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Substitution Rules for Cut and Project Sequences

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Substitution Rules for Cut and Project Sequences Substitution rules for cut and project sequences Zuzana Mas´akov´a∗ Jiˇr´ıPatera† Edita Pelantov´a ‡ 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 math´ematiques, Universit´ede Montr´eal, C.P. 6128 Succursale Centre-ville, Montr´eal, Qu´ebec, 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 sub- stitution 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]. G¨ahlerand 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.
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