A dynamical classification for crossed products of fiberwise essentially minimal zero-dimensional dynamical systems

We prove that crossed products of fiberwise essentially minimal zero-dimensional dynamical systems have isomorphic $ K $-theory if and only if the dynamical systems are strong orbit equivalent. Under the additional assumption that the dynamical systems have no periodic points, this gives a classification theorem including isomorphism of the $ C^* $-algebras as well. We additionally explore the $ K $-theory of such crossed products and the Bratteli diagrams associated to the dynamical systems.


Introduction
In 1990, Ian Putnam proved in [14] that the crossed product C * -algebras associated to minimal Cantor systems are AT-algebras of real rank zero.Using the classification results of Geroge Elliott in [5] and Dadarlat-Gong in [3], one sees that such C * -algebras are classifiable by their K-theory.In 1995, Putnam, along with Thierry Giordano and Christian Skau, expanded this classification theorem to include dynamics; in [6], they showed that there is a condition on the dynamical systems, called "strong orbit equivalence", that is equivalent to isomorphism of the crossed product C * -algebras.This dynamical classification was motivated by Krieger's theorem (see [9] and [10]), which says that for ergodic non-singular systems, the associated von Neumann crossed product factors are isomorphic if and only if the systems are orbit equivalent.The goal of this paper is to expand these results in the C * -setting.
In our previous paper [8], we determined a condition on a zero-dimensional dynamical system called "fiberwise essentially minimal" (see Definition 1.2 below) that guarantees that the associated crossed product is an AT-algebra.As its name suggests, this class is a broadening of minimal (and also essentially minimal).If we further restrict the system to have no periodic points, this crossed product has real rank zero and is therefore classifiable by K-theory (due to the work of Elliott in [5] and Dadarlat-Gong in [3]).This was an expansion of work done on the minimal Cantor case in 1990 by Ian Putnam (see [13] and [14]) in which the crossed products are simple, and work done on the essentially minimal case in 1992 by Putnam and Skau along with Richard Herman (see [7]) in which the crossed products are not necessarily simple.Our result from [8] includes many more non-simple crossed products.This paper expands on the work in [8] in two major ways.The first of which is what we explore in Section 2, where we discuss some specifics about the K-theory of the crossed products.We define "large  (b) For each t ∈ {1, . . ., T }, X t is a compact open subset of X.That S is subordinate to P means that, for each t ∈ {1, . . ., T }, X t is contained in an element of P.
The following definition is introduced as Definition 1.18 in [8].Definition 1.2.Let (X, h) be a zero-dimensional system and let Z ⊂ X be a closed subset.We say that the triple (X, h, Z) is a fiberwise essentially minimal zero-dimensional system if there is a quotient map ψ : X → Z such that (a) ψ| Z : Z → Z is the identity map.
For examples of fiberwise essentially minimal zero-dimensional systems, see Examples 1.18 in [8].
The connection between Definition 1.1 and Definition 1.2 is the following theorem, which appears as Theorem 2.1 of [8].
Theorem 1.3.Let (X, h) be a zero-dimensional system.Then there exists some closed Z ⊂ X such that (X, h, Z) is fiberwise essentially minimal if and only if for any partition P of X, (X, h) admits a system of finite first return time maps subordinate to P.
The following is Theorem 2.2 in [8].
The following is a consequence of the proof of Theorem 1.4.By "circle algebra", we mean an algebra isomorphic to a finite direct sum of matrices and matrices over C(S 1 ).
Corollary 1.5.Let (X, h, Z) be a fiberwise essentially minimal zero-dimensional system, let P be a partition of X, let a 1 , . . ., a n ∈ C * (Z, X, h), and let ε > 0. Then there is a circle algebra A ⊂ C * (Z, X, h) and a partition P ′ of X that is finer than P such that (a) The diagonal matrices of A are C(P ′ ).
We now introduce the concepts important to the dynamical side of the discussion in this paper.
Let (X 1 , h 1 ) and (X 2 , h 2 ) be dynamical systems.By an orbit map, we mean a homeomorphism F : X 1 → X 2 such that for all x ∈ X 1 , we have F (orb h1 (x)) = orb h2 (F (x)), where orb h1 (x) denotes the h 1 -orbit of x (and likewise for orb h2 ).We say that (X 1 , h 1 ) and (X 2 , h 2 ) are orbit equivalent if there exists such an F .If the orbit map satisfies F • h 1 = h 2 • F , we say that (X 1 , h 1 ) and (X 2 , h 2 ) are conjugate.When we consider orbit maps between (X 1 , h 1 , Z 1 ) and (X 2 , h 2 , Z 2 ) for closed sets Definition 1.6.Let (X 1 , h 1 ) and (X 2 , h 2 ) be dynamical systems and let F : X 1 → X 2 be an orbit map.Then there are functions β, γ : The following is a generalization of Definition 1.3 of [6] from the minimal case to the fiberwise essentially minimal case.Definition 1.7.Let (X 1 , h 1 , Z 1 ) and (X 2 , h 2 , Z 2 ) be fiberwise essentially minimal zero-dimensional systems.We say that (X 1 , h 1 , Z 1 ) and (X 2 , h 2 , Z 2 ) are strong orbit equivalent if there is an orbit map F : X 1 → X 2 such that the orbit cocycles β, γ : X 1 → Z are continuous on X 1 \ Z 1 .

K-Theory
In this section we discuss the K-theory of the crossed products associated to fiberwise essentially minimal zero-dimensional systems.For some references on work done in the minimal case, see [14] and [6].For a reference on work done in the essentially minimal case, see [7].Definition 2.1.An ordered group is a pair (G, G + ), where G is a countable abelian group and G + is a subset of G, called the positive cone, that satisfies the following: (a) For all g 1 , g 2 ∈ G + , we have (c) The identity of G is the only element in both G + and −G + .
We call e ∈ G + an order unit if for all g ∈ G + , there is some n ∈ Z >0 such that ne − g ∈ G + .
Given an ordered group (G, G + ), we may write g ≥ 0 to denote g ∈ G + .The notation g 1 ≥ g 2 means that g 1 − g 2 ∈ G + .By a homomorphism of ordered groups (G 1 , G + 1 ) and (G 2 , G + 2 ), we mean a homomorphism of groups ϕ : When we fix a particular order unit e ∈ G + , we may write the triple (G, G + , e) and call this an ordered group with distinguished order unit.By a homomorphism of ordered groups with distinguished order units (G 1 , G + 1 , e 1 ) and (G 2 , G + 2 , e 2 ), we mean a homomorphism of ordered groups ϕ : We introduce notation important to the following proposition, which is Proposition 2.2 of [8], and a direct consequence of Theorem 2.4 of [11].Let T denote the Toeplitz algebra, the universal C * -algebra generated by a single isometry s.Let A be a unital C * -algebra and let α be an automorphism of A, and let u be the standard unitary of C * (Z, A, α).We denote by T (A, α) the Toeplitz extension of A by α, which is the subalgebra of C * (Z, A, α) ⊗ T generated by A ⊗ 1 and u ⊗ s.The ideal generated by A ⊗ (1 − ss * ) is isomorphic to A ⊗ K, and the quotient by this ideal is isomorphic to C * (Z, A, α).Proposition 2.2.Let (X, h) be a zero-dimensional system.Let α be the automorphism of C(X) induced by h; that is, α is defined by α(f )(x) = f (h −1 (x)) for all f ∈ C(X) and all x ∈ X.Let δ be the connecting map obtained from the exact sequence be the natural inclusion.Then there is an exact sequence Proof.Since K 1 (C(X)) = 0, this follows immediately from Theorem 2.4 of [11].
Note that K 0 (C(X)) ∼ = C(X, Z); we will use this identification throughout the paper.Let C(X, Z) + denote the subset of C(X, Z) consisting of f such that f (x) ≥ 0 for all x ∈ X.Then it is easy to check that (C(X, Z), C(X, Z) + ) is an ordered group and the function χ X is an order unit.
The following is closely related to Proposition 5.1 of [7]; although the hypotheses are broadened, the proof is essentially the same.Adopting the notation of Proposition 2.2, we denote K 0 (C(X))/im(id − α * ) by K 0 (X, h).
Proposition 2.3.Let (X, h, Z) be a fiberwise essentially minimal zero-dimensional system and adopt the notation of Proposition 2.2.Let π : C(X, Z) → K 0 (X, h) denote the quotient map.Define ) is an ordered group with distinguished order unit.
Proof.We check the conditions of Definition 2.1.Conditions (a) and (b) follow from surjectivity of π.For condition (c), let g ∈ K 0 (X, h) + ∩ −K 0 (X, h) + .This means that there is Since f − α * (f ) ≥ 0, we must have h(E) ⊂ E. Let ψ be as in Definition 1.2 and let z ∈ ψ(E) and define E z = E ∩ ψ −1 (z).Since h(E z ) ⊂ E z , E z is invariant so must intersect the minimal set.But then by Theorem 1.1 of [7], since E z = ∅, we have n∈Z ≥0 h n (E z ) = ψ −1 (z), and so E z = ψ −1 (z).
Since this holds for all z ∈ ψ(E), we see that f is constant on ψ −1 (z) for all z ∈ Z. Since ψ −1 (z) is invariant for all z ∈ Z, we must have f = α * (f ), and so f 1 + f 2 = 0, and since f 1 , f 2 ≥ 0, we see , and finally we see g = 0.This proves condition (c).
Finally, the fact that π(1) is an order unit is also clear from the surjectivity of π.
Proof.Since ker(i * ) = im(id − α * ), and since i : C(X) → C * (Z, X, h) is the natural inclusion, the map i * induces a map ϕ : K 0 (X, h) → K 0 (C * (Z, X, h)) which is an isomorphism of groups and satisfies Let p ∈ C * (Z, X, h) be a projection.By applying Corollary 1.5 with a 1 = p and ε = 1/2, p is unitarily equivalent to χ U for some compact open U ⊂ X.Let q be the image of χ U under the Finally, that ϕ(1) = 1 is clear, proving the theorem.
What we have also shown in the previous proof is the following.
Corollary 2.5.Let (X, h, Z) be a fiberwise essentially minimal zero-dimensional system.Let i : is surjective as a map between ordered groups.
The following definition is from Section 2 of [12], later studied in the minimal case in [13].These have been referred to as "large subalgebras" in the literature.
Definition 2.6.Let (X, h, Z) be a fiberwise essentially minimal zero-dimensional system.We define A Z to be the C * -algebra generated by C(X) and uC 0 (X \ Z).
The following theorem is contained in Theorem 2.3 of [12].We provide a direct proof in our context for the reader, which helps give an idea of the AF structure of the large subalgebra.
Theorem 2.7.Let (X, h, Z) be a fiberwise essentially minimal zero-dimensional system.Then A Z is an AF-algebra.
Proof.Let (P (n) ) be a generating sequence of partitions of X.For each n ∈ Z >0 , we inductively define systems )) of finite first return time maps.First, let S (1) = (T (1) , (X t,k )) be any system of finite first return time maps subordinate to P (n) such that P 1 (S (1) ) is finer than P (1) and such that T (1)  t=1 X (1) t ⊃ Z (the former is possible by Proposition 1.14 of [8] and the latter is possible by Lemma 4.12 of [8]).Now, let n ∈ Z >0 and suppose we have chosen )) be any system of finite first return time maps subordinate to P (n+1) such that P 1 (S (n+1) ) is finer than P (n+1) and finer than P 1 (S (n) ) and such that t }, and i, j ∈ {0, . . ., J t,k − 1}.
We see and then notice that uC(X \ Z (n) ) ⊂ A (n) as the superdiagonal matrices, so A (n)   is generated by C(P 1 (S (n) )) and uC(X \ Z (n) ).
Notice that A (n) ⊂ A (n+1) , so we get a directed system of finite dimensional C * -algebras, whose limit A (∞) contains C(X) since (P 1 (S (n) )) is a generating sequence of partitions and since ) is generated by C(X) and uC 0 (X \ Z), and is therefore equal to A Z .
The following is Lemma 4.2 from [13].
Lemma 2.8.Adopt the notation of Theorem 2.7 and its proof.Let p be a projection in C(X) ∩ A (n)   and suppose that p = 0 on Z We finally have the following theorem, which tells us enough about the K 0 structure of the crossed product to be able to prove Theorems 4.2 and Theorem 4.3.The proof follows that of Theorem 4.1 in [13].
Theorem 2.9.Let (X, h, Z) be a fiberwise essentially minimal zero-dimensional system.Then ) denote the canonical inclusion, and let (i 1 ) * and (i 2 ) * denote the induced maps on K 0 .
We then clearly have i ) is a surjective map between ordered groups, and therefore so is i * .By Proposition 2.2, ker((i 2 ) * ) = ran(id − α * ).Thus, since (i 2 ) * = (i 1 ) * • i * , we have (i 1 ) * (ran(id − α * )) ⊂ ker(i * ).Now suppose that a ∈ ker(i * ).Because (i 1 ) * is surjective, we can find g ∈ C(X, Z) We adopt the notation of the proof of Theorem 2.7.That is, we have a generating sequence of partitions (P (n) ) and a sequence of systems of finite first return time maps (S (n) ) and a sequence of subalgebras (A (n) ) of C * (Z, X, h) as in the proof.We now claim that (i 1 ) * (ran(id ) is a generating sequence of partitions and for each n ∈ Z >0 , we have P 1 (S (n) ) is finer than and such that , and so Then f | Z = g| Z , and so by the above paragraph, we have , and so (i 1 ) * (f − α(f )) = 0. Thus, we have (i 1 ) * (ran(id − α * )) = 0. Combining this with (1), we see ker(i * ) = 0. Altogether, we have shown that i * is an isomorphism of ordered groups.
) is an essentially minimal zero-dimensional system, by Theorem 1.1 of [7], there is an and so f = α n (f ), and so , and so f = α(f ), and so f ∈ ker(id − α).
Thus, we have as desired.

Bratelli Diagrams
In this section, we explore the construction of ordered Bratelli diagrams associated to fiberwise essentially minimal zero-dimensional systems.This correspondence is used to prove Theorem 4.2.For work done in the minimal case, see [2] and [6].For work done in the essentially minimal case, see [1] and [7].
V n where V 0 contains a single point v 0 and V n is finite and nonempty for all n ∈ Z ≥0 .For each n ∈ Z ≥0 , we call V n the set of vertices of B at level n.

(b)
The set E is called the set of edges of B. We can write E = ∞ n=1 E n where E n is finite and nonempty for all n ∈ Z >0 .For each n ∈ Z >0 , we call E n the set of edges at level n.
(c) There are maps r, s : is the set of all vertices in V n+1 that are connected to v by an edge, and S(v) is the set of all vertices in V n−1 connected to v by an edge.In a reasonable sense, this gives us range and source maps for vertices.Definition 3.3.An ordered Bratteli diagram B is a Bratteli diagram (V, E) together with a partial order ≤ on E such that e, e ′ ∈ E are comparable if and only if r(e) = r(e ′ ).We write B = (V, E, ≤).
Let B = (V, E, ≤) be an ordered Bratteli diagram.We define E min (E max ) to be the set of all edges that are minimal (maximal, resp.) with respect to ≤.We define V min (V max ) to be the set of all v ∈ V such that there is an e in E min (E max , resp.) with s(e) = v.Definition 3.4.Let B = (V, E, ≤) be an ordered Bratteli diagram.We define a partial Vershik Let y k denote the successor of x k in E and let (y 1 , y 2 , . . ., y k−1 ) be the unique path from v 0 to s(y k ) such that y j ∈ E min for all j ∈ {1, . . ., k − 1}.
Let B = (V, E, ≤) be an ordered Bratteli diagram.For each k, k ′ ∈ Z >0 with k < k ′ , we define P k,k ′ to be the set of all paths from V k to V k ′ .Formally, P k,k ′ is the set of (e k+1 , . . ., e k ′ ) such that for all j ∈ {k + 1, . . ., k ′ }, e j ∈ E j and for all j ∈ {k + 1, . . ., k ′ − 1}, we have r(e j ) = s(e j+1 ).
Definition 3.5.Let B = (V, E, ≤) be an ordered Bratteli diagram.We define the infinite path space X B to be the set of all sequences x = (x 1 , x 2 , . ..)where x n ∈ E n and r(x n ) = s(x n+1 ) for all n ∈ Z >0 together with the topology generated by sets of the form U (e 1 , . . ., e k ), which is the set of all x = (x 1 , x 2 , . ..) with x j = e j for all j ∈ {1, . . ., k}.
Let B = (V, E, ≤) be an ordered Bratteli diagram.It is easy to see that the infinite path space is a zero-dimensional space.We define X B,min (X B,max ) to be the set of all x = (x 1 , x 2 , . ..) ∈ X B such that x j is in E min (E max , resp.) for all j ∈ Z >0 .
The following terminology appears in Definition 2.18 of [?], although we restate it to give more clarity as to when the definition applies.
Definition 3.6.Let B = (V, E, ≤) be an ordered Bratteli diagram and let h B be its partial Vershik transformation.We say that the ordering on B is perfect if for every e ∈ X B,min , orb hB (e) ∩ X B,max contains a single element, and if for every e ∈ X B,max , orb hB (e) ∩ X B,min contains a single element.
In this case, we define the Vershik transformation of X B , denoted by h B , to be the extension of h B which, for each e ∈ X B,min , sends the unique element of orb hB (e) ∩ X B,max to e.
Thus, given an ordered Bratteli diagram B = (V, E, ≤) with a perfect ordering, the system (X B , h B ) is a zero-dimensional system.
Lemma 3.7.Let (X, h) be a fiberwise essentially minimal zero-dimensional system, let P and P ′ be partitions of X, and let S = (T, (X t ), (K t ), (Y t,k ), (J t,k )) be a system of finite first return time maps subordinate to P such that for each t ∈ {1, . . ., T }, we have ψ(X t ) ⊂ X t .Then there is a system )) of finite first return time maps subordinate to P and a system )) of finite first return time maps subordinate to P ′ such that: (a) The partition P 1 (S ′ ) is finer than P ′ and P 1 (S 0 ).(b) The partition P 1 (S 0 ) is finer than P 1 (S).
We now verify the conclusions of the lemma.Conclusions (a), (b), and (c) have already been verified.2) and from the fact that P 1 (S 0 ) is finer than P 1 (S).Conclusion (e) follows from (2) and the fact that that P 2 (S 0 ) is finer than P 2 (S).Conclusion (f) is shown by (4).This proves the lemma.Proposition 3.8.Let (X, h, Z) be a fiberwise essentially minimal zero-dimensional system.There is an ordered Bratteli diagram B = (V, E, ≤) with a perfect ordering such that (a) The system (X B , h B , X B,min ) is conjugate to (X, h, Z).

Conclusion (d) follows from (
(b) For each v in V min (or V max ) there is a v ′ in V min (V max , resp.) and an edge e in E min (E max , resp.) such that s(e) = v and r(e) = v ′ .(c) For each v in V min (or V max ) and each e in E min (E max , resp.) with r(e) ∈ R(v) satisfies s(e) = v.
Proof.Let Z and ψ correspond to (X, h) as in Definition 1.2, and let (P (n) ) be a generating sequence of partitions of X.We will construct an ordered Bratteli diagram B = (V, E, ≤) such that (X B , h B ) is conjugate to (X, h) via a map F : X → X B that satisfies F (Z) = X B,min .
First, we construct a sequence (S (n) ) of finite first return time maps subordinate to (P (n) ).First, let S (1)′ be any system of finite first return time maps subordinate to P (1) such that P 1 (S (1)′ ) is finer than P (1) (such a system exists by Proposition 1.9 of [8]).We construct the other systems inductively.
For each n ∈ Z >0 , we apply Lemma 3.7 with S (n)′ in place of S, P (n) in place of P, P (n+1) in place of P ′ , and get S (n) (that is S 0 in the lemma) and S (n+1)′ (that is S ′ in the lemma) satisfying the conclusions of the lemma.Thus, to construct the sequence of systems, we only need to define S (0) by 1,1 = 1, and J (0) 1,1 = 1.Now we begin to define B. For each n ∈ Z ≥0 , define t,k .Note that this is well defined; by assumption, h j (Y (n+1) t ′ ,k ′ ) is a subset of an element of P 1 (S (n) ), so we do not need to include t and k in the tuple defining this edge.We define an order on the edges r −1 ((n, t, k)) by (n, t, k, j 1 ) ≤ (n, t, k, j 2 ) if j 1 < j 2 .
We now construct the orbit map F : X → X B .Let x ∈ X.Then for each n ∈ Z >0 , there is precisely t }, and one j ∈ {0, . . ., J (this follows from Lemma 3.7(c) and (d)), which is not possible since by definition h i (Y t,k − 1}.This, combined with the fact that P 1 (S (n+1) ) is finer than P 1 (S (n) ), tells us that h j ′ −j (Y t,k , and therefore there is an edge from (n, t, k) to (n + 1, t ′ , k ′ ); namely, (n + 1, t ′ , k ′ , j ′ − j).Thus, this gives us an infinite path in X B associated to x.We define F by sending x to this infinite path.
We now show that F is injective.Suppose x, x ′ ∈ X and F (x) = F (x ′ ) = (e 1 , e 2 , . ..)where we write e n = (n, t n , k n , j n ) for n ∈ Z >0 .First, by definition is it clear that for each n ∈ Z >0 , there are tn,kn ).First, notice that i 1 = j 1 and i ′ 1 = j 1 by definition of F .Then, by definition of F we have j 2 = i 2 − j 1 and j 2 = i ′ 2 − j 2 ; in particular, i 2 = i ′ 2 .Proceeding like this, we see that i n = i ′ n for all n ∈ Z >0 .Since (P (n) ) is a generating sequence of partitions, so is (P 1 (S (n) )), and therefore ∞ n=0 h in (Y tn,kn ) contains at most one element of X.Thus, x = x ′ , and therefore F is injective.
Next, we show that F is surjective.Let e = (e 1 , e 2 , . ..) ∈ X B and write e n = (n, t n , k n , j n ) for n ∈ Z >0 .We construct a sequence (i n ) with i n ∈ {0, . . ., J tn,kn ) is nonempty and contains the element of X that F maps to e. First, let i 1 = j 1 .Then, for all n ∈ Z >1 , let i n = j n + i n−1 (note that this can be rewritten as i n = n k=1 j k ).The claim now follows from the definition of B, and F is therefore surjective.So far we have shown that F is bijective.We now show that F is a homeomorphism.Let U (e 1 , . . ., e N ) be an element of the basis of the topology of X B .For each n ∈ {1, . . ., N }, write tN ,kN ).First notice that by definition of e N , Thus, Similarly, we have Thus, for every n ∈ {1, . . ., N }, we have tn,kn ), and so since j ′ n − j ′ n−1 = j n , the nth edge of F (x) is indeed e n , and F (x) ∈ U (e 1 , . . ., e N ) as desired.Next, we claim that if x ∈ X satisfies F (x) ∈ U (e 1 , . . ., e N ), then x ∈ h j ′ N (Y tN ,kN ).So let x ∈ X satisfy F (x) ∈ U (e 1 , . . ., e N ).Then for each n ∈ {1, . . ., N }, x ∈ h in (J It is clear that i 1 = j 1 .Then, notice that i 2 is such that j 2 = i 2 −j 1 , and so i 2 = j 1 +j 2 = j ′ 2 .Repeating this process inductively, we see that i N = j ′ N , and so tN ,kN ) as desired.Altogether, this shows that F is a homeomorphism.
tn,kn .By definition of the order on B, this means that F (x) ∈ X B,min .Conversely, suppose x ∈ X satisfies F (x) ∈ X B,min .
Write F (x) = (e 1 , e 2 , . ..) and for n ∈ Z >0 write e n = (n, t n , k n , i n ).Since e n is minimal, i n is the minimal element of {0, . . ., J tn,kn .Thus, F (Z) = X B,min .Also notice that if x ∈ h −1 (Z), there are sequences of integers (t n ) and tn,kn ).By definition of the order on B, this means that F (x) ∈ X B,max .Similarly, the converse holds, and so F (h −1 (Z)) = X B,max .

We now show that (F
tn }, and let j n ∈ {0, . . ., K tN ,kN − 1.We have and so by Lemma 3.7(d), there is a Inductively, we can find for each n ∈ {1, . .
tN ,kN and since for each n ∈ Z >N we have tN , it follows that j n +1 < J 1 , e ′ 2 , . ..) and for each n ∈ Z >0 write e n = (n, s n , l n , i n ) and e ′ n = (n, s ′ n , l ′ n , i ′ n ).By definition of F , for all n ∈ Z >0 we have s n = t n , l n = k n , and i n = j n − j n−1 where j 0 = 0. We also see that N is the smallest element of Z >0 such that e N / ∈ E max , so e ′ N is the successor of e N , (e ′ 1 , . . ., e ′ N −1 ) is the minimal path such that r(e ′ N −1 ) = s(e ′ N ), and e ′ n = e n for all n ∈ Z >N .In particular we see that s . Thus, this combined with From ( 5) and ( 6) we see Altogether, we see F (h(x)) = h B (F (x)), and so We now show that the order on B is perfect.For each x ∈ X, z is in the minimal set of the essentially minimal zero-dimensional system (ψ −1 (Z), h| ψ −1 (Z) ), orb(x) contains exactly one element of Z and exactly one element of h −1 (Z).Now, (7) combined with F (Z) = X B,min and tells us that the ordering on B is perfect.It is now clear that F • h = h B • F .This proves conclusion (a) of the proposition.
Before we prove the rest, we first prove two claims that will be used a few times in the proof.
We now prove conclusion (c) of the proposition.Let v ∈ V min and e ∈ E min satisfy r(e) ∈ R(v).
It is clear that v ∈ (S m • R m )(v), and so , meaning there is an path (e 1 , . . ., e m ) with s(e 1 ) = (n, . But then by Lemma 3.7(d) (applied m times), we must have tn,kn .Therefore w ′ ∈ R m (v) (by a minimal path), as desired.An identical argument using Lemma 3.7(e) in place of (d) shows that this equation also holds when v ∈ V max .This proves conclusion (d) of this proposition and therefore finishes the proof of the proposition.Proposition 3.9.Let (X, h) be a fiberwise essentially minimal zero-dimensional system and let B = (V, E, ≤) be an ordered Bratteli diagram that satisfies the conclusions of Proposition 3.8.If ) is a telescoping of B, then B ′ also satisfies the conclusions of Proposition 3.8.
We now show that conclusion (b) holds.Let v ∈ V ′ n,min .What we are looking for is a v ′ ∈ V ′ n+1,min and e ∈ E ′ n+1,min such that s(e) = v and r(e) = v ′ .We can all of this as happening in B instead of B ′ , so that we have v ∈ V kn,min , and we are looking for v ′ ∈ V kn+1,min and a minimal path (e kn+1 , . . ., e kn+1 ) ∈ P kn+1,kn+1 .By Proposition 3.8(b), there is a e kn+1 ∈ E kn+1,min and a v kn+1 ∈ V kn+1,min with s(e) = v and r(e) = v kn+1 .Proceeding inductively, we construct the desired result, with v ′ = v kn+1 .The same argument works for V ′ max in place of V ′ min .This proves that conclusion (b) holds.
Next, we prove that conclusion (c) holds.Let v ∈ V ′ n,min and let e ∈ E ′ n+1,min satisfy r(e) ∈ R(v).We want to show that s(e) = v.We once again regard all of this as happening in B instead of B ′ .
That conclusion (d) holds is immediate; replace m with k n+m − k n .This completes the proof of the proposition.

The Dynamical Classification Theorem
We now prove our main theorems, Theorem 4.2 and Theorem 4.3.where (e ′ 1 , . . ., e ′ n1−1 ) is the minimal path from v 0 ∈ V 0 to s(f n1 ).Now, h M B (e) and f pass through the same vertex at level n 1 − 1.So let n 2 be the largest element of {1, . . ., n and Proof.(⇐).Let B 1 and B 2 be the Bratteli diagrams satisfying the conclusions of Proposition 3.8 for (X 1 , h 1 , Z 1 ) and (X 2 , h 2 , Z 2 ), respectively.By Theorem 2.4, we have By Proposition 3.8(a), we have K 0 (X B1 , h B1 ) ∼ = K 0 (X B2 , h B2 ).By a slight but trivial extension of Theorem 5.4 in [7], we have K 0 (B 1 ) ∼ = K 0 (B 2 ).By [4], we have B 1 ∼ B 2 (in the equivalence class of Bratteli diagrams generated by telescoping and isomorphism), which tells us that there is a (non-ordered) Bratteli diagram B such that telescoping B to odd levels yields a telescoping of B 1 and telescoping B to even levels yields a telescoping of B 2 .By replacing B 1 and B 2 with their telescopings (which can be done without changing the above due to Proposition 3.9), we may assume that telescoping B to odd levels yields B 1 and telescoping B 2 to even levels yields B 2 .Let B ′ be the telescoping of B by the sequence (3n − 2), so that telescoping B to odd levels yields a telescoping of B 1 by (3n − 2) and telescoping B to even levels yields a telescoping of B 2 by (3n − 1).Note that by Proposition 3.9, these telescopings of B 1 and B 2 also satisfy the conclusions of Proposition 3.8.
We denote by V ′ min (V ′ max ) the minimal (resp.maximal) vertices as inherited by B 1 and B 2 .We claim that B ′ has the following properties: There is precisely one v ′ ∈ V ′ min with v ′ ∈ S(v).We first prove property (i).Let v ∈ V ′ min,n .The statement is trivially true for n = 0 so suppose n > 0 and without loss of generality suppose that n is odd.View v as a vertex in V min,3n−2 , so that the statement we are trying to prove becomes R 3 (v) ∩ V min = ∅.Notice that for every vertex min with s(e) = w.Thus, it follows that (S max is analogous.Altogether, this proves property (i).We now prove property (ii).Let v ∈ V ′ min,n and without loss of generality suppose that n is odd.The case n = 1 is trivial so suppose n > 1.View v as a vertex in V min,3n−2 , so that the statement we are trying to prove is that S 3 (v) contains precisely one element.Let w ∈ S(v) and let (e 1 , e 2 ) be a minimal path with r(e 2 ) = w.Since s(e 1 ) ∈ V min , we have S 3 (v) ∩ V min = ∅.Now, suppose But now notice that by Proposition 3.8(d), we have R 4 (v 1 ) = R 2 (R 2 (v 1 ) ∩ V min ) and R 4 (v 2 ) = R 2 (R 2 (v 2 ) ∩ V min ).Thus, we see that (10) and ( 11) yield a contradiction.Therefore, S 3 (v) ∩ V min contains precisely one element.The proof for v ∈ V ′ max is analogous.This proves property (ii).For convenience, replace B (1) with its telescoping by (3n − 2) and replace B (2) with its telescoping by (3n − 1).Now, let e = (e 1 , e 2 , . ..) ∈ X B (1) ,min and let (v 1 , v 2 , . ..) be its associated vertices.Let n ∈ Z >1 .We view v n as a vertex in V ′ 2n−1 .By property (ii), there is a unique vertex v ′ n−1 ∈ V ′ min,2n−2 such that v ′ n−1 ∈ S(v n ).We claim that v n−1 ∈ S(v ′ n−1 ).If not, then by property (ii) there is some w ∈ S(v ′ n−1 ) such that w = v n−1 .But then R 2 (w) ∩ R 2 (v n−1 ) = ∅, which is a contradiction again by Proposition 3.8(c).Thus, for each n ∈ Z >1 , v ′ n−1 is connected by a path to v ′ n , there is an e ′ n ∈ E 2n for all even n ∈ Z >0 .Since this respects the vertices and the range and source maps, we are free to define F 1 (e) = f and F 2 (e ′ ) = f .We now show that the above pairing is a bijection between X 1 , e ′ 2 , . ..) ∈ X B (2) ,min and (v ′ n ) be as above.By property (ii), for each n ∈ Z >0 there is precisely one w n ∈ V ′ min with w n ∈ S(v ′ n ).Since v n ∈ S(v ′ n ), we must have w n = v n .Thus, the bijection is established, and so F 1 (X B (1) ,min ) = F 2 (X B (2) ,min ).We now repeat the process for maximal vertices, choosing edges which are not minimal if the number of edges between the vertices is more than one.This extends F 1 and F 2 so that F 1 (X B (1) ,max ) = F 2 (X B (2) ,max ).
We now extend F 1 and F 2 by any bijection between E Since f ∈ U (e 1 , . . ., e k+1 ) was arbitrary, this shows that α is continuous at e.
This research was supported by the Israel Science Foundation grant no.476/16.