Pure discrete spectrum for a class of one-dimensional substitution tiling systems

We prove that if a primitive and non-periodic substitution is injective on initial letters, constant on final letters, and has Pisot inflation, then the R-action on the corresponding tiling space has pure discrete spectrum. As a consequence, all beta-substitutions for beta a Pisot simple Parry number have tiling dynamical systems with pure discrete spectrum, as do the Pisot systems arising, for example, from the Jacobi-Perron and Brun continued fraction expansions.


MARCY BARGE
letter of the alphabet A occurs as a first letter of a word φ(a) for some a ∈ A and all such words end in the same letter. It is the main theorem of this article that the tiling dynamical system of any (primitive) Pisot substitution with this property has pure discrete spectrum.
Given a substitution φ on the finite alphabet A, say φ(a) = a 1 · · · a t(a) for a ∈ A, we will say that φ is injective on initial letters if the map a → a 1 is injective, and we will say that φ is constant on final letters if a t(a) = b t(b) for all a, b ∈ A. Examples of families of substitutions that are both injective on initial letters and constant on final letters include β-substitutions for β a simple Parry number, and substitutions arising in connection with continued fraction algorithms (see Section 4). The following theorem is the main result of this article.
Theorem (Theorem 3.12 in text): If φ is a primitive Pisot substitution that is injective on initial letters and constant on final letters then the tiling dynamical system (Ω φ , R) has pure discrete spectrum.
In outline, the argument will be as follows:

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(1) Every tiling dynamical system (Ω, R) has a maximal equicontinuous factor (Ω max , R) and (Ω, R) has pure discrete spectrum if and only if the factor map π max : Ω → Ω max is a.e. one-to-one.
(2) Given T, T ′ ∈ Ω φ we write T ≈ s T ′ provided π max (T ) = π max (T ′ ) and, for a dense set of t ∈ R, there is n(t) ∈ N so that so that Φ n(t) (T −t) and Φ n(t) (T ′ −t) have exactly the same tiles at the origin. The relation ≈ s is a closed equivalence relation.
(3) If the system (Ω φ , R) does not have pure discrete spectrum, the quotient system (Ω φ / ≈ s , R) is isomorphic with a tiling dynamical system (Ω φs , R) for a primitive, non-periodic Pisot substitution φ s that is also injective on initial letters and constant on final letters. The relation ≈ s is trivial on Ω φs .
(4) If ψ is a primitive, non-periodic, Pisot substitution that is injective on initial letters and constant on final letters, then ≈ s is nontrivial on Ω ψ .
The proof now follows by contradiction: Were (Ω φ , R) not to have pure discrete spectrum, the relation ≈ s would be trivial on Ω φs . But item (4) would apply with ψ = φ s to say that ≈ s is nontrivial on Ω φs .
Many of the ingredients of the proof have been developed elsewhere: (1) holds quite generally and is a consequence of the Halmos -von Neumann theory (see Chapter 3 of [W], for example); that Φ is a homeomorphism is a result of Mossé ([M], or, more generally, [S2]); much of (2) and (3) appears in [B] in arbitrary dimension; and (4) is derived from the main idea of [BD1]. The details are presented in Section 3 and applications are given in Section 4.

Background and Notation
Given an alphabet A, A * will denote the set of all finite, nonempty, words in A. (For convenience, we will sometimes confuse ρ i with [0, ω i ] × {i}.) A tile is a translate of a prototile, ρ i − t := ([−t, ω i − t], i), with support spt(ρ i − t) := [−t, ω i − t] and type i. A tiling is a collection T of tiles whose supports cover R with the property that any two distinct tiles in T have supports with disjoint interiors. A patch P is a finite subset of a tiling with support equal to the union of the supports of its constituent tiles. Given where φ(i) = i 1 · · · i k . Extend Φ to patches P by Φ(P ) := ∪ τ ∈P Φ(τ ), and likewise to tilings. Notice that spt(Φ(P )) = λ · spt(P ).
Given a primitive substitution φ as above, there are k ∈ N and a, b ∈ A so that φ k (a) = a · · · , φ k (b) = · · · b, and ba is in the language, L := {w : w is a factor of φ n (i) for some n ∈ N and i ∈ A}, of φ.
with the closure taken in the local topology in which two tilings are close if a small translate of one agrees with the other in a large neighborhood of the origin. This is a metric topology (we will denote the metric by d), in which Ω φ is compact and connected. Furthermore, the space Ω φ does not depend on T and the R-action by translation, (S, t) → S − t, on Ω φ is minimal and uniquely ergodic (see, for example, [AP]).
For φ that is primitive and non-periodic (that is, there are no translation periodic tilings in Ω φ ), the map Φ : Ω φ → Ω φ is a homeomorphism ( [M], [S2] Given T ∈ Ω φ and R ≥ 0, the R-patch of T at 0 is defined as (HereB R (0) is the closed ball of radius R at the origin.) Thus d(T, T ′ ) is small if there is a t with |t| small, and a large R, so that The substitution φ is a Pisot substitution if its inflation λ is a Pisot number. For such φ, the maximal equicontinuous factor (Ω max , R) of the tiling dynamical system (Ω φ , R), which is unique up to topological isomorphism, is a (non-trivial) torus or solenoid of dimension equal to the algebraic degree of λ, the factor map π max : Ω φ → Ω max is uniformly finite-to-one, measure preserving (with respect to the unique invariant measure on Ω φ and Haar measure on Ω max ), and almost everywhere r-to-one for an r < ∞ called the coincidence rank of φ ([BKw, BBK]). The system (Ω φ , R) has pure discrete spectrum if and only if the coincidence rank, r, equals 1. Besides semiconjugating R-actions, π max also semi-conjugates Z-actions: there is a hyperbolic and . These two relations are equal for Pisot φ (this is a consequence of the 'Meyer property' -see [BK]) and clearly π max (T ) = π max (T ′ ) if T and T ′ are proximal. The equicontinuous structure relation for

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Pisot φ is strong regional proximality, ∼ srp , defined by: T ∼ srp T ′ if and only if for each [BK]).

Proof of the Main Theorem
Suppose that φ is a (primitive and non-periodic) Pisot substitution on the alphabet then there is ǫ > 0 so that T, T ′ are eventually coincident at t ′ for all t ′ ∈ (t − ǫ, t + ǫ).
We will say that T and T ′ are densely eventually coincident if T, T ′ are eventually coincident at t for a set of t dense in R (hence for an open dense set of t). Define ≈ s on Ω φ by T ≈ s T ′ if and only if T and T ′ are strongly regionally proximal and densely eventually coincident.
The following theorem is a compilation of one-dimensional restrictions of several results of [B]. We include a proof for completeness.
Proof. As mentioned in the previous section, T ∼ srp T ′ is equivalent to π max (T ) = π max (T ′ ), so ∼ srp is a closed equivalence relation. That 'T is densely eventually coincident with T ′ ' really means 'T is open-and-densely eventually coincident with T ′ ' implies that dense eventual coincidence is an equivalence relation, so ≈ s is an equivalence relation. To see that it's closed, suppose that T n → T , T ′ n → T ′ and T n ≈ s T ′ n for each n. Then T ∼ srp T ′ . Fix R > 0 and let t n , t ′ n → 0 be so that (for large n) . By Corollary 5.8 of [BK] there are, up to translation, only finitely many pairs of the form (B R follows that t n = t ′ n for infinitely many n. Since, for |t| < R, eventual coincidence of T n and T ′ n at t depends only on the R-patches of T n and T ′ n at the origin, and T n , T ′ n are densely eventually coincident, we see that T and T ′ are eventually coincident at a set of t dense in B R (0). Thus T ≈ s T ′ and ≈ s is a closed equivalence relation.
The Φ-and R-invariance of ≈ s is clear. Also, since the relation ≈ s is contained in the relation ∼ srp , π max factors through Ω φ / ≈ s .

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is the maximal equicontinuous factor of (Ω φ , R) by showing that strong regional proximality implies dense eventual coincidence. Suppose that T ∼ srp T ′ and suppose that there are t 0 ∈ R and ǫ > 0 so that T and T ′ are not eventually coincident at any t ∈ (t 0 − There are then (for large again used that there are, up to translation, only finitely many pairs (B 0 T and T ′ are eventually coincident at t 0 + (t i /λ n i ). We have a contradiction when i is large enough so that t i /λ n i < ǫ.
As there are only finitely many pairs (B 0 is bounded away from 0. Then by minimality of the R-action and closed-ness of ∼ srp , π max : Ω φ → Ω max is at least 2-to-1 everywhere. But π max is a.e. 1-to-1 since (Ω φ , R) has pure discrete spectrum. Hence T must be densely eventually coincident with T ′ . This proves that if (Ω φ , R) has pure discrete spectrum then ∼ srp is the same as ≈ s ; that is, (Ω φ / ≈ s , R) is the maximal equicontinuous factor.
Conversely, suppose that (Ω φ , R) does not have pure discrete spectrum. Then the coincidence rank of φ is r ≥ 2 and there are z ∈ Ω max and T 1 , . . . , T r ∈ π −1 max (z) with T i and T j disjoint for i = j (this is from [BK] -see Theorem 4 of [B]). By minimality (and the fact that, up to translation, there are only finitely many pairs (B 0 this is true for all z ∈ Ω max . Such T i and T j are not proximal, and therefore Φ n (T i ) and Φ n (T j ) are not proximal for any n ∈ N (this is because Φ is a homeomorphism with Φ(S − t) = Φ(S) − λt, so Φ n preserves proximality for all n ∈ Z). Then by Lemma 5.12 of [BK], Φ n (T i ) and Φ n (T j ) are disjoint for all n ∈ N and i = j. It follows that the factor map from Ω φ / ≈ s to Ω max is at least r-to-1 everywhere. Since π max is almost everywhere r-to-1, the quotient map from Ω φ to Ω φ / ≈ s must be almost a.e. 1-to-1 (and, furthermore, (Ω φ / ≈ s , R) is not the maximal equicontinuous factor).
Assume now that (Ω φ , R) does not have pure discrete spectrum. We will push the Lemma 3.2. : Suppose that φ is Pisot and is constant on final letters. There is then ǫ > 0 so that if (x n , i n )R(x n+1 , i n+1 ] for n = 1, . . . N − 1 and P 1 and P N are and P 1 and P N are densely eventually coincident at s for Proof. Suppose l ∈ A = {1, . . . , d} is such that φ(a) = · · · l for all a ∈ A. Then the tiles ρ i −ω i and ρ j −ω j are eventually coincident at s for all s ∈ (−ω l /λ, 0) for each i, j ∈ A.

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Let ǫ = ω l /λ. The conclusion of the lemma is certainly true if N = 1. Suppose it to be true for some N ≥ 1 and suppose that (x n , i n )R(x n+1 , i n+1 ) for n = 1, . . . N. Then P 1 and P N are densely eventually coincident on [−ǫ, 0] and there are T, T ′ ∈ Ω φ and t, t ′ ∈ R so that: Given a patch Q with support [a, b], letQ be the periodic tilingQ := ∪ k∈Z (Q + k(b − a)). The main technical tool that we use in this article is the following theorem (which has analogues in all dimensions -see [BSW], Theorem 3.1).  Proof. Given tilings T, T ′ made of tiles for φ (but not necessarily in Ω φ ) let us write We claim that if U v is nonempty for some v ∈ R, then U v is dense in R. To see this, suppose that x ∈ U v and fix w ∈ W . There is then x ′ + w ∈ U v . Since x ′ is as close to x as we wish, this shows that x + W ⊂ cl(U v ), and thus U v is dense in R, as claimed.
Suppose now that (Ω φ , R) does not have pure discrete spectrum. There are k ∈ N and

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Let P ⊂ S and P ′ ⊂ S ′ be patches with supports containing [−1, t+1]. We may assume that P − s and P ′ − s ′ are sub patches of Q for some s, s ′ ∈ R (otherwise, replace Q by Φ m (Q) for sufficiently large m, so that, by primitivity, translates of P and P ′ occur Corollary 3.4. Suppose that φ is a primitive Pisot substitution and suppose there is a tile τ for φ and a sequence t n → 0, t n = 0, so that for each n the set of x so that τ and τ − t n are coincident at x is dense in spt(τ ) ∩ spt(τ − t n ). Then (Ω φ , R) has pure discrete spectrum.
Proof. Let a be the type of τ and let w be a word in the language of φ of the form There is m ∈ N so that w occurs in φ m (a) and then there is a sub patch Q of Φ m (τ ) that follows the pattern of the word bu. Let N ∈ N be such that λ m |t n | < ω b for all n ≥ N. The tilingsQ andQ − λ m t n are densely eventually coincident for n ≥ N. Theorem 3.3 applies with W = {λ m t n : n ≥ N}.
The following is Lemma 10 of [B] with slightly weaker hypotheses.
Proof. Let ǫ > 0 be as in Lemma 3.2. If there is no such B then, using Lemma 3.2, there are i ∈ A and y n → y ∈ R, y n = y, so that the patches P = {ρ 1 − ω 1 , ρ 1 } and P n = {ρ 1 − ω 1 − y n , ρ i − y n } are eventually coincident at a dense set of points in an interval I n ⊂ spt(P 1 ) ∩ spt(P n ) of length ǫ. There is then a subsequence and an interval In either case, we have a tile τ , a sequence x k → 0, x k = 0, and a nontrivial interval J ⊂ spt(τ ) ∩ spt(τ − x k ) so that, for each k, τ and τ − x k are eventually coincident at a dense subset of J. If m ∈ N is large enough so that λ m times the length of J is larger than the length of the largest prototile then there is a tile τ ′ ∈ Φ m (τ ) so that τ ′ and τ ′ − λ m x k are eventually coincident at a dense set of points in spt(τ ′ ) ∩ spt(τ ′ − λ m x k ). But then (Ω φ , R) would have pure discrete spectrum by Corollary 3.4.

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Lemma 3.6. Suppose that φ is Pisot, primitive, and constant on final letters and suppose that (Ω φ , R) does not have pure discrete spectrum. ThenR is a closed equivalence relation.
Proof. ClearlyR is an equivalence relation. That R is closed follows from ≈ s being closed; by Lemma 3.5,R is also closed.
Under the hypotheses of the two previous lemmas, let  Proof. Let [(x, i)] denote theR equivalence class of (x, i) and suppose that ( t (x,i) (y, j) so that t min (y,j) = t min (x,i) + t (x,y) (y, j), and t min (y,j) = t min (x,i) + t (x,y) (y, j). It follows that for any t with t min (x,i) < x−t < t max (x,i) and any (y, j) ∈ [(x, i)] we have (x−t, i)R(y−t, j) and t min (y,i) < y −t < t max (y,i) . (This is clearly true if (x, i)R(y, j); then apply this in finitely many steps.) If x ∈I j i , y ∈I n k and (x, i)R(y, j), then I j i = [t min (x,i) , t max (x,i) ], I n k = [t min (y,j) , t max (y,j) ], and we have: Let us say that intervals I j i and I n k as in the Lemma above are equivalent and let be the map that locally stretches length by a factor of λ and follows the pattern of φ, and let π : ∪ d i=1 ρ i → X φ be the quotient map. Note that if π 1 : Ω φ → X φ is given by ≈ s is invariant under Φ,R, pushed forward to X φ (which we continue to callR) is invariant under f φ . It follows from Lemma 3.7 that X φ /R is also a wedge of circles, one for each J k , and f φ induces a continuous map f φs on X φs := X φ /R satisfying: If π s : X φ → X φs is the quotient map, then π s • f φ = f φs • π s . Now let φ s be the substitution on the alphabet {1, . . . , m} with the property that φ s (k) = k 1 k 2 · · · k r if f φs maps π s (α k ) first around π s (α k 1 ), then around π s (α k 2 ), ..., finally around π s (α kr ).

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Lemma 3.8. Suppose that φ is primitive, Pisot, and constant on final letters and suppose that (Ω φ , R) does not have pure discrete spectrum. Then φ s is primitive, Pisot, and constant on final letters and if φ is injective on initial letters, so is φ s .
Proof. If J k is an equivalence class of intervals, each I j i ∈ J k has the property that its endpoints a < b satisfy (a, i) and (b, i) are eachR-equivalent to an endpoint of some prototile: say (a, i)R(0, r) and (b, i)R(ω t , t). Then I 1 r , I m(t) t ∈ J k . Let l ∈ A be such that φ(c) = · · · l for each c ∈ A and let l ′ be such that I On the other hand, α • φ(t) = α(· · · l) = · · · l ′ .
If φ is injective on initial letters there is n ∈ N so that φ n (c) = c · · · for all c ∈ A.
and, from Lemma 3.2, there is ǫ > 0 so that the patches {ρ 1 − ω 1 − x, ρ i − x} and This means that T − t and T ′ − t are densely eventually coincident at a dense set of  j). We are assuming that , and x n −x ′ n is constant. Since Φ −k 1 (T ) and Φ −k 1 (T ′ ) are densely eventually coincident, there is s ∈ spt(ρ i −x 1 )∩spt(ρ j −x ′ 1 ), N ∈ N, and a tile η so that Thus T and T ′ are strongly proximal, and hence strongly regionally proximal. We have shown thatᾱ(T ) =ᾱ(T ′ ) ⇔ T ≈ s T ′ . Since the R-action on Ω φs is minimal and ᾱ is surjective, and hence induces an isomorphism of (Ω φ / ≈ s , R) with (Ω φs , R).

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that tilings T, T ′ in either space have the same image under the maximal equicontinuous factor map if and only if T ∼ srp T ′ . Sinceᾱ(T ) =ᾱ(T ′ ) implies that T and T ′ are strongly regionally proximal, π max factors through Ω φs . Hence, by maximality, and by uniqueness up to isomorphism of the maximal equicontinuous factor, (Ω max , R) and (Ω s max , R) are isomorphic and we may take Ω s max = Ω max and π max = π s max •ᾱ. Now Together withᾱ(T ) ≈ sᾱ (T ′ ) implies T and T ′ are densely eventually coincident, this For each z ∈ Ω max = Ω s max there are T, T ′ ∈ π −1 max (z) with T and T ′ nowhere eventually coincident ( [BK]). Thenᾱ(T ) =ᾱ(T ′ ) but π s max (ᾱ(T )) = π s max (ᾱ(T ′ )).
Thus, π s max is at least 2-1 everywhere and (Ω φs , R) does not have pure discrete spectrum.
Lemma 3.10. Suppose that φ is primitive, non-periodic, Pisot, injective on initial letters, and constant on final letters. Then there are i = j ∈ A so that the prototiles ρ i and ρ j are eventually coincident at a dense set of points in spt(ρ i ) ∩ spt(ρ j ).
Proof. By replacing φ by φ k for an appropriate k ∈ N, we may assume that φ(a) = a · · · for all a ∈ A. Let r be the coincidence rank of φ. There are then T 1 , . . . , T r ∈ Ω φ with the properties: T i ∼ srp T j for all i, j ∈ {1, . . . , r}; Φ n (T i ) ∩ Φ n (T j ) = ∅ for all i = j ∈ {1, . . . , r} and all n ∈ N; and the T i are Φ-periodic (see [BK], or Theorem 4 24 MARCY BARGE of [B]). Replacing φ by an appropriate power, we may assume that the T i are all fixed by Φ. It follows from the constancy of φ on final letters that, for i = j ∈ {1, . . . , r}, the set of endpoints of T i (that is, the set of endpoints of supports of tiles in T i ) is denote the union of the sets of endpoints of the T i , i = 1, . . . , r. For each k ∈ Z and i ∈ {1, . . . , r} let τ k i denote the tile in T i that has (v k + v k+1 )/2 in its support. We will call C k := {τ k 1 , . . . , τ k r } a configuration. Up to translation, there are only finitely many distinct configurations (see [BK]), and since the T i are not (translation) periodic, the translation equivalence of C k is not a periodic function of k. Therefore, there are the configurations C n−1 and C n differ only in the tile of C n−1 with terminal endpoint v n and the tile of C n with initial endpoint v n . Hence there are i = j with ρ i ∈ C k − v k , Suppose (by way of contradiction) that ρ i and ρ j are not eventually coincident at a dense set of points in the intersection of their supports. Note that because φ(i) = i · · · and φ(j) = j · · · , eventual coincidence of ρ i and ρ j at a dense set of points in an interval [0, δ], with δ > 0, would imply eventual coincidence of those prototiles at a dense set of points in the intersection of their supports. Thus there are 0 ≤ t 0 < t 2 ≤ min{v k+1 − v k , v l+1 − v l } so that ρ i and ρ j are not eventually coincident at any point of

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the interval [t 0 , t 2 ]. Let i ′ , j ′ be such that ρ i ∈ T i ′ −v k and ρ j ∈ T j ′ −v l and let t 1 := t 0 +t 2 2 .
By compactness, there are We claim that the S q , q = 1, . . . , r + 1, are pairwise strongly regionally proximal. Indeed, strong regional proximality is closed and Φ-invariant, so the S q , q = 1, . . . , r, are strongly regionally proximal. Pick Then Φ np (T q ′ − v l − t 1 ) → S q ′′ , and again, since ∼ srp is closed and Φ-invariant, S r+1 is strongly regionally proximal with S q ′′ . Hence the S q , q = 1, . . . , r + 1, are all strongly regionlly proximal, as claimed. We further claim that the S q , q ∈ {1, . . . , r + 1}, are pairwise disjoint. For q ∈ {1, . . . , r} this is clear: since the T q are fixed by Φ, the set of configurations formed by the Φ np (T q − v k − t 1 ) equals the set of configurations formed by the T q and hence the set of configurations formed by the S q is contained in the set of configurations formed by the T q . In particular, each configuration formed by the S q , q ∈ {1, . . . , r}, consists of r distinct tiles, so these S q are pairwise disjoint.
The same argument shows that S r+1 is disjoint from S q for q ∈ {1, . . . , r} \ {i 1 }, It remains to show that S r+1 ∩ S i 1 = ∅. For this, consider the tilings U i , U j ∈ Ω φ that are fixed by Φ with ρ c − ω c , ρ i ∈ U i and ρ c − ω c , ρ j ∈ U j , where c is such that φ(a) = · · · c for all a ∈ A. Then S i 1 = lim p→∞ Φ np (U i −t 1 ), and S r+1 = lim p→∞ Φ np (U j − t 1 ). As U i ∼ srp U j (since U i and U j are proximal), there are, up to translation, only For p large enough that |t/λ np | < (t 0 + t 2 )/2, we have s := t 1 − (t/λ np ) ∈ (t 0 , t 2 ) with ρ i and ρ j eventually coincident at s, a contradiction. This proves that S i 1 and S r+1 are disjoint and hence the claim that the S q , q ∈ {1, . . . , r + 1}, are pairwise disjoint.
We now have the situation that there are r + 1 tilings S 1 , . . . , S r+1 that are strongly regionally proximal and pairwise disjoint. Suppose that T ′ = T ′ 1 ∈ Ω φ . By minimality of the R-action, there are s n such that S 1 − s n → T ′ 1 , by passing to a subsequence, we may assume that S q − s n → T ′ q ∈ Ω φ for q = 1, . . . , r + 1. Then the T ′ q are strongly regionally proximal and pairwise disjoint (the latter uses again that, up to translation, there are only finitely many pairs . But then the coincidence rank of φ is at least r + 1, not r. Thus ρ i and ρ j must be eventually coincident at a dense set of points in the intersection of their supports.
Corollary 3.11. Suppose that φ is primitive, non-periodic, Pisot, injective on initial letters, and constant on final letters. Then the relation ≈ s is nontrivial on Ω φ .

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Proof. Say φ(a) = · · · c for all a ∈ A. Let ρ i , ρ j be as in Lemma 3.10 and let U i , U j ∈ Ω φ be the Φ-periodic tilings with ρ c − ω c , ρ i ∈ U i and ρ c − ω c , ρ j ∈ U j . Then U i = U j and Theorem 3.12. Suppose that φ is primitive, Pisot, injective on initial letters, and constant on final letters. Then (Ω φ , R) has pure discrete spectrum.
Proof. Suppose that (Ω φ , R) does not have pure discrete spectrum. Then φ s is also primitive, Pisot, injective on initial letters, and constant on final letters by Lemma 3.8. By Theorem 3.9, ≈ s is trivial on Ω φs . If there were a translation periodic tiling in Ω φs , it would follow from primitivity of φ s that (Ω φs , R), being simply translation on a circle, has pure discrete spectrum, which it doesn't by Theorem 3.9. Thus φ s is non-periodic. But now Corollary 3.11 says ≈ s is nontrivial on Ω φs . Thus (Ω φ , R) must have pure discrete spectrum.
Remark 3.13. By replacing φ by φ n for appropriate n ∈ N, the hypothesis in Theorem 3.12 that φ is constant on final letters can be weakened to φ being eventually constant on final letters. Also, if φ satisfies the other hypotheses but is not necessarily primitive, let c be the (eventual) last letter of all φ(a) and let A ′ := {a ∈ A : ∃n ∈ N with a occurring in φ n (c)}. Then the theorem applies to φ ′ := φ| A ′ to conclude that the 'minimal core' (Ω φ ′ , R) of (Ω φ , R) has pure discrete spectrum. under T β is finite and a simple Parry number if T n β (1) = 0 for some n ∈ N. All Pisot numbers are Parry numbers ( [Ber, Sc]).

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substitution-induced homeomorphism on Ω φ β with the shift homeomorphismT β on the inverse limit space lim ← − T β .
Each non-negative real number x has a greedy expansion in base β: . .) defines a metric isomorphism that conjugates the shift σ on Σ β withT β on lim ← −Tβ . The map where π max : Ω φ β → Ω max is the maximal equicontinuous factor map, gives a continuous and bounded-to-one semi-conjugacy between the β-shift σ and the hyperbolic automorphism (Φ β ) max : Ω max → Ω max . This map g has the nice property of being an arithmetical coding 2 : If x and x ′ are non-negative real numbers with sequences x, x ′ and x + x ′ in Σ β corresponding to the greedy β-expansions of x, x ′ and x + x ′ , then 30 MARCY BARGE g(x + x ′ ) = g(x) + g(x ′ ). Since φ β is primitive, injective on initial letters, and eventually constant on final letters (see Remark 3.13), it follows from Theorem 3.12 that, for Pisot simple Parry numbers, the map π max , and hence the coding g, is a.e. one-to-one.
(This result appears in [BBK] under the additional hypothesis that the algebraic degree of β is greater than n/p, p the smallest prime divisor of n.) Corollary 4.1. If β is a Pisot simple Parry number, then the system (Ω φ β , R) has pure discrete spectrum.
Remark 4.2. For β a Pisot simple Parry number and a unit, the substitutive system (Σ φ β , Z) also has pure discrete spectrum provided the substitution φ β is irreducible (that is, n = d, d the algebraic degree of β - [BKw] or [CS]). But if φ β is reducible this is not necessarily the case as counterexamples of Ei and Ito show ([EI]). In any event, if β is a unit, (Σ φ β , Z) is measurably conjugate to the system of a map induced by a rotation on the (d − 1)-torus ( [BBK], Proposition 8.1).

4.2.
Arnoux-Rauzy, Brun, and Jacobi-Perron substitutions. The substitutions of this section arise as finite products of elements taken from certain collections of basic substitutions. The Arnoux-Rauzy substitutions generalize the two-letter Sturmian substitutions and the three-letter Rauzy substitution (see [BFZ] and [CC] for general accounts of these) and the Brun and Jacobi-Perron substitutions come from multidimensional continued fraction algorithms (see [Be] and [S]).

PURE DISCRETE SPECTRUM FOR A CLASS OF ONE-DIMENSIONAL SUBSTITUTION TILING SYSTEMS 31
Let A = {1, . . . , d} and for each i ∈ A, let σ i be the substitution on A defined by σ i (i) = i and σ i (j) = ji for j = i. Given a word w = w 1 · · · w k ∈ A * that contains at least one occurrence of each letter of A, let σ w = σ w 1 • · · · • σ w k . Such a σ w is called an Arnoux-Rauzy substitution. Arnoux and Ito ([AI]) first proved that all Arnoux-Rauzy substitutions on 2 or 3 letters are irreducible Pisot and recently Avila and Delecroix have proved that Arnoux-Rauzy substitutions on any number of letters are irreducible Pisot ( [AD]). The following (assuming Pisot) is proved in [BSW] by analyzing balanced pairs and a rather different proof, for d = 3, has been given by Berthé, Jolivet, and Siegel ([BJS]). It is easily verified that Arnoux-Rauzy substitutions are primitive, injective on initial letters, and constant on final letters.
The results for the Brun and Jacobi-Perron substitutions follow immediately from Theorem 3.12 and Remark 3.13.