1. Introduction
The study of long-range aperiodic order has played an important role in understanding the structures of physical models like “quasicrystals”. As an important class of models for the long-range aperiodic order, substitution tilings have been the subject of a great deal of study. Among these substitution tilings, Pisot or Pisot family substitutions get more attention due to their well-ordered properties. A Pisot number is an algebraic integer
for which all of the other algebraic conjugates of
lie strictly inside the unit circle. If the largest eigenvalue of the corresponding substitution matrix is a Pisot number, we call the substitution a Pisot substitution. There is a well-known conjecture called the “Pisot substitution conjecture”. The associated substitution system always has a pure point spectrum [
1]. For special cases of the Pisot substitutions, the conjecture has been answered affirmatively [
2,
3]. Extending the idea of a Pisot number, one can also look at a Pisot family, a set of algebraic integers whose Galois conjugates with modulus larger than or equal to 1 are all in the same set [
4]. There has been a great deal of of study on Pisot substitution sequences or Pisot family substitution tilings on
which characterizes the property of pure point spectrum (see [
1] and therein). There are two natural cut-and-project schemes (CPSs) arising in this study. One CPS is constructed with a Euclidean internal space using the Pisot property [
5]. We extend the idea of constructing the CPS from the Pisot property to the Pisot family property. The other CPS is made by constructing an abstract internal space from the property of pure point spectrum [
6]. These two CPSs were developed independently from different aims of study. It is not yet known if these two CPSs have any relation to each other. Here we would like to provide how these two CPSs are related, showing that two internal spaces are basically isomorphic to each other in Theorem 2.
In this article, we consider mainly primitive substitution tilings in
with pure point spectrum. It is proven in [
7] that primitive substitution tilings with pure point spectrum always show finite local complexity (FLC). So, it is not necessary to make an assumption of FLC in the consideration of pure point spectrum.
In
Section 2, we visit the basic definitions of the terms that we use. In
Section 3, we construct a natural CPS which arises from the property of Pisot family substitution. In
Section 4, we recall the other CPS constructed from the property of pure point spectrum. We show in Theorem 2 that the two CPSs are closely related, demonstrating that there is an isomorphism between two internal spaces of the CPSs under certain model set conditions. In
Section 6, we raise a few questions for later study.
2. Preliminary
2.1. Tilings
We consider a set of types (or colours) . A tile in is a pair where (the support of T) is a compact set in with and is the type of T. Let for . A set P of tiles is a patch if the number of tiles in P is finite and the interiors of the tiles are mutually disjoint. A tiling of is a set of tiles for which and the interiors of distinct tiles are disjoint. Given a tiling , -patch is a finite set of tiles of . We always assume that any two -tiles with the same type are translationally equivalent. Thus, up to translations, there is a finite number of -tiles.
We will use the following notation:
A van Hove sequence for is a sequence of bounded measurable subsets of satisfying
2.2. Delone -Sets
A Delone set in is a point set which is relatively dense and uniformly discrete in . We call a Delone κ-set in if each is Delone and is Delone. A Delone -set is called representable if there exist tiles , so that is a tiling of .
2.3. Substitutions
Definition 1. Let us consider a finite set of tiles in with , which we will call prototiles.
We denote by the set of patches which are formed by tiles that are translates of ’s. is a tile-substitution (or substitution) with expansion map ϕ if there are finite sets for , for whichwithHere all sets in the union have disjoint interiors. It is possible for some of the to be empty. The substitution matrix is defined by . If for some , then the corresponding substitution tiling is primitive.
A set of algebraic integers is a Pisot family if for any . Every Galois conjugate of , with , is contained in . For , with real and , this reduces to being a real Pisot number, and for , with non-real and , to being a complex Pisot number. We say that is a Pisot substitution tiling if the expansive map is a Pisot expansive factor , and a Pisot family substitution tiling if the eigenvalues of the expansive map form a Pisot family.
2.4. Cut-and-Project Scheme
Definition 2. A cut and project scheme (CPS) is a collection of spaces and mappings as follows:
where
is a real Euclidean space,
H is a locally compact Abelian group,
and
are the canonical projections,
is a lattice in
(i.e., a discrete subgroup such that the quotient group
is compact),
is injective, and
is dense in
H.
For a subset , we denote .
We define a model set in as a subset in for which, up to translation, , W is compact in H, . If the boundary of W is of (Haar) measure 0, we say that the model set is regular. We say that is a model -set (resp. regular model -set) if each is a model set (resp. regular model set) with regard to the same CPS.
We assume, without loss of generality, that
H is generated by the windows
’s, where
for all
. When
H satisfies the following:
we say that the windows
’s have irredundancy.
2.5. Pure Point Spectrum
Let
be the set of all primitive substitution tilings in
such that clusters of each tiling are translates of a
-patch. We give a usual metric
in tilings in such a way that two tilings are close if there is a large agreement on a big area with small shift (see [
8,
9,
10]). Then,
, where we take the closure in the topology induced by the metric
. We consider a natural action of
by translations on the dynamical hull
of
and get a topological dynamical system
. Assume that
is a measure-preserving dynamical system with a unique ergodic measure
. We look at the associated group of unitary operators
on
:
Every defines a function on by , which is positive definite on . So, its Fourier transform is a positive measure on and we call it the spectral measure corresponding to g. We say that the dynamical system has pure point spectrum if is pure point for each . If the dynamical system has a pure point spectrum, we also say that has a pure point spectrum.
3. Cut-and-Project Scheme for Pisot Family Substitution Tilings
We consider a primitive substitution tiling on with expansion map . There is a standard way to choose distinguished points in the tiles of primitive substitution tiling so that they form a -invariant Delone set. They are called control points. A tiling is called a fixed point of the substitution if .
Definition 3 ([
11,
12])
. Let be a fixed point of a primitive substitution with expansion map ϕ. For every -tile T, we choose a tile on the patch . For all tiles of the same type, we choose with the same relative position. This defines a map called the tile map. Then, we define the control point for a tile by The control points satisfy the following:
- (a)
, for any tiles of the same type;
- (b)
, for .
For tiles of any tiling
, control points have the same relative position as in
-tiles. The choice of control points is non-unique, but there are only finitely many possibilities, determined by the choice of the tile map. Let
be a set of control points of the tiling
in
. Let
where
is the set of control points of tiles of type
i.
Let us assume that
is diagonalizable over
and the eigenvalues of
are algebraically conjugate with multiplicity one. For a complex eigenvalue
of
, the
diagonal block
is similar to a real
matrix
where
,
, and
. So we can assume, by appropriate choice of basis, that
is diagonal with the diagonal entries equal to
corresponding to real eigenvalues, and diagonal
blocks of the form
corresponding to complex eigenvalues
.
We assume, without loss of generality, that is a diagonal matrix.
We recall the following theorem. The theorem is not in the form as shown here, but one can readily note that from the proof of [
4] (Theorem 4.1).
Theorem 1 ([
4] Theorem 4.1)
. Let be a primitive substitution tiling on with expansion map ϕ. We assume that has FLC, ϕ is diagonalizable, and all the eigenvalues of ϕ are algebraically conjugate with multiplicity one. Then, there exists an isomorphism such thatwhere and .
Let us assume now that
has FLC,
is diagonalizable, the eigenvalues of
are all algebraically conjugate with multiplicity one, and there exists at least one other algebraic conjugate different from eigenvalues of
. Suppose that
has
e number of real eigenvalues, and
f number of
blocks of the form of complex eigenvalues where
. Let all the algebraic conjugates of eigenvalues of
be real numbers
and complex numbers
. Let
and write
for
for convenience. Let us consider a space
, where
Let us consider the following map:
where
is a polynomial over
. Let us build a new cut and project scheme:
where
and
are canonical projections,
, and
. It is clear to see that
is injective. We now show that
is dense in
and
is a lattice in
.
Lemma 1. is a lattice in .
Proof. Since is a free -module of rank m and matrix is non-degenerate by the Vandermonde determinant, the natural embedding combining all conjugates; gives a lattice in . Consequently, is isomorphic to a free -submodule of due to the theory of elementary divisors. From Theorem 1, is isomorphic to a full rank -submodule of , that is, a sub-lattice of . Thus, the claim is shown. The case with complex conjugates can be shown in a similar manner, taking care of embeddings to . □
Lemma 2. is dense in .
Proof. We showed that
is a sub-lattice of
in the proof of Lemma 1. So it suffices to prove that
is dense in
. We prove the totally real case, that is,
for all
i. By [
13] (Theorem 24),
is dense if
implies
for
. The condition is equivalent to
with
in the terminology of Lemma 1. Multiplying the inverse of
A, we see that entries of
must be Galois conjugates. As
has at least one zero entry, we obtain
, which shows
for
. In fact, this discussion uses the Pontryagin duality that the
has a dense image if and only if its dual map
is injective (see also [
14,
15,
16] [Chapter II,
Section 1]). The case with complex conjugates is similar. □