We consider Pisot family substitution tilings in whose dynamical spectrum is pure point. There are two cut-and-project schemes (CPSs) which arise naturally: one from the Pisot family property and the other from the pure point spectrum. The first CPS has an internal space for some integer defined from the Pisot family property, and the second CPS has an internal space H that is an abstract space defined from the condition of the pure point spectrum. However, it is not known how these two CPSs are related. Here we provide a sufficient condition to make a connection between the two CPSs. For Pisot unimodular substitution tiling in , the two CPSs turn out to be same due to the remark by Barge-Kwapisz.
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 . 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 . 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  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 . 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 . 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  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.
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 .
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 which
Here 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
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 .
([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:
, for any tiles of the same type;
, 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  (Theorem 4.1).
( 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 that
where 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 .
is a lattice in .
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 . □
is dense in .
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  (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. □
4. Two Cut-and-Project Schemes
Let be a primitive substitution tiling on with expansion map . Let
be the group generated by , , where is a control point set of and
be the set of periods of . We say that admits an algebraic coincidence if there exist and for some for which . It is known in  that admits an algebraic coincidence if and only if has a pure point spectrum.
With the assumption that admits an algebraic coincidence, we define a topology on L and construct a completion H of the topological group L such that the image of L is a dense subgroup of H. This enables us to construct a cut-and-project scheme (CPS) such that each point set , , arises from the CPS. From the following lemma, we understand that the system satisfies the topological properties for the group L to be a topological group [18,19,20].
( (Lemma 4.1)).Let be a primitive substitution tiling with an expansive map ϕ. Suppose that admits an algebraic coincidence. Then, the system serves as a neighbourhood base for of the topology on L relative to which L is a topological group.
For the topology on L with the neighbourhood base , we name ϕ-topology. Let be the space L with -topology.
Let . From  (Sections 3.4 and 3.5) and Lemma 3, there exists a complete Hausdorff topological group (H) of for which is isomorphic to a dense subgroup of the complete group H (see [6,21]). Moreover, there is a uniformly continuous mapping which is the composition of the canonical injection of into H and the canonical homomorphism of L onto . Here is dense in H and the mapping from L onto is an open map, where is with the induced topology of the completion H. We can directly consider H as the Hausdorff completion of L vanishing .
There is another topology on L which is equivalent to -topology under the assumption of algebraic coincidence.
Let be a van Hove sequence and be two tilings in , where and are representable Delone -sets of the tilings . We define
Here Δ is the symmetric difference operator. Let for each . For any , is relatively dense, if admits an algebraic coincidence, then, [6,8,22,23]. In this case, the system serves as a neighbourhood base for of the topology on L where L becomes a topological group. We call this topology -topology on L and indicate the space L with -topology by (see [6,22] for -topology with the name of autocorrelation topology).
( (Propositions 4.6 and 4.7)).Let be a primitive substitution tiling. Assume an algebraic coincidence on , then the map from onto is topologically isomorphic.
From Proposition 1, is topologically isomorphic to . Then, the completion of is topologically isomorphic to the completion H of . We will identify the completion of with H. Thus, is uniformly continuous, is dense in H, and the map from onto is an open map where the latter is with the induced topology of the completion H. Hence, we can consider the CPS (5) with an internal space H that is a completion of . We note that since is repetitive, and in .
We observe that and are all topologically isomorphic when the control point set is a regular model -set in CPS (10).
Let be a primitive Pisot family substitution tiling in with an expansive map ϕ. Suppose that ϕ is diagonalizable, all the eigenvalues of ϕ are algebraic conjugates with multiplicity one, and there exists at least one algebraic conjugate λ of eigenvalues of ϕ for which . If is a regular model κ-set in CPS (10), then the internal space H which is the completion of with ϕ-topology is isomorphic to the internal space , which is constructed from using the conjugation map Ψ in (8).
Proof. Since is an expansive map and satisfies the Pisot family condition, we first note that there is no algebraic conjugate of eigenvalues of with .
We show that if is close to 0 in for , then is close to 0 in H. Since every point set is a regular model set from the assumption where and in the CPS (10), for
where is a Haar measure in (see  [Theorem 1]).
We note that
is uniformly continuous in  [Section 1]. So if converges to 0 in ,then converges to 0 in .
On the other continuity, suppose that is a sequence such that as . Then, for every
Note that for large enough , , and so for all . From the fact that is compact, has a converging subsequence . For any such sequence, we define . Then
and thus for each . Hence , and it implies . On the other inclusion, and . So, . Hence and . This equality is for every . Since is isomorphic to , each model set has irredundancy. Therefore . So all converging subsequences converge to 0 and .
This establishes the equivalence of the two topologies. By  (Proposition 5, Chapter 3, Section 3.3), there exists an isomorphism between H onto . □
The above theorem shows that the internal space H constructed from with -topology is isomorphic to Euclidean space (i.e., ).
It is known in [1,5,26] [Theorem 3.6] that unimodular irreducible Pisot substitution tilings in with pure point spectrum give rise to regular model sets. We give a precise statement below.
( Remark 18.5).Let be a primitive substitution tiling in with expansion factor β being a unimodular irreducible Pisot number. Then has a pure point spectrum if and only if for any , each is a regular model set in CPS (10).
Let be a primitive Pisot substitution tiling in with an expansion factor β. Assume that there exists at least one algebraic conjugate λ of β for which . If has a pure point spectrum, then the internal space H which is the completion of with ϕ-topology can be realised by Euclidean space , where m is the degree of the characteristic polynomial of β.
Proof. By Theorem 3, it is known that for a primitive Pisot substitution tiling in , if has a pure point spectrum, then is a regular model -set in CPS (10).
We constructed a natural cut-and-project scheme (10) where the control point sets are in the form of a module . We showed that for 1-dimensional primitive Pisot substitution tilings, if they have pure point spectrum, then the abstract internal spaces given in  can actually be realised as Euclidean spaces. The questions still remains as to whether it holds for n-dimensional primitive substitution tilings. Substitution tilings are often used as mathematical models to understand the structures of physical materials and cut-and-project schemes with Euclidean spaces are used in simulation experiments in order to understand the pure point part of the spectrum. Here the results show that if one works on the simulation of 1-dimensional primitive Pisot substitution tilings with pure point spectrum, the corresponding representative point sets are always realised as model sets in a cut-and-project scheme with a Euclidean internal space.
6. Further Study
We are left with the following questions extending Theorem 2.
Can we replace the assumption of regular model κ-set by pure point spectrum? In other words, for a primitive Pisot family substitution tiling in with an expansion map ϕ, does the pure point spectrum of imply that is a regular model κ-set with a Euclidean internal space?
Can the theorem be extended into the case where the multiplicity of eigenvalues of ϕ is not one?
Writing—original draft, J.Y.L., S.A. and Y.N.
This research was funded by [Local University Excellent Researcher Supporting Project through the Ministry of Education of the Republic of Korea and National Research Foundation of Korea (NRF)] grant number , [Japan Society for the Promotion of Science (JSPS)] grant number [17K05159, 17H02849, BBD30028].
The first author would like to acknowledge the support by the NRF grant and is grateful to the Korea Institute for Advanced Study (KIAS), where part of this work was done during her sabbatical year. The second author is partially supported by the JSPS grants. The third author was supported by the project I3346 of the Japan Society for the Promotion of Science(JSPS) and the Austrian Science Fund (FWF).
Conflicts of Interest
The authors declare that they have no competing interests.
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