Tilings from some non-irreducible, Pisot substitutions

A generating method of self-afﬁne tilings for Pisot, unimodular, irreducible substitutions, as well as the fact that the associated substitution dynamical systems are isomorphic to rotations on the torus are established in [AI01]. The aim of this paper is to extend these facts in the case where the characteristic polynomial of a substitution is non-irreducible for a special class of substitutions on ﬁve letters. Finally we show that the substitution dynamical systems for this class are isomorphic to induced transformations of rotations on the torus.


Introduction
In this paper we want to discuss tilings and dynamical systems generated by the following substitutions given by σ : (1.1) The characteristic polynomial of the incidence matrix L σ is Since it is non-irreducible, its factor x 3 − Kx 2 − (K + 1)x − 1 is a minimal polynomial of some Pisot number β. Furthermore | det(L σ )| = 1; we thus say that the above substitution σ is of the non-irreducible, Pisot, unimodular type.The aim of this paper is to discuss how we obtain tilings and dynamical systems generated by non-irreducible, Pisot, unimodular substitutions for the special class (1.1) which is coming from Pisot β-expansions.
Let us recall some results in the irreducible, Pisot, unimodular case.(See [AI01].)For example we consider the following substitution σ on 3 letters: (1. 3) The substitution σ has the incidence matrix and its characteristic polynomial x 3 −x 2 −x−1.So this substitution is of the irreducible, Pisot, unimodular type.
For the contractive plane P of the matrix L σ , a stepped surface which is a discrete plane approximation of P is determined.Thus we have a tiling of the plane P with three prototiles, which are parallelograms, by using the projection π defined as the map from R 3 to P along the eigenvector u of L σ corresponding to the Pisot eigenvalue β.
This substitution σ has a unique fixed point and we denote it by ω = s 0 s 1 • • • s n • • • .Then we obtain the sets X i (i = 1, 2, 3) given by the closure of {π n−1 k=0 e s k | s n = i, n = 1, 2, • • • } and X = ∪ 3 i=1 X i , where {e i } i=1,2,3 is the canonical basis of R 3 .These sets X i , X are called atomic surfaces of σ.
On the other hand, it is known that the tiling and the atomic surfaces can be generated by the so called tiling substitution E * 1 (σ) on the Z-module G * 1 defined by Here we identify (x, i * ) ∈ Z 3 × {1 * , 2 * , 3 * } with the unit square {x + se j + te k ∈ R 3 |{j, k} = {1, 2, 3}\{i}, 0 ≤ s ≤ 1, 0 ≤ t ≤ 1}, and summation "+" in an element in G * 1 means the union of these unit squares.More generally we consider a substitution σ denoted by σ(i) = W l (i) −1 .By using the canonical homomorphism f from the free monoid on 3 letters to Z 3 defined by f (i) = e i (i = 1, 2, 3) and the notations P (i) k and S (i) k stand for respectively the prefix of length k and suffix of length l (i) − 1 − k of σ(i), we define the endomorphism E * 1 (σ) on G * 1 as follows: k ), j * ) .
On this setting we can generate the stepped surface of the plane P by E * n 1 (σ)((e 1 , 1 * ) + (e 2 , 2 * ) + (e 3 , 3 * )) (n → ∞ ) and the atomic surfaces by where the right side converges in the sense of the Hausdorff metric and we denote lim n→∞ L n σ πE * n 1 (σ)(e i , i * ) by X i .Furthermore, we have the following property: Property 1.1 (1) The pieces in the union ∪ 3 i=1 X i denoted by X do not overlap up to a set of Lebesgue measure 0, (2) the following set equation holds: where the sets in the right side of the equation do not overlap up to a set of Lebesgue measure 0, and moreover, the transformation F : X → X given by is well-defined and it is a Markov transformation with matrix structure L σ , (3) the transformation E : X → X, called the domain exchange transformation, given by is well-defined and the transformation E is measure-theoretically isomorphic to a rotation on the 2-dimensional torus, and moreover, the orbit of the origin point by Property (2) and (3) are not known to hold for any irreducible Pisot substitution.Property (2) holds provided that (1) holds if the substitution σ has the strong coincidence condition (cf.[AI01]).Moreover, for the transformation F to be well-defined, the pieces must not overlap.The isomorphism with a rotation (3) is equivalent with the tiling property.The aim of the paper is to obtain the analogous property for the non-irreducible, Pisot, unimodular substitutions given by (1.1).
This paper is sketched as follows.
In Section 2, we define a projection map π from R 5 to the contractive 2-dimensional plane of the incidence matrix L σ of σ.A substitution given by (1.1) has a unique fixed point.Therefore, using the projection we obtain atomic surfaces X and X i (i = 1, 2, 3, 4, 5) with respect to the letter i in the same way as in the irreducible case.(See Fig. 1.) In Section 3, 4 and 6, we define the tiling substitution τ * of a substitution σ according to some modification of the method by [AI01].Since we deal with the non-irreducible case, we can not use the same method as in the irreducible case.However, we introduce new polygonal tiles instead of parallelograms, a tiling substitution τ * and the concept of a stepped surface; and try to construct atomic surfaces by using a map τ * and tilings T τ * with five polygonal prototiles and T b X τ * with the atomic surfaces X i (i = 1, 2, 3, 4, 5).(See Fig. 2 and Fig. 3.) In Section 5, we introduce two dynamical systems on X := −X associated with non-irreducible substitutions, that is, a Markov transformation and a domain exchange transformation related to Property 1.1 (2) and (3).The main theorems in this paper are Theorem 5.2 and Theorem 5.3, and these theorems say the following: The domain exchange transformation E : X → X is defined by and the orbit of the origin point by the transformation The transformation E is not measure-theoretically isomorphic to a rotation on the 2-dimensional torus, but isomorphic to the induced transformation of a rotation on the torus.
2 Substitutions and atomic surfaces

General results on atomic surfaces
Let A be an alphabet consisting of d letters {1, 2, • • • , d} .The free monoid on the alphabet A with the empty word ǫ is denoted by A * = ∪ ∞ n=0 A n and A N denotes the collection of all right infinite sequences of symbols from A. Let σ be an endomorphism on A * such that σ(i) ∈ A * \ {ǫ} for all i ∈ A, which is called a substitution.By defining σ(U V ) = σ(U )σ(V ) for a concatenation U V of words, the substitution σ is extended to an endomorphism on A * and A N .Put = ǫ for any i ∈ A and any positive integer n.We call P (i) k (resp., S (i) k ) the k-prefix (resp., the k-suffix) of a word σ(i).We define the canonical homomorphism f : There is a matrix L σ on Z d for a substitution σ satisfying the following commutative diagram: The matrix L σ is called the incidence matrix of the substitution and its entry L σ (i, j) is equal to the number of occurrences of the letter i in σ(j).
Recall that a Pisot number is an algebraic integer greater than 1 and whose conjugates have modulus strictly less than 1.Before discussing atomic surfaces of substitutions given by (1.1), we will define atomic surfaces on a general setting with the following assumption: Assumption Throughout this paper, we assume that for a substitution σ (3) the characteristic polynomial of L σ is not irreducible and is decomposed as f (x)g(x) such that f (x) is a minimal polynomial of some Pisot number β, and the roots of g(x) have modulus 1.
A substitution with Assumption (2) and ( 3) is referred to be of unimodular, non-irreducible, Pisot type.From Assumption (1), there exists a fixed point ω of the substitution σ: From Assumption (2) and (3), we define the expanding subspace L(u) spanned by the eigenvector u corresponding to the eigenvalue β and the contractive subspace P = {0}(⊂ R d ) corresponding to the other conjugate eigenvalues of β.Then we have a direct sum R d = P ⊕L(u)⊕P ′ , where P ′ corresponds to the other eigenvectors coming from g(x).Let us define the projection π : R d → P by where p ∈ P , x + p ′ ∈ L(u) ⊕ P ′ .
Definition 2.1 From sets Z i , Z ′ i of prefixes of a fixed point ω: we define sets Y i , Y ′ i in P as follows: The sets X, X i , X ′ i in P are defined by where cl.S means the closure of a set S.
We call the sets X, X i (i ∈ A) atomic surfaces of σ.Note that the equality Proposition 2.1 The following set equations hold: (1) Remark 1 We will see the following property in Theorem 4.1 (4): the sets = i in the equation (1) (resp., (2)) do not overlap up to a set of Lebesgue measure 0.
Proof: From the property cl.( (2.1) Consequently there exists an integer t such that where W (sm) t = i.Let f act on both sides of the above equality, ).
) with W (sm) t = i.This shows ⊆ is true for the equality (2.1) by choosing j = s m and k = t.
On the other hand, for any s 0 s 1 • • • s m ∈ Z j with s m+1 = j and W (j) k = i, we have Since s m+1 = j, then it is clear that This leads to the other direction of the equality (2.1).The second set equation is shown by (1) and

Atomic surfaces generated by the labeled graph G *
To obtain a numerical representation of X or X i as we will see in Theorem 2.1, we introduce the new alphabet B and the subset B * σ of the monoid B * by using the prefix automaton as follows: where |U | means the length of a word U .(See Fig. 4.) We define a labeled graph G * such that the set of vertices is V = {1, 2, • • • , d} and the set of edges is = i, that is, the letter i occurs in σ(j) as k-th letter, then one edge from the vertex i to the vertex j named k is drawn.(See Fig. 5. cf.
is the set of all finite paths whose initial vertex is i in G * .
Theorem 2.1 For any substitution σ satisfying Assumption, the following equalities hold: and is monotone decreasing with respect to l, at last we have the following equality: , σ .This shows We obtain the converse inclusion as in the proof of Proposition 2.1.Thus the first equality is proved.Analogously we prove the second one.✷ From the above decomposition of f (s 0 s 1 • • • s m ), we have the following corollary: Corollary 2.1 For any substitution which satisfies Assumption, the atomic surface X is bounded.

Atomic surfaces corresponding to a substitution given by (1.1)
From now on, let us deal with the non-irreducible, Pisot, unimodular substitutions σ given by (1.1): In this case, We see that the characteristic polynomial of the incidence matrix L σ is given by (1.2) and it is also the characteristic polynomial of β (cf.[PAR60]).It is easy to check that the equation x 3 − Kx 2 − (K + 1)x − 1 = 0 gives one real root β and two imaginary roots , and λ 4 , λ 5 are the imaginary roots of x 2 − x + 1 = 0. Choose any eigenvector u i associated with the eigenvalue , and let us define the 5 × 5 matrix V by The real vectors v i (i = 1, 2, 3, 4, 5) and the real matrix V satisfy the following relation: where ℜ[a] (resp., ℑ[a]) means the real (resp., imaginary) part of a and The space P <v1,v2> spanned by vectors v 1 , v 2 is an invariant contractive space of the linear transformation L σ .More precisely, we know , where P ′ <v4,v5> is spanned by v 4 , v 5 ; and define the projection map π : R 5 → P <v1,v2> by

It is easy to see that
.
By Proposition 2.1, in this case we obtain equations in concrete terms as stated in the following corollary: Corollary 2.2 For the substitution σ given by (1.1), the sets {X i } i=1,2,3,4,5 given in Theorem 2.1 satisfy the following set equations: (1) For the substitution σ, the alphabet B and the graph G * are as follows: From the definition of σ given by (1.1), we have f (P k ) = ke 1 for any j ∈ {1, 2, 3, 4, 5}.Hence it is enough to take only the path Then the set X is written as follows: where t(e) ∈ V (resp., i(e)) means the terminal (resp., the initial) vertex of an edge e.The condition ≺ ω in the first line of the above equality means i N k N ∈ B \ { 2 0 , 3 0 , 4 K } by the definition of σ, but we can omit this condition for the following reason: Let us consider the case i From the graph G * in Fig. 6, the path is determined Moreover, from the labeled graph G * , the sets {X i } i=1,2,3,4,5 are given by This leads to the following corollary: Corollary 2.3 An element in the atomic surface X has an infinite expansion, that is, Example 2.1 Let us consider the case of K = 0.The substitution σ is σ : .
By the labeled graph G * in Fig. 6, the following admissible conditions hold for the sequence of digits {k n }: 3 A tiling with polygonal tiles of the plane P <v 1 ,v 2 >

A tiling substitution τ *
In this subsection we introduce a tiling substitution τ * associated with σ according to [AI01].From now on let σ be a substitution given by (1.1) in Section 1. Then we have the direct sum R 5 = P <v1,v2> ⊕ L(v 3 ) ⊕ P ′ <v4,v5> , and the map π is a projection from R 5 to the 2-dimensional plane P <v1,v2> .Lemma 3.1 The following relations hold:

Proof:
We check that we can choose the vector t (−1, 0, 1, 1, 0) = −e 1 + e 3 + e 4 as the vector v 5 .Since v 5 is mapped to 0 by the projection π, we have From the fact that L σ • π = π • L σ , we have the second and third equalities.✷ We introduce Z-modules F and F * using finite integer combinations of elements of Z 5 × {1, 2, 3, 4, 5} and Z 5 × {1 * , 2 * , 3 * , 4 * , 5 * } as follows: From Lemma 3.1 we can introduce the equivalence relation ∼ on F u (resp., F * u ) defined by ). F will be used in Section 4 mainly.To give a geometrical meaning of (x, i * ), first we define the map π 1 : F → P <v1,v2> , which gives a one dimensional geometric representation of the symbolic object (x, i), by  Proof: We show {πe 2 , πe 3 , πe 4 } is a hexa-generator.The other part can be shown analogously.
, that is, a normal vector of x.We can calculate the coordinates of πe i (i = 1, 2, 3, 4, 5) and easily check that every n(π(e 2 )) • π(e 4 ), n(π(e 4 )) • π(e 3 ) and n(π(e 3 )) • π(e 2 ) has the same signature, where a • b means the inner product for a, b ∈ R 2 .Therefore {πe 2 , πe 3 , πe 4 } is a hexa-generator.(See Fig. 8.) ✷ From Lemma 3.1 and Lemma 3.2, we can consider the map π 2 : F * → P <v1,v2> , which gives a two-dimensional geometric representation of the symbolic object (x, i * ), by (See. We remark that the sign is plus here in this definition with respect to the sign minus in Formula (2) in Proposition 2.1.(See Remark 4 in Section 4.) Remark 2 For the substitution σ given by (1.1), the tiling substitution τ * is defined explicitly by (See Fig. 10.) Remark 3 The formula of the tiling substitution τ * can be found in [AI01] under the notation E * 1 (σ).But the geometrical meaning of (x, i * ) (i = 1, 2, 3, 4, 5) is different.In [AI01] we mean by (x, i * ) a unit cube of dimension four in R 5 and a cube (x, i * ) projected by π is a good prototile for the tiling of R 4 for an irreducible substitution, however, for a non-irreducible substitution which we deal with now, a cube (x, i * ) projected by π does not work as prototile of a tiling of the contractive space P <v1,v2> .In this section we will see that the tiles π 2 (x, i * ) given by Fig. 9 where for V, W ∈ F * , V ⊃ W means that π 2 (V) ⊃ π 2 (W), and that there exists Proof: From the definition of τ * , we see The replacing and re-dividing method Observe the two domains π 2 (τ * (x, i * )) and L −1 σ (π 2 (x, i * )), then we have the following three cases (See Fig. 10.): In the case where i = 1, 3, 4, each of these domains π 2 (x, i * ) (i = 1, 3, 4) contains at least one edge of the form π 1 (y, 2) and one edge of the form π 1 (y ′ , 5), so we introduce the following "replacing and re-dividing" method to get the domain π 2 (τ * (x, i * )) from the domain L −1 σ (π 2 (x, i * )).
First, we "replace" every edge what is more, in the case of K ≥ 1, replace every edge By this procedure, we have π 2 (τ * (x, i * )) in the case of i = 3, 4.
, that is, π 2 (τ * n (U)) covers the plane as n goes to ∞.The proof of this proposition is long and not easy.So the detail of this proof is put in Section 6.In the irreducible case, we can prove the property in Proposition 3.3 by using the notion of stepped surface of a substitution σ (cf.[AI01]), but here we must prove it without such a notion.That is the reason why the proof is difficult.
The following proposition is deduced from Proposition 3.2 and Proposition 3.3: Proposition 3.4 The sets τ * n (U) generate a tiling of the plane P <v1,v2> , that is, is a tiling of P <v1,v2> .(See Fig. 2 in Section 1.) Finally we discuss the periodicity of the tiling T τ * .Let us introduce the following notation: and we also introduce the new prototiles (x, i * ) based on πx and the tiling associated to G * defined by So we have the ordinary tiling T τ * by dividing the prototiles in G * following the method in Subsection 3.1.We say G * (or T τ * ) is periodic if there exists at least one non-zero period p ∈ R 5 such that (x, i * ) ∈ G * implies (x + p, i * ) ∈ G * .
Lemma 3.3 If p is a non-zero period of G * , then G * also has the period p.
Proof: Assume G * has a non-zero period p.At first we consider the periodicity of the tiles of the form (x, 1 * ) in G * .Note that the only τ * (x, 1 * ) in the images by τ * includes a tile of the form (y, 5 * ), that is, (x, 1 * ) ∈ G * if and only if (x, 5 * ) ∈ G * .Suppose (x, 1 * ) ∈ G * , then (x, 5 * ) ∈ G * , and so (x + p, 5 * ) ∈ G * by the assumption, and finally we have (x + p, 1 * ) ∈ G * .It means that p is also a period for the tiles of the form (x, 1 * ).Now we want to observe the periodicity for the tiles in G * except the tiles (x, 1 * ).Put From the above discussion we know that D has a period p.This means that D is closed for the translation by p, and has the same period p.Thus G * − D also has a period p by the assumption.After projection by π 2 , G * − D and G * − D provide the same covering of P <v1,v2> with many holes by the equality So G * − D has a period p.Therefore, G * has a period p. ✷ Theorem 3.1 The tiling T τ * is not periodic.
Proof: Suppose the tiling T τ * is periodic, that is, G * is periodic.Since G * is a discrete set, there exists a non-zero and minimum period p of G * , where a minimum period is a period whose norm πp is minimum.From Lemma 3.3, p is also a period of G * .Define the map ι * : By the definition of G * , ι * (x, i * ) is in G * , that is, ι * is well-defined.Moreover, it is a bijection and the inverse is given by This means π(L σ p) is a period of the tiling T τ * .On the other hand, L σ is contractive on P <v1,v2> .Therefore, it contradicts the minimality of the period p. ✷ Definition 3.1 A tiling T = {T λ | λ ∈ Λ, T λ is a tile on P } of the space P is called a quasi-periodic tiling if for any r > 0 there exists R > 0 such that any patch γ = λ ′ ∈Λ ′ ⊂Λ # Λ ′ <∞ T λ ′ whose diameter is smaller than r occurs somewhere in a neighbourhood of radius R of any point.
For γ, δ ∈ F * , γ ≻ δ denotes that there exists z ∈ Z 5 such that M z γ ⊃ δ, where M z is the translation map given by Theorem 3.2 The tiling T τ * is quasi-periodic.
Therefore, we have where R = max i=1,2,3,4,5 diam.(π 2 (τ * M (e i , i * ))) and U R (x) means the neighbourhood of x with the radius R. Thus, U R (x) contains any configuration of π 2 (γ) whose diameter is smaller than r.✷ 4 Atomic surfaces given by τ * and a second tiling In Section 2 we constructed atomic surfaces X, X i (i = 1, 2, 3, 4, 5) from the fixed point of a substitution and the projection map π.In Subsection 4.1 we generate the atomic surfaces by using the tiling substitution τ * ; and by the virtue of this construction, we can observe the boundaries of atomic surfaces in Subsection 4.2; and in Subsection 4.3 we obtain a second tiling with atomic surfaces by replacing the polygonal tiles on the first tiling T τ * by atomic surfaces.

Atomic surfaces given by
n , (resp., ) as follows: Theorem 4.1 We take a renormalization of the domains D n , D n , then (1) the following limit sets exist in the sense of the Hausdorff metric: and they satisfy the relations: (2) the following inequality holds: moreover, the vector of volumes t (µ( X 1 ), µ( X 2 ), • • • , µ( X 5 )) is an eigenvector of L σ with respect to the maximum eigenvalue β, where µ is the Lebesgue measure, (3) the following set equations hold: The proof of the theorem can be obtained by a quite similar way as in [AI01] following Lemma 11, Lemma 12 and Corollary 2.

Remark 4
We can find a relationship between X i and the atomic surfaces X i .By Proposition 2.1, we have the following equation for the atomic surfaces X i : This means −X i and X i (i = 1, 2, 3, 4, 5) satisfy the same set equations.Since L σ is a contractive transformation on the plane P <v1,v2> and from the uniqueness of self-affine sets (See Theorem 1 in [MW88] for the uniqueness.),we have the following relation Remark 5 We are interested in the disjointness of the partitions X i (i = 1, 2, 3, 4, 5) of X. From the property τ * 4 (e 1 , 1 * ) ⊃ U, any X i is included in the right side of the equation thus by Theorem 4.1 (4), for any i, j This property of disjointness holds for substitutions satisfying the strong coincidence property in [AI01].

Boundaries of atomic surfaces
In this subsection we will observe the boundaries of atomic surfaces X, X i .One of our aims here is to obtain Proposition 4.1, which says that the origin is an inner point of the domain X 1 .For that we will show that the distance between the origin point and the boundary is positive by studying the boundary.
We introduce an endomorphism τ on F as follows (See Fig. 13.):

Fig. 13: Geometrical meaning of τ
The following diagram is commutative: , and from the definition of the maps π 1 , π 2 , we have , where ∂D denotes the boundary of the domain D.
From the two diagrams, we deduce that n } converge towards the same set as n goes to ∞ in the sense of the Hausdorff metric.

Proof:
Existence of the limit set of {∂(L n σ D It is enough to show that the following limit set exists: where d H is the Hausdorff metric.In general the following property holds: for sets A, B, C, D. It is easy to check that τ n (x, i) does not have cancellation for any (x, i) and any positive integer n.Therefore we see where This means the sequence {L n σ π 1 (τ n (0, i))} ∞ n=1 is a Cauchy sequence and it has a limit set in the sense of the Hausdorff metric.
Analogously we see that the sequence n and a simple approximation argument, we see these limit sets are equal.✷ Let B (i) (resp., B) denote the limit set Lemma 4.2 µ( X i ∩ U r (x)) > 0 (i = 1, 2, 3, 4, 5) for any x ∈ X i and any r > 0.
By the boundedness of X i (See Corollary 2.1.),for any x ∈ X i and any r > 0, there exist positive integers n, j, k such that x ∈ L n σ X j − πf (P Thus we have Lemma 4.3 says that the boundary of X i is constructed by the map τ .Proof: and the other cases are shown analogously.To see n ⊂ ∂ X 1 for any n.Let N be the collection of tiles which consists of (e 1 , 1 * ) and its neighbour tiles: n , then there exists (e j , j * ) ⊂ N , j = 1 such that x ∈ X 1 and x ∈ X j .Suppose that x is an inner point of X 1 , that is, there is r > 0 such that U r (x) ⊂ X 1 .By x ∈ X j and Lemma 4.2, we have which leads to a contradiction and it implies B (1) n ⊂ ∂ X 1 for any n.Conversely, suppose that x ∈ ∂ X 1 ⊂ X 1 .Then there are sequences {x n } ∞ n=1 with x n ∈ L n σ D (1) n and {y n } ∞ n=1 with y n ∈ X 1 such that d(x, x n ) < 1 n and d(x, y n ) < 1 n for any positive integer n, where d is the usual Euclidean distance on the plane P <v1,v2> .By y n ∈ X 1 , we have sequences {y n,m } ∞ m=1 : lim m→∞ y n,m = y n , and for any m there exists M ≥ m such that y n,M ∈ L M σ D (1) M .Therefore, we can choose y n,kn ∈ L kn σ D (1) and this means lim n→∞ y n,kn = x.On the segment between x kn ∈ L kn σ D (1) kn and y n,kn ∈ L kn σ D (1) kn , there exists c kn ∈ ∂L kn σ D (1) kn , and lim n→∞ c kn = x.This implies n and ∂ X 1 ⊂ B (1) .✷ Proposition 4.1 The origin is an inner point of X 1 .
Proof: By Lemma 4.3, we see for any positive integer N .This means that L −N σ ∂( X i ) is constructed by replacing each edge π 1 (x, j) (j = 1, 2, 3, 4, 5) on ∂D (i) N with I j + x, where I j = lim n→∞ L n σ π 1 (τ n (0, j)).By the inequality (4.1), we have the following inequality where c is some positive number.Therefore, By the fact that π 2 (τ * n (U)) is covering the plane P <v1,v2> as n goes to ∞ (See Proposition 3.3 and Section 6.), there exists a positive integer N such that From the construction of L −N σ ∂( X 1 ) and (4.2), we have Thus we see that for some positive number c This implies the origin is an inner point of X 1 .✷ Proposition 4.2 The Hausdorff dimension of the limit set B of the boundaries where and λ θ is the maximum eigenvalue of the matrix M θ , that is, the maximum solution of the equation x 5 − Kx 4 − (K + 1)x − 1 = 0.
Proof: At first we show that T b X τ * is a tiling of P <v1,v2> .From Proposition 4.1, there exists a positive number δ such that U δ (0) ⊂ X 1 .
Notice that for any positive integer n By Theorem 4.1 (3) where the tiles ( X j − L −n σ πf (P (n,j) k )) do not overlap for any k, j.From the union of tiles This means that the pieces of T b X τ * do not overlap and cover U β0 −n δ (0) for any n.Therefore T b X τ * is a tiling of P <v1,v2> by taking n → ∞.On the other hand, from the quasi periodicity of the tiling T τ * , T b X τ * is also quasi periodic.✷ At the beginning of this paper, we started from the substitution σ given by (1.1).And we obtained atomic surfaces {X i } i=1,2,3,4,5 (= {− X i } i=1,2,3,4,5 ) and the tiling T b X τ * .We also get tiles {T i } i=1,2,3,4,5 and a tiling T β by using the numeration system related to a Pisot number β as in [AKI99], [THU89].We plan to make the relation between {X i } i=1,2,3,4,5 (resp.T b X τ * ) and {T i } i=1,2,3,4,5 (resp.T β ) explicit with the subdivision rule in [EIR02].

Dynamical systems
We introduce two types of measured dynamical systems on X a Markov transformation and a domain exchange transformation with σ-structure.
From Theorem 4.1 (3), x ∈ X i implies that there exist integers j, k such that Therefore we get the division of X i : where Here we have the following theorem which provides a Markov transformation.
Theorem 5.1 Let us define the map F : X → X by then the map F is well-defined and The transformation F is well-defined because of Theorem 4.1 (4), and it is called a Markov transformation with matrix structure L σ with respect to partitions { X i } i=1,2,3,4,5 .
From now on we consider a domain exchange transformation with σ-structure.
Definition 5.1 Let (X, T, µ) be a measured dynamical system, σ a substitution over the alphabet A such that and consider a measurable partition {X (i) | i ∈ A} of X, a subset A of X and a measurable partition We say that the transformation T has σ-structure with respect to the pair of partitions {X (i) }, {A (i) } if T satisfies the following condition: For the transformation T with σ-structure, the induced transformation T | A on A is defined by Proposition 5.1 Let σ be a substitution satisfying (1.1) and put .
(1) The transformation E n : D n → D n given by is well-defined and preserves the Lebesgue measure µ. (See Fig. 14.) (2) The transformation E 1 : D 1 → D 1 has σ-structure with respect to the pair of partitions {D The transformation E, which preserves the Lebesgue measure µ, is well-defined on X.And E has σstructure with respect to the pair of partitions { X i }, {L σ X i }, moreover, E has σ n -structure with respect to the pair of partitions { X i }, {L n σ X i } for all n ∈ N.
Proof: From Theorem 4.1 (1), the transformation E is well-defined.From the equation given by Theorem 4.1 (3) and (4), we have This means E has σ-structure with respect to the pair of partitions { X i }, {L σ X i }.By induction, we can show the second statement.✷ This transformation E : X → X is also called the domain exchange transformation associated with a substitution σ.From this theorem and Proposition 4.1 we have the following corollaries: From Corollary 5.1 we have the following corollary: (See Lemma 6 in [AI01] and [BFMS02].)Corollary 5.2 Let (Ω σ , S) be the substitution dynamical system generated by a substitution σ given by (1.1).The dynamical system ( X, E) is a realization of (Ω σ , S), and the realization map φ from Ω σ to X is given by using φ(S k (s 0 s 1 • • • )) = E k (0) for all positive integers k.
Finally to observe ergodic property of the domain exchange transformation E, we define new domains D n (resp., X) and domain exchange transformations E n (resp., E) on the domains which are measuretheoretically isomorphic to a rotation on the 2-dimensional torus.
From Fig. 15 we have the following properties:  Thus the distance of the boundary of π 2 (τ * n (U)) from the origin tends to ∞; and from Lemma 6.2, π 2 (τ * n (U)) is a topological cell for any n.Therefore, π 2 (τ * n (U)) is covering the plane P <v1,v2> as n goes to ∞. ✷

Fig. 1 :Fig. 2 :Fig. 3 :
Fig. 1: The atomic surface X of the substitution σ in the case of K = 0

Fig. 5 :
Fig. 5:The edge k from the vertex i to the vertex j (W(j) k = i)

τ * Definition 4 . 1
Define the domains D n , D 4) the sets in the right side of the equation in (3) do not overlap up to a set of Lebesgue measure 0.

Fig. 15 : 2 Fig. 16 :Fig. 18 :
Fig. 15: The periodic tiling {π2(x, i * ) | (x, i * ) ⊂ Mz( e U ) for some z ∈ L}In Lemma 5.1 we obtained 2-dimensional fundamental domains and they were constructed by replacing every edge π 1 (x, i) on π 2 ( U) with L n σ π 1 (τ n (x, i)).Moreover, in the same way as in the proof of Proposition 4.1, we also obtain a new 2-dimensional fundamental domain by replacing every edge π 1 (x, i) on the boundary of the domains D 0 and {−D 0 + (πe 1 + πe 2 + πe 5 )} with I i + πx.Then we have the following theorem, which says the domain exchange transformation E is measure-theoretically isomorphic to the induced transformation of a rotation on the 2-dimensional torus: