HIROMI EI, SHUNJI ITO AND HUI RAO

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1 ATOMIC SURFACES, TILINGS AND COINCIDENCES II. REDUCIBLE CASE HIROMI EI, SHUNJI ITO AND HUI RAO Abstract. Unimodular Pisot substitutions with irreducible characteristic polynomials have been studied extensively in many papers. In this paper, we consider unimodular Pisot substitutions with reducible characteristic polynomials. Atomic surface, steppedsurface and dual substitution are defined. Tiling and dynamical properties arising from atomic surface are discussed. 1. Introduction 1.1. Pisot substitution. Let A = {1,, d} be an alphabet, d 2. Let A = n 0 A n be the set of finite words over A. A substitution σ is a function σ : A A. The incidence matrix of σ is M σ = M = (m ij 1 i,j d, where m ij is the number of occurrences of i in σ(j. A substitution is said to be primitive if its incidence matrix M is primitive, i.e, M N is a positive matrix for some N. By Perron-Frobenius Theorem, a non-negative primitive matrix has a simple real eigenvalue λ that is larger than all the other eigenvalues in modulus. We call λ the Perron-Frobenius eigenvalue of M. We are interested in a special type of substitution which is called Pisot substitutions. Definition 1.1. A substitution σ is a Pisot substitution if σ is primitive and the Perron- Frobenius eigenvalue of M σ is a Pisot number. An algebraic integer is a Pisot number if all its algebraic conjugates have modulus strictly less than 1. A substitution σ is unimodular if det M = ±1; it is irreducible (or of irreducible type if the characteristic polynomial of M is irreducible over Q. The atomic surfaces, also called Rauzy fractals, of irreducible and unimodular Pisot substitutions have been studied extensively by many authors([rau, Luc, AI, SW, IR1, IR2, Sie1, Sie2, BD, CS, Pra, HZ]. Recently this subject attracts many mathematicians from different areas, because it is related to and play an inter-role among many different areas such as fractal geometry, number theory, tiling theory, ergodic theory and dynamical systems. Date: August 24, Mathematics Subject Classification 52C23, 37A45 (primary, 28A80, 11B75 (secondary. The correspondence author. The third author is supported by the JSPS Postdoc fellowship. 1

2 In previous wors, the terminology Pisot substitution actually means Pisot substitution of irreducible type. The atomic surfaces of reducible Pisot substitutions have not been touched, except for some special examples studied by Ei and Ito [EI]. The goal of the present paper is to establish basic facts on atomic surfaces of unimodular Pisot substitution of reducible type. The study of the reducible type is important for several reasons. First, it can mae us gain a whole picture of the theory, and thus have a better understanding even for the irreducible case. Secondly, several new phenomena occur in the reducible case: we have only an abstract stepped-surface and do not have a geometrical stepped-surface; we do not have a domain-exchange transformation on the atomic surfaces X; the super-coincidence condition need a proper definition. Thirdly, after we generalize the atomic surfaces to the reducible case, the β-tiling becomes a special case of the atomic-surfaces tiling, and the (W-property becomes a special case of the super-coincidence condition β-substitution and β-tilings. Let β > 1 be a Pisot number. It is well nown the special β-expansion of 1 (Renyi expansion is finite or eventually periodic. The β- numeration system is called finite type in the first case and called sofic type in the second case. According to this expansion of 1, a finite automaton is defined. It is well nown that this automaton recognizes β-expansions ([T]. This automaton is associated with a substitution, which we will call β-substitution. Suppose the β-expansion of 1 is 0.a 1 a 2... a d, then the β-numeration system is governed by the following substitution σ over {1,..., d}: 1 1 a 1 2, 2 1 a 2 3, d-1 1 a d 1 d, d 1 a d. Here 1 n denotes the word consisting of n consecutive 1 s. This substitution is primitive since at least one of the a i s is positive. A similar discussion holds for β-numeration of sofic type. It has also been shown ([Par] that β is the Perron-Frobenius eigenvalue of the incidence matrix M σ. Therefore a β-substitution is a Pisot substitution whenever β is a Pisot number. However, as the following examples show, a β-substitution is usually reducible. Example 1.1 Let K 0 and let β be the dominant root of the polynomial x 3 Kx 2 (K + 1x 1, then β is a Pisot unit and the β-expansion of 1 is d β (1 = (K + 100K1. 2

3 Hence, the associated β-substitution σ and incidence matrix M σ are 1 1 (K+1 2 K K σ : 3 4 (K 0, M σ = K The characteristic polynomial of M σ is. x 5 (K + 1x 4 Kx 1 = (x 3 Kx 2 (K + 1x 1(x 2 x + 1. We will call this substitution Hoaido substitution since its atomic surface loos lie the map of Hoaido, the north part of Japan. In [EI], this example has been studied in detail. X4 X 2 X 1 X 5 X 3 Figure 1. Atomic surfaces 5 i=1 X i of Hoaido substitution. Example 1.2 Let K 2 and let β be the dominant root of the polynomial x 3 Kx 2 + Kx 1, then β is a Pisot number and d β (1 = (K 1(K 101. Hence, the associated β-substitution τ and incidence matrix M τ are 1 1 (K τ : (K 1 3 (K 2, M 3 4 τ = 4 1 The characteristic polynomial of M τ is K 1 K x 4 (K 1x 3 (K 1x 2 1 = (x 3 Kx 2 + x 1(x + 1. Through this paper, substitution τ will always refer to this substitution. 3.

4 X 1 X4 X 2 X 3 Figure 2. Atomic surfaces i=1,2,3,4 X i of substitution τ. Thurston [T] constructed a self-similar tiling from β-numeration system when β is a Pisot number and an algebraic unit (we will call it a Pisot unit in short. This β-tiling system has been studied in detail by Aiyama ([A1] [A2]. We will show in a sequel paper [EIR] that the β-tiling is a special case of atomic surfaces tiling. Precisely, the collection J defined in Section 6 is a refinement of the β-tiling. This is one of the main motivation of investigating the atomic surface of the reducible case Main results. Let σ be an unimodular Pisot substitution over d letters. Let λ be the Perron-Frobenius eigenvalue of the incidence matrix M. Let g(x be the minimal polynomial of λ, then m = deg g(x < d. 2. Projections. It is nown that in the irreducible case, there is a natural projection from R d to the stable space of M, and the atomic surfaces are defined by projection methods ([Rau, AI, IR2]. However, in the reducible case, the first difficulty is how to define projections. Let λ 1 = λ, λ 2,, λ m be the roots of g(x, then they are eigenvalues of M as well as A 1. Let V be the eigenspace corresponding to the eigenvalues λ 1, and let P be the direct sum of the eigenspaces corresponding to the eigenvalues λ 2,..., λ m. Denote P 1 = P V. Proposition 1.2. There is a unique M-invariant subspace V 1 satisfying R d = P 1 V 1. The spaces P, V and V 1 are M-invariant subspace, where P is stable and V is unstable.according to the direct sum R d = V 1 P V, we define projections φ : R d P 1, π : R d P, π : R d V. 3. Atomic surface and self-similarity. The following notations are used throughout this paper. Let e 1,..., e d be the canonical basis of the Euclidean space R d. First let us define a map f : A Z d which sends a finite word to an integer in R d as follows: (i f(ɛ = 0, where ɛ denotes the empty word; (ii f(i = e i for 1 i d and f(uv = f(u + f(v for U, V A. Clearly f σ = M f. 4

5 We write σ(i = W (i 1... W (i l i, i = 1, 2,..., d. Let P (i = W (i 1... W (i 1 be the prefix of σ(i with length 1, and denote by S (i = W (i W (i l i the suffix of σ(i. Then σ(i can be written as σ(i = P (i W (i S(i. Now in the same manner as the irreducible case, the atomic surfaces are defined by projection methods. Let ω = s 1 s 2... be a periodic point of σ. Let Y = {f(s 1... s 1 : 1}. The atomic surface of σ is the closure X = π(y of the projection of Y onto the contractive space P. Furthermore, let Y i = {f(s 1... s 1 : s = i, 1}, and set X i = π(y i, 1 i d. We call the family {X i } 1 i d the partial atomic surfaces of σ. Clearly X = d i=1x i. The partial atomic surfaces {X 1,..., X d } are compact subsets of the stable space P and they have a self-similar structure. Theorem 1.3. Let σ be an unimodular Pisot substitution. Then the partial atomic surfaces {X i } d i=1 are compact and satisfy the following set equations M 1 X i = d j=1 W (j =i ( X j + M 1 π(f(p (j. ( Stepped-surface. The stepped-surface and dual substitution are powerful tools to study the set equations (1.1, as they do in the irreducible case. But the structure of the stepped-surface in the reducible case is much more complicated. The stepped-surface S is defined as a set of colored points on the lattice L = φ(z d, where these points are very close to the stable space P. We will use [x, i ] to denote an element of S, where x is an point in L and i indicates the color of x. Now the number of colors is larger than the ran of L. This fact causes many difficulties. First, we do not have an obvious geometrical representation of the stepped-surface, and we do not now whether such a representation exists in general. (Because of this, we do not now whether the atomic surfaces can tile the space P periodically or not. So we have only a abstract stepped-surface. Nevertheless, we could show that it is quasi-periodic. Theorem 1.4. The stepped-surface S of a Pisot substitution is quasi-periodic. 5. Dual substitution. According to this set equations (1.1, we define the dual substitution σ. An important fact is the dual substitution eeps the stepped surface S invariant. Theorem 1.5. If σ is an unimodular Pisot substitution, then the stepped surface is invariant under the action of the dual substitution σ. Precisely, (i σ (S = S. (ii σ [x, i ] σ [y, j ] =, for [x, i ], [y, j ] S with [x, i ] [y, j ]. 5

6 6. Self-similar tiling system. Based on the above discussion, now we can prove that the partial atomic surfaces {X 1,..., X d } form a self-similar tiling system. Precisely, Theorem 1.6. Let σ be an unimodular Pisot substitution. Then (i The interiors of the partial atomic surfaces X i are not empty. (ii The right side of (1.1 consists of non-overlapping unions. (iii X i = X i and X i has Lebesgue measure 0, where X i denotes the boundary of X i. In the irreducible case, this theorem is proved in several different ways ([AI, SW]. Here we use a result of Lagarias and Wang [LW] on substitution Delone set to give a quic proof. The idea of [LW] is to construct a dual system of the set equations (1.1. If a family of Delone sets satisfies the dual system, then Theorem 1.6 holds for the original system. We will show that, by Theorem 1.5, the projection of the stepped-surface is a solution of the dual system. Combining the atomic surfaces and the stepped-surface together, we define the following collection J := {π(x + X i ; [x, i ] S}. We will see that J is a self-replicating collection(see Section 5. From the quasi-periodicity of the stepped-surface and the self-replicating property of J, one has Theorem 1.7. There exists a constant, such that almost every point of the space P is covered by pieces of tiles in J. Figure 3. Non-periodic tiling J of Hoaido substitution. The problem whether J is always a tiling is an open problem. In a sequel paper [EIR], we will introduce the super-coincidence condition for reducible Pisot substitution and show that J is a tiling if and only if σ satisfies super-coincidence condition. We will also show there that for β-substitution, the super-coincidence condition is equivalent to a (W-property introduced by Aiyama [A2]. Several classes of Pisot numbers are proved to have (W-property ([FS, ARS] and hence the associated collections J are tilings. 6

7 Figure 4. Non-periodic tiling J of substitution τ. 7. Collection J 2 and Marov partition. As an analogue to the reducible case, we construct a family { ˆX 1,..., ˆX d }. Let ˆX = d i=1 ˆX i. Set It is shown that J 2 := { ˆX + z : z L}. Theorem 1.8. The collection J 2 is a tiling of P 1 if and only if J is a tiling of P. Since the matrix M is unimodular, it can be regarded as a group automorphism of the m-dimensional torus P 1 \ L. Theorem 1.9. If J 2 is a tiling, then the collection { ˆX 1,..., ˆX d } is a Marov partition of group automorphism M on the torus P 1 \ L. In the irreducible case, this result has been proved for some given substitutions in [Rau, IO, Pra]; it is announced in [AI] and proved in unpublished [Sie2]. For completeness, we give a proof for the reducible case, in spite that it is very similar to the proof of the irreducible case. 8. Can X tile P periodically? In the irreducible case, the atomic surface X can tile P periodically. Actually this tiling is a lattice tiling. If σ satisfies a strong coincidence condition, then a domain exchange transformation is defined on X ([AI]. Furthermore, if σ satisfies a super-coincidence condition, then X can be regarded as a (d 1-dimensional torus and the domain exchange transformation E(x is a rotation on the torus ([IR2]. In the reducible case, the existence of geometrical representation of the stepped-surface is unclear. Hence in general, we do not now can X tile P periodically or not. This is the second difference between the irreducible case and the reducible case. Ei and Ito have studied this problem for the substitutions in Example 1.1 and Example 1.2. By trial and error method, they found geometrical representation of the steppedsurface of these substitutions, and constructed domain exchange transformation. 7

8 2. Projections 2.1. Direct sum decomposition. Recall that M = M σ is the incidence matrix of σ, g(xh(x is the characteristic polynomial of M, and g(x is the minimal polynomial of a Pisot number λ. By a standard result about integral matrix (See Newman [New], there is an integral d d matrix Q with det Q = ±1 such that ( Q 1 A1 0 MQ =, (2.1 C A 2 where A 1, A 2, and C are integral matrices and the characteristic polynomials of A 1 and A 2 are g(x and h(x respectively. Let λ 1 = λ, λ 2,, λ m be the roots of g(x, then they are eigenvalues of M as well as A 1. We arrange them in such an order: the real roots are ahead of the complex roots, and for a complex root λ i we put its complex conjugate λ i next to it to mae a pair. Let v = v 1, v 2,, v m be the eigenvectors of M corresponding to the eigenvalues λ 1, λ 2,, λ m. Corresponding to each real eigenvalue there is a 1-dimensional, M-invariant real eigenspace. For a pair of complex eigenvalues λ i and λ i+1 = λ i, we choose v i+1 = v i ; hence there is a 2-dimensional, M-invariant real eigenspace {c i v i + c i+1 v i+1 : c i+1 = c i, c i C}. Let us denote by P 1 the direct sum of these eigenspaces, i.e., P 1 = {c 1 v c m v m : c i R if λ i R; c i+1 = c i C if λ i+1 = λ i } (2.2 Hereafter we will use v 1,, v m as a basis of P 1 in the above sense. The following Proposition is a stronger version of Proposition 1.2. Proposition 2.1. There is a unique M-invariant subspace V 1 satisfying R d = P 1 V 1. Furthermore, if Q is a matrix satisfying (2.1, then ( 0 V 1 = Q R d m. Proof. Let Q be a matrix satisfying (2.1 and let M = Q 1 MQ. Then P 1 = Q 1 P 1 is the eigenspace of M with respect to the eigenvalues λ1,..., λ m. Set Ṽ 1 = {(x 1, x 2,..., x d T R d ; x 1 = = x m = 0}. Clear Ṽ1 is M-invariant and R d = P 1 Ṽ1 is a direct sum. To prove the proposition, we need only to show that Ṽ1 is the only one satisfying these two requirements. Suppose R d = P 1 V and V is M-invariant. Let s 1, s 2,..., s be a basis of V, where = d m. Then s i, 1 i, can be unique written as It is seen that w 1, w 2,..., w is independent. 8 s i = u i + w i, u i P 1, w i Ṽ1. (2.3

9 Since V is M-invariant, the vector Ms i (1 i is a linear combination of s 1,..., s. Hence there is a matrix L such that Hence, by (2.3 and R d = P 1 Ṽ1, we have It follows that M(s 1, s 2,..., s = (s 1, s 2,..., s L. M(w 1, w 2,..., w = (w 1, w 2,..., w L. A 2 (w 1, w 2,..., w = (w 1, w 2,..., w L, where w i is the vector in R such that wi T = (0,..., 0, (w i T. Now the linear independence of w i implies that L is similar to A 2. Suppose the eigenvector of M corresponding to the eigenvalue λi is β i. Then s i can be written uniquely as s i = c i1 β 1 + c i2 β c im β m + w i. On one hand, On the other hand, Ms i = λ 1 c i1 β 1 + λ 2 c i2 β λ m c im β m + Mw i. Ms i = (s 1, s 2,..., s Le i. By comparing the coefficients of β j in s 1,..., s, we conclude that λ j (c 1j, c 2j,..., c j = (c 1j, c 2j,..., c j L. If (c 1j, c 2j,..., c j is not zero, then λ j is an eigenvalue of L as well as an eigenvalue of A 2, which is a contradiction. Hence we have c ij = 0 for all 1 i, 1 j m. Therefore s i = w i and so that V = Ṽ1. We may decompose P 1 into P 1 = P V, where V is the eigenspace corresponding to λ, and P is the direct sum of the eigenspaces corresponding to λ 2,..., λ m. Clearly both P and V are M-invariant Projections. According to the direct sum R d = P 1 V 1, a natural projection from R d to P 1 is defined: φ : R d P 1. According to the direct sum P 1 = P V, we define two natural projections We define projections π : R d P and π : R d V by π 1 : P 1 P, π 1 : P 1 V. (2.4 π := π 1 φ, π := π 1 φ. Since P 1, V 1, P and V are M-invariants subspace, all the above projections are commutative with M. 9

10 Lemma 2.2. The relation ϕ M = M ϕ holds for ϕ {φ, π 1, π 1, π, π }. Let e 1,, e d be the canonical basis of R d, let, denote the inner product of the Euclidean space R d. Let v be a positive vector of t M σ with respect to the eigenvalue λ. Then we have the following each facts. Lemma 2.3. The following properties hold. (i v (P V 1. (ii φ(e i, v > 0 holds for 1 i d. Proof. (i For any vector y, we have y, v = y, λ ( t M v = λ M y, v. If y P, since λ 2,..., λ m have modulus less than 1, it is easy to show that lim λ M y = 0. Now we show it is still true when y ( V 1. Let Q be a matrix satisfying (2.1 and let z 0 be a vector in R d m such that y = Q. Then z λ M y = λ Q M ( 0 z ( = Q 0 λ A 2z Since the spectrum radius ρ(λ 1 A 2 < 1 by Definition 1.1, we have that lim (λ 1 A 2 = 0. (ii. Let y = e i φ(e i V 1. Then φ(e i, v = e i y, v = e i, v > Atomic surfaces: definition and self-similarity 3.1. Inflation and substitution map. For x Z d, we denote by (x, i := {x + θe i ; θ [0, 1]} the segment from x to x + e i. Define y + (x, i := (x + y, i. Let G = {(x, i : x Z d, 1 i d}. The term segment will refer to any element of G. The inflation and substitution map F σ is defined as follows on the set of subsets of G. Define and for K G, F σ (0, i := l i =1 {(f(p (i (i, W }, 1 i d. F σ (x, i := {Mx + : F σ (0, i}, F σ (K := {F σ ( : K}. Rigorously we should use F σ {(x, i} instead of F σ (x, i. By the definitions of the incidence matrix and f, one has M σ n = M n σ and f(σ(u = M σ f(u for every word U, and so that F σ n = F n σ. 10

11 3.2. Atomic surfaces. A infinite word over {1, 2,..., d} is a fixed point of σ if σ(s = s, is a periodic point of σ if σ (s = s holds for some > 0. A primitive substitution has at least one periodic point. For a finite or infinite word s = s 1... s n..., a broen line s starting from the origin is define as follows: s = i 1{(f(s 1... s i 1, s i }. Recall that π = π 1 φ is a projection from R d to the contractive subspace P. Let ω = s 1 s 2... be a periodic point of σ. Let Y = {f(s 1... s 1 : 1} be the set of integer points located on ω. Then the atomic surface of σ is the closure X = π(y of the projection of Y onto the contractive space P. Furthermore, let Y i = {f(s 1... s 1 : s = i, 1}, and X i = π(y i, 1 i d. We call the family {X i } 1 i d the partial atomic surfaces of σ. As we have expected, the partial atomic surfaces have a self-similar structure. Theorem 1.3 Let σ be an unimodular Pisot substitution. Then the partial atomic surfaces {X i } d i=1 are compact and satisfy the following set equations M 1 X i = d j=1 W (j =i ( X j + M 1 π(f(p (j. (3.1 The formula of the above set equations has exactly the same form as the irreducible case. Proof. (i By the definition of F σ, we see that every element of Y can be expressed as y = N =0 M a, where a belongs to the finite set {f(p (i j : 1 i d, 1 j l i }. (If s is a periodic point satisfying σ (s = s, we replace σ by σ. By the Pisot assumption, we can choose a real θ such that λ j < θ < 1 holds for 2 j m. Since π(a are points of the contractive eigenspace P, we have π(y = N M π(a < C =0 θ = =0 C 1 θ, where C > 0 is a constant depends on the set {f(p (i j : 1 i d, 1 j l i }. So X is bounded in the ball B(0, C/1 θ P and that X i are compact. (ii First we consider the case that s is a fixed point of σ. In this case, we show that Y i = d j=1 W (j =i (MY j + f(p 11 (j. (3.2

12 Let us use the notation (Y i, i to denote the set {(y, i; y Y i }. Then the stair of the periodic point can be expressed as s = d i=1 (Y i, i and (Y i, i = {(y, i : (y, i s} = {(y, i : (y, i F σ ( d j=1 (Y j, j} = d j=1 {(y, i : (y, i F σ(y j, j} = d (j j=1 =i{(my + f(p, i : y Y j}. W (j Taing the starting points of the line segments in the above equation, we get (3.2. Formula (3.1 is obtained by taing the projection π and then taing a closure on both sides of (3.2. (iii Suppose s is a periodic point of σ, say σ (s = s. Hence by (ii, X 1,..., X d defined by s satisfy the set equation (3.1 for the substitution σ. Notice that the -th iteration of the set equation (3.1 for σ coincides with the set equation (3.1 for σ, hence the set equations for σ and that for σ have the same invariant sets (see [Fal]. So X 1,..., X d satisfy the set equation (3.1 for σ. Since the compact solution of (3.1 is unique ([Fal], it is seen that the atomic surfaces do not depend on the choices of the periodic points. 4. Stepped-surface In the reducible case, the stepped-surface and dual substitution are powerful tools to study the atomic surfaces, as they do in the irreducible case. In this section, we investigate the stepped-surface Definition of the stepped-surface. Set L := φ(z d ; it is seen that L is a full lattice (of ran m in P 1 and {φ(e j } d j=1 is a group of redundant generators of L. Let P + := {x P 1 : x, v 0} and P := {x P 1 : x, v < 0}. From its definition, P + denotes the half space of P 1 above the subspace P, according to the direction given by v. We now intend to associate to any point on L a color described by a letter on A. Formally, let us denote G := {[x, i ] : x L, 1 i d} the set of such colored points. We define S, the stepped-surface of P, to be the set of nearest colored points [x, i ] above P, meaning that x belongs to P + whereas x φ(e i doesn t. S := {[x, i ] G : x P + and x φ(e i P }. (4.1 Notice that a point x in L may have several colors. For u R d, denote by P + u the translation of P. That is, P + u is a hyperplane which is parallel to P and contains u. The space in P 1 between P and P + u, including 12

13 P and excluding P + u, is denoted by [P, P + u. As a consequence of the definition of S, [x, i ] S if and only if x [P, P + φ(e i Geometrical representation of the stepped-surface. In the irreducible case, there is a natural geometrical representation of S. To any colored integer point [x, i ], one associates a face of a unit cube of R d whose lowest vertex is in x and that is orthogonal to the direction e i. We denote this face by [x, i ], namely [x, i ] := {x + c 1 e c i e i + + c d e d R d : c i = 0, 0 c 1 if i}. Then S := [x, i ]. [x,i ] S is an approximation of the space P, and sometimes it is also called the stepped-surface of P. Let us denote [i ] = [0, i ] the faces of the unit cube placed in zero. Projecting the stepped-surface S to P, we obtain a polygonal tiling of P, J := {π[x, i ] : [x, i ] S}, (4.2 which uses d polygons π[1 ],..., π[d ] as prototiles. Unfortunately all these nice properties are uncertain in the reducible case. This is the first big difference between the irreducible case and the reducible case. Here we present a possible definition of a geometrical stepped-surface. If there exist polyhedrons of P such that {T 1, T 2,..., T d } {π(x + T j ; [x, j ] S} (4.3 is a tiling of P, then we say this tiling is a geometrical representation of the stepped-surface S. It is very unclear that whether there always exists a geometrical stepped-surface. However, Ei-Ito found very interesting geometrical representation for Hoaido tiling (cf. [EI] Quasi-periodicity of the stepped-surface. Let Λ be a subset of R d. Λ is said to be uniformly discrete if there exists a positive real number r 0 such that, for any x, y Λ, x y > r 0. Let E be a subset of R d. Λ is relatively dense in E if there is a positive real number R 0 such that every sphere with center in E and with radius greater than R 0 contains at least one point of Λ in its interior. A set Λ is said to be a Delone set in E, if it is uniformly discrete and it is relatively dense in E. (See for example [Sen]. Let J be a translation tiling of R d with finite prototiles. Denote by B(x, r the ball centered on x and with radius r. We call P(B(x, r = {T : T J and T B(x, r } 13

14 Figure 5. Geometrical representation of stepped-surface of Hoaido substitution. Figure 6. Geometrical representation of stepped-surface of substitution τ. the local arrangement in B(x, r. If for any r > 0, there exists R > 0 such that for any x, y R n, the local arrangement of B(x, r appears (up to translation in the ball B(y, R, then we say that J is quasiperiodic. Now we define the quasi-periodicity for a stepped-surface defined in Section 4. Let S 0 = {x : [x, i ] S} be the base set of S. For z S 0, we define the local arrangement of a ball B(z, r in S as P(B(z, r = {[x, i ] : [x, i ] S and x B(z, r}. Definition 4.1. A stepped-surface S is said to be quasi-periodic provided that it satisfies the following property: for any r > 0, there exists R > 0 such that for any x, y S 0, the local arrangement of B(x, r appears (up to translation in the local arrangement of the ball B(y, R in S. Theorem 1.4 The stepped-surface S of a Pisot substitution is quasi-periodic. In the irreducible case, this theorem has been proved in [ABI] by associating the stepped-surface with a certain dynamical system on the real line. We do not now whether their method still wors for the reducible case. Here we give a direct proof. 14

15 Proof. We use D(A, B = inf{ a b ; a A, b B} to denote the distance between two sets in R d. First, we rearrange these hyperplanes P, P + φ(e 1,..., P + φ(e d from below to above, and denote them by P 0, P 1,..., P d after arrangement. The hyperplanes P, 0 d, divide the space into d + 2 parts R 0, R 1,..., R d+1 from below to above. Notice that R 0 = P and R d+1 is the space above (and including the hyperplane P d. Since [y, i ] S if and only if y [P, P + φ(e i, we deduce that all points of L in a region R have the same color-configuration. Precisely, an integer point in R 0 and R d+1 has no color, an integer point in R, 1 d, has exact d + 1 colors. For any y L, there is a unique j {0, 1,..., d + 1} such that y R j ; we set { D(y, Pj if j < d + 1 (y := + if j = d + 1. By definition (y > 0 always holds. P 0 P 1 P 2 P 3 V (y o R 0 R 1 R 2 R 3 R 4 Let z S 0 and r > 0. We consider the distribution of the local arrangement of P(B(z, r in the stepped-surface. Let ɛ(z, r := min{ (y : y B(z, r L}. Then ɛ(z, r > 0 since B(z, r L is a finite set. Suppose P d = P + ẽ max, i.e., ẽ max is the vector in φ(e 1,..., φ(e d such that π (ẽ max attains the maximal value. Choose N large enough such that the distance between P and P +ẽ max /N is less than ɛ(z, r. We claim that B(z, r and B(y, r have the same local arrangements for y [P + z, P + z + ẽ max /N L. Set T (x = x + (y z; it is an one-one from B(z, r L to B(y, r L. Tae an integer x B(z, r. Let be the unique number such that x R. If = d + 1, then clearly T (x still belongs to the region R since y z is in the positive direction (according to v. If < d + 1, then D(P + x, P + T (x < D(P + x, P + x + ẽ max /N < ɛ(z, r < D(P + x, P, which means that T (x is between P + x and P and so that T (x is still in the region R. Hence x B(z, r has the same color as T (x B(y, r. Our claim is proved. 15

16 We remain to show that [P + z, P + z + ẽ max /N L is relatively dense in S 0. Since S 0 = N 1 =0 [P + ẽ max/n, P + ( + 1ẽ max /N is a finite union of translations of [P, P + ẽ max /N, we have [P, P + ẽ max /N is relatively dense in S 0. Therefore [P + z, P + z + ẽ max /N L is a Delone set, and this completes the proof. 5. Dual substitution For [y, i ] G, we define x + [y, i ] := [x + y, i ]. According to set equations (3.1, we define the dual substitution σ for every subset of G as follows. and for any K G, σ [0, i ] := d j=1 {[M 1 φ f(p (j, j ]}, 1 i d, W (j =i σ [x, i ] := {M 1 x + : σ [0, i ]}, σ (K := {σ ( : K}. For simplicity, we use σ [x, i ] to denote σ {[x, i ]}. Using the dual substitution, formula (3.1 is equivalent to M 1 X i = π(x + X. (5.1 [x, ] σ [0,i] Arnoux and Ito ([AI] showed that the dual substitution eeps the stepped-surface invariant in the irreducible case. Here we generate this remarable property to the reducible case. Theorem 1.5 If σ is an unimodular Pisot substitution, then the stepped surface is invariant under the action of the dual substitution σ. Precisely, (i σ (S = S. (ii σ [x, i ] σ [y, j ] =, for [x, i ], [y, j ] S with [x, i ] [y, j ]. The proof is very similar to that of the irreducible case in [AI]. Proof. We decompose the proof into the following three claims. Claim 1. If [x, j ], [x, j ] S and [x, j ] [x, j ], then σ [x, j ] σ [x, j ] =. Suppose on the contrary that [y, i ] belongs to the intersection of σ [x, j ] and σ [x, j ]. Then, by definition of σ, there exist, such that It follows that x + φ f(p (i y = M 1 (x + φ f(p (i = M 1 (x + φ f(p (i. = x + φ f(p (i. 16

17 If x = x, then φ f(p (i P (i = 0, and so that f(p (i P (i = 0 by Lemma 2.3 (ii. It follows that = and that j = W (i = W (i = j. Hence we get [x, j ] = [x, j ], which is a contradiction. If x x, then by the above discussion. Let us assume that < without loss of generality. Then Notice that W (i x φ f(w (i W (i 1 = x. = j and f(w (i = e j, so x, v = x φ f(w (i W (i 1, v = x φ(e j, v φ(f(w (i +1 W (i 1, v The first term of the last line is negative since [x, j ] S, the second term is negative by Lemma 2.3(ii. So (x, v < 0 and x P. Again we get a contradiction and our claim is proved. Claim 2. σ (S S. It suffices to show that if [x, i ] S and W (j = i, then [M 1 (x + φ f(p (j, j ] S. On one hand, x P + implies x+φ f(p (j P + and so that M 1 (x+φ f(p (j P +. On the other hand, from M 1 (f(σ(j = e j we have M 1 ( x + φ f(p (j φ(e j = M 1 ( x + φ f(p (j φ f(σ(j = M 1 ( x φ(e i φ f(s (j. Now [x, i ] S implies x φ(e i P, so the vector in the above formula belongs to P. Therefore [M 1 (x + φ f(p (j, j ] S. Claim 3. S σ (S. Suppose [x, i ] S. Then x = x φ(e i P and that M(x P. Since f(σ(i = Me i, we deduce that M(x {φ f(p (i 1,, φ f(p (i l i, φ M(e i } determines a path (on L from M(x to M(x, i.e, from P + to P. This path must go through the hyperplane P. Suppose it intersects P at the -th step, then we have [Mx φ f(p (i, j ] S where = j. Hence by the definition of σ we have W (i [x, i ] σ [Mx φ f(p (i, j ] σ (S. 17

18 6. Self-similar tiling system In this section, we will give the first application of the stepped-surface and the dual substitution. That is, we prove that the atomic surfaces X 1,..., X d, as the unique invariant sets of (1.1, have non-empty interior and the right sides of (1.1 consist in non-overlapping union. Such system is called self-similar tiling system. (See Vince [Vin] Substitution Delone set. First, let us cite the definition of multi-tile functional equation and inflation functional equation of Lagarias and Wang [LW]. Let A be an expanding n n real matrix, i.e. all its eigenvalues λ > 1. Let be finite sets in R n. {D ij ; 1 i, j d} Multi-Tile Functional Equation. The family of compact sets (X 1,..., X d, X i R n, satisfy the system of equations d AX i = (X j + D ji (6.1 for 1 i d. j=1 The subdivision matrix associated to (6.1 is where D ij denotes the cardinality. B = [ D ji ] 1 i,j d, Inflation Functional Equation. The family (Z 1,..., Z d, Z i R n, satisfy the system of equations d Z i = (A(Z j + D ij, 1 i d. (6.2 j=1 Actually in the paper [LW], they define the inflation functional equations for a multiset family (Z 1,..., Z d. However, for our purpose, a normal family (Z 1,..., Z d is sufficient. A solution (Z 1,..., Z d to the inflation functional equation is a substitution Delone set family if each set Z i is a Delone set. It is seen that the subdivision matrix of (6.2 is B T. Let us denote by ρ(b the absolute value of the maximal eigenvalue of B. The following theorem is a partial result of Theorem 5.1 in [LW]. Theorem 6.1. Let (6.2 be an inflation function equation that has a primitive subdivision matrix B T such that ρ(b T = det A. If there exists a family of Delone sets (Z 1,..., Z d which is a solution of (6.2, Then 18

19 (i the unique compact solution (X 1,..., X d of (6.1 consists of sets X i that have positive Lebesgue measure, 1 i d. (ii The system (6.1 satisfies an open set condition, i.e., the right side of (6.1 consists in non-overlapping union. (iii X i = X i and X i has Lebesgue measure 0, where X i denotes the boundary of X i Atomic surface. Now we apply Theorem 6.1 to the atomic surface. First, the action of M 1 on the subspace P is a linear map, and hence it is equivalent to a (m 1 (m 1 real matrix. Let us denote this matrix by A. Since M is an unimodular Pisot matrix, we have that A is expanding and det A = λ is the Perron-Frobenius eigenvalue of M. Let us define D ji := {M 1 π(f(p (j (j ; W = i}, 1 i, j d. The sets D ji are determined completely by the substitution σ. Then X 1,..., X d are the unique invariant sets of (6.1. It is seen that the subdivision matrix B = M T. Now a solution of (6.2 is constructed by projecting the stepped-surface to the contractive space P. Let {Z 1,..., Z d } be the family defined by Z i := {π(x; [x, i ] S}. Then Theorem 1.5 is amount to say that {Z 1,..., Z d } is a solution of (6.2. Therefore by Theorem 6.1, we have immediately Theorem 1.6 Let σ be an unimodular Pisot substitution. Then (i The interiors of the partial atomic surfaces X i are not empty. (ii The right side of (3.1 consists in non-overlapping unions. (iii X i = Xi and X i has Lebesgue measure 0, where X i denotes the boundary of X i A self-replicating collection J. Combining the atomic surfaces and the steppedsurface together, we define the following collection J : = {π(x + X i ; [x, i ] S} = d i=1 {z + X i; z Z i }. We say J is a self-replicating collection in the following sense: Let us blow up the collection J by M 1, then M 1 J := {M 1 T : T J } is a collection with the prototiles {M 1 X i } d i=1. Subdividing M 1 X i into small pieces according to the set equations (3.1, we obtain a new collection which coincides with J by Theorem 1.5. If J is a tiling of P, then it is a self-replicating tiling. (See [Ken]. One may as that is J always a tiling. This problem is still open, in both the irreducible case and the reducible case. However, by using the quasi-periodicity of the stepped-surface and the self-replicating property of J, we could show that 19

20 Theorem 1.7. There exists a constant, such that almost every point of the space P is covered by pieces of tiles in J. The proof is exactly the same as that of the irreducible case, which can be found in [IR2]. Similar to the irreducible case, we can construct several graph or matrix to obtain algorithms to chec whether the collection J is a tiling. The constructions in the irreducible case can be found in [Sie1, Thu]. 7. Collection J 2 and Marov partition In this subsection, we show how to construct Marov partition by using atomic surface. Recall π is the projection from R d to V defined in Section 2. Let and and let ˆX := d i=1 ˆX i. Set ˆX i := {X i θπ (e i : 0 θ < 1}, 1 i d, J 2 := {z + ˆX : z L} = ˆX + L. Theorem 1.8. The collection J 2 is a tiling of P 1 if and only if J is a tiling of P. Proof. We consider the intersection of P and ˆX + L. By the definition of ˆX, x + ˆX intersects P if and only if x is on the stepped surface. Precisely, x + ˆX does not intersect P if [x, i ] S for any 1 i d; [x, i ] S implies π (x < π (e i, and so (π(x+x i P (x + ˆX. But [x,i ] Sπ(x + X i is a tiling of P, so P ˆX + L and in fact P is tiled by ˆX + L in the sense that almost every point of P is covered but only once by ˆX + L. Now it is ready to see for any integer point z L, P + z is covered and tiled by ˆX + L. Since π (L is dense in V, for any x P 1, P + x is covered and tiled by ˆX + L. Therefore ˆX + L is a tiling of P 1. In this case ˆX is a m-dimensional torus. In the same manner we can prove the inverse is also true. Since the matrix M is unimodular, it can be regarded as a group automorphism of the d-dimensional torus P 1 \ L. Let I i = {θπ (e i : 0 θ 1}, 1 i d, be subsets of the expanding eigenspace of M. It is easy to see that {I i } satisfies the set equations MI i = l i =1 I W (i Let us rewrite (7.1 as MI i = l i =1 I i and define ˆX i = {x z : x X i, z I i }. + f(p (i (7.1 Theorem 1.9. If J 2 is a tiling, then the collection { ˆX i : 1 i d, 1 l i } is a Marov partition of group automorphism M on the torus P 1 \ L. 20

21 Proof. Write σ(i = P (i W (i S(i, 1 i d. We construct a directed graph H σ ( with i vertices {1,..., d}. we draw an edge from i to j and label this edge by whenever W (i = j. The label set is ( i E = { : 1 i d, 1 l i }. The adjacency matrix of H σ is M T. Let Ω be the set of two-side sequences which is admissible by the graph, that is, ( i Ω = {ω = 1 i 0 i 1 : W (in 1 0 n = i n+1 }. 1 Let S be the shift operator (Sω n = ( in+1 n+1. Then the symbolic dynamical system (Ω, S is a shift of finite type because the labels of the edges are distinct. We are going to show that if ˆX is a d-dimensional torus, then (Ω, S is a symbolic representation of the dynamical system ( ˆX, M. Consequently the realization of the cylinders ( { ˆX i : 1 i d, 1 l i } is a Marov partition of group automorphism M. i0 i Let 1, be a 1-side path which is admissible by H 0 σ, we say it is a forward 1 sequence starting from i 0. Denote M 1 x by x. Then M I i = { n=0 π f(p (in n M n+1 : i 0 = i} (7.2 establishes a coding from forward sequences to interval I i. This coding is bijective almost everywhere. ( i On the other hand, let 2 i 1 be a sequence which is admissible by the graph 2 1 H σ. We say it is a bacward sequence stopping at i = W (i 1 1. Then by set equation (3.1, we have X i = { n= 1 M n+1 πf(p (in n : W i 1 1 = i}. (7.3 This establishes a coding from bacward sequences to X i and it is also bijective almost everywhere. 21

22 ( ( i If 2 i 1 i0 i is an admissible bacward sequence stopping at i, and 1 is ( 1 i an admissible forward sequence starting from i, then 1 i 0 i 1 is an admissible two-side sequence. We define a realization map ψ : Ω R d ( i ψ 1 i 0 i 1 = M n+1 πf(p (in π f(p (in 1 0 n n. 1 M n+1 n= 1 By (7.2 and (7.3 we have ψ(ω = ˆX. If X = d i=1x i is a non-overlapping union, then ψ is bijective for almost every point of ˆX. We will show the following graph is commutative. Let ω = ( ψ Ω S ˆX M (mod L i 1 i 0 i 1 Ω, then ψ S(ω = M ψ(ω = n= 1 n= 1 Ω ψ M n+2 πf(p (in n + πf(p (i 0 ˆX 0 M n+2 πf(p (in n π f(p (i 0 0 n=0 n=1 n=1 π f(p (in n M n, π f(p (in n M n. In the above two equations only the middle terms are different. Hence ψ S(ω M ψ(ω = πf(p (i π f(p (i 0 0 = φ(f(p (i 0 0 L, which means ψ S(ω = M ψ(ω (mod L. 8. Can X tile P periodically? In the irreducible case, the atomic surface X can tile P periodically. Actually this tiling is a lattice tiling, i.e., the translation set Γ is a discrete subgroup, where Γ is generated by the vectors π(e 2 π(e 1,..., π(e d π(e 1. If σ satisfies a strong coincidence condition, then a domain exchange transformation can be defined on X as: E(x = x π(e i, x X i. (8.1 (See [AI]. For a point x on the common boundaries of X i, we can set x belongs to only one of X i to avoid ambiguity. Furthermore, if σ satisfies a super-coincidence condition, then X can be regarded as a (d 1-dimensional torus and the domain exchange transformation E(x is a rotation on the torus ([IR2]. In this case, the rotation on the torus is 22

23 topologically conjugate to the substitution dynamical system associated to σ and hence the later one has discrete spectrum. To show that X + Γ is a lattice tiling of P, the geometrical representation of the stepped-surface plays a crucial role. As we has pointed out in Section 4, the existence of geometrical representation is unclear. Hence in general, we do not now can X tile P periodically or not. This is the second difference between the irreducible case and the reducible case. Ei, Furuado and Ito have studied this problem for the substitutions in Example 1.1 and Example 1.2. By trial and error method, they found geometrical representation of the stepped-surface of these substitutions. For the substitution in Example 1.2, Ei, Furuado and ITO shows that X can tile P by a lattice translation set. Hence X is a two-dimensional torus and a domain exchange transformation can be defined by formula (8.1. Actually the translation set is Γ = {mπ(e 2 e 1 + nπ(e 4 e 1 : m, n Z}. Figure 7. Periodic tiling by polygons of substitution τ. Figure 8. Periodic tiling by atomic surface X of substitution τ. The situation is different for the Hoaido substitution. Ito and Ei ([EI] show that X can not tile P by a lattice translation set. But they prove that there is a domain U consists of X and a translation of reflection of X, such that U adopts a lattice tiling of P. Exactly set U = X ( X + π(e 1 + e 2 + e 3, Γ = {mπ(e 2 e 4 + nπ(e 1 + e 2 2e 3 : m, n Z}, 23

24 then U + Γ is a tiling of P. They also managed to define a domain exchange on the set U. Figure 9. Periodic tiling by polygons of Hoaido substitution. Figure 10. Periodic tiling by atomic surface X of Hoaido substitution. We conjecture that X can always tile P periodically if reflection and rotation are allowed. We guess domain exchange transformation can always be defined on a set consisting of copies of X. Acowlegement The authors lie to than S. Aiyama for many helpful discussions and we than P. Arnoux and V. Berthe for many valuable comments. 24

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26 [Sie1] A. Siegel, Pure discrete spectrum dynamical system and periodic tiling associated with a substitution. Ann. Inst. Fourier (Grenoble 54 (2004, no. 2, [Sie2] A. Siegel, Représentations géométrique, combinatoire et arithmétique des systèmes substitutifs de type Pisot, Thèsis de Doctorat, Université de la Méditérranée, [SW] V. Sirvent and Y. Wang, Self-affine tiling via substitution dynamical systems and Rauzy fractals. Pacific J. Math. 206 (2002, no. 2, [T] W. P. Thurston, Groups, tilings, and finite state automata. AMS Colloquium Lectures. AMS, Providence, RI, [Thu] J. Thuswaldner, Unimodular Pisot substitutions and their associated tiles. Preprint [Vin] A. Vince, Digit tiling of Euclidean space. Center de Recherches Mathematiques CRM Monograph Series 13 (2000, Hiromi Ei: Department of Information and System Engineering, Chuo University, Kasuga, Bunyo-u, Toyo, Japan ei@ise.chuo-u.ac.jp Shunji Ito: Mathematical Department of Kanazawa University, Kanazawa, Japan ito@t.anazawa-u.ac.jp Hui Rao: Department of Mathematics, Tsinghua University, Beijing, China and Tsuda college, Kodaira, Toyo, Japan hrao@math.tsinghua.edu.cn 26