Symmetric and congruent Rauzy fractals

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Symmetric and congruent Rauzy fractals Klaus Scheicher a , V´ıctor F. Sirvent b , Paul Surer a,∗ a Institut f¨ur Mathematik, Universit¨at f¨ur Bodenkultur, Gregor-Mendel-Straße 33, A-1180 Vienna, Austria b Departamento de Matem´aticas, Universidad Sim´on Bol´ıvar, Apartado 89000, Caracas 1086-A, Venezuela

Received 18 March 2015; received in revised form 18 December 2015; accepted 15 January 2016 Communicated by S.J. van Strien

Abstract Two Rauzy fractals are congruent if they differ by an affine transformation only. We give conditions on unimodular Pisot substitutions in order to ensure the congruence of the Rauzy fractals. We use these results to characterise a large family of substitutions that yield central symmetric Rauzy fractals in terms of the induced language. c 2016 Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG). ⃝

Keywords: Rauzy fractals; Substitution dynamical systems; Palindromic words; Mirror-invariant languages

1. Introduction Rauzy fractals are important geometrical objects in the study of dynamical systems associated with Pisot substitutions. Since the first emergence in [28] the geometric, topological and dynamical properties of Rauzy fractals have been analysed extensively; see the recent surveys [14,32] and within references. We want to outline the motivation to the present research with the aid of Fig. 1which shows the Rauzy fractal associated with the tribonacci substitution 1 → 12, 2 → 13, 3 → 1 as well as the Rauzy fractal associated with the reversed substitution 1 → 21, 2 → 31, 3 → 1. ∗ Corresponding author.

E-mail addresses: [email protected] (K. Scheicher), [email protected] (V.F. Sirvent), [email protected] (P. Surer). URL: http://www.palovsky.com (P. Surer). http://dx.doi.org/10.1016/j.indag.2016.01.011 c 2016 Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG). 0019-3577/⃝

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Fig. 1. The (original) Rauzy fractals associated with the tribonacci substitution is central symmetric with respect to the point c while the Rauzy fractal associated with the reversed tribonacci substitution is central symmetric with respect to the point −c.

According to [36,37] both Rauzy fractals are central symmetric with respect to the points c and −c, respectively. Actually, by [30], the second Rauzy fractal corresponds to the first one by a reflection with respect to the origin 0. In that article the symmetric intersection has been studied. However, one Rauzy fractal can obtained by the other one via a translation. The purpose of the present article is to analyse these observations in a general way. More precisely, we study how certain properties of the language generated by an (irreducible) unimodular Pisot substitution influence the geometry of the corresponding Rauzy fractal. In particular, we show that if two irreducible unimodular Pisot substitutions generate the same language then the corresponding Rauzy fractals coincide up to a translation (Theorem 3.3). For reducible substitutions this result only holds under additional conditions. In case of conjugacy (as for the two substitutions from above) we give an explicit formula for the corresponding translation vector. As an application of these results we show that for unimodular Pisot substitutions having a symmetric (or mirror invariant) language implies the corresponding Rauzy fractal to be symmetric with respect to a point (Corollary 3.6). In Theorem 5.1 we prove that if a unimodular Pisot substitution is conjugate to its reversed substitution then the Rauzy fractal is symmetric with respect to a point and we give a characterisation of this centre of symmetry. This completes the results obtained by the second author in the already cited papers [36,37] who studied symmetries of some particular families of Rauzy fractals. Using these facts we show in Section 6 that important families of substitutions have symmetric Rauzy fractals, among them the k-bonacci substitutions and Arnoux–Rauzy substitutions. However, our research suggests a lot of open questions. Especially, we will show links to up to now unsolved problems, namely, the Pisot conjecture (see, for example, [1]) and the class P-conjecture (cf. [19]). The present article is organised as follows: In Section 2 we present the basic definitions and constructions related to substitutions and Rauzy fractals where we use the approach via the so-called broken line. Section 3 contains the main results – Theorem 3.3 and Corollary 3.6 as described above – in the most general form. In Section 4 we introduce the concept of conjugacy of substitutions which ensures two substitutions to generate the same language. The main result of this section is Theorem 4.3, where we characterise the translation vector that relates the Rauzy fractals associated with the two conjugated substitutions. In Section 5, we apply the results of

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Section 4 to substitutions that are conjugated to their respected reversed substitution. It will turn out that these substitutions are closely related to the class P described in [19]. In Section 6 we give several examples of symmetric Rauzy fractals, some of them associated with important families of substitutions, as the Arnoux–Rauzy substitutions. The last section, Section 7, is dedicated to open questions and remarks for further developments of the ideas expounded in the article. 2. Notations and definitions 2.1. Substitution dynamical systems  Let A = {1, . . . , m} be a finite set (alphabet) and A∗ = i≥0 Ai the set of finite words over ∗ A, we shall denote the empty word by e. For A ∈ A and a ∈ A we denote by |A| the length of the word A and by |A|a the number of occurrences of the letter a in A. Furthermore, for a ˜ i.e. if finite word A ∈ A∗ , we denote the reversed word (sometimes called mirror-word) by A, A = a1 . . . an , then A˜ = an . . . a1 , and e˜ = e. An infinite (or one-sided infinite) word over A is a sequence v = (vn )n≥0 ∈ AN . For 0 ≤ i ≤ j we denote by v[i, j] the finite word of length j − i + 1 that is formed by the letters from the index i to the index j. For convenience we set v[i, j] = e for i > j. Consider a morphism σ over A that respects the concatenation of words, i.e. σ (AB) = σ (A)σ (B), for A, B ∈ A∗ , and σ (e) = e. The morphism σ is called a substitution if σ (a) ̸= e for all a ∈ A and there exists an a ∈ A such that the sequence (|σ n (a)|)n≥1 is unbounded. The map σ extends naturally to the set of infinite words over A. In the majority of the article we consider primitive substitution which means that there exists an n ∈ N such that for all (a, b) ∈ A2 we have |σ n (a)|b ≥ 1. For a substitution σ over the alphabet A we denote by σ˜ the substitution over the same alphabet A defined by σ˜ (a) := σ (a) for a ∈ A and, following [40], call it mirror substitution or reversed substitution. Note that σ˜ is primitive if and only if σ is primitive. The language induced by σ is the set Lσ ⊂ A∗ that consists of all of σ n (a) for a ∈ A and n ∈ N. Trivially, Lσ = Lσ n for all n ≥ 1. Following [26] each substitution σ induces a periodic point, that is an infinite word v ∈ AN such that σ n (v) = v for some n ≥ 1. Now consider a primitive substitutions σ . In this case the language Lσ is given by the finite factors of any periodic point v ∈ AN . We define Ωσ ⊂ AN to be the set of one-sided infinite words over A whose finite factors are all contained in Lσ . We endow Ωσ with the product topology. Let S : Ωσ → Ωσ be the left shift, i.e. (Sw)i = wi+1 . The pair (Ωσ , S) is called the substitution dynamical system induced by σ . The dynamical system (Ωσ , S) is minimal, i.e. every shift orbit is dense in Ωσ . Actually, in the literature Ωσ is defined as the closure of the shift orbit of a periodic point. For A ∈ A∗ define the cylinder [A] := {w ∈ Ωσ | w[0,|A|−1] = A}. The collection of cylinders forms a basis of the topology of Ωσ . Note that it is also possible to consider substitution dynamical systems that consist of the two-sided infinite sequences over A whose finite factors are all contained in L(σ ). Actually, in some contexts (especially, in combinatorial ones) it is inevitable to consider two-sided infinite sequences, see for instance [25]. However, in the present paper we will consider one-sided systems only.

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2.2. Rauzy fractals For a substitution σ let Mσ := (|σ ( j)|i )0≤i, j≤m be the incidence matrix of σ . Due to Perron–Frobenius Theorem the incidence matrix of a primitive substitution has a dominant real eigenvalue β. We call a substitution σ a (unimodular) Pisot substitution if σ is primitive and the dominant eigenvalue β is a Pisot number (unit). The substitution σ is called irreducible if the characteristic polynomial of Mσ is irreducible. We introduce the abelianisation map l : A∗ −→ Zm , A −→ (|A|1 , . . . , |A|m )T . Note that l(σ (A)) = Mσ l(A) for all A ∈ A∗ . Now consider a unimodular Pisot substitution σ and denote by vβ ⊂ Q(β)m a right eigenvector corresponding to the dominant root β. By Perron–Frobenius Theorem and primitivity we may assume vβ to be strictly positive. Furthermore, if we normalise vβ such that the sum of the entries is equal to 1, then the entries correspond to the frequencies of the symbols in the periodic point(s) of σ (cf. [26]). Let d + 1 be the algebraic degree of β. Since β is a Pisot number we have |α| < 1 for each Galois conjugate different from β. We now decompose the Euclidean space Rm into a direct sum of subspaces determined by the eigenvectors of Mσ . In particular, we have three subspaces: The expanding or unstable line E u . It is the one-dimensional subspace of Rm spanned by the eigenvector vβ associated to the dominant eigenvalue of Mσ . The contractive or stable space E s . It is the linear subspace space spanned by the right eigenvectors corresponding to the Galois conjugates different from β. We therefore have Es ∼ = Rd . The complementary or central space E c . It is the linear subspace of dimension m − d − 1 spanned by the remaining eigenvectors. For σ irreducible we have E c = {0}. We thus have Rm = E u ⊕ E s ⊕ E c . Denote by P E u , P E s and P E c the m × m matrices that realise the projections onto the respective subspaces (with respect to the Canonical real basis) such that each point x ∈ R can be uniquely represented with respect to the decomposition of Rm as x = P E u x + P E s x + P E c x. Since we only need P E s in our further proceeding we set Pσ := P E s for convenience. Note that Mσ is a contraction on E s since we have ∥Mσ x∥ ≤ |α| ∥x∥ for all x ∈ E s , where α is the Galois conjugate different from β which is the largest in modulus. Furthermore, a power of the product Pσ Mσ is a contractive matrix on Rm . Now let v be a periodic point of σ . The sequence l(v[0,n] )n≥0 of integer vectors is called the broken line. The closure of the projection Pσ of the broken line yields the Rauzy fractal Rσ induced by σ . In particular, we have Rσ := {Pσ l(v[0,n] )| n ∈ N} ⊂ E s .

(2.1)

Note that Rσ does not depend on the actual choice of v: each periodic point determines the same Rauzy fractal (cf. [13]). Observe that a Rauzy fractal is a subset of Rm that is contained in the subspace E s . In this way it is uniquely determined since the projection is uniquely determined by the decomposition of

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the Euclidean space Rm . On the other hand, for all topological considerations concerning Rauzy fractals (interior points, measures and so on) we refer to the d-dimensional subspace topology of E s and the respecting d-dimensional Lebesgue measure without mention this explicitly. Note that the embedding of E s into the real space Rd and the graphical representation (usually in the case d = 2) is not unique and depends on the choice of a basis of E s . For realising the figures in the present paper (in the case d = 2 only) we embedded the Rauzy fractals into R2 with respect to the basis B defined as follows: let α1 and α2 be the Galois conjugates of modulus less than 1. If they are real than B is formed by the normalised (right) eigenvectors associated to α1 and α2 (such that α1 > α2 ). If α1 and α2 are complex conjugates then B is formed by the normalised real part and normalised imaginary part of the (right) eigenvector associated with α1 . The Rauzy fractal naturally admits a decomposition into |A| = m subsets Rσ (a), for a ∈ {1, . . . , m}, defined by (2.2) Rσ (a) := {Pσ l(v[0,n] )| n ∈ N, vn = a}.  We obviously have Rσ = a∈A Rσ (a). This decomposition is called the natural decomposition of Rσ , (cf. [38]). Note that this union is measure-theoretically disjoint provided that σ satisfies the condition of strong coincidence. A primitive substitution over the alphabet A satisfies the condition of strong coincidence if • for all (a, a ′ ) ∈ A there exist n ∈ N, b ∈ A and P, P ′ , S, S ′ ∈ A∗ such that σ (a) = PbS, σ (a ′ ) = P ′ bS ′ and l(P) = l(P ′ ) or • for all (a, a ′ ) ∈ A there exist n ∈ N, b ∈ A and P, P ′ , S, S ′ ∈ A∗ such that σ (a) = PbS, σ (a ′ ) = P ′ bS ′ and l(S) = l(S ′ ). For more details see [5]. It is conjectured that each irreducible unimodular Pisot substitution satisfies the condition of strong coincidence (coincidence conjecture). However, the Rauzy fractal and the elements of its natural decomposition are compact subsets of E s and the closure of their interior (see [38]). In particular, Rauzy fractals do not contain isolated points. 3. Congruent and symmetric Rauzy fractals In the first part of the present section we are interested in (unimodular Pisot) substitutions that induce the same language. We want to know whether this coincidence in the language is reflected in the Rauzy fractals. In particular, we will see that for irreducible substitutions the Rauzy fractals are congruent. In the reducible case additional conditions are required. We say that a set A ⊂ Rd is symmetric (with respect to the point c), if there exists c ∈ Rd , such that for any point x ∈ A, there exists a point x′ ∈ A, satisfying c − x = x′ − c. The point c is called the centre of symmetry of A. A symmetric set with respect to a point, is also called central symmetric. In the second part of the section we will use the latter results to give a sufficient condition in terms of the language of a unimodular Pisot substitution that ensures the associated Rauzy fractal to be central symmetric. We need several preliminary considerations. Lemma 3.1. Let σ and σ ′ be two primitive substitutions over the alphabet A such that Lσ = Lσ ′ . Then the following properties hold. (a) Ωσ = Ωσ ′ . (b) Let β and β ′ the dominant eigenvalues of Mσ and Mσ ′ , respectively, and vβ and vβ ′ the corresponding right eigenvectors. Then vβ = vβ ′ (up to a constant factor).

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Proof. Statement (a) follows immediately by the definition of Ωσ and Ωσ ′ . In order to prove Statement (b), suppose that the sum of the entries of vβ and vβ ′ , respectively, equals 1. Let v be a periodic point of σ and v ′ be a periodic point of σ ′ . As already indicated, vβ contains the frequencies of symbols in v while vβ ′ contains the frequencies of symbols in v ′ . Since Ωσ is the closure of the shift orbit of v, the frequencies of symbols of each element of Ωσ is given by vβ . By (a), v ′ ∈ Ωσ and, hence, vβ = vβ ′ .  Note that the incidence matrices Mσ and Mσ ′ as well as the eigenvalues β and β ′ do not necessarily coincide (for example, if σ ′ is a power of σ ). In order to properly compare the Rauzy fractals of the two substitutions σ and σ ′ (in the unimodular Pisot case) we have to carefully study the decomposition of the real space and the associated projections Pσ and Pσ ′ , respectively. By suitable normalisation of the dominant eigenvectors (for example, by assuming that the sum of entries equals 1) we can ensure that the entries are contained in a uniquely determined algebraic extension field. Hence, algebraic conjugation of the dominant eigenvectors yields the same eigenvectors associated with the algebraic conjugates of β and β ′ , respectively. Thus, for irreducible substitutions we immediately obtain the following result. Lemma 3.2. Let σ and σ ′ be irreducible unimodular Pisot substitutions over the alphabet A with Lσ = Lσ ′ . Then the eigenvectors of Mσ and Mσ ′ induce the same decomposition into expanding and contractive subspace and Pσ = Pσ ′ . Now we suppose that σ and σ ′ are reducible. Of course, the expanding as well as the contractive space still coincide. Furthermore, for the corresponding complementary subspaces E c and E c ′ we have E c ∼ = E c ′ , too. However, in general we may not expect that Pσ = Pσ ′ . More precisely, for an arbitrary set S ⊂ Rm we cannot guarantee that the images Pσ S and Pσ ′ S differ by a linear transformation only. As important theorem of the present section we will show that the Rauzy fractals of unimodular Pisot substitutions that induce the same language coincide up to an affine transformation provided that the respective projections are the same. By Lemma 3.2 this is always the case for irreducible substitutions. For reducible substitutions we need an extra condition on the projections. Theorem 3.3. Let σ and σ ′ be unimodular Pisot substitutions over the alphabet A (|A| = m) with Lσ = Lσ ′ , and suppose that there exists an m × m matrix T such that Pσ ′ = TPσ . Then the associated Rauzy fractals are congruent, that is there exists an affine bijection T : Rm → Rm such that Rσ ′ = T (Rσ ). Proof. Let v, v ′ be periodic points of σ and σ ′ , respectively. Denote by (ki )i≥0 a monotonically ′ . In other words, the increasing sequence of non-negative integers such that (S ki v)[0,i] = v[0,i] ′ subword v[0,i] appears in v at the position ki . The existence of such a sequence is guaranteed by primitivity and the equality of language. Now observe that any subsequence of (ki )i≥1 satisfies this condition and Pσ l(v[0,ki −1] ) ∈ Rσ for all i ≥ 0, hence, is contained in a compact set. Thus, without loss of generality, we may   assume that the sequence Pσ l(v[0,ki −1] ) i≥0 converges. Let t := lim Pσ l(v[0,ki −1] ) ∈ E s . i→∞

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Now observe that for each n ≥ 0 ′ Pσ ′ l(v[0,n] ) = TPσ l(v[ki ,ki +n] ) = TPσ l(v[0,ki +n] ) − TPσ l(v[0,ki −1] ) ′ holds for all i ≥ n. Thus, Pσ ′ l(v[0,n] ) ∈ T(Rσ − t) for each n ≥ 0 and therefore

T−1 (Rσ ′ ) ⊆ −t + Rσ .

(3.1)

Finally, we show that equality must hold in (3.1). First, observe that with the same strategy as above we can show the existence of a translation vector t′ ∈ E s with T(Rσ ) ⊆ −t′ + Rσ ′ . Now the proof runs indirectly. Suppose that the inclusion in (3.1) were strict. This would yield Rσ $ −T−1 (t′ ) − t + Rσ , which is impossible, since Rauzy fractals are compact. We therefore have T(t) = −t′ and Rσ = t + T−1 (Rσ ′ ). Thus, Rσ ′ = T (Rσ ) for the affine bijection T : x → T(t + x).  Remark 3.4. Note that for each sequence (k  i )i≥0 as defined in the beginning of the proof of Theorem 3.3 we must have that Pσ l(v[0,ki ] ) i≥0 converges to t. Otherwise the proof would yield a contradiction. Observe that the condition on the projection can be omitted for irreducible substitutions only. In Example 7.7 we will see that reducible substitutions that induce the same language may yield Rauzy fractals that are not related via an affine transformation. In the rest of the section and, in fact, in the most part of the rest of the entire paper, we will compare Rauzy fractals whose underlying substitutions have the same incidence matrix. In this case the substitutions, of course, induce the same decomposition of Rm and the respective projection matrices coincide. We now turn to the problem of symmetric Rauzy fractals. Let A be a finite alphabet and L ⊂ A∗ . We call L a mirror-invariant or symmetric set if for each A ∈ L the mirror-word A˜ is contained in L, too. This definition was used in [39]. We will see that substitutions that induce mirror-invariant languages yield symmetric Rauzy fractals. In Section 5 we will meet a whole class of such substitutions that is quite well understood. One type of substitution that we can see immediately to induce a mirror-invariant language are those who coincide with their mirror-substitution. Actually, for such substitutions the symmetry of the Rauzy fractal follows by the following lemma. Lemma 3.5 (Cf. [30, Corollary 2.1]). Let σ be a unimodular Pisot substitution. Then Rσ = −Rσ˜ . As a consequence of Theorem 3.3 we immediately obtain the symmetry of the Rauzy fractal for a substitution with mirror-invariant language. Corollary 3.6. Let σ be a unimodular Pisot substitution over the alphabet A such that Lσ is mirror-invariant. Then there exists a c ∈ E s such that Rσ is symmetric with respect to c. Proof. Since Lσ is mirror-invariant we immediately see that the mirror-substitution σ˜ induces the same language, i.e. Lσ˜ = Lσ . From Mσ = Mσ˜ we deduce that Pσ = Pσ ′ . Application of Theorem 3.3 yields that there exists a translation t such that Rσ = t + Rσ˜ .

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By considering Lemma 3.5 and setting c = 12 t we obtain −c + Rσ = c − Rσ .



Note that a reversed statement does not hold in general. In Example 7.5 we will see a (reducible) substitution that has a central symmetric Rauzy fractal without inducing a mirror-invariant language. 4. Conjugacy In the present section we will study one specific way to modify substitutions without changing the language, the so-called conjugacy. Two substitutions σ, τ over the same alphabet A are conjugated if either 1. there exists a word A ∈ A∗ such that Aσ (a) = τ (a)A for all a ∈ A or 2. there exists a word B ∈ A∗ such that σ (a)B = Bτ (a) for all a ∈ A. For σ, τ conjugated we denote σ ∼ τ . The following example illustrates this notion of conjugacy. Example 4.1. Let σ and τ be substitutions over the alphabet A = {1, 2, 3} defined by σ : 1 → 1211 2 → 311 3 → 1

τ : 1 → 1112 2 → 113 3 → 1.

Then σ ∼ τ since 11σ (a) = τ (a)11 for all a ∈ A. We summarise the most important properties concerning conjugacy and conjugated substitutions in the following lemma: Lemma 4.2. The following statements hold: (a) The relation ∼ is an equivalence relation on the class of primitive substitutions. (b) For two primitive substitutions σ , τ with σ ∼ τ we have Lσ = Lτ . (c) For two primitive substitutions σ , τ with σ ∼ τ we have Mσ = Mτ . Proof. Statement (a) has been shown in [18, Lemma 2.4]. Statement (c) follows from [3, Lemma 3]. In order to prove Statement (c), notice that the incidence matrix Mσ is composed of the column vectors l(σ (1)), . . . , l(σ (m)). Thus, it suffices to show that l(σ (a)) = l(τ (a)) for all a ∈ A. Indeed, suppose first that there exists an A with Aσ (a) = τ (a)A for all a ∈ A. Then l(A) + l(σ (a)) = l(Aσ (a)) = l(τ (a)A) = l(τ (a)) + l(A). For conjugacy, according to (2) of the definition, the proof follows analogously.



Observe that in the most cases it is not necessary to consider both types of conjugacy since one shape can be transferred into the other shape by switching the role of τ and σ . From Lemma 4.2 we immediately see that conjugacy maintains the important properties of a substitution: primitivity and (unimodular) Pisot. Furthermore, by Theorem 3.3, unimodular Pisot substitutions that are conjugated induce congruent Rauzy fractals. Since the incidence matrices do not differ the respective projections matrices are the same. The Rauzy fractals coincide up to translation. As main result of this section we want to explicitly describe this translation.

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Consider two (primitive) substitutions σ and τ such that there exists A, B ∈ A∗ with Aσ (a) = τ (a)A for all a ∈ A. Observe that τ (a) does not necessarily start with A for all a ∈ A (cf. Example 4.1). This fact makes our proof a little more difficult. However, it is easy to see that τ (a) starts with the same symbol for all a ∈ A, namely, with A1 , the first symbol of A. Thus, τ has only one periodic point. From Aσ (a) = τ (a)A we obtain for a, b ∈ A Aσ (ab) = Aσ (a)σ (b) = τ (a)Aσ (b) = τ (a)τ (b)A = τ (ab)A and, more generally, ∀W ∈ A∗ :

Aσ (W ) = τ (W )A.

(4.1)

With this formula we also can express powers of σ in terms of powers of τ . Aσ (A)σ 2 (a) = Aσ (Aσ (a)) = Aσ (τ (a)A) = τ (τ (a)A)A = τ 2 (a)τ (A)A. Applying this idea several times yields ∀n ∈ N, a ∈ A :

Aσ (A) · · · σ n−1 (A)σ n (a) = τ n (a)τ n−1 (A) · · · τ (A)A.

(4.2)

Combining (4.1) and (4.2) immediately gives ∀n ∈ N, W ∈ A∗ :

Aσ (A) · · · σ n−1 (A)σ n (W ) = τ n (W )τ n−1 (A) · · · τ (A)A.

(4.3)

Theorem 4.3. Let σ, τ be two conjugated unimodular Pisot substitutions over the alphabet A such that Aσ (a) = τ (a)A for all a ∈ A. Then Rτ = t + Rσ with t = Pσ (Im − Pσ Mσ )−1 l(A), where Im denotes the m-dimensional identity matrix. Proof. The strategy will be the following: denote by w the periodic point of τ (as already observed, τ has only one periodic point). According to Theorem 3.3 we will explicitly construct a sequence (ki )i≥0 such that (S ki w)[0,i] = v[0,i] for a periodic point v of σ . The actual transformation is then given by t = lim Pσ (l(w[0,ki −1] )). i→∞

Denote by (A(i) )i≥0 and (B (i) )i≥0 the sequence of finite words given by A(i) := Aσ (A)σ 2 (A) · · · σ i−1 (A),

B (i) := τ i−1 (A) · · · τ 2 (A)τ (A)A.

Let (n i )i≥0 be an (arbitrary) sequence of strictly monotonically increasing non-negative integers such that σ n i (w0 )[0,i] = v[0,i] for a periodic on (n i )i≥0 and (A(i) )i≥0  n point v of σ . Dependent  (n )     i i we can find sequence ( ji )i≥0 with τ (w[0, ji ] ) ≥ A + i + 1. Finally, define (ki )i≥0  another  by ki :=  A(ni ) . Using (4.3) we easily calculate A(n i ) σ n i (w[0, ji ] ) = τ n i (w[0, ji ] )B (n i ) . Since w is the only periodic point of τ we have that w starts with τ n i (w[0, ji ] ). Thus, by the choice of ji and n i , we see that w[0,ki +i] = A(n i ) v[0,i] . Hence, we really have (S ki w)[0,i] = v[0,i] , as desired.

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We proceed to the calculation of the translation vector t: we have w[0,ki −1] = A(n i ) , hence, Pσ (l(w[0,ki −1] )) = Pσ (l(A(ni ) )) =

n i −1

Pσ Msσ l(A).

s=0

Now observe that by the choice and idempotence of Pσ we have P2σ = Pσ as well as Pσ Msσ = (Pσ Mσ )s for all s ≥ 1. Thus, t = lim Pσ (l(w[0,ki −1] )) = lim i→∞

i→∞

n i −1

Pσ Msσ l(A)

s=0

∞  =Pσ (Pσ Mσ )s l(A) = Pσ (Im − Pσ Mσ )−1 l(A) s=0

where we used the facts that (n i )i≥0 is a strictly monotonically increasing integer sequence and that a power of Pσ Mσ is a contraction.  As last result of this section we want to study how pairs of conjugate substitutions (over the same alphabet) behave with respect to the composition. Note that it was already shown in [18] that for substitutions σ, σ ′ , τ, τ ′ with σ ∼ σ ′ and τ ∼ τ ′ we have σ ◦ τ ∼ σ ′ ◦ τ ′ . However, in order to calculate the corresponding translation with respect to Theorem 4.3 (in the case of unimodular Pisot substitutions) we reconsider this problem and show the following two composition lemmas. Lemma 4.4. Let σ, σ ′ and τ, τ ′ be substitutions over the same alphabet A such that σ (a)A = Aσ ′ (a) and τ (a)B = Bτ ′ (a) for each a ∈ A. Then, for all a ∈ A, σ ◦ τ (a)σ (B)A = σ (B)Aσ ′ ◦ τ ′ (a). If σ ◦ τ is a unimodular Pisot substitution then Rσ ◦τ = t + Rσ ′ ◦τ ′ with t = Pσ ◦τ (Im − Pσ ◦τ Mσ ◦τ )−1 (l(σ (B)) + l(A)). Proof. At first recall that σ (W )A = Aσ ′ (W ) for all W ∈ A∗ by (4.1). Now, for each a ∈ A we easily calculate σ ◦ τ (a)σ (B)A =σ (τ (a)B)A = Aσ ′ (τ (a)B) =Aσ ′ (Bτ ′ (a)) = Aσ ′ (B)σ ′ ◦ τ ′ (a) = σ (B)Aσ ′ ◦ τ ′ (a). The second assertion follows directly by Theorem 4.3 and the additivity of the abelianisation map l.  Lemma 4.5. Let σ, σ ′ and τ, τ ′ be substitutions over the same alphabet A such that σ (a)A = Aσ ′ (a) and Bτ (a) = τ ′ (a)B for each a ∈ A. The σ ◦ τ ∼ σ ′ ◦ τ ′ . In particular, if |σ (B)| ≥ |A| then X σ ◦ τ (a) = σ ′ ◦ τ ′ (a)X with AX = σ (B) holds for all a ∈ A, otherwise we have σ ◦ τ (a)X = X σ ′ ◦ τ ′ (a) for all a ∈ A, where A = X σ ′ (B). If σ ◦ τ is a unimodular Pisot substitution then Rσ ◦τ = t + Rσ ′ ◦τ ′ with t = Pσ ◦τ (Im − Pσ ◦τ Mσ ◦τ )−1 (l(A) − l(σ (B))). Proof. At first suppose that |σ (B)| ≥ |A|. From the identity σ (B)A = Aσ ′ (B) we conclude that σ (B) = AX for some X ∈ A∗ . This immediately yields σ (B)A = AX A = Aσ ′ (B)and, hence,

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σ ′ (B) = X A. Therefore, for all a ∈ A we have AX σ ◦ τ (a)A = σ (B)σ ◦ τ (a)A = σ (Bτ (a))A = σ (τ ′ (a)B)A = Aσ ′ (τ ′ (a)B) = Aσ ′ ◦ τ ′ (a)σ ′ (B) = Aσ ′ ◦ τ ′ (a)X A, which shows the lemma for this case. The case |σ (B)| ≤ |A| runs analogously: from σ (B)A = Aσ ′ (B) we see that σ (B)X = A and X σ ′ (B) = A for some X ∈ A∗ . Thus, σ (B)σ ◦ τ (a)X σ ′ (B) = σ (B)σ ◦ τ (a)A = σ (Bτ (a))A = σ (τ ′ (a)B)A = Aσ ′ (τ ′ (a)B) = Aσ ′ ◦ τ ′ (a)σ ′ (B) = σ (B)X σ ′ ◦ τ ′ (a)σ ′ (B). The second assertion follows from the additivity of l.



When the conjugacy of σ and σ ′ is of the second shape one easily obtains similar results by switching the role of σ and σ ′ and τ and τ ′ , respectively. Observe that the these composition lemmas do not require the primitivity of the involved substitutions. 5. Substitutions that are conjugate to their mirror substitution In the present section we consider substitutions that are conjugate to their respective mirror substitutions. By Lemma 4.2 such a substitution can be easily identified to induce a mirrorinvariant language and, hence, yields a symmetric Rauzy fractal (provided that it is unimodular Pisot). In this case we can even give an explicit formula for the point of symmetry. Theorem 5.1. Let σ be a unimodular Pisot substitution over the alphabet A with Aσ˜ (a) = σ (a)A, for some word A ∈ A∗ and for all a in A. Then Rσ is central symmetric with respect to c = 12 Pσ (Im − Pσ Mσ )−1 l(A). Proof. By Theorem 4.3 we have Rσ = Pσ (Im − Pσ Mσ )−1 l(A) + Rσ˜ and by Lemma 3.5 we have Rσ˜ = −Rσ . Thus, Rσ = Pσ (Im − Pσ Mσ )−1 l(A) − Rσ and, hence, 1 1 − Pσ (Im − Pσ Mσ )−1 l(A) + Rσ = Pσ (Im − Pσ Mσ )−1 l(A) − Rσ .  2 2 We remark that if A is the empty word then c = 0. We will see that the class of substitutions that is conjugate to its mirror substitution is quite well known. Following [19] we define the class P of substitutions in the following way: σ over the alphabet A is contained in P if there exists a palindrome U ∈ A∗ and for each a ∈ A a palindrome Va ∈ A∗ such that σ (a) = U Va . A primitive substitution in the class P is known to induce a palindromic language, that is the language contains infinitely many palindromes (see [19]). It is conjectured that for each primitive substitution σ with Lσ palindromic there exists a σ ′ ∈ P such that Lσ = Lσ ′ (see Section 7.3 for further remarks). We denote by P ′ the substitutions that are conjugate to some substitution in P. In the following proposition we show that the class P ′ consists exactly of the substitutions that are conjugate to their mirror substitution. Observe that we do not require primitivity here. Proposition 5.2. Let σ be a substitution over the alphabet A. Then σ ∈ P ′ if and only if σ ∼ σ˜ .

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Proof. First we prove sufficiency. Let σ ∼ τ with τ ∈ P. For each a ∈ A we have τ (a) = U Va for palindromes U, Va ∈ A∗ . Hence, τ (a)U = U τ˜ (a) which shows that τ ∼ τ˜ . By the transitivity of the relation ∼, we get σ ∼ σ˜ . We prove necessity. Let σ ∼ σ˜ and suppose there exists a word A ∈ A∗ with σ (a)A = Aσ˜ (a) (for conjugacy of the other shape just switch the role of σ and σ˜ ). We distinguish two cases: Case 1. ∀a ∈ A : |A| ≤ |σ (a)|. We immediately see that for each a ∈ A there exists a word Va ∈ A∗ such that σ (a) = AVa . From a = A V˜a A˜ AVa A = σ (a)A = Aσ˜ (a) = A AV we conclude that A = A˜ and Va = V˜a . Therefore, A and Va are palindromes for all a ∈ A and σ ∈ P. Case 2. ∃a ∈ A : |A| > |σ (a)|. Let b ∈ A such that |σ (b)| ≤ |σ (a)| for all a ∈ A. For sure we have |σ (b)| < |A|. Thus, A = σ (b)A(1) for a word A(1) ∈ A∗ . Moreover, for each a ∈ A there exists a word Ya ∈ A∗ such that σ (a) = σ (b)Ya (where Yb = e). Inserting this into the conjugacy we obtain σ (b)Ya σ (b)A(1) = σ (b)A(1) Y˜a σ˜ (b) and, hence, Ya σ (b)A(1) = A(1) Y˜a σ˜ (b).

(5.1)

Define another substitution σ (1) by σ (1) (a) = Ya σ (b) for each a ∈ A. Note that σ ∼ σ (1) since (1) (a) for all a ∈ A. σ (a)σ (b) = σ (b)σ     Now we have  A(1)  < |A| and for all a ∈ A we have σ (1) (a) = |σ (a)| as well as σ (1) (a)A(1) = A(1) σ˜ (1) (a) by (5.1).     If A(1)  ≤ σ (1) (a) does still not hold for all a ∈ A then we repeat this process until we (n) (n) obtain a substitution  σ(n) conjugate   (n) to σ (by the transitivity of ∼) and a word A such that for     all a ∈ A we have σ (a) ≥ A and σ (n) (a)A(n) = A(n) σ˜ (n) (a). By Case 1 we have that σ (n) ∈ P and, again by the transitivity of the relation ∼, σ ∈ P ′ .



Observe that this Proposition, together with Theorem 4.3, implies that for each unimodular Pisot substitution σ with σ ∼ σ˜ there exists a (unimodular Pisot) substitution τ ∈ P such that Rτ coincides with Rσ , up to translation. In [18] it was shown that the class P ′ is closed under composition. The two composition lemmas (Lemmas 4.4 and 4.5) yield a complimentary result. Corollary 5.3. Let σ and τ be substitutions over the same alphabet A such that σ (a)A = Aσ˜ (a) and τ (a)B = B τ˜ (a) for each a ∈ A. If σ ◦ τ is a unimodular Pisot substitution then Rσ ◦τ is symmetric with respect to the centre c= and Rσ ◦τ˜ c=

1 Pσ ◦τ (Im − Pσ ◦τ Mσ ◦τ )−1 (l(A) + Mσ l(B)) 2 is symmetric with respect to the centre 1 Pσ ◦τ˜ (Im − Pσ ◦τ˜ Mσ ◦τ˜ )−1 (l(A) − Mσ l(B)). 2

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Proof. This follows by the observation that σ ◦ τ = σ˜ ◦ τ˜ and by Lemma 4.4, Lemma 4.5 as well as Theorem 5.1. Note that σ ◦ τ is a unimodular Pisot substitution if and only if σ ◦ τ˜ is a unimodular Pisot substitution.  With the aid of these results concerning composition we will be able to give explicit formulas for the centre of symmetry of Rauzy fractals associated with Arnoux–Rauzy substitutions in Section 6.4. 6. Examples and applications 6.1. The k-bonacci substitutions The k-bonacci substitution σk (with k ≥ 3) is defined by σ : 1 → 12, 2 → 13,

...,

(k − 1) → 1k, k → 1.

This is a generalisation of the tribonacci case (k = 3) introduced by Rauzy in [28]. The k-bonacci substitution satisfies the conditions of Theorem 5.1 with A = 1. Therefore the corresponding Rauzy fractal is symmetric. The symmetries of this set have already been studied in [36] (cf. Section 1). 6.2. Beta-substitutions Beta-substitutions are substitutions that are closely connected with the beta-expansion (see [8] for a more recent summary). As a principle property, all beta-substitutions (or at least powers of them) possess conjugates. By Theorem 4.3 the huge number of existing results concerning Rauzy fractals associated with beta-substitutions can be transferred to these conjugates. As an example we consider the unimodular beta-substitutions σ p,q : 1 → 1 p 2, 2 → 1q 3, 3 → 1

· · 1) (where 1 p = 1 · · · 1, 1q = 1 · p

q

for integers p, q with p ≥ q ≥ 1. The Rauzy fractals of these substitutions have been studied in [23,24,41]. We immediately see that for each positive integer r ≤ q the substitution τ p,q,r : 1 → 1 p−r 21r , 2 → 1q−r 31r , 3 → 1 is conjugate to σ p,q since σ p,q 1r = 1r τ p,q,r . Hence the associated Rauzy fractals Rσ p,q and Rσ p,q,r coincide up to translation and the translation vector t can be computed by Theorem 4.3. In Fig. 2 we see this for the example p = 3, q = r = 2 (cf. Example 4.1). Note that the intersection of the two fractals can be described by the results obtained in [29,34]. 6.3. Symmetric beta-substitutions Consider again the class σ p,q of beta-substitutions and suppose that p = q. In this case we have σ p, p (a)A = Aσ˜ p, p (a) with A = 1 p , hence, the Rauzy fractal Rσ p, p is central symmetric and we can use Theorem 5.1 to calculate the centre. The symmetries have already been studied in [37] from the point of view of numeration.

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6.4. Arnoux–Rauzy substitutions In [6] an important family of substitutions were introduced in the following way. Let    1 → 1, 1 → 21, 1 → 31, σ1 : 2 → 12, σ2 : 2 → 2, σ3 : 2 → 32,    3 → 13; 3 → 23; 3 → 3. We consider σ := σi1 ◦ · · · ◦ σin such that σ1 , σ2 and σ3 appear at least once. The resulting substitution is unimodular Pisot and irreducible (cf. [6]). However, each substitution σ j is not primitive. If we take σ = σ1 ◦ σ2 ◦ σ3 , we obtain σ = τ 3 , where τ is the tribonacci substitution from the Introduction (see also Section 6.1). The dynamics and the Rauzy fractals related to Arnoux–Rauzy substitutions have been intensively studied in various research papers (see, for example, [12,11,15,16,35]). With the theory of this article we are able to show as a complementary result that all Rauzy fractals induced by Arnoux–Rauzy substitutions are symmetric and we can determine the centre of symmetry. Indeed, it is easy to see that for each i ∈ {1, 2, 3} we have σi (a)i = i σ˜ i (a) for all a ∈ A = {1, 2, 3}. Thus, by Lemma 4.4, there exists an A ∈ A∗ such that σ (a)A = Aσ˜ (a) and we can even state A explicitly. In particular, we have  X k = i1 for k = 1, A = X n · · · X 1 with X k = σi1 ◦ · · · ◦ σik−1 (i k ) otherwise. The centre of symmetry of the Rauzy fractal Rσ can now be calculated directly by using Theorem 5.1. Fig. 3 shows the Rauzy fractal for a particular Arnoux–Rauzy substitution. Following [6] the class of Arnoux–Rauzy substitutions can be generalised to an alphabet of k-symbols (k ≥ 3) by defining    1 → 1, 1 → 21, 1 → k1,          2 → 12, 2 → 2, 2 → k2, σ1 : . σ2 : . ··· ; σk : . .. .. ..             k → 1k; k → 2k; k → k. The results expounded here can be extended in a straightforward manner to this large family of substitutions.

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Fig. 3. The Rauzy fractal associated to σ = σ2 ◦ σ1 ◦ σ2 ◦ σ2 ◦ σ3 . It is symmetric with respect to the point c which can easily be calculated by Theorem 5.1 since we have σ (a)A = Aσ˜ (a) with A = σ2 (σ1 (σ2 (σ2 (3)2)2)1)2 = 2122122123212212212.

Fig. 4. The Rauzy fractal associated to the symmetric 6 Narayana substitution (k = 3). It is symmetric with respect to the origin 0.

6.5. Symmetric 2k Narayana For k ≥ 1 define the substitution σ : 1 → 1k 21k ,

2 → 3, 3 → 1.

It is called symmetric 2k Narayana substitution which are studied, for example, in [27]. These substitutions are called after the XIV century Indian mathematician Narayana Pandit, since the corresponding linear recurrence is related to the so called Narayana sequence (cf. [4]). These substitutions are obviously unimodular Pisot substitutions (since the characteristic polynomial of Mσ equals x 3 − 2kx 2 − 1) and satisfy the conditions of Theorem 5.1, with A = e, the empty word. Therefore, the corresponding Rauzy fractals are symmetric and the centre of symmetry is the origin (see example in Fig. 4). 7. Remarks, open questions and (counter)-examples The majority of the paper was dedicated to results that, in fact, concern sufficiency in some form. Therefore, in this last section we will concentrate on the arising questions that concern

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necessity. Especially for reducible substitutions we give some examples that show that our results cannot be generalised without additional analysis. Without going into too much details we note that in these examples the substitutions have a non-trivial coboundary in the sense of [21]. Such substitution have quite special properties. It was already observed in [31] that several results concerning Rauzy fractals cannot be applied on them in full generality. In this context we also want to refer to [33]. At the end of the section we also will give some suggestions for further investigations concerning the symmetry of Rauzy fractals. 7.1. Symmetry of Rauzy fractals and mirror-invariance In Corollary 3.6 we saw that substitutions with mirror-invariant languages induce symmetric Rauzy fractals. The question now is whether mirror-invariance is necessary for the Rauzy fractal be symmetric. Here we have a partly answer. Let σ be an irreducible unimodular Pisot substitution over the alphabet A (with |A| = m) that satisfies the condition of strong coincidence (which always holds when the coincidence conjecture is true). Due to [5] the Rauzy fractal Rσ induces a multiple lattice tiling of the contractive space E s , where the lattice is given by Γ := {Pσ (z 1 , . . . , z m ) : (z 1 , . . . , z m ) ∈ Zm , z 1 + · · · + z m = 0}. We say that σ has the tiling property when this tiling is a proper tiling, i.e., the family {Rσ + t : t ∈ Γ } covers E s and int (Rσ + t) ∩ int (Rσ + t′ ) = ∅ for t, t′ ∈ Γ with t ̸= t′ . In the following considerations we concentrate on irreducible substitutions that have the tiling property (which implies the condition of strong coincidence by [2]). In this case we are able to show that the symmetry of the Rauzy fractal implies the induced language to be mirror-invariant. Lemma 7.1. Let σ be an irreducible unimodular Pisot substitution over the alphabet A that has the tiling property, and A = a1 a2 · · · an ∈ A∗ . If for an interior point x ∈ int (Rσ ) we have x + Pσ l(a1 a2 · · · ai ) ∈ int (Rσ ) for all i ∈ {1, . . . , n} then A ∈ Lσ . Proof. At first we claim the following: let y ∈ E s and a ∈ A such that y + Pσ l(a) ∈ int (Rσ ). Then, for each a ′ ∈ A with y + Pσ l(a ′ ) ∈ Rσ we have a = a ′ . We prove it by the absurd and suppose that a ̸= a ′ . This immediately implies that Rσ ∩ int (Rσ ) + Pσ (l(a ′ ) − l(a)) ̸= ∅. Since Rσ is the closure of its interior this condition is equivalent to the fact that int (Rσ ) ∩ int (Rσ ) + Pσ (l(a ′ ) − l(a)) ̸= ∅ which violates the tiling property since Pσ (l(a ′ ) − l(a)) ∈ Γ . Now observe that by the assumption of the lemma we actually have an entire open neighbourhood X ⊂ int (Rσ ) that contains x such that X + Pσ l(a1 a2 · · · ai ) ⊂ int (Rσ ) holds for all i ∈ {1, . . . , n}. By the definition of Rσ there must be an k ∈ N such that Pσ l(v[0,k] ) ∈ X where v is a periodic point of σ . Furthermore, we obviously have for all i ∈ {1, . . . , n} Pσ l(v[0,k+i] ) = Pσ l(v[0,k] ) + Pσ l(v[k+1,k+i] ) ∈ Rσ . By the claim from above this shows that we necessarily have ai = vk+i for all i ∈ {1, . . . , n} and, hence, a1 a2 · · · an = A ∈ Lσ .  Proposition 7.2. Let σ be a unimodular Pisot substitution over the alphabet A, v a periodic point of σ , and A ∈ L. Then the set Rσ (A) := {Pσ l(v[0,n] ) : n ∈ N, v[n−|A|+1,n] = A} is a compact subset of Rσ with non-empty interior.

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Proof. This can be proved without difficulties by considering Rauzy fractals as graph directed iterated function system. In fact, the actual proposition can be shown analogously to [38, Theorem 4.1]. We will not execute this explicitly here.  Theorem 7.3. Let σ be a unimodular Pisot substitution over the alphabet A that has the tiling property. If the Rauzy fractal Rσ is central symmetric then the language Lσ is mirror-invariant. Proof. Let A = a1 a2 · · · an be an arbitrary word contained in Lσ . For each i ∈ {1, . . . , n} we now consider Rσ (a1 · · · ai ) as defined in Proposition 7.2. By construction we have Rσ (a1 · · · ai ) − Pσ l(ai ) ⊂ Rσ (a1 · · · ai−1 ) for each i ∈ {2, . . . , n} and Rσ (a1 ) − Pσ l(a1 ) ⊂ Rσ . Let x be an interior point of Rσ (a1 · · · an ) and define y := x − Pσ l(a1 · · · an ). Then y ∈ int (Rσ ) and for each i ∈ {1, . . . , n} we analogously have y + Pσ l(a1 · · · ai ) ∈ int (Rσ (a1 · · · ai )) ⊂ int (Rσ ). Now use need the fact that Rσ is symmetric with respect to a point, say, c ∈ E s . Thus, for each i ∈ {0, . . . , n} we have 2c−y−Pσ l(a1 · · · ai ) ∈ int (Rσ ). Hence, 2c−x+Pσ l(an an−1 · · · an−i ) ∈ int (Rσ ) for all i ∈ {0, . . . , n − 1} and, by Lemma 7.1, we conclude that an an−1 · · · a1 = A˜ ∈ Lσ .  Observe that there exist a lot of conditions equivalent to the tiling property. A detailed overview can be obtained from [14]. Actually, it is conjectured that all unimodular Pisot substitutions satisfy the condition of strong coincidence and have the tiling property. This conjecture, which is known as Pisot conjecture, is probably the most famous unsolved problem in context with Rauzy fractals. For details and partial results we refer to the recent articles [1,9]. In view of this, Theorem 7.3 states that in the irreducible case the symmetry of the Rauzy fractal implies the mirror-invariance of the induced language provided that the Pisot conjecture holds. For the two-letter case (for which the coincidence conjecture as well as the Pisot conjecture has already been settled in [10,20]) Theorem 7.3 has a nice consequence concerning the connectedness of the Rauzy fractal. Corollary 7.4. Let σ be an irreducible unimodular substitution over a two-letter alphabet. If the Rauzy fractal Rσ is connected (hence, it is a line segment) then the language Lσ is mirror-invariant. Proof. A line segment is always central symmetric.



For the reducible case the situation is more tricky. On one hand, a reducible substitution that satisfies the so-called quotient mapping condition also induces a lattice multiple tiling (see [32]). In case of a proper tiling we can analogously apply Theorem 7.3 to show that the symmetry of the Rauzy fractal implies the mirror-invariance of the language. On the other hand, the following example shows that there exist reducible substitutions with a symmetric Rauzy fractal that do not have a mirror-invariant language. Example 7.5. Consider the Fibonacci substitution σ : 1 → 12, 2 → 1 over the alphabet A = {1, 2}. The Rauzy fractal Rσ is central symmetric (since σ satisfies the conditions of Theorem 5.1). Denote by Lσ (2) the elements of Lσ of length 2. We have Lσ (2) = {11, 12, 21}. Following [26], there exists a natural way of extending a primitive substitution σ to a (primitive and reducible) substitution σ ′ over the alphabet Lσ (2) by defining σ ′ (ab) = (a1 a2 )(a2 a3 ) · · · (an−1 an )(an b1 ) where σ (a) = a1 · · · an , σ (b) = b1 . . . bk .

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In particular, in our case this yields σ ′ : 1′ → 2′ 3′ , 2′ → 2′ 3′ , 3′ → 1′ where 1′ := 11, 2′ := 12, 3′ := 21. The incidence matrix M′σ has the same dominant eigenvalue β as Mσ and the corresponding right eigenvalue v′β (normalised such that the sum of the entries equals 1) corresponds to the frequencies of the elements of Lσ (2). Especially, σ ′ is also a unimodular Pisot substitution. We will show that the associated Rauzy fractal Rσ ′ is central symmetric although, Lσ ′ is not mirror-invariant since Lσ ′ (2) = {1′ 2′ , 2′ 3′ , 3′ 1′ , 3′ 2′ }. Let v be the (only) periodic point of σ and v ′ the (only) periodic point of σ ′ . It is easy to see ′ that for each n ∈ N we have l(v[0,n] ) = Rl(v[0,n] ) for the projection   1 1 0 R= . 0 0 1 Furthermore one verifies that we have Rv′β = vβ and, therefore, Rv′α = vα for the eigenvectors associated to each Galois conjugates α of β. This shows that Pσ R = RPσ ′ . Hence, for all n ∈ N we have ′ ′ Pσ l(v[0,n] ) = Pσ Rl(v[0,n] ) = RPσ ′ l(v[0,n] )

and therefore Rσ = RRσ ′ . This shows that Rσ ′ is also central symmetric. As already indicated, the substitution σ ′ in the above example has a non-trivial coboundary (see [17]). We do not know examples of reducible substitutions without non-trivial coboundary that yield a central symmetric Rauzy fractal without having a symmetric language. Furthermore, the Rauzy-fractal associated with σ ′ is actually a linear transform of the Rauzy fractal Rσ . We can summarise our observations and results in the following two questions. Open question 1. Does there exist a reducible unimodular Pisot substitution σ that has no nontrivial coboundary such that Rσ is central symmetric but Lσ is not mirror-invariant? (As already mentioned, such a substitution must not induce a proper lattice tiling.) Open question 2. Let σ be a unimodular Pisot substitution with central symmetric Rauzy fractal Rσ . Does there always exist a unimodular substitution τ with mirror-invariant language Lτ such that Rσ = T (Rτ ) for some linear bijection T ? 7.2. Mirror-invariance vs. palindromicity of the language In Section 5 we mentioned the notion of palindromic languages: the language Lσ of a primitive substitution σ is palindromic if it contains infinitely many palindromes. Besides the already cited ones there exist several articles about palindromic languages, especially concerning their complexity (see, for example, [7] and the references within). It seems to be less known about mirror-invariant languages. It is an easy exercise to show that the palindromicity implies Lσ to be mirror-invariant. The converse seems to be more tricky. In fact, it was shown in [39] that a mirror-invariant language Lσ is also palindromic in the case of a two-letter alphabet. For larger alphabets we cannot exclude the existence of a primitive substitution that induces a mirror-invariant language that is not palindromic. However, we do not know any (counter)example. Open question 3. Does, for primitive substitutions, the mirror-invariance of the language implies its palindromicity?

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Fig. 5. The natural decomposition of the Rauzy fractals induced by σ1 , σ2 and σ3 . We see that the elements of the natural decomposition are line segments of the same shape, independent of the substitution (where the largest one corresponds to the letter 1 and the smallest one to the letter 2). However, each substitution yields a different Rauzy fractal. Note that σ1 does not satisfy the condition of strong coincidence. Since all substitutions induce a mirror-invariant language, the Rauzy fractals are central symmetric by Corollary 3.6.

7.3. The class P conjecture The class P conjecture is a reformulation of a question stated in [19]. Conjecture 7.6 (Class P Conjecture, cf. [18,19]). Let σ be a primitive substitution with palindromic language Lσ . Then there exists a primitive substitution σ ′ ∈ P such that Lσ = Lσ ′ . As a partial result it was shown in [39] that for a primitive two-letter substitution σ that induces a palindromic language, we actually have σ 2 ∈ P ′ . On the other hand, in [22] it was shown that there exist primitive substitutions with palindromic language such that no power of it is contained in P ′ . Example 7.7 (Cf. [18,22]). Consider the substitutions σ1 : 1 → 131 2 → 312 3 → 2

σ2 :

1 → 13 2 → 1312 3 → 12

σ3 : 1 → 12 2 → 1313 3 → 13.

We have Lσ1 = Lσ2 = Lσ3 are palindromic. Furthermore, the periodic point of σ1 starting with the symbol 1 (we call it v) coincides with the periodic point of σ2 while S 2 v is the periodic point of σ3 . One also finds that v[2k,2k+1] = 1a with a ∈ {2, 3} for all k ≥ 0 (in fact, by removing the letter 1 from v we obtain the periodic point of the two-letter substitution 2 → 332, 3 → 32). All substitutions √ are reducible and have the same dominant eigenvalue (which is the quadratic Pisot unit 1 + 2), but only σ3 is contained in P ′ . Since the additional eigenvalue yields different eigenvectors in all three cases, we cannot apply Theorem 3.3. For all three substitutions s the contractive space √ √ ET is given by the one-dimensional subspace spanned by the vector u := (− 2, 2 − 2, 2) . Furthermore, we easily verify that Pσ1 l(12) = Pσ2 l(12) = Pσ3 l(12) as well as Pσ1 l(13) = Pσ2 l(13) = Pσ3 l(13), while Pσ1 l(1), Pσ2 l(1) and Pσ3 l(1) are pairwise different. Thus, from the definition of the Rauzy fractal (more precisely, from the definition of the natural decomposition), we immediately see that Rσ1 (2) = Rσ2 (2) = Rσ3 (2) − Pσ3 l(13) and Rσ1 (3) = Rσ2 (3) = Rσ3 (3) − Pσ3 l(13). On the other hand, we have Rσi (1) = (Rσi (2) ∪ Rσi (3)) + Pσ3 l(1) for all i ∈ {1, 2, 3}. Since Rσi (2) ∪ Rσi (3) is a compact set (actually, it is a line segment), we see that the Rauzy fractals cannot coincide up to translation. In fact, the sets Rσ1 (2) ∪ Rσ1 (3) and Rσ1 (1) do not have disjoint interiors (σ1 does not satisfy the condition of strong coincidence), Rσ2 (2) ∪ Rσ2 (3) and Rσ2 (1) share a common point, and Rσ3 (2) ∪ Rσ3 (3) and Rσ3 (1) are disjoint sets (see Fig. 5). Let us remark again that the three substitutions σ1 , σ2 and σ3 from the example have a non-trivial coboundary. Note that the example does not contradict the class P-conjecture.

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However, in context with symmetric Rauzy fractals the class P ′ seems to be somehow incomplete (except for the two letter case, due to the cited result from [39]). For irreducible substitutions we can adapt and rewrite the class P-conjecture in the following way. Conjecture 7.8. Let σ be an irreducible unimodular Pisot substitution with palindromic language Lσ . Then there exists a (unimodular Pisot) substitution σ ′ ∈ P such that Rσ coincides with Rσ ′ up to translation. 7.4. Symmetry of the elements of the standard partition In the actual paper we study the symmetry of the entire Rauzy fractal only. Note that the results do not depend on the condition of strong coincidence, hence, the theory holds independently of whether the elements of the standard partition overlap or not. It would be interesting to analyse the symmetric behaviour of the elements of the standard partition. Actually, for all examples of symmetric Rauzy fractals that we had so far the elements of the standard partition also seem to be symmetric but, of course, with different centres of symmetry. 7.5. Other types of symmetries In the present paper we deal only with symmetries, that are reflections through a point. Further development of the theory could be if some Rauzy fractals show some more complex symmetries. Also, it could be worth to extend this theory to Pisot substitutions that are not unimodular. 7.6. Consequences for the induced self-replicating tiling For substitutions that satisfy the condition of strong coincidence the natural decomposition of the Rauzy fractal gives rise to an aperiodic (multiple) tiling of the contractive space E s , which is frequently called the self-replicating tiling. Details can be found, for example, in [32,38]. It could be interesting to compare the self-replicating tilings of substitutions whose Rauzy fractals are congruent. Do the tilings also differ by an affine transformation? Acknowledgements The authors want to thank the anonymous referee for several useful suggestions that made it possible to improve a lot the quality and readability of the article. We want to especially mention that the statement of Theorem 7.3 was formulated as open question in a former version. The idea for the proof is due to the referee. The actual research was supported by the FWF project No. P23990. It was started during the stay of the second author in 2014 at the Universit¨at f¨ur Bodenkultur. References [1] S. Akiyama, M. Barge, V. Berth´e, J.-Y. Lee, A. Siegel, On the Pisot substitution conjecture, in: Mathematics of Aperiodic Order, in: Progress Mathematics, vol. 309, Birkh¨auser, 2015, pp. 33–72. [2] S. Akiyama, J.-Y. Lee, Overlap coincidence to strong coincidence in substitution tiling dynamics, European J. Combin. 39 (2014) 233–243. [3] J.-P. Allouche, M. Baake, J. Cassaigne, D. Damanik, Palindrome complexity, Theoret. Comput. Sci. 292 (1) (2003) 9–31. Selected papers in honor of Jean Berstel.

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