Spectral measures and Cuntz algebras

0.31 MB·English

by  Dorin Ervin Dutkay

SPECTRAL MEASURES AND CUNTZ ALGEBRAS
DORINERVIN DUTKAYANDPALLEE.T. JORGENSEN
Abstract. We consider a family of measures µ supported in Rd and generated in the sense of
0
Hutchinson by a finite family of affine transformations. It is known that interesting sub-families
1 of thesemeasures allow for an orthogonal basis in L2(µ) consisting of complex exponentials, i.e., a
0
FourierbasiscorrespondingtoadiscretesubsetΓinRd. Hereweoffertwocomputationaldevicesfor
2
understandingthe interplay between the possibilities for such sets Γ (spectrum) and the measures

n µ themselves. Our computations combine the following three tools: duality, discrete harmonic
Ja analysis, and dynamical systems based on representations of theCuntzC∗-algebras ON.

5
2

]
A Contents
F 1. Introduction 1
.
h 2. Representations associated to minimal words 8
t 3. Representations associated to extreme cycles 10
a
m 4. Beyond cycles 13
[ References 24

1
v
5
6 1. Introduction
5
4 The idea of selfsimilarity is ubiquitous in pure and applied mathematics and has many incarna-
. tions, and even moreapplications. Inthisgenerality themethodoften goes bythenamemulti-scale
1
0 theory, see for example [ZHSS09]. We are faced with a fixed system S, say finite, of transforma-
0 tions. The mappings from S are applied to the data at hand, and we look for similarity. A single
1
measure, say µ, may be transformed with the mappings in S. If µ is a convex combination of the
:
v resulting measures, we say that µ is a selfsimilar measure. Here we aim for a harmonic analysis
i
X of the Hilbert space L2(µ) in the event µ is selfsimilar. However this is meaningful only if further
r restrictions are placed on the setup. For example, we will assume that S consists of affine and
a contractive transformations in Rd for some fixed dimension d; and we will assume that the affine
transformations arise from scaling with the same matrix for the different mappings in S. In this
case µ is the result of an iteration in the small, and its support is a compact subset in Rd. We
explore algorithms for generating Fourier bases of complex exponentials in L2(µ) by iteration in
the large.
2000 Mathematics Subject Classification. 28A80, 42B05, 46C05, 46L89.
Key words and phrases. Spectrum, Hilbert space, fractal, Fourier bases, selfsimilar, iterated function system,
operator algebras.
With partial support by theNational Science Foundation.
1
2 DORINERVINDUTKAYANDPALLEE.T.JORGENSEN
A word about the Cuntz algebras from the title of our paper: These are infinite algebras on
N
O
a finite number of generators, the following relations
N
(1.1) S∗S = δ 1, (i,j 1,...,N ), S S∗ = 1.
i j ij ∈ { } i i
i=1
X
They are intrinsically selfsimilar and therefore ideally serve to encode iterated function systems
(IFSs) from geometric analysis, and their measures µ. At the same time, their representations
offer (in a more subtle way) a new harmonic analysis of the associated L2(µ)-Hilbert spaces. Even
though the Cuntz algebras initially entered into the study of C∗-algebras [Cun77] and physics, in
recent years these same Cuntzalgebras, and their representation, have foundincreasing usein both
pure and applied problems, wavelets, fractals, signals; see for example [Jor06].
Earlier work along these lines include the papers [DJ06a, DJ07, Fug74, HL08a, Jør82, Ped04,
Tao04]; as well as an array of diverse applications [ACHM07, BV05, CM07, DLC09, HL08b,
MZ09, OS05, YALC03]. Recent papers by Barnsley et al [Bar09, BHS05] show that even the
restricted family of affine IFSs suffices for such varied applications as vision, music, Gabor trans-
form processing, and Monte Carlo algorithms. The literature is large and we direct the reader
to [DJ07] for additional references. Our present study suggests algorithms for this problem based
on representations of a certain scale of C∗-algebras (the Cuntz algebras [Cun77]); see also [Jor06].
In addition we shall make use of Hadamard matrices [AdLM02, CHK97, Dit04, GR09, KS93];
number theory and tilings [BL07, FLS09, LL07, MP04]; and of tools from symbolic dynamics
[Den09, Tho05, DFdGtHR04, MM09].
Our starting point is a system (R,B,L) in Rd subject to the conditions from Definitions 1.3, i.e.,
a Hadamard triple. With the given expansive matrix R and the vectors from the finite set B we
build one affine iterated function system (IFS) (1.2), and the associated Hutchinson measure µ .
B
Our objective is to set up an algorithmic approach for constructing orthogonal families of complex
exponentials in L2(µ ), so called Fourier bases.
B
With the third part of the triple (R,B,L), the finite set of vectors from L and the transposed
matrix RT we build a second IFS (1.13), the L-system, now with RT as scaling matrix and the
vectors from L as translations. The set L, together with the transpose matrix RT will be used to
construct the frequencies associated to the Fourier basis. Due to the Hadamard property which
is assumed for the combined system (R,B,L), we then get a natural representation of the Cuntz
algebra (where N = #L = #B) with the generating operators S :l L acting as isometries
N l
O { ∈ }
in L2(µ ), see (1.10) in Proposition 1.4.
B
Our objective is to use this Cuntz-algebra representation in order to recursively construct and
exhibit an orthogonal family of complex exponentials in L2(µ ). For this purpose we introduce
B
a family of minimal and finite cycles C for the L-system, which are extreme in the sense made
precise in Definition 3.1. For each of the extreme cycles C, with the L-Cuntz representation, we
then generate an infinite orthogonal family Γ(C) of complex exponentials in L2(µ ). With the
B
cycle C fixed, we prove that the corresponding closed subspace (C) now reduces the Cuntz-
H
algebra representation, and further that the restricted representation is irreducible. For distinct
minimal cycles we get orthogonal subspaces (C) and corresponding disjoint representations (see
H
Theorem 3.4).
A key tool in the study of these representations of the Cuntz algebra is a transfer operator R
B,L
(Definition 1.7), analogous to a transfer operator used first by David Ruelle in a different context.
SPECTRAL MEASURES AND CUNTZ ALGEBRAS 3
In Corollary 1.13 we show that the reduction of the L-Cuntz representation (1.10) is accounted for
by a family of harmonic functions h for the transfer operator R .
C B,L
Finally we prove that the sum of the functions h equals to the constant function 1 if and only
C
if the union of the sets Γ(C) forms an orthogonal basis of Fourier exponentials in L2(µ ).
B
In section 1, we define the representation of the Cuntz algebra (S ) associated to a Hadamard
l l∈L
pair and we present some computational features of this representation. We show in Proposition
1.12 and Corollary 1.13 how the canonical endomorphism on (L2(µ )), constructed from the
B
B
representation:
α(T) = S TS∗, (T (L2(µ )))
l l ∈ B B
l∈L
X
is connected to the transfer operator associated to the “dual” IFS (τ ) .
l l∈L
In section 2 we revisit the permutative representations defined in [BJ99]. They are needed for
the decomposition of our representation (S ) into irreducible representations. We will see in
l l∈L
section 3, that extreme cycles (Definition 3.1) will generate irreducible atoms in this decomposition
(Theorem 3.4). In dimension 1, we know from [DJ06b] that the extreme cycles offer a complete
description of the picture. However, in higher dimensions, more complicated decompositions might
appear. We analyze these possibilities in section 4, and we illustrate it in Example 4.7.
Definition 1.1. We will denote by e the exponential function
t
e (x) = e2πit·x, (x,t Rd)
t
∈
Let µ be a Borel probability measure on Rd. We say that µ is a spectral measure if there exists
a subset Λ of Rd such that the family E(Λ) := e : λ Λ is an orthonormal basis for L2(µ). In
λ
{ ∈ }
this case Λ is called a spectrum for the measure µ and we say that (µ,Λ) is aspectral pair.
Definition 1.2. We will use the following assumptions throughout the paper. Let R be a d d
×
expansive integer matrix, i.e., all its eigenvalues have absolute value strictly bigger than one, and
let B be a finite subset of Zd, with 0 B. We denote by N the cardinality of B. Define the maps
∈
(1.2) τ (x) = R−1(x+b), (x Rd,b B)
b
∈ ∈
We call (τ ) the (affine) iterated function system (IFS) associated to R and B.
b b∈B
By [Hut81], there exists a unique compact set X called the attractor of the IFS (τ ) such
B b b∈B
that
(1.3) X = τ (X ).
B b B
b∈B
[
In our case, it can be written explicitly
∞
(1.4) X = R−kb : b B for all k 1 .
B k k
∈ ≥
( )
k=1
X
We will use also the notation X(B) for X .
B
There exists a unique Borel probability measure µ such that
B
1
(1.5) µ (E) = µ (τ−1(E)) for all Borel subsets E of Rd.
B N B b
b∈B
X
4 DORINERVINDUTKAYANDPALLEE.T.JORGENSEN
Equivalently
1
(1.6) fdµ = f τ dµ for all bounded Borel function f on Rd.
B b B
N ◦
Z b∈BZ
X
The measure µ is called the invariant measure of the IFS (τ ) . It is supported on X .
B b b∈B B
We say that the measure µ has no overlap if
B
(1.7) µB(τb(XB) τb′(XB)) = 0, for all b = b′ B.
∩ 6 ∈
If the measure µ has no overlap, then one can define the map :X X
B B B
R →
(1.8) (x) = Rx b, if x τ (X ).
b B
R − ∈
The map is well defined µ -a.e. on X .
B B
R
Definition 1.3. Let R be a d d integer matrix, and B and L two subsets of Zd of the same
×
cardinality N, with 0 B and 0 L. We say that (B,L) forms a Hadamard pair if the following
∈ ∈
matrix is unitary:
(1.9) 1 e2πiR−1b·l .
√N b∈B,l∈L
(cid:16) (cid:17)
Throughout,theHilbertspaceconsideredisL2(µ )unlessotherwisespecified. Andweintroduce
B
specificfamilies of operatorsacting there, starting withtheoperator system(S ) in (1.10)below.
l l∈L
This system (S ) will be fixed, and it defines a representation of the Cuntz algebra with
l l∈L N
O
N = #L.
Proposition 1.4. Let (τ ) be as in Definition1.2 and assume the invariant measur µ has no
b b∈B B
overlap. Let L be a subset of Zd with the same cardinality N as B. Define the operators on L2(µ ):
B
(1.10) (S f)(x)= e2πil·xf( x), (x X ,f L2(µ ))
l B B
R ∈ ∈
(i) The operator S is an isometry for all l L.
l
∈
(ii) The operators (S ) form a representation of the Cuntz algebra if and only if (B,L)
l l∈L N
O
forms a Hadamard pair.
Proof. We will need the following lemma:
Lemma 1.5. If the measure µ has no overlap then for all b B and all integrable Borel functions
B
∈
f:
1
(1.11) fdµ = f τ dµ
B b B
N ◦
Zτb(XB) Z
Proof. From the invariance equation
N χ gdµ = χ τ′g τ′dµ .
τb(XB) B τb(XB)◦ b ◦ b B
Z b′∈BZ
X
Since there is no overlap χτb(XB) ◦τb′ is 1 if b = b′, and is 0 if b 6= b′, µB-almost everywhere (since
µ is supported on X ).
B B
This leads to the conclusion. (cid:3)
SPECTRAL MEASURES AND CUNTZ ALGEBRAS 5
We prove (i). Take f L2(µ ), l L. Then, with Lemma 1.5,
B
∈ ∈
e2πil·xf( x)2dµ = f(Rx b)2dµ (x)
B B
| R | | − |
Z b∈BZτb(XB)
X
1
= f(Rτ (x) b)2dµ (x)= f 2dµ .
b B B
N | − | | |
b∈B Z Z
X
This shows that S is an isometry.
l
We prove now (ii). For this, we have to compute S∗. We have, for f,g L2(µ ):
l ∈ B
S f , g = e2πil·xf( x)g(x)dµ (x) = e2πi·xf(Rx b)g(x)dµ (x), and with Lemma 1.5,
l B B
h i R −
Z b∈BZτb(X)
X
1
= e2πi·τb(x)f(x)g(τ (x))dµ (x).
b B
N
b∈BZ
X
This shows that
1
(1.12) (S∗g)(x) = e−2πil·τb(x)g(τ (x)), (x X ,g L2(µ ))
l N b ∈ B ∈ B
b∈B
X
Then
Sl∗Sl′f(x)= N1 e−2πil·τb(x)e2πil′τb(x)f(Rτb(x)−b)= N1 e2πi(l′−l)·R−1b e2πi(l′−l)·R−1xf(x).
!
b∈B b∈B
X X
Therefore Sl∗Sl′ =δl,l′IL2(µB) if and only if the matrix in (1.9) is unitary.
Also
1
A := S S∗f(x)= e2πil·x e−2πil·τb(Rx)f(τ ( x)).
l l N b R
l∈L l∈L b∈B
X X X
If x τ (X ), then τ ( x) = x+b b and we have further
∈ b0 B b R − 0
1
A = f(x+b b ) e2πil·(b−b0).
0
− N
b∈B l∈L
X X
If (B,L) is a Hadamard pair then we get A= f(x), which proves the other Cuntz relation
S S∗ = I .
l l L2(µB)
l∈L
X
(cid:3)
Definition 1.6. Suppose (B,L) form a Hadamard pair. We denote by RT, the transpose of the
matrix R. We consider the dual IFS
(1.13) τ(L)(x) =(RT)−1(x+l), (x Rd,l L)
l ∈ ∈
We denote the attractor of this IFS by X and its invariant measure by µ .
L L
(L)
To simplify the notation, we will use τ instead of τ , and make the convention that when the
l l
(L)
subscript is an l then we refer to τ , and when the subscript is a b we refer to τ .
l b
6 DORINERVINDUTKAYANDPALLEE.T.JORGENSEN
Definition 1.7. For the set B define the function
1
(1.14) χ (x) = e2πib·x, (x Rd)
B
N ∈
b∈B
X
The transfer operator R is defined on functions f on Rd by
B,L
(1.15) (R f)(x) = χ (τ (x))2f(τ (x)) = χ ((RT)−1(x+l))2f((RT)−1(x+l)).
B,L B l l B
| | | |
l∈L l∈L
X X
The operators S∗ behave well on exponential functions: the next lemma will be helpful for our
l
computations.
Lemma 1.8. The following assertions hold:
(i) For all l L and t Rd
∈ ∈
(1.16) S∗e = e χ ( τ(L)( t)).
l t −τ(L)(−t) B − l −
l
(ii) Let be a reducing subspace of L2(µ ) for the representation (S ) . If e and
B l l∈L t
H ∈ H
(L)
χ ( τ ( t)) = 0, then e .
B − l − 6 −τ(L)(−t) ∈ H
l
Proof. We have
(S∗e )(x) = 1 e−2πil·R−1(x+b)e2πit·R−1(x+b) = 1 e−2πi(RT)−1(−t+l)·(x+b)
l t N N
b∈B b∈B
X X
= e−2πi(RT)−1(−t+l)·x 1 e−2πi(RT)−1(−t+l)·b
N
b∈B
X
= e−2πiτl(L)(−t)·xχB(−τl(L)(t)) = e−τ(L)(x)χB(−τl(L)(−t)).
l
(ii) follows from (i). (cid:3)
Definition 1.9. We will use the following notation for a finite word w = w ...w Ln,
1 n
∈
S = S ...S .
w w1 wn
Proposition 1.10. For a subspace H of L2(µ ) denote by P the orthogonal projection onto H.
B H
Let be a subspace which is invariant for all the maps S∗, l L, i.e., S∗ for all l L. Let
K l ∈ lK ⊂ K ∈
:= ,
0
K K
:= span S :w Ln ,
n w
K { K ∈ }
:= span S :w Ln,n N .
∞ w
K { K ∈ ∈ }
Then
(i) For all n N, is invariant for S∗, l L and S∗ = for all l L, n 0.
(ii) P P ∈ foKrnall n N. l ∈ lKn+1 Kn ∈ ≥
(iii) FoKrna≤ll nKn+N1 ∈
∈
P = α(P ) = S P S∗.
Kn+1 Kn l Kn l
l∈L
X
(iv) The projections P converge to P in the strong operator topology.
Kn K∞
SPECTRAL MEASURES AND CUNTZ ALGEBRAS 7
Proof. (i) Take w = w ...w Ln and k . Then for any l L
1 n 0
∈ ∈ K ∈
S∗S S k = δ S k = δ S S S∗k.
l0 w1 w2...wn l0w1 w2...wn l0w1 w2...wn l l
l∈L
X
But S∗k so S∗S k .
l ∈ K l0 w ∈ Kn
This computation implies also that S∗ . The other inclusion follows from the Cuntz
l0Kn ⊂ Kn−1
relations, and this implies (i).
(ii) is immediate from (i).
Since is also invariant under the maps S∗, it is enough to prove (i) for n = 0. Take l L
Kn l 0 ∈
and k . Then
∈ K
α(P )S k = S P S∗S k = S P k = S k.
K l0 l K l l0 l0 K l0
l∈L
X
Therefore is contained in the range of the projection α(P ).
1 K
K
Also, for any α(P )v = S P S∗v since P S∗v .
(iv) is clear sinceK spal∈nLs l K. l ∈K1 K l ∈ K (cid:3)
n n ∞
∪ K P K
Definition 1.11. For an operator T on L2(µ ) define the following function
B
h (t)= Te , e , (t Rd).
T −t −t
h i ∈
Proposition 1.12. Let T be an operator on L2(µ ). Then
B
h = R h .
α(T) B,L T
The function h is entire analytic.
T
Proof. We have, with Lemma 1.8:
h (t)= S TS∗e , e = TS∗e , S∗e = χ (τ (t))2 Te , e
α(T) l l −t −t h l −t l −ti | B l | −τl(t) −τl(t)
* +
l∈L l∈L l∈L
X X X (cid:10) (cid:11)
= χ (τ (t))2h (τ (t)) = R h (t).
B l T l B,L T
| |
l∈L
X
Since the operator T is bounded, it is easy to check that the function h is entire analytic. (cid:3)
T
Corollary 1.13. Using the notations above:
(i) If A is a bounded operator in the commutant of the representation (S ) then the function
l l∈L
h is an entire analytic harmonic function for the transfer operator R , i.e.,
A B,L
(1.17) R h = h
B,L A A
(ii) Suppose is a subspace which is invariant under all the maps S∗, l L. Then the function
K l ∈
h is an entire analytic subharmonic function for the transfer operator R i.e.,
PK B,L
R h h .
B,L PK ≥ PK
Proof. If A commutes with the representation then α(A) = S AS∗ = A S S∗ = A. If is
l l l l l l K
invariant under the maps S∗ then α(P ) P by Proposition 1.10. Then the corollary follows
l K ≥ K P P
directly from Proposition 1.12.
(cid:3)
8 DORINERVINDUTKAYANDPALLEE.T.JORGENSEN
Corollary 1.14. Let be a subspace of L2(µ ) which is invariant under all the maps S∗, l L.
K B l ∈
With the notation in Proposition 1.10, the following limit exists uniformly on compact sets:
lim Rn h = h .
n→∞ B,L PK PK∞
Proof. With Propositions 1.12 and 1.10 we have
Rn h = h = h .
B,L PK αn(PK) PKn
Since P converges in the strong operator topology to P , we obtain that h converges to
Kn K∞ PKn
h pointwise. Using Corollary 1.13, we see that this is an increasing sequence of functions; then,
PK∞
with Dini’s theorem we obtain the conclusion. (cid:3)
2. Representations associated to minimal words
We now turn to a dynamical systems feature which determines both the algorithmic and the
analytic part of the problem of L2(µ). This feature in turns divides up into two parts, periodic and
non-periodic. The precise meaning of these terms is fleshed out in Definitions 2.1 and 2.2 (in the
present section), and in Definition 4.2 insidethe paper. In both cases, we deal with a random walk-
dynamical system. The first case is especially easy to understand in terms of a natural encoding
with finite and infinite code-words. By contrast, the second case involves invariant sets (for the
walk, Definition 4.2) which can have quite subtle fractal properties. Some of the possibilities (case
2) are illustrated by examples in section 4.
We begin with a discussion of the minimal words in the encoding for case 1.
Definition 2.1. Consider a finite word over a finite alphabet L with #L = N. We say that w is
minimal if there is no word u such that w = uu...u for some p 2.
≥
p times
We denote by w = ww..., the infinite word obtained by the repetition of w infinitely many
| {z }
times. For two words u, w with u finite, we denote by uw the concatenation of the two words.
We define the shift σ on infinite words σ(ω ω ...) = ω ω ....
1 2 2 3
Definition 2.2. Let w be a minimal word. Let Γ(w) be the set of infinite words over the alphabet
L that end in an infinite repetition of the word w, i.e.,
(2.1) Γ(w) := uw : u is a finite word over L
{ }
We will use the Dirac notation for vectors in l2(Γ(w)): for ω l2(Γ(w))
∈
1, ξ = ω
ω =δ , δ (ξ) =
| i ω ω 0, ξ = ω
(cid:26) 6
We define the represetation ρ of the Cuntz algebra by
w N
O
(2.2) ρ (S )ω = lω , (ω l2(Γ(w)))
w l
| i | i ∈
Moreover, if u,v arevectors in a Hilbertspace, wewill usethenotation u v for thecorresponding
| ih |
rank-one operator.
Theorem 2.3. The following assertions hold:
(i) The operators in (2.2) defines an irreducible representation of the Cuntz algebra .
N
O
SPECTRAL MEASURES AND CUNTZ ALGEBRAS 9
(ii) Define the subspace spanned by the vectors w , σ(w) ,..., σp−1(w) , where p is the
w
K {| i | i | i}
length of w. Then is invariant for the operators ρ (S )∗, l L and it is cyclic for the
w w l
K ∈
representation ρ .
w
(iii) If w and w′ are two minimal words that are not a cyclic permutation of each other, then
the representations ρw and ρw′ are disjoint.
Proof. It is easy to check that
0, if ω = l
ρw(Sl)∗|ωi = σ(ω) , if ω11 6= l
(cid:26) | i
for ω = ω ω .... From this it follows after a simple computation that ρ is a representation of the
1 2 w
Cuntz algebra.
Since σp(w) = w, it follows that ρ (S )∗ . Since every word in Γ(w) ends in w, it can
w l w w
K ⊂ K
be easily seen that is cyclic for the representation.
w
K
It remains to check that the representation is irreducible and assertion (iii). For this we will use
Theorem from [BJKW00], applied to our situation (see also [BJ97b, BJ97a]):
Theorem 2.4. There is a bijective correspondence between
(i) Operators A that intertwine the representations ρw and ρw′, i.e.,
ρw′(Sl)A = Aρw(Sl), (l L).
∈
(ii) Fixed points of the map
Φ(C)= Vl′CVl∗, (C ∈ B(Kw,Kw′)),
l∈L
X
whereVl = Pρw(Sl)P, Vl′ = P′ρw′(Sl)P′, l ∈ L, withP =projectionontoKw, P′ =projection
onto w′. The correspondence from (i) to (ii) is given by
K
C = P′AP.
Returning to the proof of Theorem 2.3, we will compute the fixed points of the map Φ.
Let u = σj(w) , j = 0,...,p 1, where p =length of w, and let u′ = σi(w′) , i = 0,...,p′ 1,
j | i − i | i −
where p′ =length of w′. The space B(Kw,Kw′) is spanned by the rank one operators |u′iihuj|. We
have
Φ(u′ u )= V′u′ V u
| iih j| | l iih l j|
l∈L
X
But
V u = Pρ (S )u = P lσj(w) = P lw w ...w w = δ σj−1(w)
l j w l j | i | j+1 j+2 p i l,wj| i
(we use here a notation modp, or modp′ when required, so σ−1(w) will mean σp−1(w))
Then
Φ(|u′iihuj|) = δl,wi′δl,wj|σi−1(w′)ihσj−1(w)| = δwi′,wj|σi−1(w′)ihσj−1(w)|.
l∈L
X
Now suppose C = p′−1 p−1c u′ u is a fixed point for Φ. Then
i=0 j=0 i,j| iih j|
P p′−P1p−1
ci,j|u′iihuj|= ci,jδwi′,wj|u′i−1ihuj−1|.
i=0 j=0 i,j
XX X
10 DORINERVINDUTKAYANDPALLEE.T.JORGENSEN
Then
ci,j = ci+1,j+1δwi′+1,wj+1, for all i,j.
Thus, c = 0 only if w′ = w for all k, which means that σi−j(w′) = w. But this implies, since
i,j 6 i+k j+k
w,w′ are minimal, that w is a cyclic permutation of w′. Thus, if w, w′ are not cyclic permutations
of each other, then the only fixed point of Φ is C, and with Theorem 2.4, this implies that there
are no nonzero intertwining operators.
To check that the representation ρ is irreducible, we use the same computation, now with
w
w = w′. We saw that c = 0 only if σi−j(w′) = w. But since i j < p and w is minimal this
i,j
6 | − |
implies that i = j. In the case i = j, the same computation implies that c = c , therefore
i,i i+k,i+k
C is a scalar multiple of the identity. Using again Theorem 2.4, we obtain that the only operators
in the commutant of ρ are scalar multiples of the identity.
w
(cid:3)
3. Representations associated to extreme cycles
We mentioned in section 2, that the algorithmic and the analytic part of the problem of L2(µ)
involves a dynamical system. Its nature divides up into two parts, periodic and non-periodic. Case
1 has an algorithmic part (code-words), and an analytic part taking the form of extreme cycles,
and their associated representations. This is worked out below.
Definition 3.1. A finite set of distinct points C = x ,x ,...,x is called an L-cycle if there
0 1 p−1
{ }
exist l ,...,l L such that
0 p−1
∈
τ (x ) = x ,τ (x )= x ,...,τ (x ) = x ,τ (x ) =x .
l0 0 1 l1 1 2 lp−2 p−2 p−1 lp−1 p−1 0
We call w(C) := l l ...l the word of the cycle C. The points in C are called L-cycle points.
0 1 p−1
An L-cycle is called B-extreme if
(3.1) χ (x) = 1, for all x C.
B
| | ∈
Theorem 3.2. [DJ07, Theorem 4.1] Under the conditions above, assume (B,L) is a Hadamard
pair. Suppose there exist d linearly independent vectors in the set
n
(3.2) Γ(B):= Rkb : b B,n N .
k k
∈ ∈
( )
k=0
X
Define
(3.3) Γ(B)◦ := x Rd : β x Z for all β Γ(B) .
∈ · ∈ ∈
ThenΓ(B)◦ isalattice that containns Zd, isinvariant underRT, andif l,ol′ L with l l′ RTΓ(B)◦
∈ − ∈
then l = l′. Moreover
(3.4) Γ(B)◦ X = C :C is a B-extreme L-cycle .
L
∩ { }
Definition 3.3. Let (B,L) be a Hadam[ard pair. We say that the Hadamard pair is regular if the
IFS (τ ) has no overlap and there exist d linearly independent vectors in the set Γ(B) defined
b b∈B
by (3.2).
Define the maps σ on Rd by
l
(3.5) σ (x) = RTx+l, (x Rd,l L)
l
∈ ∈