Substitution Rules for Cut and Project Sequences

Substitution rules for cut and project sequences

Zuzana Masáková∗ Jiˇr´ıPatera† Edita Pelantová ‡

CRM-2640 November 1999

∗Department of Mathematics, Faculty of Nuclear Science and Physical Engineering, Czech Technical University, Trojanova 13, Prague 2, 120 00, Czech [email protected] †Centre de recherches mathématiques, Universitéde Montréal, C.P. 6128 Succursale Centre-ville, Montréal, Québec, H3C 3J7, [email protected] ‡Department of Mathematics, Faculty of Nuclear Science and Physical Engineering, Czech Technical University, Trojanova 13, Prague 2, 120 00, Czech [email protected]

Abstract We describe the relation between substitution rules and cut-and-project sequences. We consider aperiodic sequences arising from the cut and project scheme with quadratic unitary Pisot numbers β. For a sequence Σ(Ω) with a convex acceptance window Ω, we prove that a substitution rule exists, iff Ω has boundary points in the corresponding quadratic field Q[β]. In this case, one may find for arbitrary point x Σ(Ω) a substitution generating the sequence Σ(Ω) starting from x. We provide an∈ algorithm for construction of such a substitution rule.

1 Introduction

A substitution rule is an alphabet, together with a mapping which to each letter of the alphabet assigns a finite word in the alphabet. A fixed point of the substitution is an infinite word w which is invariant with respect to the substitution. It can be studied from a combinatorial point of view (configurations of possible finite subwords), or from a geometrical point of view [1, 11]. In the later case we consider the letters of w to be tiles of given length put in a prescribed order on the real line. In such a way, the tiling determines a Delone point set Λ R. (A set Λ R is said to satisfy the Delone property if it has uniform lower and upper bounds⊂ on the distances⊂ between adjacent points.) In order that the set Λ, obtained from the substitutitution be self-similar, i.e. that there exists a factor s, such that sΛ Λ, one has to choose the tiles to have suitable lengths. The substitution sequences⊂ may be both periodic or aperiodic. A standard way to construct quasiperiodic point sets is the ‘cut and project’ method. One of the major applications of cut and project point sets is the modelling of physical quasicrystalline materials. The point sets obtained by the cut and project scheme, called here simply ‘quasicrystals’, have many remarkable properties. Among them especially the diffraction properties [5] and symmetries are of interest for quasicrys- tallographers. A very important attribute of cut and project sets is the presence of a rich structure of self-similarities [2, 7]. A substitution rule carries a self-similarity factor for the corresponding Delone point set as a Perron-Frobenius eigenvalue of the substitution matrix. One may ask about the relation of cut and project quasicrystals and Delone sets generated by substitutions. In this paper we provide an answer to this question for large family of 1-dimensional quasicrystals. We decide about the existence of a substitution rule generating the given quasicrystal, and if there exists one, we give a prescription how to find it. The connection of substitutive sequences to quasicrystals has been recognized for example by Bombieri and Taylor. In [3] they show that if the characteristic polynomial of the substitution rule is the minimal polynomial of a Pisot number, then the point set given by such a substitution is a subset of a cut and project set. The most frequently mentioned example of the Fibonacci chain, constructed by the rule a ab, b a, is a 1-dimensional cut and project sequences, see [6]. Gählerand Klitzing study→ the→ Fourier transform of Delone sets generated by substitution rules. They show in [4] that a substitution determines a set with non trivial Bragg spectrum iff the largest eigenvalue of the substitution matrix (=scaling factor of the substitution tiling) is a Pisot number. In this paper we consider those cut and project sets which arise by projection of chosen points of a two dimensional grid Z2 to a straight line. In such a case the requirement of a self-similarity forces that the cut and project sequence is embedded into an extension of rational numbers by a quadratic irrationality. In this article we study 1-dimensional cut and project sets based on the 1 √ golden mean irrationality τ = 2 (1 + 5). However, our result can be extended for other quadratic unitary Pisot numbers as well. The algebraic definition of cut and project quasicrystals used in this paper (Definition 2.2) follows the analogy of [9]. A quasicrystal is a subset of the ring of integers Z[τ] of the quadratic field Q[√5]. A point x Z[τ] belongs to the quasicrystal if its Galois conjugate ∈ x0 lies in a chosen ‘acceptance interval’ Ω. In order that a Delone set Λ R with a finite number of tiles (distances between adjacent points) could be generated from a⊂ point x Λ by a substitution with the eigenvalue s, Λ has to satisfy s(Λ x) Λ x. (Of course, it is not∈ a sufficient condition for the existence of a substitution rule.) A cut− and⊂ project− quasicrystal Σ has the property that for any x Σ there are infinitely many different factors s, such that s(Σ x) Σ x. Such points x as centers∈ of scaling symmetry are called s-inflation centers. A complete− description⊂ − of inflation centers is given in [7]. In this article we find necessary and sufficient condition on the acceptance interval Ω for the existence of a substitution rule which enables to generate the quasicrystal with the acceptance

3 window Ω, starting from a given point x of the quasicrystal. If this condition is satisﬁed, then for any quasicrystal point y there exists a substitution θy, depending on y, which generates the quasicrystal Σ from y. Note that the proof of this statement is constructive. We provide an algorithm for construction of the substitution rules for given Ω and y.

2 Preliminaries

A substitution is a rule that assigns to each letter of an alphabet A a concatenation of letters, which is called a word. The set of words with letters in A is denoted by A∗. Iterations of the substitution starting from a single initial letter leads to words of increasing length. Certain assumptions on the substitution rule ensure that the words give raise to an inﬁnite sequence of letters which is invariant with respect to the substitution.

Definition 2.1. Let A be an alphabet. A map θ : A∗ A∗ is called a substitution, if θ(a) is non → empty for all a A and if it satisfies θ(vw) = θ(v)θ(w) for all v, w A∗. An infinite sequence ∈ ∈ u A∗ is called a fixed point of θ, if θ(u) = u. ∈Let u be a fixed point of a substitution and ω : A V be a mapping of the alphabet A into → another alphabet V . The mapping ω can be extended into ω : A∗ V ∗. The sequence v = ω(u) is called the ω-image of u. →

If θ : A∗ A∗ is a substitution and for a letter a A, one has θ(a) = aw for some non empty → n ∈ word w A∗, then the sequence of words θ (a) converges to an infinite word u, which is a fixed ∈ point of θ. We use the notation u = θ∞(a). Similarly, if there exists b A, such that θ(b) =wb ˜ for ∈ some non emptyw ˜ A∗, we can construct−−−→ a bidirectional infinite sequence starting the substitution rule from the pair b∈a. As an example,| let us mention the well known Fibonacci substitution rule θ on the alphabet formed by two letters, A = a, b , given by a ab, b a. The second iteration of θ is a substitution { } 7→ 7→ a aba (1) b 7→ ab 7→ therefore we can generate from the pair b a a bidirectional infinite word |

θ∞(a) θ∞(a) = . . . abaababaabaababaababa abaababaabaababaababa . . . (2) | | ←−−− −−−→ We can associate lengths to letters a and b, and imagine the inﬁnite sequence (2) as a labeling the tiles of a tiling of R. If we identify each tile with its left end point starting with the delimiter at the origin, then we obtain a Delone point set Λ R. Using for a and b the lengths determined by| the standard analysis of the corresponding substitution⊂ matrix, the Delone set has a self-similarity sΛ Λ, where s is the dominant eigenvalue of the matrix. The suitable lengths of tiles are given⊂ as components of the eigenvector associated with s. Consider the substitution of (1). The corresponding matrix, its dominant eigenvalue and the associated eigenvector are given by

2 1 τ M = , λ = τ 2 , v = , 1 1 1

where τ is the golden mean irrationality. Setting the lengths l(a), l(b) to be τ and 1, it can be shown that the Delone set arising from the substitution coincides with a 1-dimensional cut and project quasicrystal. The deﬁnition is given below.

4 2 Let τ and τ 0 be the roots of x = x + 1, i.e. ττ 0 = 1 and τ + τ 0 = 1. We have − 1 + √5 1 √5 τ = , τ 0 = − . 2 2 Recall the ring of integers Z[τ] = a + bτ a, b Z of the quadratic field Q[√5] and the Galois { | ∈ } automorphism 0 defined by x0 = (a + bτ)0 := a + bτ 0. Definition 2.2. Let Ω R be a bounded interval with non empty interior. A cut and project quasicrystal is the set ⊂ Σ(Ω) := x Z[τ] x0 Ω , { ∈ | ∈ } and Ω is called the acceptance window (interval) of the quasicrystal Σ(Ω). Let us formulate some of the properties of quasicrystals which will be used in this paper. Their proof is a direct consequence of the definition. Note that for (v) of Lemma 2.3 one needs to know that τ k, k Z, are divisors of unity in the ring Z[τ], which ensures that Z[τ] = τ kZ[τ]. ± ∈ Lemma 2.3. Let Ω1, Ω2, and Ω R be intervals. Then ⊂ (i) Σ(Ω1) Σ(Ω2) if Ω1 Ω2, ⊂ ⊂ (ii) Σ(Ω1) Σ(Ω2) = Σ(Ω1 Ω2), ∩ ∩ (iii) Σ(Ω1) Σ(Ω2) = Σ(Ω1 Ω2), ∪ ∪ (iv) Σ(Ω) + η = Σ(Ω + η0), for any η Z[τ], ∈ k k (v) τ Σ(Ω) = Σ(τ 0 Ω), for k Z. ∈ Consequently, we shall be interested only in quasicrystals with acceptance windows Ω = [c, d), such that 1 d c < τ. If the length d c of Ω is not within desired bounds, the property (v) of Proposition≤ 2.3−allows us to find geometrically− equivalent quasicrystal (rescaling the acceptance window properly), which satisfies the required condition. The restriction made on the interval to be semiclosed, [c, d), may influence only presence or absence of two particular points, namely c0 and d0 in the quasicrystal. This remark applies, of course, only if c, d belong to Z[τ]. Otherwise, including the boundary of the acceptance interval is not relevant for the quasicrystal. It is obvious that adding or removing a single point to a point set may strongly influence the form of the substitution rule. We deal with this problem in Section 5, providing examples of substitution rules for quasicrystals with closed and open acceptance windows. Construction of a substitution rule for a given quasicrystal Σ(Ω) from its particular point x ∈ Σ(Ω) is, according to (iv) of Proposition 2.3, a task equivalent to finding a rule for Σ(Ω x0) starting from 0. Therefore we can limit ourselves for quasicrystals containing the origin, which− gives us the conditions c 0, d > 0, on the acceptance interval Ω = [c, d). Further important≤ property of quasicrystals, which will be used in this paper, is formulated in the following proposition. Its proof can be found in [8]. Proposition 2.4. Let Ω be an interval of length l, 1 l < τ. The distances between adjacent points of the quasicrystal Σ(Ω) are 1, τ and τ 2. ≤ We denote the distances 1, τ and τ 2 by S (=short), M (=middle) and L (=long) respectively. The letters will stand for both the tiles and their lengths. Following lemma allows to divide the elements of a given quasicrystal into three groups, according to tiles which follow them. The lemma is a straightforward consequence of Proposition 2.4.

5 Lemma 2.5. Let c, d R, 1 d c < τ. The sets ΣS, ΣM and ΣL of points of the quasicrystal Σ[c, d) which are followed∈ be the≤ distance− 1, τ, and τ 2 respectively, are given by

ΣS = x Z[τ] x0 [c, d 1) , ∈ ∈ −

1 ΣM = x Z[τ] x0 c + , d , ∈ ∈ τ

1 ΣL = x Z[τ] x0 d 1, c + . ∈ ∈ − τ

In particular, ΣS is empty if d c = 1. − 3 Construction of substitutions - idea and examples

In this Section we explain the main ideas of the paper about the construction of a substitution for a given quasicrystal. We illustrate it on two examples. In the first of them (Example 3.1) we take the well known Fibonacci substitution and explain the fact that the corresponding Delone set is a cut and project quasicrystal. In the second Example 3.2, we start with a chosen quasicrystal and demonstrate the method how to find a substitution rule for it. A quasicrystal with tiles S, M, L is a bidirectional infinite word in these letters. Fixing a point in a quasicrystal corresponds to fixing the delimiter between two particular letters of the word. Construction of a substitution for such quasicrystal with| respect to a chosen point stems in finding a rule θ, such that the word of the quasicrystal with a given delimiter , is its fixed point. In general only for a certain family of quasicrystals we find a substitution rule| having the alphabet with three letters S, M, L . For other quasicrystals, the tiles can be divided into groups S1,...,Sk , { } S M1,...,MkM , and L1,...,LkL . Then there exists a substitution θ with alphabet A = S1,...,LkL such that the quasicrystal is its fixed point. More formally, it is possible to find a substitution{ }θ with an alphabet A and a mapping ω : A S, M, L such that the quasicrystal is an ω-image of some fixed point of θ. → { }

τ 4 τ τ 4 + τ 2 τ 5 + τ 2 τ 5 − − τ 4 τ 3 τ 2 0t τ 2 τ 3 τ 4 τ 5 τ 5 + τ 3 τ 6 − − − − a b a b a a b a a b a b a

t t t t t t

a b a b a a b a a b a b a t t t t t t 2 2 | τ{za |} τ{zb } Figure 1: How to understand the construction of substitution rules: In the example we stretch the quasicrystal by a suitable scaling factor and fill in the stretched tiles by tiles of the original quasicrystal. Stretched tiles of the same length should be filled in by the same sequence of original tiles. We shall construct the substitution rule by dividing the short, middle and long distances into several groups in such a way that tiles in one group are, after rescaling by the factor τ 2, filled by the same sequence of distances. It is illustrated on Figure 1.

6 Determination of points followed by tiles of length S, M, or L consists, according to Lemma 2.5, in splitting the acceptance window of the quasicrystal into three disjoint subintervals. Similarly,

determination of points followed by S1,...,SkS , M1,...,MkM , or L1,...,LkL corresponds to a split-

ting of the acceptance window Ω into kS +kM +kL disjoint intervals Ω1,..., ΩkL . We show that two 2 quasicrystal points which have their 0 image in the same subinterval Ωi are, after rescaling by τ , followed by the same sequence of distances filling the corresponding tile Si, Mi, or Li. The points determining the division of Ω into Ωi’s are called splitting points. We provide an explicite prescription for finding the splitting. It is obvious that if one has a substitution for a given quasicrystal and its point, then it is possible to find another substitution with larger alphabet for the same scaling factor. For example instead of a letter a we may write a1, a2 in a random order. Our division of Ω using splitting points provides a substitution rule with minimal alphabet for the scaling factor τ 2. It can be recognized from the form of the acceptance window Ω, whether the set of splitting points is finite or infinite, see Proposition 4.9. If it is infinite, then for a given quasicrystal a substitution rule with factor τ 2 does not exist. We show that then one cannot find neither a substitution with another scaling factor, (Theorem 4.10). If the set of splitting points is finite, it determines the number of letters of the alphabet. The size of the alphabet maight be reduced using another scaling factor, for example by iteration of the given substitution. This prolonges the word assigned to a single letter by the substitution. At the end of Section 4 we illustrate such a possibility on a specific example. Example 3.1. Recall the Fibonacci substitution rule (1). Consider the letters a, b in its fixed point (2) to be tiles dividing the real axis. Set the length of the tile represented by a to be τ 2, and the length of the tile b to τ. Let us show that the tiling sequence θ∞(a) θ∞(a) produces the quasicrystal | Σ[ 1/τ 2, 1/τ). The procedure is illustrated on Figure 1. We←−−− shall−−−→ use the self-similarity property (v)− of Lemma 2.3, 1 1 1 1 1 1 τ 2Σ , = Σ , Σ , . −τ 2 τ −τ 4 τ 3 ⊂ −τ 2 τ The last inclusion is valid due to (i) of the lemma. All together this means that the quasicrystal point set rescaled with respect to 0 by the scaling factor τ 2 is subset of the original quasicrystal. The quasicrystal Σ[ 1/τ 4, 1/τ 3) has the same ordering of tiles as Σ[ 1/τ 2, 1/τ), but the lengths are τ 2 times rescaled, i.e.− τ 4 and τ 3. The points of Σ[ 1/τ 2, 1/τ) divide− the tiles of Σ[ 1/τ 4, 1/τ 3). Clearly, the rescaled tiles (lengths τ 4, τ 3) are divided− into tiles of lengths τ 2 and τ. It− is possible to do only in such a way, that τ 4 = 2τ 2 + τ is cut into two of length τ 2 and one of length τ, similarly the tile of length τ 3 splits into one of length τ 2, and one of length τ (recall that τ 3 = τ 2 + τ). Therefore the longer tile a after rescaling is replaced by the concatenation of twice a and b, the rescaled b becomes a concatenation of a and b. We will show that each of letters a are replaced by a, b, a in the same order, and similarly for the replacement of b, i.e. we will show that Σ[ 1/τ 2, 1/τ) is generated by the substitution θ˜(a) = aba, θ˜(b) = ab. − Note that the substitution θ˜ is given by the action of several affine mappings. First, we rescale 2 2 the points by the factor τ . It corresponds to the mapping t(1)x := τ x. Then we split the enlarged tiles by adding new points into them. Let x be followed by the tile a. The splitting of the enlarged tile with left end point τ 2x according to θ˜(a) = aba means to insert new successors to the point τ 2x, by mappings 2 2 t(2)x := τ x + τ , 2 2 t(3)x := τ x + τ + τ . Let now y be followed by the tile b. The splitting of the rescaled tile b with the left end point being 2 ˜ 2 2 t(1)y = τ y according to θ(b) = ab is done by adding one new point t(2)y := τ y + τ .

7 Now it is important to decide, which quasicrystal points are left end points of tiles a with length τ 2, and which of them are left end points of tiles of the type b with length τ. Using Lemma 2.5 for 2 3 the given quasicrystal Σ[ 1/τ , 1/τ) we ﬁnd that y ΣM = Σ[1/τ , 1/τ) are left end points of tiles 2− 3 ∈ b and x ΣL = Σ[ 1/τ , 1/τ ) are the left end points of tiles a. In order∈ to show− that the substitution θ˜ generates our quasicrystal we have to prove that 1 1 Σ , = t(1)ΣM t(2)ΣM t(1)ΣL t(2)ΣL t(3)ΣL . (3) −τ 2 τ ∪ ∪ ∪ ∪ Applying (iv) and (v) of Lemma 2.3, we obtain

2 1 1 t ΣM = τ ΣM = Σ 5 , 3 (1) τ τ 2 1 1 t(1)ΣL = τ ΣL = Σ 4 , 5 − τ τ 2 2 1 1 1 t ΣM = τ ΣM + τ = Σ 5 + 2 , (2) τ τ τ 2 2 1 1 1 t ΣL = τ ΣL + τ = Σ 3 , 5 + 2 (2) τ τ τ 2 3 1 1 t(3)ΣL = τ ΣL + τ = Σ 2 , 4 − τ − τ The statement (iii) of Proposition 2.3 then gives us the result (3).

In the previous example we have shown how the Fibonacci substitution (1) generates the 2- tile quasicrystal Σ[ 1/τ 2, 1/τ). In the following example we ﬁnd a substitution rule for the 3-tile quasicrystal Σ[0, 1 +− 1/τ 2). Although the quasicrystal has only three types of tiles, we will use an alphabet formed by four letters. It will be clear that with the chosen scaling factor τ 2, one cannot ﬁnd a substitution rule with smaller alphabet.

2 Example 3.2. Let us split the acceptance window Ω = [0, 1 + 1/τ ) into intervals Ωi, i = 1,..., 4 as follows.

2 2 2 Ω1 = [0, 1/τ ) , Ω2 = [1/τ , 1/τ) , Ω3 = [1/τ, 1) , Ω4 = [1, 1 + 1/τ ) .

Recall the sets ΣS, ΣL, and ΣM of points followed by the tile S, M, L, respectively, (cf. Lemma 2.5). We have ΣS = Σ(Ω1), ΣL = Σ(Ω2), ΣM = Σ(Ω3) Σ(Ω4). Let us assign the letters to intervals Ωi, S ∪ to Ω1, L to Ω2, M1 to Ω3, and M2 to Ω4. In the construction of the substitution for our quasicrystal, we proceed in the following way: We stretch the quasicrystal by the factor τ 2 with respect to the origin. We obtain the quasicrystal τ 2Σ[0, 1 + 1/τ 2) = Σ[0, 1/τ 2 + 1/τ 4) Σ[0, 1 + 1/τ 2). The tiles of the stretched quasicrystal have lengths τ 2 times greater than S, M, L⊂, namely τ 2, τ 3 and τ 4. Let us take the left end points yI of a stretched tile with length τ 2S = τ 2. We shall look for right neighbours yII , yIII , . . . of yI in Σ[0, 1 + 1/τ 2), which ﬁll the corresponding tile τ 2S. We I 2 I 2 2 have y τ ΣS. Therefore the 0 image x = (y )0 belongs to 1/τ Ω1. Since 1/τ Ω1 Ω1, the I ∈ II ⊂ I point y belongs to ΣS, and thus its smallest right neighbour in Σ(Ω) is the point y = y + S. II 2 II Therefore (y )0 1/τ Ω1 + 1 Ω4. This means that y ΣM . Its smallest right neighbour is III II ∈ III⊂ 2 2∈ III y = y + M2. We have (y )0 1/τ Ω1 + 1 1/τ 1/τ Ω. Thus y is the right end point of the stretched tile τ 2S. Symbolicaly∈ it can be written− as⊂

1 S 1 M 1 1 1 1 S : 0, 1, 1 + 2 , + Ω . τ 4 −→ τ 4 −→ τ 2 τ 2 τ 4 ⊂ τ 2

2 We see that a stretched tile τ S is filled by tiles S and M2, respectively. Formally, we have the rule S SM2. → 8 Similarly we can proceed for left end points of other stretched tiles. It suffices to realize that the 2 2 4 3 0 images of the left end points of stretched tiles τ L belong to 1/τ Ω2 = [1/τ , 1/τ ). For left end 2 2 3 2 2 2 2 2 4 points of τ M1 we have 1/τ Ω3 = [1/τ , 1/τ ), and for τ M2 we have 1/τ Ω4 = [1/τ , 1/τ + 1/τ ). The filling of the stretched tiles is schematically described by the following formulas and illustrated on Figure 2.

1 1 S 1 1 M 1 1 1 L L : , 1 + , 1 + 2 + , τ 4 τ 3 −→ τ 4 τ 3 −→ τ 2 τ 4 τ −→

L 2 1 M 1 1 1 1 + , 1 1 + , Ω . −→ τ 2 τ 4 −→ τ 3 τ 6 τ 2 ⊂ τ 2

1 1 S 1 1 M2 1 2 M1 1 1 M1 : , 1 + , 1 + , 0, Ω . τ 3 τ 2 −→ τ 3 τ 2 −→ τ τ 2 −→ τ 4 ⊂ τ 2

1 1 1 L 2 2 1 M1 1 1 1 1 M2 : , + , + , + Ω . τ 2 τ 2 τ 4 −→ τ 2 τ 2 τ 4 −→ τ 4 τ 3 τ 6 ⊂ τ 2

S L M1 M2 x x x x x 1 1 2 1 1 1 1 1 2+ 4 1 2 2+ 4 1 1+ 3 1 0 τ τ τ τ 1 +1 τ 1+ τ 4 τ 3 τ 2 τ τ 2 τ 4 τ 2

I II S S S SM2 7→

I III IV II L L L L L SM2LM1 7→ I III II M M M M1 SM2M1 1 1 1 7→ I II M M M2 LM1 2 2 7→

IV III V III M1 M2 L S

Figure 2: Schematic representation of the construction of a substitution rule for the quasicrystal Σ[0, 1 + 1/τ 2).

Let us now make several comments to the table on Figure 2. We can see that always the first 2 and the last iterations of a given letter X,(X S, L, M1,M2 ), belong to 1/τ Ω. The second up to last iterations XII , XIII , . . . of all letters together∈ { cover disjointly} the entire acceptance window Ω. Note that the first iteration XI divide the interval 1/τ 2Ω in a way different to the division by the last iteration of letters. The points of XI correspond to left end points of stretched tiles τ 2X, and the points of last iterations correspond to right end points of the stretched tiles τ 2X. Note that it I I was necessary to split M into M1 and M2, since points from M1 and M2 jump according to different prescriptions.

9 4 Justiﬁcation of the method

In the previous Section, we have explained on examples the steps which have to be done, in order to construct a substitution rule for a given quasicrystal. In this Section, we provide an algorithm for finding a suitable splitting of the acceptance window to groups of points which will possess the same letter. We shall justificate the procedure of construction of the substitution rules from such splittings, illustrated in Example 3.2. Throughout this Section, we shall consider the acceptance window Ω = [c, d), where c 0, d > 0, 1 d c < τ. Using Lemma 2.5 above, we are able to give a prescription, how to generate,≤ one after≤ another,− the points of a quasicrystal. In each step we go from a point y Σ[c, d) to its ∈ closest right neighbour in Σ[c, d). Considering the same procedure for 0 images of points x = y0 in the acceptance window, we may define a function f :Ω Ω by → x + 1 for x [c, d 1) ,  ∈ −  1 1  x for x c + , d , f(x) :=  − τ ∈ τ (4) 1 1  x + for x d 1, c + .  τ 2 ∈ − τ  Note that if x Z[τ], then f(x) Z[τ]. Teh function f corresponds to walking through the ∈ ∈ quasicrystal. The closest right neighbour of a quasicrystal point y is f(y0) 0. Generally, the k-th (k) neighbour of y is f (y0) 0. Note that f is in fact defined not only for numbers in Z[τ], but for all x [c, d). For x / Z[τ], the image f(x) does not have such a nice meaning. However, if x Q[τ], ∈ 1 Z ∈ N 1 Z ∈ i.e. x q [τ], fro some q , then also f(x) q [τ]. The image of f(x) under the Galois automorphism∈ is well defined∈ and we may write ∈

2 x0 + 1 f(x) 0 x0 + τ . (5) ≤ ≤ For determination of the substiution rule, we shall work with the given quasicrystal Σ(Ω), and with its s times rescaled copy sΣ(Ω), which we denote by Dense and Sparse, respectively. If s is the self-similarity factor of Σ(Ω), we have Sparse Dense. It can be for example any s Z[τ], with ⊂ 2 ∈ 0 < s0 < 1. In the following, we shall work with the least of such factors, s = τ . Recall that the distances between adjacent points of Dense are 1, τ and τ 2, therefore the lengths of tiles in Sparse are τ 2, τ 3, and τ 4. Walking on points of Dense from left to right step-by-step, we come time to time to a point which belongs also to Sparse. Since τ 4 = 3τ + 2, the number of steps needed to move from one point of Sparse to another one is 5. On the other hand, any point y from Dense either belongs to Sparse, or we arrive to y from a≤ point in Sparse after at most 4 steps. Since the corresponding motion in the acceptance window Ω is described by iterations of the function f, we can formulate previous observations into the following propositions. The assertions are transparent only for x, y Z[τ]. The proof for x, y / Z[τ] follows from the fact that Z[τ] Ω is dense in Ω, and f is right continuous∈ on Ω. ∈ ∩

Proposition 4.1. For any x Ω/τ 2 there exists a positive integer n(x) 5, such that ∈ ≤ 1 f (n(x))(x) Ω . ∈ τ 2 Proposition 4.2. For any y Ω, there exists a unique x Ω/τ 2, and a unique 0 i < n(x) such that f (i)(x) = y. ∈ ∈ ≤

10 ( 1) The function f is bijective, therefore the inverse f − is well defined. Thus for a given y Ω, ( 1) ∈ the x from Proposition 4.2 can be found by iteration of f − . Let us define a function g :Ω Ω by the prescription → 2 ( i) g(y) := τ f − (y) , (6) ( i) 2 where i is the minimal non negative integer such that f − (y) Ω/τ . Note that 0 i 4, due to Propositions 4.1 and 4.2. The function g will be essential further∈ on. ≤ ≤ Now assume that we have a substitution rule with the scaling factor s, whose fixed point is the given quasicrystal Σ(Ω). The prescription of the substitution corresponds to filling of the tiles sS, sL, sM of the stretched quasicrystal sΣ(Ω), by the tiles of the original Σ(Ω), according to the function f. Consider two points x, y Σ(Ω) followed by tiles which are assigned with the same letter, say a. It means that in an arbitrary∈ iteration, say r-th, of the substitution, the sr times stretched tiles, following the points srx and sry are filled by the same sequence of distances. This r r implies that the iterations of the function f behave in the same way on points (s x)0,(s y)0. It means that for any z Σ(Ω), such that x0 < z0 < y0, the function f behaves in the same way, and therefore we can assign∈ the tile following z by the letter a. Without loss of generality, we may assume that the points assigned to the same letter form an interval in Ω Z[τ]. The acceptance window Ω is split into a finite disjoint union of intervals, corresponding to∩ letters of the alphabet. The above considerations are valid for a general self-similarity factor s. The following definition uses Proposition 4.8, thus we are back to s = τ 2.

Deﬁnition 4.3. Let α0, α1, . . . , αk be points such that c = α0 < α1 < < αk = d, and there exist ··· kS, kL, 0 kS < kL < k, for which αk = d 1 and αk = c + 1/τ. We denote by Ωi the intervals ≤ S − L Ωi := [αi 1, αi), i = 1, . . . , k. The points αi, i = 0, . . . , k 1, are called the splitting points. We − − deﬁne a mapping ϕ :Ω A, where A is an alphabet A = a1, a2, . . . , ak , by → { }

ϕ(x) = ai x Ωi . ⇐⇒ ∈ 2 2 Further, we deﬁne a mapping w :Ω/τ A∗, which to any x Ωi/τ associates the word → ∈ (2) (n(x) 1) w(x) = ϕ(x) ϕ f(x) ϕ f (x) . . . ϕ f − (x) .

We say that a splitting Ω1,..., Ωk, corresponding to splitting points α0, α1, . . . , αk 1, is proper, if 2 −2 the mapping w is constant on each Ωi/τ . The word w(x), common for all x Ωi/τ is denoted by ∈ wi. A proper splitting deﬁnes naturally a substitution rule ζ : A A∗ by ζ(ai) = wi. → Let us make several remarks to the deﬁnition above: Remark 4.4.

1. A splitting Ω1,..., Ωk is proper, if and only if for all 1 i k, the following is true: For any 2 ≤ ≤ x Ωi/τ there is a common value of n(x) =: ni, and there exist indices s0, s1, . . . , sni 1 such that∈ − 1 1 1 1 (2) (ni 1) Ω Ω , f Ω Ω , f Ω Ω , . . . , f − Ω Ω . 2 i s0 2 i s1 2 i s2 2 i sni 1 τ ⊂ τ ⊂ τ ⊂ τ ⊂ −

2. For any 1 i k, the interval Ωi is a subset of either [c, d 1), or [c+1/τ, d), or [d 1, c+1/τ). ≤ ≤ − − 2 Thus the function f is continuous on each of the intervals Ωi. It means that Ωi/τ Ωs0 2 (i) ⊂2 being an interval implies that f Ωi/τ Ωs1 is an interval. Similarly, images f Ωi/τ of 2 ⊂ Ωi/τ under all iterations of f are intervals.

11 3. Note that if 0 belongs to Ωj0 for some j0, then ϕ(0) = aj0 and therefore the word w(0) starts 2 2 with aj0 . Since 0 belongs also to the interval Ωj0 /τ , and w is constant on Ωj0 /τ , one has ( 1) w(0) = wj0 = ζ(aj0 ). Similarly, let z = f − (0), i.e. z0 is the predecessor of 0 in Dense. Denote (ni 1) 2 by i0 the index for which z Ωi . Since f 0 − z/τ = z, the word w(z) = wi = ζ(ai ) ∈ 0 0 0 ends with ai0 .

Deﬁnition 4.5. We say that a substitution θ with the alphabet B = b1, . . . , bk generates the { } quasicrystal Σ[c, d) from 0, if there exist indices i0, j0 1, . . . , k , such that ∈ { }

θ∞(bi ) θ∞(bj ) 0 | 0 ←−−−− −−−−→ is a fixed point of the substitution θ, and it is possible to assign lengths to letters b1, . . . , bk in such a way that the Delone set corresponding to the bidirectional infinite word θ∞(bi ) θ∞(bj ), with the 0 | 0 delimiter fixed at the origin, is the quasicrystal Σ[c, d). ←−−−− −−−−→ | Remark 4.6. Suppose that we have a proper splitting of the acceptance window Ω. This splitting

assigns the letters, say aj0 , ai0 , to the origin and its predecessor in the quasicrystal. According to

point 3 of Remark 4.4, for the substitution ζ given by a proper splitting, we have ζ(aj0 ) = aj0 w, for some w A∗, and ζ(ai ) =wa ˜ i , for somew ˜ A∗. Therefore ζ∞(ai ) ζ∞(aj ) is a fixed point of the ∈ 0 0 ∈ 0 | 0 substitution ζ. If we assign to letters a1, . . . , akS , the length 1,←−−−− to letters−−−−→akS +1, . . . , akL , the length 2 τ , and to letters akL+1, . . . , ak, the length τ, the fixed point is the quasicrystal Σ(Ω). The entire quasicrystal Dense may arise by filling the distances in Sparse correctly (according to the function f) by tiles of Dense. It means that except of the points of Sparse, we take also a suitable number of their right neighbours in Dense. A point y of Sparse corresponds to x Ω/τ 2 Z[τ], (0) (2) ∈ (n(x) 1)∩ where x = y0 = f (x). The neighbours of y correspond to points f(x), f (x), . . . , f − (x) in 2 the acceptance window. Recall that the function n(x) is constant on the intervals Ωj/τ of a proper 2 splitting, i.e. we can denote n(x) = nj, for any x Ωj/τ . Therefore we have the disjoint union ∈

k nj 1 − (i) 1 f Ωj Z[τ] = Ω Z[τ] . [ [ τ 2 j=1 i=0 ∩ ∩

From the right continuity of the function f, we may write also

k nj 1 − (i) 1 f Ωj = Ω , [ [ τ 2 j=1 i=0

where again the union is disjoint. According to point 1 of Remark 4.4, all splitting points are (i) 2 boundary points of some of the intervals f (Ωj/τ ). Let now the splitting point αr be a boundary (i) 2 1 of an interval f (Ωj/τ ) = [αr, ξ), for some 1 j k and 0 i nj 1. Then τ 2 [αj 1, αj) = ( i) 2 ( i) ≤ ≤ ≤ ≤ − − f − [αr, ξ), and hence αj 1 = τ f − (αr). Note that − 2 ( i) τ f − (αr) = g(αr) = αj 1 , (7) − where g was deﬁned by (6). The following properties of g will be important. Remark 4.7.

1. For x Z[τ], g(x) is also an element of Z[τ], i.e. if x0 Σ(Ω), we have (g(x))0 Σ(Ω). ∈ ∈ ∈ 12 2. If x Ω/τ 2, then g(x) = τ 2x. ∈ 3. For c / Ω/τ 2, i.e. c = 0, we have g(c + 1/τ) = g(c). If c = 0, then g(c) = c = 0, and ∈ 6 g(c + 1/τ) = g(1/τ) = 1/τ, since 1/τ 0 = τ is the predecessor of 0 in the quasicrystal Σ[0, d). − 4. Note that the equation (7) says that the set of splitting points is invariant with respect to g.

Proposition 4.8. Let the set Γ of splitting points of Ω be ﬁnite. Then the splitting is proper, if and only if Γ is invariant with respect to g. Proof. The implication is done by 4 of Remark 4.7. For the other implication ( ) recall that ⇒ (i) 2 ⇐ splitting points are always boundary points of intervals f (Ωj/τ ), and therefore the entire interval (i) 2 (i) 2 f (Ωj/τ ) is situated between splitting points. This implies that f (Ωj/τ ) belongs to some Ωp. According to point 1 of Remark 4.4, this has to say that the splitting is proper. Proposition 4.9. Let Ω = [c, d), c, d Q[τ]. Denote ∈

(j) 1 (j) Γ1 := g c + j N0 and Γ2 := g (d 1) j N0 . (8) τ ∈ − ∈

Then the set Γ := c Γ1 Γ2 { } ∪ ∪ is ﬁnite, in particular, Γ is a proper splitting of Ω. Note that c, c + 1/τ and d 1 have to be splitting points according to the deﬁnition. Due to point 3 of Remark 4.7, Γ is the− minimal set containing c, c + 1/τ and d 1, which is invariant with respect to the function g. −

Proof. We first prove that Γ1 is finite, similar argumentation then can be used to show that also (j) Γ2 is finite. Denote the sequence of points γj := g (c + 1/τ). Note that γ0 = c + 1/τ, and thus γ0 = c0 τ. For all j, the point γj [c, d). From− the relation (5), it follows∈ that for any x Q[τ] [c, d) one has ∈ ∩ 2 ( r) x0 rτ f − (x) 0 x0 , for r 0 . − ≤ ≤ ≥ Using the definition of functions f and g and the inequality above, one has

2 2 2 ( i) 0 γj0 1 4τ τ (g(γj 1))0 = τ γj0 = f − (γj 1) γj0 1 . − − ≤ − − ≤ − (Note that the index i comes from the deﬁnition of g in (6) and is always less or equal to 4.) By induction it is easy to show the following inequality for j = 1, 2,... , 1 1 γ0 4τ γ0 γ0 . (9) τ 2j 0 − ≤ j ≤ τ 2j 0 Q N 1 Z Now c [τ], thus there exists q , such that c q [τ], and according to properties of f and g, ∈ 1 ∈ ∈ also γj Z[τ]. Since γj [c, d), we have qγj [qc, qd). Thus qγj [qc, qd) Z[τ], i.e. (qγj)0 = qγ0 ∈ q ∈ ∈ ∈ ∩ j belongs to the quasicrystal Σ[qc, qd). Such set is Delone and therefore in a bounded interval, it has only ﬁnitely many points. It follows from (9) that

1 q q ( c0 τ 4τ) q (c0 τ) 4τ qγ0 (c0 τ) q c0 τ . −| − | − ≤ τ 2j − − ≤ j ≤ τ 2j − ≤ | − |

13 Therefore qγ0 belongs to q c0 τ 4qτ, q c0 τ , and we can conclude that there are only finitely j − | − |− | − | many values in the set Γ1. (j) Denote δj := g (d 1). This sequence has similar properties as γj and therefore also the set − Γ2 = δj j N0 is finite. We{ have| ∈ shown} that Γ is finite. Due to point 3 of Remark 4.7, it is invariant with respect to g. Thus we may use Proposition 4.8 to conclude that Γ defines a proper splitting of Ω. The previous proposition states that there are only finitely many splitting points. Their number determines the cardinality of the alphabet in the substitution. The main result of the paper is formulated as the following theorem.

Theorem 4.10. There exists an alphabet A = a1, . . . , ak , k 2, and a substitution θ : A∗ A∗, reproducing the quasicrystal Σ[c, d) starting from{ 0 if and} only≥ if c, d Q[τ]. → ∈ Proof. If c, d Q[τ], the substitution exists. We construct the set Γ1 Γ2, which is a proper splitting of Ω,∈ according to Propositions 4.8 and 4.9. It was shown in Remark∪ 4.6, that a proper splitting of the acceptance window defines a substitution with a finite alphabet, which generates the quasicrystal Σ(Ω) starting from the origin. From the other hand, suppose that there exists a substitution rule θ with an alphabet A = a1, . . . , ak , generating the quasicrystal Σ(Ω). Necessarily, the Perron-Frobenius eigenvalue s of {the substitution} (scaling factor of the quasicrystal) belongs to Z[τ], and is in absolute value greater than 1. Without loss of generality we may assume that s > 1, otherwise we consider the second 2 iterate of θ, which has the eigenvalue s . Since sΣ(Ω) Σ(Ω), we have s0Ω Ω, and thus s0 < 1. ⊂ ⊂ | | Note that sΣ(Ω) = Σ(s0Ω) (sZ[τ]). ∩ The substitution determines a splitting of Ω into k intervals according to letters a1, . . . , ak of the alphabet. Let us denote the splitting points by c = α0 < α1 < α2 < < αk = d. Similarly as in Example 3.1, the substitution corresponds to the action of a finite number··· of affine mappings

t(j)x := sx + pj ,

where s is the scaling factor and pj is a translation in Z[τ]. Corresponding action in the acceptance window is done by mappings

t(∗j)x := s0x + pj0 .

For the substitution of a particular letter ai we use only certain of the mappings t(j). For those t(j), there exists an index n, such that

t∗ s0(Ωi Z[τ]) Ωn . (10) (j) ∩ ⊂ Since the substitution θ generates from sΣ(Ω) the entire quasicrystal Σ(Ω), necessarily

Ω Z[τ] = t∗ s0(Ωi Z[τ]) , (11) [ (j) ∩ j,i ∩ where this is a disjoint union over suitable indices i, j. From (10) and (11) we ﬁnd that for any splitting point αj, j = 0, 1, . . . , k, there exists an αi, i = 0, 1, . . . , k, and an index ri,j, such that α = s α + p . i 0 j r0 i,j The latter is a system of linear equations with parameters p for the variables α , i = 0, . . . , k. r0 i,j i The matrix of the system is diagonally dominant and has entries in Z[τ]. Such a system of linear equations has a unique solution (α0, . . . , αk) given by the Cramer’s rule, i.e. each αi equals to the ratio of two determinants with values in Z[τ]. Therefore αi Q[τ] for i = 0, . . . , k. In particular, ∈ both c = α0, and d = αk belong to Q[τ].

14 Finding a substitution for Σ(Ω) from its point x is equivalent to ﬁnding a rule for Σ(Ω x0) from the origin. The existence of such a substitution rule implies according to the above theorem− that Ω x0 has boundaries in Q[τ]. Taking any y Σ(Ω), the set Ω y0 has also boundaries in − ∈ − Q[τ], since x0, y0 Z[τ]. Consequently, we have the following corollary. ∈ Corollary 4.11. Let x Σ(Ω). If there exists a substitution generating the quasicrystal Σ(Ω) from the point x, then for any∈y Σ(Ω), there exists a substitution generating Σ(Ω) from y. ∈ Generally the alphabet of the substitution is large, dependingly on the position of the quasicrystal acceptance window [c, d). A number of examples with c, d Z[τ] shows that the cardinality of the alphabet depends on the length of τ-expansions of c, d. For∈ the details on β-expansions of numbers see [10]. In certain cases it is possibile to reduce the alphabet of a given substitution. Let us illustrate this possibility on the substitution of the Example 3.2. Consider the quasicrystal Σ[0, 1 + 1/τ 2). Its substitution ζ may be found on Figure 2. The second iteration of ζ is

S SM2LM1 7→ L SM2LM1SM2LM1SM2M1 7→ M1 SM2LM1SM2M1 7→ M2 SM2LM1SM2M1 7→

It is obvious that the letters M1 and M2 are substituted by the same sequence of letters and therefore one may identify them. We obtain the substitution with the alphabet S, L, M , { } S SMLM L 7→ SMLMSMLMSMM M 7→ SMLMSMM 7→ It would be interesting to determine in which cases such a reduction of the alphabet of the substitution is possible, namely, to answer the following question: For which acceptance windows Ω there exists a three-letter substitution generating the corresponding quasicrystal Σ(Ω)?

5 Types of acceptance intervals

In previous sections we have explained how to construct substitution rules for quasicrystals with acceptance windows [c, d), c, d Q[τ]. Let us examine, how the presence or absence of boundary points in the acceptance window∈ may inﬂuence the substitution rule. If the boundary point d Q[τ] does not belong to the ring Z[τ], then Σ[c, d) = Σ[c, d]. Similarly, if c Q[τ], c / Z[τ], we∈ have Σ[c, d) = Σ(c, d). Thus if at least one of the boundary points of the∈ acceptance∈ interval is not an element of Z[τ], it is always possible to use the presented method, considering the acceptance window Ω = [c, d), or Ω = (c, d]. In both cases, the function f from (4) is continuous at least from one side, right continuous, or left continuous, respectively. If c, d are both elements of the ring Z[τ], and the given quasicrystal has an open or closed acceptance window (Ω = (c, d) or Ω = [c, d]), we need a special approach. The discontinuities of the function f in points c + 1/τ and d 1 are of diﬀerent types. These points must be assigned by additional letters. The resulting substitution− rule is not primitive. The letters of the alphabet are divided into two groups, a1, . . . , ak , corresponding to interiors of the intervals Ωi of the splitting, { } and b1, . . . , br , corresponding to splitting points. Note that r = k + 1 for Ω = [c, d], and r = k 1 for Ω{ = (c, d).} Clearly, by the same procedure as for the case of semiclosed intervals, we assign− to

15 all letters ai words formed only by letters aj. Together with words assigned to letters bi, we have a substitution rule with the alphabet a1, . . . , ak b1, . . . , br . Its matrix has the block form { } ∪ { } P11 0 P = . P21 P22 Since any power M n has the same form as M, the substitution is not primitive. Example 5.1. In here, we illustrate the diﬀerence between substitution rules for quasicrystals with acceptance windows [ 1/τ 2, 1), ( 1/τ 2, 1], ( 1/τ 2, 1), [ 1/τ 2, 1]. Using the algorithm, described in Sections 3 and 4, for− the quasicrystal− Σ[ 1/τ−2, 1), we obtain− the following splitting of the acceptance window −

a1 a2 a3 a4 1 1 1 0 1 −τ 2 τ 3 τ 2 3 Quasicrystal points x, such that x0 [ 1/τ , 0), are assigned with the letter a1, similarly, [0, 1/τ ) 3 ∈ − → a2, [1/τ , 1/τ) a3, and [1/τ, 1) a4, as illustrated on the picture. Using this assignment of letters, we have→ the substitution rule→ζ from Deﬁnition 4.3

a1 a1a4 7→ a2 a2a3a1a4 7→ (12) a3 a2a3 7→ a4 a3a1a4 7→ 2 2 Using the lengths 1 for a1, τ for a2 and τ for a3, a4, the quasicrystal Σ[ 1/τ , 1) corresponds to − the fixed point ζ∞(a4) ζ∞(a2). The starting points were chosen according to point 3 of Remark 4.4. | For the acceptance←−−−− −−−−→ window ( 1/τ 2, 1], the splitting is the same as before. However, the assign- − 2 3 3 ment of letters is different: ( 1/τ , 0] a1, (0, 1/τ ] a2, (1/τ , 1/τ] a3, and (1/τ, 1] a4. The substitution rule coincides− with that→ of (12), but the→ starting points have→ to be chosen differently.→ 2 The quasicrystal Σ( 1/τ , 1] is the fixed point ζ∞(a3) ζ∞(a1). − | 2 Consider now the open interval ( 1/τ , 1)←−−−−. The letters−−−−→a1,..., a4 are assigned to the open − intervals. For their boundaries, we have to introduce three additional letters b1, b2, b3, 0 b1, 3 → 1/τ b2, and 1/τ b3, → →

a1 a2 a3 a4 b1 b2 b3 1 1 1 n 0n n n 1n −τ 2 τ 3 τ The substitution rule (12) has to be extended by

b1 b1a3a1a4 7→ b2 a2a3a1a4 (13) 7→ b3 b2b3 7→ where the words were obtained in a standard way using the function f of (4). If we assign the 2 2 lengths τ to b1, b2, and τ to b3, the quasicrystal Σ( 1/τ , 1) is the ﬁxed point ζ∞(b3) ζ∞(b1) of the − | substitution ζ given by (12) and (13), with the alphabet a1, . . . , a4, b1, b2, b3 . ←−−−− −−−−→ Finally, let us consider the quasicrystal Σ[ 1/τ 2, 1].{ The boundary points} of the intervals are 2 3 − assigned by letters 1/τ c1, 0 c2, 1/τ c3, 1/τ c4, and 1 c5, − → → → → → 16 a1 a2 a3 a4 c1 c2 c3 c4 c5 1 1 1 n 0n n n 1n −τ 2 τ 3 τ Using the function f we obtain the substitution rule

c1 a1a4 7→ c2 c2c5 7→ c3 a2a3 (14) 7→ c4 c3c1c4 7→ c5 a3a1a4 7→ which together with (12) gives the fixed point ζ∞(c4) ζ∞(c2). If the length of c1, c2 is 1, and the | 2 length of c3, c4, c5 is τ, the later fixed point coincides←−−−− −−−−→ with the quasicrystal Σ[ 1/τ , 1]. If all the splitting points have their own letters, we obtain a substitution− rule which generates the given quasicrystal. However, in majority of cases, we may reduce the number of elements in the alphabet, if we examine carefully, whether while iterating the function f the boundary point does not behave in the same way as an interval adjacent to it. For example, in the rule given by (12) 2 and (14), generating the quasicrystal Σ[ 1/τ , 1], one may identify c1 a1, c3 a3, and c5 a4. − ≡ ≡ ≡ Then, it suffices to add to (12) the rules c2 c2a4, c4 a3a1c4. 7→ 7→ Recall that the substitution process starts with the pair c4 c2. From the reduced prescription, it | is clear that the letters c4, c2 occur only once in the entire infinite word, namely around the origin, denoted by the delimiter . It means that the densities of letters c2, c4 in the infinite word vanish. This can be also justified| using the substitution matrix of the rule,

1 0 0 1 0 0  1 1 1 1 0 0   0 1 1 0 0 0  M =   .  1 0 1 1 0 0     0 0 0 1 1 0     1 0 1 0 0 1  First, note that the matrix M is decomposable. It means that any of its powers contains zero elements. Such a matrix is not primitive. Both M and M T have the eigenvalue τ 2, (scaling factor T of the quasicrystal). Corresponding eigenvector of M has zero components in positions c2, c4. These are the densities of c2, c4 in the ﬁxed point.

6 Conclusion

In this paper we have shown that in addition to the well known Fibonacci chain, there is an infinite family of non equivalent quasicrystals based on the golden mean τ, which may be generated by substitution rules. It is straightforward to generalize the results of the article to the so-called quadratic unitary Pisot numbers. These are the roots of the equation x2 = mx + 1, m N, or 2 ∈ x = nx 1, n 3, n N. Denote by β and β0 the two roots of such an equation, β > 1 > β0 . − ≥ ∈ | | Similarly as for the golden mean, we have the Galois automorphism β β0, on the field Q[β]. It is possible to define the quasicrystal as a subset of the ring Z[β] := Z + Z7→β, by

Σ(Ω) := x Z[β] x0 Ω , { ∈ | ∈ } 17 where Ω is a bounded interval. It is known that such a 1-dimensional quasicrystal is a Delone set in R with at most three diﬀerent tiles, S, M, L, say. Moreover, the sets ΣS,ΣM ,ΣL of left end points of tiles S, M, L, respectively, are quasicrystals with acceptance windows ΩS,ΩM , and ΩL - three disjoint intervals. Since β is a unitary Pisot number, we have β2Z[β] = Z[β]. Similarly as for τ, this property is used for construction of substitutions. Walking by quasicrystal points step by step may be described in terms of iterations of the function f analogous to the one of (4). Hence, it is possible to generalize the whole construction procedure for any quadratic unitary Pisot number β.

Acknowledgements

We are grateful to Dr. R. V. Moody for useful suggestions and comments. We acknowledge partial support from the National Science and Engineering Research Council of Canada, FCAR of Quebec, by FR VSˇ 767/1999 of the Czech republic and by the NATO Collaborative Research Grant CRG 974230. Two of the authors (Z.M. and E.P.) are grateful for the hospitality of the Centre de recherches mathématiques,Universitéde Montréal,Québec, Canada, where the work was done.

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