ,
(cid:13) 1 1 (cid:13) 1 1 1
(cid:13) (cid:13)∞
which co(cid:13)mpletes the pr(cid:13)oof for C = C′a.
(cid:13) (cid:13)
7
Proposition 3.4. There are constants C ,c ,...,c > 0, that depend only
1 2 n
on H, such that for every 0 < ǫ < λ λ there exists an N such that
1 2 1
−| |
every m N satisfies
1
≥
(m)
t
(4) i c C δm
i 1
(cid:12)t(m) − (cid:12)≤
(cid:12) 1 (cid:12)
(cid:12) (cid:12)
for every patch P of τ , whe(cid:12)re δ as in(cid:12)(2).
m (cid:12) (cid:12)
Proof. By Proposition 3.2, λ > 1 and it has an associated eigenvector
1
1
c
2
v1 =  ...  (c1 = 1) with ci > 0 for i = 2,...,n. Denote v1′ = kvv11k2. Fix
 
c 
n
 
0 < ǫ< λ λ , then there is an N such that for every m N and u Rn
1−| 2| ≥ ∈ +
we have (3) with v instead of v .
1′ 1
We pick N N such that every m N satisfies
1 1
≥ ≥
1
(5) C v δm .
k 1k2 ≤ 2
For an arbitrary m N and a patch P of τ , consider the vector
1 m
≥
(0)
t
1
u=  ... , where t(i0) is the number of tiles from P which are equivalent to
(0)
t
 n 
ξmT. Obviously u Rn r 0 , then u satisfies (3) with v instead of v .
i ∈ + { } 1′ 1
Hence
v Amu
k 1k2· H v C v δm.
α (u)λm − 1 ≤ k 1k2
(cid:13) 1 1 (cid:13)
(cid:13) (cid:13)∞
(cid:13) b(m) (cid:13)
(cid:13) 1 (cid:13)
If we denote kvα11k(2u·)AλmHm1u =  ...  then for i = 2,...,n we have
(m)
b
 n 
 
(6) b(m) c C v δm and b(m) 1 C v δm.
i − i ≤ k 1k2 1 − ≤ k 1k2
(cid:12) (cid:12) (cid:12) (cid:12)
In particu(cid:12)lar, by (5(cid:12)), 1 b(m) 11, for ev(cid:12)ery m (cid:12)N . Notice that by the
(cid:12) (cid:12) 2 ≤ 1 ≤ 2 (cid:12) ≥(cid:12) 1
definitions of u, P and A we have
H
(m)
t
1
(7) AmHu=  ... .
(m)
t
 n 
 
8
Therefore for i = 2,...,n we have
(m) (m)
ti c (=7) bi c 1 b(m) c + c b(m)c
(cid:12)t(m) − i(cid:12) (cid:12)b(m) − i(cid:12)≤ b(m) i − i i− 1 i
(cid:12) 1 (cid:12) (cid:12) 1 (cid:12) 1 (cid:16)(cid:12) (cid:12) (cid:12) (cid:12)(cid:17)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(5),(6) (cid:12) (cid:12)
2C v (1+c ) δm C δm,
≤ k 1k2 i · ≤ 1·
where C = 2C v (1+max c ), as required.
1 k 1k2 i{ i}
We denote by s ,...,s the areas of T ,...,T respectively. Define
1 n 1 n
{ }
n n
a
1
(8) a = c , a = c s and α =
1 i 2 i i
a
2
i=1 i=1
X X
Proof of Proposition 3.1. Let 0 < ǫ < λ λ . By Proposition 3.4 there
1 2
−| |
exists an N = N such that (4) holds for every m N and a patch P in
1
≥
τ , for some constant C . Then for i= 2,...,n we have
m 1
t(m)(c C δm) t(m) t(m)(c +C δm)
1 i− 1· ≤ i ≤ 1 i 1·
and
t(m)(c s C δms ) t(m)s t(m)(c s +C δms ).
1 i i− 1· i ≤ i i ≤ 1 i i 1· i
Therefore
n n n
t(m) c nC δm t(m) t(m) c +nC δm
1 i− 1· ≤ i ≤ 1 i 1·
! !
i=1 i=1 i=1
X X X
n n n n n
t(m) c s C δm s t(m)s t(m) c s +C δm s .
1 i i− 1· i ≤ i i ≤ 1 i i 1· i
! !
i=1 i=1 i=1 i=1 i=1
X X X X X
Thus, according to (8), for C = max C n,C n s we get (1) as re-
2 { 1 1 i=1 i}
quired.
P
4 The Main Results
We prove Theorem 1.2 by showing that substitution tilings create separated
nets that satisfy the conditions of the following theorem:
9
Theorem 4.1 (Burago and Kleiner [BK02]). Let Y be a separated net in
R2. For a real number α > 0 and a square B with integer coordinates define:
α µ (B) #(B Y)
2
e (B)=max · , ∩
α
#(B Y) α µ (B)
(cid:26) ∩ · 2 (cid:27)
E (2i)=sup e (B): B as above with an edge of length 2i .
α α
If there exists an α(cid:8)> 0 such that the product ∞j=1Eα(2j) converg(cid:9)es, then
Y is biLipschitz to Z2.
Q
Proof of Theorem 1.2. ItsufficestoshowthatthereareconstantsC ,k > 0
1 1
and ω < 1 such that for every square B with an edge of length 2j = k k
1
≥
we have
(9)
α µ (B) #(B Y) α µ (B) #(B Y)
| · 2 − ∩ | C ωj and | · 2 − ∩ | C ωj.
1 1
#(B Y) ≤ · α µ (B) ≤ ·
2
∩ ·
Then we get, for all large enough j,
E (2j) 1 C ωj,
α 1
− ≤ ·
(cid:12) (cid:12)
which implies the convergence of the product.
(cid:12) (cid:12)
By Proposition 3.2 we can pick an ǫ > 0 such that λ > λ +ǫ. By
1 2
| |
Proposition 3.1 there is an N such that for every m N and a patch P
1 1
≥
in τ , (1) holds. Let N N such that for every m N we have
m 1
≥ ≥
1
(10) C δm min a ,a (a ,a as in (8),δ as in (2)).
2 1 2 1 2
· ≤ 2 { }
We pick k = ξ2N.
′
Let B be an arbitrary square in R2 with an edge of length k k . Let
′
≥
m N such that ξ2m k < ξ2m+2, then m N. Consider the patch
∈ ≤ ≥
P = T τ :T B ,
m
{ ∈ ⊆ }
then P satisfies (1), where V = supp(P) as before.
Let R and r be constants such that every ball of diameter R contains a
tile of τ and every tile of τ contains a cube of area r. From the definition
0 0
of P, for every x B that satisfies d(x,∂B) R ξm, the tile of τ that
m
∈ ≥ ·
covers x must be in P. Then V contains a square with an edge of length
k 2R ξm. Since R ξm R √k, V contains a square with an edge of
− · · ≤ ·
length k 2R √k. If so, there is a k such that for every k k we have
′′ ′′
− · ≥
10