key: cord-0628974-jmf1fyy0
authors: He, Zhuofeng; Wei, Sihan
title: A note on the nuclear dimension of Cuntz-Pimsner $C^*$-algebras associated with minimal shift spaces
date: 2021-12-31
journal: nan
DOI: nan
sha: 3443bbccce440b83d8064a727260038024b102d7
doc_id: 628974
cord_uid: jmf1fyy0

For every one-sided shift space $X$ over a finite alphabet, left special elements are those points in $X$ having at least two preimages under the shift operation. In this paper, we show that the Cuntz-Pimsner $C^*$-algebra $mathcal{O}_X$ has nuclear dimension 1 when $X$ is minimal and the number of left special elements in $X$ is finite. This is done by describing thoroughly the cover of $X$ which also recovers an exact sequence, discovered before by T. Carlsen and S. Eilers.

The Cuntz-Pimsner C * -algebra O X is an invariant of conjugacy associated to any shift space X. This interplay between shift spaces and C * -algebras starts from the study of the C * -algebra O A of a two-sided shift of finite type represented by a {0, 1}-matrix A in a canonical way, see [11] . In the next thirty years, the C * -algebra O X , to every shift space X, is constructed and studied in [1] , [5] , [7] , [8] , [9] , [13] , [14] , [16] , [17] , [18] by several authors (for example, K. Matsumoto, S. Eilers, T. Carlsen, K. Brix and their collaborators, to name a few), but in different manners for their own virtues.

Among these approaches, the cover (X, σ X ), of a one-sided shift space X, is a dynamical system defined by T. Carlsen in [4] , and used to construct the O X as the groupoid C *algebra of the cover. In particular, the reason why Carlsen considers the groupoid C *algebra of the cover but not the shift space X itself is that every such cover defines a dynamical system whose underlying map is a local homeomorphism, while this is not always the case for a one-sided shift. Actually, a one-sided shift on an infinite space is a local homeomorphism if and only if it is of finite type, as in [19] .

In [6] , it is shown that for every shift space X with the property (*), there is a surjective homomorphism ρ : O X → C(X) ⋊ σ Z, which sends the diagonal subalgebra D X onto the canonical commutative C * -subalgebra C(X), with X the corresponding two-sided shift space of X and σ the natural two-sided shift operation. Besides, if X has the property (**), then

where n X is a positive integer related to the structure of the left special elements in X, namely, the number of right shift tail equivalence classes of X containing a left special element. Consequently, for every minimal shift space X, if it has the property (**), which is equivalent to X having finitely many left special elements, then its Cuntz-Pimsner C * -algebra O X is an extension of a unital simple AT-algebra by a finite direct sum of the compact operators. Also note that this extension makes O X falls into a class of C *algebras considered by H. Lin and H. Su in [15] , called the direct limits of generalized Toeplitz algebras.

In [3] , K. Brix considers the C * -algebra O α of a one-sided Sturmian shift X α for α an irrational number, by describing the cover of X α . In particular, he proves that the cover X α of X α is a union of the two-sided Sturmian shift X α and a dense orbit consisting of isolated points. The unique dense orbit corresponds to the unique point ω α in X α which has two preimages under the shift operation. This is the first concrete description of covers of non-sofic systems, whereas the cover of a sofic system is a specific class of shifts of finite type. We remark here that the uniqueness of ω α benefits from the well-known fact that X α has the smallest complexity growth for shift spaces with no ultimately periodic points: p X (n) = n + 1 for all n ≥ 1.

There are two corollaries from the concrete description of the cover of a Sturmian system in [3] : one for a reducing of the exact sequence in [6] to its simplest form, that is, O α is an extension of C(X α ) ⋊ σ Z by K; one for the precise value of dynamic asymptotic dimension of the associated groupoid. The latter together with the exact sequence make the O α be of nuclear dimension 1, where the nuclear dimension is a concept that plays a key role in the classification programs for C * -algebras.

In this note, we generalize this interesting approach and show that for every minimal one-sided shift X with finitely many left special elements, the Cuntz-Pimsner algebra O X has nuclear dimension 1. More specifically, with our concrete description, the cover of each such space will be a finite disjoint union: a copy of the corresponding minimal two-sided shift space X (induced from the projection limit of the original one-sided shift), and n X dense orbits, each consisting of isolated points. This also recovers the whole situation of the exact sequence in [6] . We also hope that with this description, more K-information can be read out from the groupoid for many other minimal shifts, such as non-periodic Toeplitz shifts X with lower complexity growth (which is to sufficiently make X have finitely many left special elements, or equivalently, have the property (**)).

Outline of the paper. The paper is organized as follows. Section 2 will provide definitions, including basic notions of one-sided shift spaces, the corresponding two-sided shift spaces and C * -algebras. In Section 3, we recall definitions of past equivalence, right tail equivalence, covers and their properties. A couple of technical preparations will also be presented for the later use. Section 4 is devoted to the main body of the paper, in which we give a concrete description to the cover of a minimal shift with finitely many left special elements. We divide the description into three parts: (i) for isolated points in the cover, see Theorem 4.1.1; (ii) for the surjective factor π X , see Theorem 4.2.4 and Theorem 4.2.5; (iii) for non-isolated points in the cover, see Theorem 4.3.1. Finally, we conclude our main result for the nuclear dimension of O X in Section 5.

1.2. Acknowledgements. The authors were partially supported by Shanghai Key Laboratory of PMMP, Science and Technology Commission of Shanghai Municipality (STCSM), grant #13dz2260400 and a NNSF grant (11531003). We would like to thank members of Research Center for Operator Algebras, for providing online discussions weekly during the difficult period of COVID-19. The first author would also like to thank his advisor Huaxin Lin from whom he is learning a lot as a doctoral student.

Throughout the paper, we denote by N the set of nonnegative integers. For a finite set S, we will always use #S to denote its cardinality.

2.1. Shift spaces. Let A = {0, 1}. Endowed with the product topology, the spaces A Z and A N are homeomorphic to the Cantor space, i.e., the totally disconnected compact metric space with no isolated point. Note that A Z and A N can be given the following metrics:

We use A * and A ∞ to denote the moniod of finite words and the set of infinite one-sided sequences with letters from A, that is,

where ǫ is the unique empty word in A * . For any word µ ∈ A * , we use |µ| to denote the length of µ and write |µ| = n if µ ∈ A n . For the empty word, we usually define |ǫ| = 0. Besides, the length of any element µ in A ∞ is defined to be ∞. Let µ ∈ A * and ν ∈ A * ⊔ A ∞ , we say µ occurs in ν, if there exists a ∈ A * and b ∈ A * ⊔ A ∞ such that ν = aµb.

If µ occurs in ν, we also say µ is a factor of ν.

A full-shift is a continuous map σ :

A one-sided (two-sided, respectively) shift space is a nonempty compact σ-invariant subspace X of A N (or A Z respectively) together with the restriction σ| X . Note that by σ-invariant, we mean σ(X) ⊂ X. Any two-sided shift is a homeomorphism, and any onesided shift σ : X → X is injective if and only if X is finite. Throughout the paper we will only consider one-sided shifts on infinite compact spaces. If X is a shift space, x ∈ X and −∞ < n ≤ m < ∞, we define

to denote the natural infinite positive and negative parts of x respectively.

For any two-sided shift space X, we use X + to stand for the corresponding one-sided shift space, that is,

If X is a one-sided shift space, then X is used in this paper, to denote the inverse limit of the projective system

Note that X is a two-sided shift space under a canonical identification.

For any shift space X, its language L(X) will play a central role, whose elements are those finite words over A occurring in some x ∈ X. A language uniquely determines a shift space, or in other words, x ∈ X if and only if any factor µ of x is an element of L(X).

This fact implies that for any two-sided shift space Y , σ(Y ) = Y , and therefore for any one-sided shift space X, σ(X) = X if and only if X = (X) + . Any topologically transitive one-sided shift is automatically surjective since its image is a dense compact subset. Definition 2.1.1. Let X be a one-sided shift space and x ∈ X. We define the positive and negative orbits of x to be Orb + σ (x) = {σ n (x) : n ≥ 0} and Orb − σ (x) = {y ∈ X : ∃n > 0(σ n (y) = x)} respectively, and the whole orbit of x to be Orb σ (x) = Orb + σ (x) ∪ Orb − σ (x). 2.2. C * -algebras and groupoids. Definition 2.2.1 (cf. [21] , Definition 2.1). Let A and B be C * -algebras. A * -homomorphism π : A → B is said to have nuclear dimension at most n, denoted dim nuc (π) ≤ n, if for any finite set F ⊂ A and ε > 0, there is a finite-dimensional subalgebra F and completely positive maps ψ : A → F and ϕ : F → B such that ψ is contractive, ϕ is n-decomposable in the sense that we can write

satisfying ϕ| F (i) is c.p.c order zero for all i, and for every a ∈ F, π(a) − ϕψ(a) < ε.

The nuclear dimension of a C * -algebra A, denoted dim nuc (A), is defined as the nuclear dimension of the identity homomorphism id A .

We now recall the definitions of groupoid and its dynamic asymptotic dimension. Definition 2.2.2 (cf. [20] , (3.1)). Let X be a local homeomorphism on a compact Hausdorff space X. We then obtain a dynamical system (X, T ). The corresponding Deaconu-Renault Groupoid is defined to be the set

x ∈ X} identified with X, range and source maps r(x, n, y) = x and s(x, n, y) = y, and operations (x, n, y)(y, m, z) = (x, n + m, z) and (x, n, y) −1 = (y, −n, x).

In this paper, the groupoids G X under consideration will all be locally compact, Hausdorff andétale, where X is the cover of X in the sense of Definition 3.4.3. They are also principal since all such X have no periodic point, as is shown in Section 4.

The Cuntz-Pimsner C * -algebra O X of a one-sided shift space X is defined to be the groupoid C * -algebra C * (G X ). The diagonal-subalgebra D X is defined to be C( X) ⊂ O X .

Finally we recall the definition of dynamic asymptotic dimension forétale groupoids.

Definition 2.2.3 (cf. [12] , Definition 5.1). Let G be anétale groupoid. Then G has dynamic asymptotic dimension d ∈ N if d is the smallest number with the following property: for every open relatively compact subset K of G there are open subsets U 0 , U 1 , · · · , U d of G 0 that covers s(K) ∪ r(K) such that for each i, the set {g ∈ K : s(g), r(g) ∈ U i } is contained in a relatively compact subgroupoid of G.

It is known that for a minimal Z-action on a compact space, the associated groupoid has dynamic asymptotic dimension 1, see Theorem 3.1 in [12] .

From now on, to avoid invalidity or triviality, we only consider infinite one-sided shift space X with σ(X) = X. We use X to denote the associated two-sided shift space.

3.1. Left special elements and past equivalence.

We use Sp l (X) and Sp l (X) to denote the collections of left special elements in X and X respectively.

We say x ∈ X has a unique past if #(σ k ) −1 ({x}) = 1 for all k ≥ 1. Moreover, we say x ∈ X has a totally unique past if σ n (x) has a unique past for all n ≥ 1.

It is clear from the definition that x ∈ X is left special precisely when x has at least two preimages under σ, that is, #σ −1 ({x}) ≥ 2. We note that for any one-sided shift on an infinite space, left special element always exists, or σ will be injective which implies that X is finite. It is also immediate that x has a totally unique past if and only if x / ∈ Orb σ (ω) for any ω ∈ Sp l (X).

is finite, then we can take x ∈ Sp l (X) with infinitely many preimages in Sp l (X) under π + . Denote this infinite preimage by F . Since A is finite, the Pigeonhole principle ensures the existence of an infinite subset

This means that there is some z 1 ∈ F 1 such that (z 1 ) [n 1 ,∞) ∈ Sp l (X). Similarly, choose an infinite subset F 2 ⊂ F 1 , an integer n 2 ≤ n 1 − 1 with the same property as the first step and a point z 2 ∈ F 2 such that (z 2 ) [n 2 ,∞) ∈ Sp l (X). Repeating this procedure, we have a strictly decreasing sequence of negative integers {n k } k≥1 and an infinite sequence {z k } k≥1 ⊂ Sp l (X) with the following property:

Note that it follows from the latter condition that (z k ) [n k ,∞) all lie on a single orbit in X. Since Sp l (X) is finite, it has to contain a periodic point, which is a contradiction.

Notation. Let S ⊂ X be a set and l ∈ N. We define S [0,l] to be the set whose elements are the prefixes of x ∈ S of length l + 1.

. For x, y ∈ X, we say x and y are l-past equivalent and write x ∼ l y, if P l (x) = P l (y). In particular, x and y are said to be past equivalent if x ∼ l y for some l ≥ 1.

We call x isolated in past equivalent if there exists l ≥ 1 such that x ∼ l y implies x = y.

Lemma 3.1.4. Suppose that x ∈ X has a unique past. Then for every l ≥ 1, there exists

Proof. Assume that there exists l 0 ≥ 1 such that for every n ∈ N, we can always find some y n ∈ X with y n [0,n] = x [0,n] but #P l 0 (y n ) ≥ 2. We are then given a sequence {y n } n≥0 which is easily seen to converge to x as n → ∞.

Note that the alphabet A is finite, we now claim that there exists two distinct words µ, ν in L(X) of length l 0 such that two sequences of natural numbers {n k } k≥0 and {m k } k≥0 can be chosen, satisfying µy n k ∈ X and νy m k ∈ X. In fact, from the Pigeonhole principle, there is at least one word µ with |µ| = l 0 such that µ can be a prefix of infinitely many y n , say, y n k for k ≥ 1. However, if µ is the unique word with such property, then all others in L(X) with length l 0 can only be prefixes of finitely many of y n , and which means that for some natural number N , y n will only have the unique prefix µ whenever n ≥ N . This is then a contradiction.

Finally, note that since y n → x as n → ∞, every finite word occurring in µx and νx is an element of L(X). This proves µx, νx ∈ X and hence x does not have a unique past.

Right tail equivalence and j-maximal elements.

Proposition 3.2.2. Suppose that Sp l (X) is finite and contains no periodic point of X. Then every j ∈ J X has a unique j-maximal element. In particular, an element ω ∈ Sp l (X) is j-maximal if and only if

Proof. Let η ∈ j be arbitrary. Since Sp l (X) is finite and contains no periodic point, we

. However, this contradicts to the assumption that σ k (η) / ∈ Sp l (X) for all k ≥ K + 1. Consequently, M = 0, in other words, ω = σ M ′ −1 (ω ′ ). This proves the existence of j-maximal elements.

The uniqueness follows directly from the absence of periodic point in Sp l (X). Finally, the above argument verifies the second assertion at the same time.

Definition 3.2.3. Let X be a one-sided shift space with finite left special elements. From now on, for any j ∈ J X , we will always denote the unique j-maximal element by ω j . For every j ∈ J X , define

Note that for all ω ∈ U j , αω has a unique past whenever αω ∈ X for some α ∈ A.

for all y, y ′ ∈ X, y, y ′ ∈ Sp l (X) and

is isolated in N ′ + 1-past equivalence, and therefore for every l > n ≥ N , σ n (ω) is isolated in l-past equivalence as well.

3.3. Property (*) and (**). Definition 3.3.1 (cf. [8] , Definition 3.1). A one-sided shift space X has property (*) if for every µ ∈ L(X), there exists x ∈ X such that P |µ| (x) = {µ}. We will also say X has property (*) if X does so. Actually, the proof is basically the same as that of Example 3.6 in [8] for the minimal case, which goes like follows. Since X is transitive, take x 0 ∈ X with a dense orbit, which follows that every word in L(X) occurs in x 0 . Therefore it suffices to show that, for every word µ occurring in x 0 , there exists y 0 such that P |µ| (y 0 ) = {µ}. Now since x 0 is a transitive point, µ appears in x 0 infinitely many times. Consider the intersection

Since Sp l (X) is finite and contains no periodic point, this intersection has to be finite, which means that there exists N ≥ 1 such that σ n (x 0 ) / ∈ Sp l (X) for all n ≥ N . This indicates that for all n ≥ N , σ n (x 0 ) has only one preimage. Upon taking L > N + |µ| with (x 0 ) [L−|µ|+1,L] = µ, we conclude that σ L+1 (x 0 ) has only one preimage of length |µ|, and which is exactly µ. Example 3.3.5. Every non-regular Toeplitz shifts has property (*), as is shown in [8] .

We now prove that this is the case for every non-periodic Toeplitz shift. The same notations as in [22] will be used in the following proposition. Proposition 3.3.6. Let η be a non-periodic Toeplitz sequence. Then X η has property (*).

Proof. Let µ ∈ L(X η ). Without loss of generality, assume η [0,m−1] = µ for m = |µ|. We show that ∞) can be approximated by a sequence σ n k (η). Write µ = µ 1 µ 2 · · · µ |µ| . We then note that η m−1 = µ |µ| .

Consider the p m−1 -skeleton of η, say,η ∈ (A ∪ {∞}) N . Thenη is a periodic sequence with period orbit {η, σ(η), · · · , σ p m−1 −1 (η)}. From the Pigeonhole principle, there is 0 ≤ l ≤ p m−1 − 1 such that there exists infinitely many n k j (j = 1, 2, · · · ) satisfying

for some l ∈ {0, 1, · · · , p m−1 − 1}, which follows (σ n k j (η)) n = η m−1 for all n ∈ (l + m − 1) + p m−1 N, and therefore (µ ′ η) n = η m−1 for all n ∈ (l + m − 1) + p m−1 N. Due to the fact that the p-skeleton of a given Toeplitz sequence is the "maximal" periodic part with the given period,η plays the central role. Hence, the assumption that µ ′ η [m,∞) and µη [m,∞) has a common right infinite section indicates that l = 0. We then conclude that for all n ∈ m − 1 + p m−1 N,

and, in particular, µ ′ m = µ m .By repeatedly applying this procedure to m − 1, m − 2, · · · , 0, we therefore have µ ′ = µ. Example 3.3.7. Let X be a one-sided shift. The complexity function p X is defined on positive integers, which sends every n ≥ 1 to the number of finite words in L(X) of length n. Namely, p X (n) = #{µ ∈ L(X) : |µ| = n}.

We say that X has a bounded complexity growth if there exists K > 0 such that p X (n + 1) − p X (n) ≤ K for all n ≥ 1. Then every minimal one-sided shift space with a bounded complexity growth has property (**), as is shown in Proposition 3.3.9. Remark 3.3.8. There is a large class of minimal shifts which have bounded complexity growth, for example, minimal Sturmian shifts, which are defined to be the minimal shifts associated with irrational rotations; minimal shifts associated with interval exchange transformations, whose complexity functions are known to satisfy p X (n + 1) − p X (n) ≤ d where d is the number of subintervals; minimal shifts constructed from (p, q)-Toeplitz words in [10] , where p, q are natural numbers and p|q, whose complexity functions are shown to be linear; or also minimal shifts associated with a class of translations on 2-torus in [2] , whose complexity functions satisfy p X (n) = 2n + 1, to name a few. Proposition 3.3.9. If X is a minimal one-sided shift space with a bounded complexity growth, then X has property (**).

Proof. It suffices to show that X has only finitely many left special elements. Let K ∈ N be a growth bound of X. We actually have #Sp l (X) ≤ K.

If not, then we take K + 1 distinct points {ω 1 , · · · , ω K+1 } ⊂ Sp l (X) and an integer N ∈ N such that the following K + 1 finite words

are distinct. Note that these finite words are all of length N + 1 and each of which can be extended to the left in at least two different ways. This immediately follows that p X (N + 2) − p X (N + 1) ≥ K + 1, a contradiction. The proposition follows.

Definition 3.4.1. We use I to denote the set {(k, l) ∈ N × N : 1 ≤ k ≤ l} and D its diagonal {(k, k) ∈ I : k ≥ 1}. The partial order on I is defined by

For the later use, we prove a lemma first.

Proof. Take (k 0 , l 0 ) ∈ F satisfying l 0 − k 0 = min{l − k : (k, l) ∈ F}.

Set F 0 = {(k, l) ∈ F : k ≤ k 0 }. Then F 0 is nonempty. If F \ F 0 = ∅, take then (k 1 , l 1 ) ∈ F \ F 0 such that l 1 − k 1 = min{l − k : (k, l) ∈ F \ F 0 } and set F 1 = {(k, l) ∈ F \ F 0 : k ≤ k 1 }. By repeating this step, we are given a sequence of sets {F n } n≥0 . If each of F n is finite, then every F n is nonempty, and this is when {(k n , l n )} becomes an infinite chain. Conversely, if one of F n is infinite, say, F N , then by a partition of F N into the following k N +1 − k N parts:

, we see that there exists one of O N k being infinite, which is a chain as well.

As in [4] , for every (k, l) ∈ I, we define an equivalence relation

We write k [x] l for the k,l ∼ equivalence class of x and k X l the set of k,l ∼ equivalence classes. It is clear that k X l is finite. We then have a projective system

for all (k 1 , l 1 ) (k 2 , l 2 ). Definition 3.4.3 (cf. [4] , Definition 2.1). Let X be a one-sided shift space with σ(X) = X. By the cover X of X, we mean the projective limit lim ←− ( k X l , (k,l) Q (k ′ ,l ′ ) ). The shift operation σ X on X is defined so that k σ X (x) l = k [σ( k+1xl )] l where k+1xl is a representative of a k+1,l ∼ -equivalence relation class inx.

The following sets give a base for the topology of X:

for z ∈ X and (k, l) ∈ I. It is known that σ X is a surjective local homeomorphism, see [4] for details. Definition 3.4.4 (cf. [4] , 2.1). Let π X : X → X to be the map which sends eachx ∈ X to a point x = π(x) so that x [0,k) are determined uniquely by ( kxl ) [0,k) for every (k, l) ∈ I. Define ı X : X → X by k ı X (x) l = k [x] l for every (k, l) ∈ I.

In fact, π X is a continuous surjective factor map from ( X, σ X ) to (X, σ) and ı X is an injective map (not necessarily continuous) such that π X • ı X = id X .

Before the sequel, we recall the following lemmas.

Lemma 3.4.5 (cf. [3] , Lemma 4.2) . Let X be a one-sided shift space. Any isolated point in the cover X is contained in the image of ı X and each fibre π −1 X ({x}) contains at most one isolated point. In particular, if x ∈ X is isolated in past equivalence, then ı X (x) is an isolated point in X.

Lemma 3.4.6 (cf. [3] , Lemma 4.4) . Let X be a one-sided shift space. Suppose that x ∈ X has a unique past, then anyx ∈ π −1 X ({x}) also has a unique past. We also note the following lemmas for the later use.

Lemma 3.4.7. Let X be a one-sided shift space with property (**). Suppose that ω, ω ′ ∈ X are left special elements, {(k m , l m )} m≥1 is an infinite sequence in I such that k 1 < k 2 < · · · → ∞. Assume that to every m ≥ 1, an integer 0 ≤ n (km,lm) < l m is associated such that P lm (σ n (km,lm) (ω ′ )) = P lm (ω [km,k m+1 ) σ n (k m+1 ,l m+1 ) (ω ′ )) for all m ≥ 1. Then the sequence {n (km,lm) } m≥1 is unbounded.

Proof. Assume that {n (km,lm) } m≥1 is bounded. Then there exists an infinite subsequence {(k m i , l m i )} i≥1 and an n ∈ N such that n (km i ,lm i ) = n for all i.

By passing to the subsequence {(k m i , l m i )} i≥1 and checking the equality of P lm i , we assume, without loss of generality, that n (km,lm) = n for some n and all m ≥ 1. Note that 0 ≤ n < l m . Now that we have

for all m ≥ 1. The condition k m < k m+1 together with the property (**) then infer that

for all m ≥ 1. Also note that because ω is not periodic, there are infinitely many distinct finite words ω [km,k m+1 ] and we just assume that ω [km,k m+1 ] are all distinct without loss of generality.

The condition 0 ≤ n < l m follows that #P lm (σ n (ω ′ )) ≥ 2, and therefore

for all m ≥ 1. This immediately tells us that every ω [km,k m+1 ) σ n (ω ′ ) lies on the positive orbit of some left special element. However, since w [km,k m+1 ] σ n (ω ′ ) are distinct points lying in the negative orbit of σ n (ω ′ ), we will then have infinitely many distinct special left elements, which contradicts to the assumption that X has property (**).

Lemma 3.4.8. Let X be a minimal one-sided shift with property (**) and x ∈ X. If x has a totally unique past, then ı X (x) ∈ X is not isolated. Consequently, π −1 X ({x}) contains no isolated point for any x having a totally unique past.

Proof. Let z ∈ X and (k, l) ∈ I be so that ı X (x) ∈ U (z, k, l). Then z

Let ω j be an arbitrary j-maximal element for some j ∈ J X . Since X is minimal, then µx [0,k) occurs infinitely many times in the positive orbit of ω j . Take L ∈ N sufficiently large so that

This verifies k x l k,l ∼ z. However, it is clear that k ′x l ′ k ′ ,l ′ ≁ x for some sufficiently large l ′ , since σ L (ω j ) sits in the positive orbit of a left special element.

In this section, X is always assumed to be a one-sided minimal shift space over the alphabet A = {0, 1}, having property (**). We will, as before, still use X to denote the corresponding two-sided shift space. Note that X is also minimal. We also remark that similar conclusions can be drawn for an arbitrary finite alphabet A, but we instead restrict in this paper to the binary shifts for the simplicity of formulations.

First we point out the isolated points in X.

Theorem 4.1.1. The set of isolated points in X is dense in X, which is exactly

where ω j 's are the unique j-maximal elements.

Proof. Write I( X) for the set of isolated points in X. We know from Lemma 3.4.8 that, every isolated point of X has the form ı X (x) for some x ∈ X which does not have a totally unique past. This means x ∈ Orb σ (ω) for some ω ∈ Sp l (X). Assume now ω ∈ j 0 where j 0 is one of right tail equivalence classes. By the definition of j-maximal elements, we immediately see that x ∈ Orb σ (ω j 0 ). This implies the inclusion

Conversely, according to the proof of Lemma 3.2.4, for every j ∈ J X , there is a point z ∈ Orb + σ (ω j ) isolated in past equivalence, which makes, from Lemma 3.4.5, ı X (z) an isolated point in X. On the other hand, recall that as a local homeomorphism, σ X preserves isolatedness and non-isolatedness, which indicates that every point in ı(Orb σ (ω j )) is isolated in X. Since j is arbitrary,

This proves the second assertion. We now show that the set of isolated points in X is dense. Let z ∈ X and (k, l) ∈ I. To show the density, it suffices to take x ∈ Orb(ω j ) such that z k,l ∼ x for some j ∈ J X . We may assume z / ∈ Orb σ (ω j ) for all j ∈ J X . The argument of the existence of such x is then exactly the same as that of Lemma 3.4.8. 

The surjective factor π X . Recall that for every j ∈ J X , the set U j is defined to be U j = {ω ∈ j : Orb − σ (ω) ∩ j = ∅}. Remark 4.2.1. Elements in U j are called adjusted and j-maximal elements ω j are called cofinal in [8] . It is easy to see that U j is nonempty for every j ∈ J X . Lemma 4.2.2. For every j ∈ J X and ω ∈ U j , #π −1 X ({ω}) = 3. Proof. We first show that there are at least three distinct elements in π −1 X ({ω}). The construction below of these three preimages is similar to that of [3] .

For every α ∈ {0, 1} and (k, l) ∈ I \ D, let µ α (k,l) ∈ L(X) with |µ α (k,l) | = l − k − 1 be such that µ α (k,l) αω ∈ X. Note that such finite word µ α (k,l) is unique because ω ∈ U j . Now since X has property (**), we can take x α (k,l) ∈ X satisfying k) . Now for α = 0, 1, setx α ∈ X satisfying

It is clear that π(x α ) = ω and {ı(ω),x 0 ,x 1 } are three distinct elements. It is now enough to show thatx α are well-defined. We will only verify the case for α = 0, since the other one is exactly the same. For the simplicity of notations, we drop all the superscripts and abbreviatex 0 tox, for instance.

For every ν ∈ P l 1 (ω [k 1 ,k 2 ) x (k 2 ,l 2 ) ), νω [k 1 ,k 2 ) ∈ P k 2 −k 1 +l 1 (x (k 2 ,l 2 ) ), and since l 1 + k 2 − k 1 ≤ l 2 , νω [k 1 ,k 2 ) is the suffix of an element in P l 2 (x (k 2 ,l 2 ) ), which follows ν = ν ′ 0ω [0,k 1 ) where ν ′ is the suffix of µ (k 2 ,l 2 ) with length l 1 − k 1 − 1. However, as 0ω has a unique past, ν ′ = µ (k 1 ,l 1 ) .

(ii) Let (k 1 , k 1 ) ∈ D and (k 2 , l 2 ) ∈ I \ D with k 1 ≤ k 2 . We shall confirm that

The case for (k 1 , k 1 ), (k 2 , k 2 ) ∈ D where k 1 ≤ k 2 is quite similar to the case (ii) and hence we omit the verification. Now that we have shown #π −1 X ({ω}) ≥ 3. We next prove that these are exactly the only three elements on this fibre.

Takex ∈ π −1 X (ω) and

Claim. If there exists (k 0 , l 0 ) ∈ I such that #P l 0 (x (k 0 ,l 0 ) ) = 1, thenx ∈ {x 0 ,x 1 }. This is immediate. Suppose P l 0 (x (k 0 ,l 0 ) ) = {µ0ω [0,k 0 ) }, then every x (k ′ ,l ′ ) with (k ′ , l ′ ) (k 0 , l 0 ) are determined. Also note that for all (k ′ , l ′ ) with (k 0 , l 0 ) (k ′ , l ′ ) are also unique determined because 0ω has a unique past. Then x (k,l) are all determined, because I is directed in the sense that given any two points (k ′ , l ′ ), (k ′′ , l ′′ ) ∈ I, we can always find (k ′′′ , l ′′′ ) ∈ I with (k ′ , l ′ ) (k ′′′ , l ′′′ ) and (k ′′ , l ′′ ) (k ′′′ , l ′′′ ). Now assume that #P l (x (k,l) ) ≥ 2 for all (k, l) ∈ I. We then show thatx = ı X (ω), which will finish the proof. Fix any (k 0 , l 0 ) ∈ I. Note that this leads to the fact that, for every (k, l) ∈ I with (k 0 , l 0 ) (k, l), there exists ω (k,l) ∈ Sp l (X) and integers 0 ≤ n (k,l) ≤ l − 1 such that

x (k,l) = σ n (k,l) (ω (k,l) ).

The finiteness of Sp l (X) implies that there is ω ′ ∈ Sp l (X) and infinitely many (k m , l m ) ∈ I satisfying (k 0 , l 0 ) (k m , l m ), k 1 < k 2 < · · · → ∞ and

x (km,lm) = σ n (km,lm) (ω ′ ), for all m ≥ 1.

Upon passing to a subsequence, we may assume, according to Lemma 3.4.2, that {(k m , l m )} m≥1 is a chain in the sense that (k m , l m ) (k m ′ , l m ′ ) whenever m ≥ m ′ . By the definition of cover, we then have P lm (σ n (km,lm) (ω ′ )) = P lm (ω [km,k m+1 ) σ n (k m+1 ,l m+1 ) (ω ′ )). Now Lemma 3.4.7 applies, indicating that n (km,lm) is unbounded. Hence we may assume, without loss of generality, that n (km,lm) → ∞ as m → ∞.

On the other hand, from Lemma 3.2.4, we can take an N ∈ N such that σ n (ω ′ ) is isolated in l past equivalence whenever l > n ≥ N . Choose M ∈ N such that n (km,lm) > N whenever m > M . This follows that this contradicts to the minimality of m 0 . Therefore, z = ω and hencez ∈ {ı(ω),x 0 ,x 1 }. This shows the second assertion.

For the first assertion, it is clear that every directed path terminating at σ m 0 (ω) defines an element in π −1 X (σ m 0 (ω)). It is also not hard to see that for everyỹ ∈ π −1 X (σ m 0 (ω)), if #P l (ỹ (k,l) ) ≥ 2, thenỹ = ı X (σ m 0 (ω)). Note that these can both be approached similarly as in Lemma 4.2.2. However, if #P l (ỹ (k,l) ) = 1 for all (k, l) ∈ I, then each suchỹ corresponds to a unique branch of a left special element ω ′ ∈ Orb − σ (σ m 0 (ω)). Finally, we consider those points z ∈ X having totally unique past.

Proof. Letz ∈ X with π X (z) = z. Let us show thatz = ı X (z).

We turn to prove that z k,l ∼ z [0,k) z (k,l) for all (k, l) ∈ I. Obviously they have the same initial sections of length k. Therefore, it remains to verify that P l (z [k,∞) ) = P l (z (k,l) ). Write P l (z [k,∞) ) = {µz [0,k) } where µ is the unique prefix of length l − k. We turn to show the following claims to finish the proof.

Claim 1. µz [0,k) z (k,l) ∈ X: Since z has a unique past, so doesz. Take the uniquẽ z ′ ∈ X so that σ X (z ′ ) =z. Note that this implies

and hence π X (z ′ ) = µz. Denotez ′ = k [ kz 

which tells us z [0,k) z (k,l) = kzl k,l ∼ σ l−k ( lz ′ l ). Therefore, µz [0,k) ∈ P l (σ l ( lz ′ l )) = P l (z (k,l) ). Claim 2. #P l (z (k,l) ) = 1: Since z has a totally unique past, σ k (z) has a unique past. By Lemma 3.1.4, we can choose N 1 ∈ N with the following property:

which follows that P l (σ k ( Nzl+N −k )) = P l (z (k,l) ).

However, since σ k ( Nzl+N −k ) = z k z k+1 · · · z N −1z(N,l+N −k) , it has a prefix, of length N −k = N 1 + 1, equal to σ k (z) [0,N 1 ] . Therefore, by how we choose N 1 , we conclude that #P l (z (k,l) ) = #P l (σ k ( Nzl+N −k )) = 1.

This completes the proof.

Non-isolated points in the cover.

be the non-isolated points in the cover. ThenΛ X ∼ = X, i.e., there is a canonical conjugacy from (Λ X , σ X ) to (X, σ), where X is the two-sided shift associated with X.

Proof. Note that since the set of isolated points is open,Λ X is closed and invariant. We first show that every element ofΛ X has a unique past. For this, we only need to verify, for every k > 0, ω ∈ Sp l (X) andz ∈ π −1 X (σ k (ω)) \ {ı X (σ k (ω))}, thatz has a unique past. Let m z = min{m > 0 : ∃ ω ∈ Sp l (X) (σ m (ω) = z)}. Thenz has a unique m z -past, and for m z + 1, there is a unique element corresponding to the prefix 0 or 1. Thereforez has an m z + 1 past as well. Repeating this procedure and noting that there exists K > 0 such that x has a unique past whenever k ≤ K and x ∈ σ −k (z), we conclude thatz has a unique past. Now we construct a map from X toΛ X . This is a natural construction which is similar to that of the Sturmian case.

If x ∈ X such that σ k (x [0,∞) ) has a unique past for all k ≥ 0, then we set

wherex is the unique element in π −1 X (x [0,∞) ) by Theorem 4.2.5. If x ∈ X such that there is some k ≥ 0 making σ k (x [0,∞) ) don't have a unique past, since X has property (**), we can choose K ≥ 0 such that every element in Orb + σ (σ K (x)) is not left special anymore. Therefore, it is enough to determine Φ(σ K (x)). By abuse of notation, we denote σ K (x) by x. Let k be the smallest natural number such that

Then there is a unique element in π −1 X (x [−k,∞) ) corresponding to x −k−1 . Now by applying this argument to x [−(k+1),∞) , together with the assumption that X only has finitely many of special elements, we get a unique element Φ(x) inΛ X . It is straightforward to check that this a indeed a homeomorphism and a conjugate map. The theorem follows.

We now close this section by summarizing in the following theorem the main results for the cover in this section. Theorem 4.3.2. Let (X, σ) be a one-sided minimal shift over {0, 1} on an infinite space X with finitely many left special elements. Let X be its cover. Then we have the following.

(1) The set I( X) of isolated points in X is a disjoint union:

which forms a dense open subset of X;

(2) The subsystem ( X \ I( X), σ X | X\I( X) ) on the set of non-isolated points is invertible and conjugate to the canonical two-sided shift space X of X;

(3) For every x ∈ X \ j∈J X Orb σ (ω j ), #π −1 X (x) = 1.

Moreover, for every x ∈ j∈J X Orb σ (ω j ), #π −1 X (x) = d(x) + 1, where d(x) is the number of directed path in X terminating at x.

A commutative diagram and the nuclear dimension Proof. It suffices to show that exact sequence on the second row, since c n X 0 corresponds to the abelian C * -algebra of the space of n X discrete orbits and the commutativity of the diagram is induced by π X . From the description of the cover X, the unit space of its groupoid G X decomposes into two parts:

In particular, the groupoid restricted toΛ X is isomorphic to X ⋊ σ Z by Theorem 4.3.1, whose C * -algebra is * -isomorphic to the crossed product C(X) ⋊ σ Z, and the groupoid restricted to the open subset j∈J X ı X (Orb σ (ω j )) is the sum of full equivalence relations restricted on each discrete orbit ı X (Orb σ (ω j )) (j ∈ J X ), whose C * -algebra is * -isomorphic to the direct sum K n X . Then the exactness of the second row follows from Proposition 4.3.2 in [20] . For the nuclear dimension of O X , we first claim that

Claim. G X has dynamic asymptotic dimension 1.

To see this, let K be an open relative compact subset of G X . Denote the groupoid restricted onΛ X = X by GΛ. It has already been verified that GΛ is a minimal reversible groupoid, or in other words, a groupoid of an invertible minimal action on an infinite compact space, which follows that it has asymptotic dimension 1.

Then there are open subsetsŨ 0 ,Ũ 1 of its unit space G 0 Λ that cover s(K ∩ GΛ) ∪ r(K ∩ GΛ), and the set {g ∈ K ∩ GΛ : s(g), r(g) ∈Ũ i } is contained in a relatively compact subgroupoid of GΛ for i = 0, 1. Let

It is clear that U i are open and cover s(K)∪r(K). On the other hand, since the right most one is an discrete open set and K is relatively compact, the set {g ∈ K \ GΛ : s(g), r(g) ∈ U i } is a finite set for i = 0, 1. This implies that the groupoid generated by {g ∈ K : s(g), r(g) ∈ U i } is a relatively compact subgroupoid for i = 0, 1. This shows G X has dynamic asymptotic dimension 1. Now from Theorem 8.6 of [12] , dim nuc (O X ) ≤ 1.

However, by the exact sequence and Proposition 2.9 of [21] ,

≤ dim nuc (K n X ) + dim nuc (C(X) ⋊ σ Z) + 1 = 2.

We then conclude that dim nuc (O X ) = 1. This finishes the proof.

Dimension groups associated to β-expansions

Multidimensional continued fractions and symbolic codings of toral translations

Sturmian subshifts and their C * -algebras. preprint

C * -algebras, groupoids and covers of shift spaces

On C * -algebras associated with sofic shifts

Symbolic Dynamics, Partial Dynamical Systems, Boolean Algebras and C * -algebras Generated by Partial Isometries. preprint

Cuntz-Pimsner C * -algebras associated with subshifts

Matsumoto K-groups associated to certain shift spaces

Some remarks on the C * -algebras associated with subshifts

Toeplitz words, generalized periodicity and periodically iterated morphisms

A class of C * -algebras and topological Markov chains

Dynamic asymptotic dimension: relation to dynamics, topology, coarse geometry, and C * -algebras

Simple C * -algebras arising from β-expansion of real numbers. Ergodic Theory Dynam

Shannon graphs, subshifts and lambda-graph systems

Classification of direct limits of generalized Toeplitz algebras

K-theory for C * -algebras associated with subshifts

Relations among generators of C * -algebras associated with subshifts

Stabilized C * -algebras constructed from symbolic dynamical systems. Ergodic Theory Dynam

Symbolic dynamics and transformations of the unit interval

The primitive ideals of someétale groupoid C * -algebras

The nuclear dimension of C * -algebras

Toeplitz minimal flows which are not uniquely ergodic

Research Center for Operator Algebras

P lm (σ n (km,lm) (ω ′ )) = P lm (ω [km,k m+1 ) x (k m+1 ,l m+1 ) ) that x (km,lm) = σ n (km,lm) (ω ′ ) = ω [km,k m+1 ) x (k m+1 ,l m+1 ) for all m > M . Finally, since k 1 < k 2 < · · · < k m < k m+1 < · · · → ∞ as m → ∞, we conclude that for all m > M ,x (km,lm) = σ km (ω), and therefore kmxlm = km [ω] lm . Recall that (k m , l m ) (k 0 , l 0 ) for every m, we then haveFinally, as the above discussion can be applied to every (k 0 , l 0 ) ∈ I, x = ı X (ω), the lemma follows.It is not hard to see that for every one-sided shift space X with #Sp l (X) < ∞ and every x ∈ X, the number of directed path in X terminating at x is finite. We denote this number by d(x).Fix ω ∈ Sp l (X) and m 0 = min{m > 0 : σ m (ω) ∈ Sp l (X)}, it is clear thatHence it suffices to show that #π −1 X (σ m 0 (ω)) = 1 + ω ′ ∈Orb − σ (σ m 0 (ω))∩Sp l (X) d(ω ′ ) and, #π −1 X (ω) = #π −1 X (σ(ω)) = · · · = #π −1 X (σ m 0 −1 (ω)). We will only prove the case for which ω ∈ U j , due to the fact that we can use a similar procedure to approach other left special elements on the positive orbit of ω or on other branches of left special elements. For this, by lemma 4.2.2, we first show that #π −1 X (σ(ω)) = · · · = #π −1 X (σ m 0 −1 (ω)) = 3. Let 1 ≤ i ≤ m 0 − 1. We prove the assertion by showing thatIt is clear that these three elements are distinct and in the preimage of σ i (ω). Hence it is enough to show that there are no more elements in this fibre. Suppose thatỹ ∈ π −1 X (σ i (ω)). Since σ X is surjective, there existsz such that σ i X (z) =ỹ. Take z = π X (z). Since π X is a factor,