C*-algebras associated with interval maps

For each piecewise monotonic map tau of [0,1], we associate a pair of C*-algebras F_tau and O_tau and calculate their K-groups. The algebra F_tau is an AI-algebra. We characterize when F_tau and O_\tau are simple. In those cases, F_tau has a unique trace, and O_tau is purely infinite with a unique KMS-state. In the case that tau is Markov, these algebras include the Cuntz-Krieger algebras O_A, and the associated AF-algebras F_A. Other examples for which the K-groups are computed include tent maps, quadratic maps, multimodal maps, interval exchange maps, and beta-transformations. For the case of interval exchange maps and of beta-transformations, the C*-algebra O_tau coincides with the algebras defined by Putnam and Katayama-Matsumoto-Watatani respectively.


Introduction
Motivation. There has been a fruitful interplay between C*-algebras and dynamical systems, which can be summarized by the diagram dynamical system → C*-algebra → K-theoretic invariants.
A classic example is the pair of C*-algebras F A , O A that Krieger and Cuntz [10] associated with a shift of finite type. They computed the K-groups of these algebras, and observed that K 0 (F A ) was the same as a dimension group that Krieger had earlier defined in dynamical terms [34]. Krieger [35] had shown that this dimension group, with an associated automorphism, determined the matrix A up to shift equivalence. Later it was shown that this dimension group and automorphism determines the two-sided shift up to eventual conjugacy, cf. [43]. Cuntz [9] showed that the K-groups of O A are exactly the Bowen-Franks invariants for flow equivalence. Not only have these algebras provided interesting invariants for the dynamical systems, but the favor has been returned, in the sense that these algebras have been an inspiration for many developments in C*-algebras.
Given a piecewise monotonic map τ of the unit interval, we will associate two C*-algebras F τ , O τ , and calculate their K-theory, thus getting invariants for τ . The invariants are computable in terms of the original map and standard concepts from dynamical systems. Construction. As our notation indicates, our algebras F τ and O τ are generalizations of F A and O A respectively. They include the algebras F A and O A as special cases (Corollary 11.9), but also include simple C*-algebras that are not Cuntz-Krieger algebras (Examples 13.1, 13.2).
Before defining these algebras, we review the construction of two related algebras C * (R(X, σ)) and C * (X, σ). Suppose that X is a compact Hausdorff space, and σ : X → X is a local homeomorphism. Define an equivalence relation R(X, σ) = {(x, y) ∈ X × X | σ n x = σ n y for some n ≥ 0}, with a suitable locally compact topology. For f, g ∈ C c (R(X, σ)), define a convolution product by (f * g)(x, y) = z∼x f (x, z)g(z, y), and an involution * by f * (x, y) = f (y, x). Then C c (R(X, σ)) becomes a *-algebra, which can be normed and completed to become a C*-algebra C * (R(X, σ)). This is a special case of the construction of C*-algebras from locally compact groupoids, so one can make use of the extensive machinery developed by Renault [58]. The definition of C * (X, σ) is similar, except that R(X, σ) is replaced by G(X, σ) = {(x, n, y) ∈ X × Z × X | σ j x = σ k y and n = k − j}.
This construction is due to Deaconu [12]. It generalizes an example of Renault in [58], and is generalized further in [1,59]. It is closely related to a construction of Arzumanian and Vershik [3] in a measure-theoretic context. If this construction is applied to the local homeomorphism given by a one-sided shift of finite type associated with a zero-one matrix A, then C * (R(X, σ)) ∼ = F A and C * (X, σ) ∼ = O A , cf. [12,59].
For the construction above to work, it is crucial that σ is a local homeomorphism. Therefore, if τ : I → I is piecewise monotonic, we associate with τ a local homeomorphism σ : X → X, constructed by disconnecting the interval I = [0, 1] at points in the forward and backward orbit of the endpoints of intervals of monotonicity. This is a well known technique for one dimensional dynamical systems, cf. [30,24,62,66]. Then we define F τ = C * (R(X, σ)), and O τ = C * (X, σ). (There are alternative, equivalent, ways to define F τ and O τ . For example, F τ can be defined directly as an inductive limit of interval algebras (cf. [63,Prop. 12.3] and [14, Thm. 8.1]), or F τ and O τ can be defined as the C*-algebras F E and O E defined by Pimsner [51], where E is an appropriate Hilbert module. (For details, see [13]).
If τ is surjective, from its construction, F τ comes equipped with a canonical endomorphism Φ, so we have a C*-dynamical system (F τ , Φ). The algebra O τ is isomorphic to the crossed product F τ × Φ N, i.e., the universal C*-algebra containing a copy of F τ and an isometry implementing Φ.
Summary of results. We describe the K-groups of these algebras in dynamical terms, show that they determine the C*-algebras, and show that the properties of the C*-algebras are closely related to those of the dynamical system τ : I → I.
(For τ : I → I, an interval J is a homterval if τ n restricted to J is a homeomorphism for all n ≥ 0. The dynamical behavior of an interval map on a homterval is particularly simple, and there are no homtervals if, for example, τ is transitive. Background on homtervals can be found in §II.5] or de Melo-van Strien [44, Lemma II. 3
If (X A , σ A ) is a transitive one-sided shift of finite type, it is well known that X A decomposes into p < ∞ pieces permuted cyclically by σ A , such that σ p A is mixing on each piece. The same kind of decomposition is valid for transitive piecewise monotonic maps (cf. [64,Cor. 4.7]), and for the associated algebra F τ . However, interestingly, we have to exclude the case where τ is bijective (as can be seen from the map x → x + α mod 1, with α irrational). We say τ is essentially injective if it is injective on the complement of a finite set of points, which is equivalent to the associated local homeomorphism σ being a homeomorphism. Of course, a continuous transitive map of the interval can't be essentially injective.
Theorem (Cor. 6.13). If τ is transitive and not essentially injective, then F τ is a finite direct sum A 1 ⊕ · · · ⊕ A n of simple AF-algebras, each possessing a unique tracial state. The endomorphism Φ maps each A i to A i−1 mod n .
Theorem (Thm. 9.5). If τ is transitive and not essentially injective, then O τ is separable, simple, purely infinite, and nuclear, and it is in the UCT class.
Other C*-algebras associated with interval maps. Other authors have associated C*-algebras with particular families of interval maps that coincide with ours. For example, for interval exchange maps τ : I → I, Putnam [55,56] associated a homeomorphism σ : X → X, which is identical to the local homeomorphism we associated with τ viewed as a piecewise monotonic map. The crossed product algebra he analyzes is the same as our O τ .
Katayama-Matsumoto-Watatani [27] defined C*-algebras F ∞ β and O β for βshifts. If τ is the associated β-transformation, for all β, O τ ∼ = O β (Corollary 15.3). When the orbit of 1 under τ is finite, F τ ∼ = F ∞ β (Corollary 15.3). If the orbit of 1 is infinite, the dimension groups K 0 (F ∞ β ) and K 0 (F τ ) are isomorphic as groups, but it is unknown whether they are order isomorphic, and thus unknown whether F ∞ β and F τ are isomorphic. The algebras F ∞ β and O β are defined in quite different ways than our F τ and O τ ; it is only through their K-groups that we are able to establish these isomorphisms.
Other authors have defined C*-algebras for particular families of interval maps in ways that give different C*-algebras from those we've defined. Deaconu and Muhly [13] extended the local homeomorphism approach to branched coverings, which, for example, includes the full tent map. The C*-algebra they associate with the full tent map is not simple; thus it is not the same as our O τ . Renault [59] investigated groupoids and the associated C*-algebras associated with partially defined local homeomorphisms, which includes branched coverings as a special case. Martins-Severino-Ramos [40] associated a zero-one matrix A with quadratic maps whose kneading sequence is periodic, and then computed the K-groups of O A in terms of that kneading sequence. Since the kneading sequence can be periodic without the critical point being periodic, the algebra O τ we associate with such maps is generally different than their O A , though the computation of K 0 (O A ) in terms of the kneading sequence is quite similar. For maps on the interval that are open maps, Exel [20] has defined a C*-algebra, which turns out to be isomorphic to the Cuntz algebra O ∞ for the tent map, while our O τ is isomorphic to O 2 . Kajiwara and Watatani [26] have associated a C*-algebra with the complex dynamical systems given by a polynomial, acting on either the whole Riemann sphere, the Julia set, or its complement. In a few cases (e.g. f (z) = z 2 − 2) the Julia set is an interval, so that f on its Julia set can be viewed as an interval map. For this dynamical system the algebra of Kajiwara and Watatani is isomorphic to O ∞ , while the corresponding algebra O τ we define is isomorphic to O 2 .
Another approach that can be taken when τ : [0, 1] → [0, 1] is continuous and surjective is to form the inverse limit (X ∞ , τ ∞ ), which will be a homeomorphism, and then form the crossed product C(X ∞ ) × τ ∞ Z. This is a special case of the approach of Brenken [5], who associates a Cuntz-Pimsner C*-algebra with closed relations (such as the graph of a function). Since the inverse limits will have fixed points, the crossed product algebras won't be simple, so this also generally gives different algebras than those we consider.

Local homeomorphisms associated with piecewise monotonic maps
In this section we review a construction in [63], which to each piecewise monotonic map τ : [0, 1] → [0, 1] associates a local homeomorphism σ : X → X on X ⊂ R, where X is constructed by disconnecting [0, 1] at certain points. Definition 2.1. Let I = [0, 1]. A map τ : I → I is piecewise monotonic if there are points 0 = a 0 < a 1 < . . . < a n = 1 such that τ | (ai−1,ai) is continuous and strictly monotonic for 1 ≤ i ≤ n. We will assume the sequence a 0 , a 1 , . . . , a n is chosen so that no interval (a i−1 , a i ) is contained in a larger open interval on which τ is continuous and strictly monotonic. The sequence of points 0 = a 0 < a 1 < · · · < a n = 1 is the partition associated with τ , and the intervals {(a i−1 , a i ) | 1 ≤ i ≤ n} are called the intervals of monotonicity for τ . Note that for 1 ≤ i ≤ n, the map τ | (ai−1,ai) extends uniquely to a strictly monotonic continuous map τ i : [a i−1 , a i ] → I, which will be a homeomorphism onto its image.
If τ is not continuous, we will ignore the actual values of τ at the partition points, and instead view τ as being multivalued, with the values at a i (for 1 ≤ i ≤ n − 1) being the values given by left and right limits, i.e. the values of τ at a i are τ i (a i ) and τ i+1 (a i ). We define a (possibly multivalued) function τ on I by setting τ (x) to be the set of left and right limits of τ at x. At points where τ is continuous, If x ∈ I, the generalized orbit of x is the smallest subset of I containing x and forward and backward invariant with respect to τ . Let I 1 be the union of the generalized orbits of a 0 , a 1 , . . . , a n , and let I 0 = I \ I 1 . The set I 0 is dense in I, and both I 0 and I 1 are forward and backward invariant with respect to τ . Definition 2.2. Let I = [0, 1], and let I 0 , I 1 be as above. The disconnection of I at points in I 1 is the totally ordered set X which consists of a copy of I with the usual ordering, but with each point x ∈ I 1 \ {0, 1} replaced by two points x − < x + . We equip X with the order topology, and define the collapse map π : X → I by π(x ± ) = x for x ∈ I 1 , and π(x) = x for x ∈ I 0 . We write X 1 = π −1 (I 1 ), and X 0 = π −1 (I 0 ) = X \ X 1 .
Here X will be homeomorphic to a compact subset of R. Proposition 2. 3. Let τ : I → I be piecewise monotonic, and X as in Definition 2. 2
(2) X 0 is dense in X.

Proof. [63, Prop. 2.2]
Each set I(b 1 , b 2 ) will be clopen in X, and every clopen subset of X is a finite union of such sets [63, Prop. 2
(1) There is a unique continuous map σ : The sets X 0 and X 1 = X \ X 0 are forward and backward invariant with respect to σ. (3) π is a conjugacy from σ| X0 onto τ | I0 .
(4) The sets J 1 = I(a 0 , a 1 ), . . . , J n = I(a n−1 , a n ) are a partition of X into clopen sets such that for 1 ≤ i ≤ n, π(J i ) = [a i−1 , a i ], and σ|J i is a homeomorphism from J i onto σ(J i ).
Proof. [63,Theorem 2.3] Note that if τ is continuous, by (3) and density of I 0 in I and X 0 in X, π will be a semi-conjugacy from (X, σ) onto (I, τ ). We will call σ the local homeomorphism associated with τ . The map σ actually has a somewhat stronger property, which will play an important role in our dynamical description of K 0 (F τ ) (Corollary 7.10).
Definition 2. 5. Let X be a compact metric space. A continuous map σ : X → X is a piecewise homeomorphism if σ is open and there is a partition of X into clopen sets, each mapped homeomorphically onto its image.
If τ : I → I is piecewise monotonic, and σ : X → X is the associated local homeomorphism, then σ will be a piecewise homeomorphism by Proposition 2.4 (4).
If X is any compact metric space and is zero dimensional, then any local homeomorphism σ : X → X is a piecewise homeomorphism. (For each point x, pick a clopen neighborhood on which σ is injective, then select a finite cover of such clopen sets, and finally refine the cover to form a partition.) Property (3) of Proposition 2.4 can be used to show that σ and τ share many properties. Before being more explicit, we review some terminology.
Definition 2. 6. If X is any topological space, and f : X → X is a continuous map, then f is transitive if for each pair U, V of non-empty open sets, there exists n ≥ 0 such that f n (U ) ∩ V = ∅. We say f is strongly transitive if for every non-empty open set U , there exists n such that ∪ n k=0 f k (U ) = X. The map f is (topologically) mixing if for every pair U, V of nonempty open sets, there exists N such that for all n ≥ N , f n (U ) ∩ V = ∅, and f is topologically exact if for every non-empty open set U , there exists n such that f n (U ) = X.
Definition 2.7. If τ : I → I is piecewise monotonic, we view τ as undefined at the set C of endpoints of intervals of monotonicity, and say τ is transitive if for each pair U, V of non-empty open sets, there exists n ≥ 0 such that τ n (U ) ∩ V = ∅, and is topologically mixing if for each pair U, V of non-empty open sets, there exists N ≥ 0 such that for all n ≥ N , τ n (U ) ∩ V = ∅. We say τ is strongly transitive if for every non-empty open set U , there exists n such that ∪ n k=0 τ k (U ) = I. The map τ is topologically exact if for every non-empty open set U , there exists n such that τ n (U ) = I. (If τ is continuous and piecewise monotonic, these definitions are consistent with those in Definition 2. 6.) Clearly every strongly transitive map is transitive, and every topologically exact map is topologically mixing. The converse implications do not hold, as can be seen by considering the two-sided n-shift. However, every continuous piecewise monotonic topologically mixing map is topologically exact, cf. [54,Thm. 2.5], and [6,Thm. 45]. See also [64,Prop. 4.9] for an additional result relating mixing and exactness.
Proposition 2. 8. Let τ : I → I be piecewise monotonic and σ : X → X the associated local homeomorphism.
(4) The topological entropy of σ is equal to the topological entropy of τ .

C*-algebras associated with local homeomorphisms
Let X be a compact metric space, and σ : X → X a local homeomorphism, i.e., a continuous open map such that each point admits an open neighborhood on which σ is injective. In this section we will review the construction of two C*-algebras associated with σ. (While this construction is well known, we will need to refer to some of the details later. ) We first describe two locally compact groupoids associated with σ. (The standard reference for locally compact groupoids and their associated C*-algebras is the book of Renault [58]). These groupoids were first described in [58] for the shift map σ on the space Σ n of sequences {1, 2, . . . , n} N , for p-fold covering maps in [12], for surjective local homeomorphisms in [1] and in [2], and for general local homeomorphisms in [59]. Similar constructions in a measure-theoretic context were introduced in [3].
For each n ∈ N, let and R(X, σ) = ∪ n R n . Give each R n the product topology from X × X, and give R(X, σ) the inductive limit topology. We write x ∼ y when (x, y) ∈ R(X, σ), and define a product and involution on C c (R(X, σ)) by Each R n is a compact open subset of R(X, σ), so we can identify C(R n ) with the functions in C c (R(X, σ)) whose support is contained in R n . Thus C c (R(X, σ)) = ∪ ∞ n=0 C(R n ). Each subspace C(R n ) will be a *-subalgebra of C c (R(X, σ)). By [58, Prop. II. 4.2] there is a norm (necessarily unique) making C(R n ) a C*-algebra, which we denote by C * (R n ). Since C c (R(X, σ)) is the union of the subalgebras C * (R n ), there is a unique norm on C c (R(X, σ)) satisfying the C* norm axioms, and the completion is a C*-algebra that we denote C * (R(X, σ)). (By the uniqueness of the C* norm on C c (R(X, σ)), the full and reduced C*-algebras of R(X, σ) coincide; cf.
[59, proof of Prop. 2.4].) From the discussion above, C * (R(X, σ)) is the inductive limit of the algebras C * (R n ); cf. [ with the product (x, m, y)(y, n, z) = (x, m + n, z), inverse (x, n, y) −1 = (y, −n, x), and with the topology given by the basis of open sets where σ k and σ l are injective on the open sets U and V . Then we can define a *-algebra C c (G(X, σ)) by a convolution product, and for a suitable norm the completion is a C*-algebra, denoted by C * (G(X, σ)), or by C * (X, σ). (The reduced and full C*-algebras associated with G(X, σ) coincide, cf. [59,Prop. 2.4]. We remark that if σ is a homeomorphism, then the map (x, m) → (x, m, σ m x) is a homeomorphism and algebraic isomorphism from the groupoid X × φ Z associated with the transformation group Z acting on X, onto G(X, σ). In that context, it follows that C * (X, σ) There is a close relationship between the two C*-algebras we have just defined. Suppose that σ is surjective, and let A = C * (R(X, σ)). Then the map Φ : C c (R(X, σ)) → C c (R(X, σ)) given by where p(z) is the cardinality of σ −1 (z), extends to a *-endomorphism from A into A.
The image of Φ is qAq, where q = Φ(1). If σ is not injective, then q = 1. Thus, in the terminology of Paschke [48], Φ will be a proper corner endomorphism. Therefore we can form the crossed product A × Φ N (the universal C*-algebra generated by a copy of A and a non-unitary isometry implementing Φ), and C * (X, σ) will be isomorphic to A × Φ N, cf. [12] and [1]. Many maps of the interval that are of interest are not surjective, e.g. the members of the logistic family L k given by L k (x) = kx(1 − x). However, these maps are often eventually surjective. We say a map σ : X → X is eventually surjective if there exists an integer n ≥ 0 such that σ n+1 (X) = σ n (X). In that case we call σ n (X) the eventual range, and σ restricted to its eventual range is surjective. For example, if 2 ≤ k < 4, the map L k is not surjective but is eventually surjective with eventual range being the interval [0, L k (1/2)].
If X is a compact Hausdorff space and σ : X → X is an eventually surjective local homeomorphism, with eventual range Y , then the algebra C * (R(X, σ)) is strongly Morita equivalent to C * (R(Y, σ| Y )), and C * (X, σ) is strongly Morita equivalent to C * (Y, σ| Y ). This follows from the fact that each R(X, σ) equivalence class meets Y , cf. [41,Ex. 2.7 and Thm. 2.8]. Thus for our purposes there is not much lost by restricting consideration to surjective maps, and we will do that whenever it is convenient. 4. Simplicity of C * (R(X, σ)) and C * (X, σ).
For a groupoid G with object space X, define elements x, y ∈ X to be Gequivalent if there is an element of G whose source is x and whose range is y. This is readily seen to be an equivalence relation. A subset A of X is G-invariant if A is saturated with respect to this equivalence relation, i.e., if any element G-equivalent to an element of A is itself in A. We say a locally compact groupoid G is minimal if there are no proper G-invariant open subsets.
Let X be a compact Hausdorff space, and σ : X → X a local homeomorphism. Then C * (R(X, σ)) is simple iff R(X, σ) is minimal ([58, Prop. II. 4.6]). This can be reformulated as follows.
Proposition 4.1. Let X be a compact metric space and σ : X → X a surjective local homeomorphism. Then C * (R(X, σ)) is simple iff σ is topologically exact.
Proof. By compactness of X, R(X, σ) is minimal iff σ is topologically exact, cf. We remark that surjectivity is not necessary in order for C * (R(X, σ)) to be simple. If σ is eventually surjective, and is topologically exact on its eventual range, then a similar proof shows that C * (R(X, σ)) is simple. Similar modifications are possible for the characterizations of simplicity for C * (X, σ) that follow. (1) For every x ∈ X and every open set V , there exist n, m ∈ N such that σ n (x) ∈ σ m (V ).
(1) ⇒ (3) By (1), for every x ∈ X and every open set V of X, there exist n, m ∈ N such that x ∈ σ −n σ m (V ). This is equivalent to m,n σ −n σ m (V ) = X.
Let X be a compact metric space. A local homeomorphism σ : X → X is essentially free if for every pair m, n of distinct non-negative integers there is no open subset of X on which σ m and σ n agree. The proof of the next result relies on Renault's result [59, Prop. 2.5] that if σ : X → X is an essentially free local homeomorphism, and G(X, σ) is minimal, then C * (X, σ) is simple. 3. Let X be a compact metric space, and σ : X → X a surjective local homeomorphism. Then C * (X, σ) is simple iff X has infinite cardinality and σ is strongly transitive. Proof. Assume first that X is infinite, and that σ is strongly transitive. Then G(X, σ) is minimal by Lemma 4.2. Furthermore, since σ is surjective and transitive, there exists a point p with a dense forward orbit, cf. [67, Thm. 5.9]. If σ m = σ n on an open set V with m < n, the orbit of p must enter V , and thus is eventually periodic, and so in particular is finite. The finite orbit of p can't be dense in the infinite space X, so this is impossible. Thus σ is essentially free, and G(X, σ) is Conversely, suppose that C * (X, σ) is simple. Then there are no open G(X, σ)invariant subsets by [58, Prop. II. 4.5], so G(X, σ) is minimal, and therefore σ is strongly transitive by Lemma 4.2. Suppose that X is finite. Surjectivity of σ implies that σ is bijective. Then as observed in the previous section, the map (x, m) → (x, m, σ m x) is a homeomorphism and isomorphism from the groupoid X × σ Z associated with the transformation group Z acting on X, onto G(X, σ). Since Z does not act freely, the associated C*-algebra C * (G(X, σ)) ∼ = C(X) × σ Z is not simple, cf. [11, p. 230]. This contradiction shows that X is infinite.
We now give conditions for simplicity of F τ and O τ .
Proposition 5.2. If τ : I → I is piecewise monotonic and surjective, then F τ is simple iff τ is topologically exact.
Proof. By Proposition 2.8, σ is surjective, and τ is topologically exact iff σ is topologically exact. Now the result follows from Proposition 4.1.
Proof. By Proposition 2.8, τ is surjective, and is transitive iff σ is strongly transitive. Now the result follows from Proposition 4.3.

Structural properties of F τ
We are going to show that F τ is always an AI-algebra (Definition 6.9 below), describe when F τ is an AF-algebra, and give sufficient conditions for F τ to be the direct sum of simple AF-algebras. Notation. Let X be a compact metric space, σ : X → X, n ∈ N, and assume there exists a partition of X into clopen sets X 1 , . . . , X p , and a clopen set , so that each member g of C(Y, M n ) can be written as g = ij g ij e ij , where g ij ∈ C(Y ). Proposition 6.1. Assume X is a compact metric space, σ : X → X a local homeomorphism, and n ∈ N such that X admits a partition into clopen sets each mapped homeomorphically onto Y ⊂ X by σ n . With the notation above, the map π : C * (R n (X, σ)) → C(Y, M p ) given by Proof. π is a *-isomorphism by a straightforward calculation. Lemma 6.2. If X is a compact metric space, and σ : X → X is a piecewise homeomorphism, then for each n ∈ N, there exists a partition B 1 , . . . , B q of X into clopen sets on which σ n is injective, and whose images are pairwise equal or disjoint.
Proof. For each point x ∈ X choose a clopen set on which σ n is injective, then a finite cover of X by such sets, and then a partition of X into clopen sets A 1 , . . . , A p on which σ n is injective. Now let B 1 , . . . , B q be the partition of σ n X into clopen sets generated by the sets σ n (A i ) for 1 ≤ i ≤ p. Then the sets {σ −n (B i ) ∩ A j | 1 ≤ i ≤ q, 1 ≤ j ≤ p} form a partition P of X such that the images of the partition members under σ n are disjoint or equal. Corollary 6. 3. Let X be a compact metric space, σ : X → X a piecewise homeomorphism, n ∈ N. Choose a partition of X into clopen subsets on which σ n is injective, with the images of these sets being either equal or disjoint. Let Y 1 , . . . , Y q be the collection of the distinct images, and let n i be the number of inverse images . Now the corollary follows by repeated application of Proposition 6.1, with V i in place of X and Y i in place of Y .
By "interval" we will always mean a non-degenerate interval, i.e., not a single point.
Lemma 6. 4. If X is a compact subset of R, then there is a decreasing sequence {X n } of subsets of R such that ∩ n X n = X, and such that each X n is a finite union of closed intervals. For such a sequence, C(X) is the inductive limit of the algebras C(X n ), with the connecting maps C(X n ) → C(X n+1 ) given by the restriction map.
is an open interval with the same midpoint as J, but with length shrunk by a factor (1−1/n). Then each X n is a finite union of closed intervals. (The shrinking given by φ n is needed to insure that these intervals are non-degenerate). The sets X n are a decreasing sequence with intersection equal to X. Thus X is the projective limit X 1 ← X 2 · · · where the connecting maps are the inclusion maps. Therefore C(X) is the inductive limit of the sequence If σ : X → X is the local homeomorphism associated with a piecewise monotonic map τ , then X need not be totally disconnected. We recall from [63] the characterization of total disconnectedness for X. Definition 6. 5. An interval J is a homterval for a piecewise monotonic map τ : I → I if τ n is a homeomorphism on J for all n. Example 6. 6. A polynomial of degree ≥ 2 has homtervals iff it has an attractive periodic orbit, cf. [63,Prop. 5.7]. Proposition 6.7. Let τ : I → I be piecewise monotonic, and σ : X → X the associated local homeomorphism, and C the associated partition for τ . Then the following are equivalent.
In particular, these hold if τ is transitive.
Proof. [63, Prop. 5.8] Recall that an AF-algebra is a C*-algebra which is the inductive limit of a sequence of finite dimensional C*-algebras. The following also can be derived from [58, Prop. III.1.15]. Proposition 6. 8. If a compact Hausdorff space X is totally disconnected, and σ : X → X is a local homeomorphism, then C * (R(X, σ)) is an AF-algebra. Proof. It suffices to show C * (R n ) is AF for each n. Since X is totally disconnected, then σ is a piecewise homeomorphism. By Corollary 6.3 is an AF-algebra, and thus C * (R n ) and C * (R(X, σ)) are AF-algebras.
The converse of Proposition 6.8 is false, as was shown by Blackadar [4, Thm. 7 where A is a finite dimensional C*-algebra, and is an AI-algebra if it is an inductive limit of interval algebras. Proposition 6. 10. If τ : I → I is piecewise monotonic, then F τ is an AI-algebra. It is an AF-algebra iff τ has no homterval. In particular, this is true if τ is transitive.
, and thus is an AI-algebra. Hence {C * (R n )} is an increasing sequence of AI-algebras, and is dense in C * (R(X, σ)), so the latter is an AI-algebra. (An inductive limit of AI-algebras is an AI-algebra, since AI-algebras admit a local characterization, cf. Elliott's proof of the analogous result for AT-algebras [19, Lemma 4.2 and Thm. 4
Now suppose that τ has a homterval J. Let π : X → I be the collapse map. Then π −1 (J) ⊂ X is connected, and σ n is injective on it for all n. The same is true for the connected component J ′ of X containing π −1 (J), and J ′ is homeomorphic to [0, 1].
Thus the equivalence relation R(X, σ) restricted to J ′ is trivial. Let p be the projection corresponding to J ′ (where J ′ is viewed as a subset of the diagonal of R(X, σ)). Since no distinct points in J ′ are equivalent, pC c (R(X, σ))p is the set of functions in C c (R(X, σ)) with support in the diagonal intersected with J ′ ×J ′ . This is then isomorphic to C(J ′ ). Since this is dense in pC * (R(X, σ))p, and injections are isometric, then pC * (R(X, σ))p ∼ = C(J ′ ). If C * (R(X, σ)) were an AF-algebra, then the corner pC * (R(X, σ))p would also be AF. Since C(J ′ ) is not an AF-algebra, then neither is C * (R(X, σ)).
It is also possible to give a constructive proof of the fact that F τ is an AI-algebra, by showing that F τ is isomorphic to the algebra A τ defined in [63, §12], cf. [14, Thm. 8.1].
Definition 6. 11. A piecewise monotonic map τ : The map τ will be essentially injective iff the associated local homeomorphism σ : X → X is injective, cf. [63,Lemma 11.3]. Theorem 6.12. If τ : I → I is transitive and is not essentially injective, and σ : X → X is the associated local homeomorphism, then there is a positive integer n and a partition of X into clopen sets X 1 , . . . , X n cyclically permuted by σ, such that σ n restricted to each set X i is topologically exact.
Since σ n is topologically exact on X i , then each A i is a simple AF-algebra by Proposition 4.1 and Proposition 6. 10. Since σ maps Our goal in this section is to show that K 0 (F τ ) is isomorphic to the dynamically defined dimension group DG(τ ) defined in [63]. See [23] or [16] for background on dimension groups.
An ordered abelian group G is unperforated if for each positive integer n and g ∈ G, if ng ∈ G + then g ∈ G + . An ordered abelian group G satisfies the Riesz decomposition property if whenever g 1 , g 2 , h 1 , h 2 ∈ G with g i ≤ h j for i, n = 1, 2, then there exists f with g i ≤ f ≤ h j for i, j = 1, 2.
Definition 7.1. An ordered abelian group G is a dimension group if G is unperforated and has the Riesz decomposition property.
If G is countable, then by a result of Effros-Handelman-Shen [17], G is a dimension group iff G is an inductive limit of a sequence of ordered groups Z n k , cf. [17]. If X is any compact Hausdorff space, then C(X, Z) (with the pointwise ordering) is easily seen to be a dimension group.
Definition 7.2. Let X be a compact Hausdorff space and σ : X → X a local homeomorphism. Then the transfer map L σ : C(X, Z) → C(X, Z) is defined by (We will write L in place of L σ when the meaning is clear from the context.) By compactness of X, and the definition of a local homomorphism, each fiber σ −1 (x) is finite, so the sum in the definition of L is finite. Note that L is a positive map, and if σ is injective on a clopen set E, then The transfer map will play a central role in our dynamical description of K 0 (F τ ). Definition 7. 3. Let X be a compact metric space, and σ : X → X a piecewise homeomorphism. Define an equivalence relation on C(X, Z) by f ∼ g if L n σ f = L n σ g for some n ≥ 0, and denote equivalence class by square brackets.
The set of equivalence classes is an ordered abelian group, denoted G σ . If τ : I → I is piecewise monotonic, and σ : X → X is the associated local homeomorphism, then DG(τ ) is defined to be G σ . (We will see below that G σ and DG(τ ) are dimension groups.) Proposition 7. 5. Let X be a compact subset of R, and let σ : X → X be a piecewise homeomorphism. Then G σ is isomorphic to the inductive limit and is a dimension group if equipped with the positive cone The map L * : G σ → G σ is positive and injective, and is an order automorphism iff σ is eventually surjective.
Proof. [63, Lemma 3.8, Cor. 3.12, and Prop. 4.5] We are going to show in Proposition 7.7 that K 0 (C(X)) = C(X, Z) for X a compact subset of R. We will do this by computing appropriate inductive limits. Lemma 7. 6. If X 1 , X 2 , . . . is a decreasing sequence of compact subsets of a Hausdorff topological space, and X = n X n , then the dimension group C(X, Z) is the inductive limit of the sequence C(X 1 , Z) → C(X 2 , Z) → · · · → C(X n , Z) → · · · , where the connecting maps are the restriction maps.
Proof. Let G be the inductive limit of the sequence C(X n , Z) of dimension groups, thought of as sequences f 1 , f 2 , . . . where f i ∈ C(X i , Z), and such that f i+1 = f i | Xi+1 for all but finitely many i. Two such sequences are identified if they agree for all but finitely many terms. If f i+1 = f i | Xi+1 for all i ≥ n, we represent the equivalence class of this sequence by [f n , n], which will be the same as [g, m] if f n | X k = g| X k for some k > max{m, n}.
To see that Ψ is surjective, let p be the characteristic function of a clopen set E ⊂ X. Choose f ∈ C(X 1 ) such that 0 ≤ f ≤ 1 and such that f | X = p. Then f −f 2 = 0 on X, so by (1), there exists k ≥ 1 such that |f −f 2 | ≤ 3/16 on X k . Then which completes the proof that C(X, Z) is the desired inductive limit.
Proposition 7. 7. If X is a compact subset of R, then there is a unique isomorphism K 0 (C(X)) ∼ = C(X, Z), which for each clopen set E ⊂ X takes the class [χ E ] ∈ K 0 (C(X)) to χ E ∈ C(X, Z).
Proof. Choose X 1 ⊃ X 2 · · · so that each X i is a finite union of closed intervals, and such that ∩ n X n = X, cf. Lemma 6. 4. For a closed interval J, we have K 0 (C(J)) = Z, with a generator being the function constantly 1 on J, so we can identify K 0 (C(J)) with C(J, Z). Therefore K 0 (C(X n )) ∼ = C(X n , Z), and this isomorphism takes the class [χ E ] ∈ K 0 (C(X n )) to χ E ∈ C(X n , Z). Now the result follows from the fact that C(X) is the inductive limit of the sequence C(X 1 ) → C(X 2 ) → . . ., continuity of K 0 with respect to inductive limits, and Lemma 7. 6.
The second author would like to thank Jack Spielberg for helpful conversations regarding the proof of Proposition 7. 7.
For each clopen subset E of X, we write E for the corresponding subset {(x, x) ∈ X × X | x ∈ E} of the diagonal of R(X, σ), and [χ E ] denotes the class of the projection χ E in K 0 (C * (R n )).
Lemma 7. 8. Let X be a compact subset of R, and let σ : X → X be a piecewise homeomorphism. For each k ∈ N, there is a unique isomorphism of K 0 (C * (R k )) onto L k C(X, Z) taking [χ E ] to L k χ E for each clopen set E ⊂ X, and the elements [χ E ] generate K 0 (C * (R k )).

Proof.
We may assume without loss of generality that k = 1. (Otherwise just replace σ by σ k , and observe that L k σ = L σ k , and R k (X, σ) = R 1 (X, σ k ).) By Lemma 6.2, we can construct a partition of X into clopen sets on each of which σ is injective, and whose images are either equal or disjoint. Let Y 1 , . . . , Y p be the distinct images of these sets, and let W i = σ −1 (Y i ) for 1 ≤ i ≤ p. By construction, for each i, W i admits a finite partition into clopen sets mapped homeomorphically by σ onto Y i . It suffices to prove the statement of the lemma with W i in place of X for 1 ≤ i ≤ p. Thus without loss of generality, we may assume p = 1.
Let X 1 , . . . , X n be a partition of X into clopen subsets on which σ is 1-1, such that σ(X i ) = Y ⊂ X for 1 ≤ i ≤ n. We have isomorphisms: Here the first isomorphism is induced by the isomorphism of C * (R 1 ) and C(Y, M n ) given in Proposition 6.1. The second is induced by the inverse of the isomorphism from C(Y ) onto C(Y )e 11 given by f → f e 11 . The third isomorphism is given by Proposition 7.7. Finally, by Lemma 7.4, extending functions in C(Y, Z) to functions in X that are zero off Y gives an isomorphism C(Y, Z) ∼ = LC(X, Z).
Now suppose E is a clopen subset of the partition member X 1 . Then under the sequence of isomorphisms in (2), Since the functions Lχ E generate the group LC(X, Z), it follows that the classes [χ E ] generate K 0 (C * (R 1 )), which proves the uniqueness statement in the lemma. Theorem 7. 9. Let X be a compact subset of R, and let σ : X → X be a piecewise homeomorphism. Then there is a unique isomorphism K 0 (C * (R(X, σ))) → G σ taking [ χ E ] to [χ E ] (for each clopen subset E of X). If σ is surjective, and Φ is the canonical endomorphism of C * (R(X, σ)), then the automorphism Φ * of K 0 (C * (R(X, σ))) is carried to the automorphism ( L σ ) −1 * on G σ . Proof. If we identify X with the diagonal of R(X, σ), i.e., with R 0 (X, σ), for each n ∈ N we have the commutative diagram where the vertical arrow on the left is induced by the inclusion of C * (R 0 ) in C * (R n ), and the horizontal arrows are isomorphisms given in Lemma 7.8. Since is also surjective. It follows that the following diagram commutes: Since C * (R(X, σ)) is the inductive limit of the sequence {C * (R n (X, σ))}, then K 0 (C * (R(X, σ))) is the inductive limit of the sequence {K 0 (C * (R n ))}. By virtue of the commutative diagram above, this is the same as the inductive limit of the sequence L : L n C(X, Z) → L n+1 C(X, Z), which is in turn isomorphic to G σ , cf. Proposition 7. 5. Finally, we show that L * Φ * is the identity map on K 0 (C * (R(X, σ))). For this purpose, it suffices to prove that L * Φ * [χ Y ] = [χ Y ] for every clopen Y ⊂ X. Without loss of generality, Y is contained in one of the sets Y i in Lemma 6.3 (applied with n = 1), so that there are clopen sets E 1 , . . . , E p each mapped bijectively by σ onto , which completes the proof of the lemma.
When τ is transitive, the dimension group DG(τ ) has a unique state scaled by the canonical automorphism L * = Φ −1 * ; the scaling factor is exp(h τ ), where h τ is the topological entropy of τ , cf. [64, Thm. 5.3]. Thus one can recover h τ from the dimension group together with the canonical automorphism.
As usual, σ : X → X denotes the local homeomorphism associated with τ . If D is a finite subset of R, we will say distinct points x, y ∈ D are adjacent in D if there is no element of D between them. Recall that I(c, d) = [c + , d − ] X for c, d ∈ I 1 , where I 1 is the generalized orbit in I of the partition points a 0 , . . . , a n for τ . We will identify clopen subsets of X with their characteristic functions, so I(c, d) will be viewed as an element of DG(τ ). For the definition of τ 1 , . . . , τ n in Theorem 8.1, see Definition 2.1.

Proof.
[63, Cor. 6.3] 9. K-groups of O τ Let A be a unital C*-algebra, and α : A → A a proper corner endomorphism, i.e., q = α(1) is a proper projection of A, and α is a *-isomorphism from A onto qAq. If in addition qAq is a full corner of A, then there is a cyclic exact sequence due to Paschke [49, Thm. 4.1]: The following appears in [12] for the case where σ is a covering map and G(X, σ) is minimal.
Proposition 9.1. Let X be a compact metric space, and σ : X → X a surjective piecewise homeomorphism. Then there is an exact sequence where Φ is the canonical endomorphism of C * (R(X, σ)), cf. equation (1).
Proof. This follows from Proposition 9.1 and the definitions of F τ and O τ . Proposition 9. 3. Let G be an ordered abelian group and T : G → G a positive homomorphism. Let G ∞ = lim → (T : G → G) be the inductive limit, with canonical homomorphisms µ k from the k-th copy of G into G ∞ . Then there is a unique order automorphism T ∞ of G ∞ such that T ∞ (µ k (z)) = µ k (T z) for all k, and ker(id −T ∞ ) ∼ = ker(id −T ).

Proof.
The proof for the case G = Z n in [9, Prop. 3.1] works for general G.
Proposition 9. 4. Let τ : I → I be a piecewise monotonic surjective map, with associated local homeomorphism σ : X → X. Let Φ be the canonical endomorphism of F τ , and L : C(X, Z) → C(X, Z) the transfer map. Then Proof. We apply the exact sequence (5). Since F τ is an AI-algebra, then K 1 (F τ ) = 0, and the first isomorphisms of (8) and (9) follow. Under the isomorphism of . From this, the second isomorphism in (8) follows, and the second isomorphism of (9) follows by a similar argument. By Proposition 7.5 and Definition 7.3, DG(τ ) is isomorphic to the inductive limit lim → ( L : L n (C(X, Z)) → L n+1 (C(X, Z))). Since τ (and therefore σ and L) is surjective, this is the same as the inductive limit lim → ( L : C(X, Z) → C(X, Z)). The induced map L ∞ on DG(τ ) is L * , cf. Proposition 9. 3. Thus by (6) coker(id − L * ) ∼ = coker(id − L).
This completes the proof of the third isomorphism in (8). In a similar manner, from (7) we get the third isomorphism of (9).
Applying a result of Anantharaman-Delaroche [1], Renault [59, Prop. 2.6] has shown that for an essentially free local homeomorphism σ : X → X, if for every non-empty open set U ⊂ X, there exists an open set V ⊂ U , and m, n ∈ N such that σ n (V ) is properly contained in σ m (V ), then C * (X, σ) is purely infinite. We apply this to get a sufficient condition for O τ to be purely infinite.
Theorem 9.5. If τ : I → I is piecewise monotonic, transitive, and not essentially injective, then O τ is separable, simple, purely infinite, and nuclear, and is in the UCT class N .
Proof. Separability follows from the fact that F τ is an AF-algebra (Proposition 6.10), and the fact that O τ is the crossed product of F τ by an endomorphism. Simplicity follows from Proposition 5. 3. Nuclearity of C * (X, σ) is established in [1], or see [59,Prop. 2.4]. Furthermore, G(X, σ) is amenable [59, Prop. 2.4], so the full and reduced C*-algebras coincide. Since τ is transitive, then σ is strongly transitive (Proposition 2.8), so as shown in the proof of Proposition 4.3, σ is essentially free. We would like to show that for each open set W there is an open set V ⊂ W and positive integers m, n such that σ m (V ) is properly contained in σ n (V ). By Theorem 6.12, there is a partition X 1 , . . . , X p of X into clopen sets invariant under σ p , such that σ p is topologically exact on each X i . Choose X j so that X j ∩ W = ∅, and let V be a proper open subset of X j ∩ W . By exactness of σ p on X j , there exists k ≥ 0 such that (σ p ) k V = X j ⊃ V , which proves that O τ = C * (X, σ) is purely infinite. Since AI-algebras are in the UCT class N , and that class is closed under crossed products by N, then O τ is in N .
The hypothesis that τ is not essentially injective cannot be omitted in Theorem 9. 5. For example, if τ : I → I is the piecewise linear map given by x → x+ θ mod 1, with θ irrational, then τ is transitive and essentially injective. The associated local homeomorphism σ : X → X is a homeomorphism, and O τ ∼ = C(X) × Φ Z. Here Lebesgue measure induces a measure on X, invariant with respect to σ, so O τ has a tracial state. Since O τ is simple (Proposition 5.2), this tracial state must be faithful, so O τ can't be purely infinite.
For later reference, we recall the classification results of Kirchberg and Phillips ([33] and [50]). For an exposition, see the book of Rørdam [61, §8.4].
Theorem 9. 6. (Kirchberg and Phillips) Let A 1 and A 2 be unital C*-algebras which are separable, simple, purely infinite, and nuclear, and are in the UCT class N . If there is a unital isomorphism of K 0 (A 1 ) onto K 0 (A 2 ), and an isomorphism of K 1 (A 1 ) onto K 1 (A 2 ), then A 1 and A 2 are isomorphic.
By Theorems 9.5 and 9.6, when τ is piecewise monotonic, transitive, and not essentially injective, the C*-algebra O τ is determined by K 0 (O τ ) and K 1 (O τ ).

Traces and KMS-states
We recall the following definition from [64].
Definition 10.1. Let X be a compact metric space, ψ : X → X a map that takes Borel sets to Borel sets, and m a probability measure on X. Then ψ scales m by a factor s if m(ψ(E)) = s m(E) for all Borel sets E on which ψ is 1-1.
Lemma 10.2. Let τ : I → I be piecewise monotonic, and σ : X → X the associated local homeomorphism. Let m be a measure on X scaled by σ by a factor s. Then there is a trace on F τ = C * (R(X, σ)) which satisfies (10) tr(f ) = X f (x, x) dm for n ∈ N and f ∈ C * (R n ), and this trace is uniquely determined by (10).
Proof. By Lemma 6.2, we can construct a partition of X into clopen sets on each of which σ n is injective, and whose images are either equal or disjoint. Let Y be one of these images, and E 1 , . . . , E q a partition of σ −n Y into clopen sets mapped bijectively by σ n onto Y . Let {E ij } be the canonical matrix units in Proposition 6.1, i.e., E ij (x, y) = 1 if x ∈ E i and y ∈ E j and σ n x = σ n y. Define tr on ∪ n C * (R n ) by (10). By Proposition 6.1 and Corollary 6.3, to prove that tr(f g) = tr(gf ) for all f, g ∈ C * (R n ), it suffices to show that tr(ef ) = tr(f e) for matrix units Thus we need to verify that tr(E ij E ji ) = tr(E ji E ij ), i.e., that tr(E ii ) = tr(E jj ). We have tr(E ii ) = m(E i ), so the key requirement is that m(E i ) = m(E j ) for all i, j. Since σ n maps E i and E j bijectively onto Y , and m is scaled by σ, then m(Y ) = s n m(E i ) = s n m(E j ). This completes the proof that tr defines a trace on each C * (R n ). It is clear that tr is positive and tr(1) = 1, so tr has norm one on each C * (R n ), and therefore extends uniquely to a trace on C * (R(X, σ)).
We now describe one way to find scaling measures.
Definition 10. 3. A map τ : I → I is uniformly piecewise linear if there is a number s > 0, and a partition 0 = a 0 < a 1 < · · · a n = 1, such that τ is linear with slope ±s on each interval (a i , a i+1 ).
Note that if τ : I → I is uniformly piecewise linear with slopes ±s, then Lebesgue measure m is scaled by τ by the factor s. Let σ : X → X be the associated local homeomorphism, and define a measure µ on X by µ(A) = m(π(A ∩ X 0 )). Then it is readily verified that µ is a probability measure on X scaled by σ by a factor s, cf. [64,Prop. 3.3], and so induces a trace on F τ = C * (R(X, σ)) satisfying (10) (with µ in place of m).
Uniformly piecewise linear maps occur more frequently than might be thought. In the following result, τ is assumed to be right continuous at 0, left continuous at 1, and either left or right continuous at any other points of discontinuity. Recall that the topological entropy of τ and σ are the same (Proposition 2.8). Proof. The conjugacy result is in [46] and [64,Cor. 4.4]. If τ is continuous, then so is the conjugate uniformly piecewise linear map, and the fact that ln s = h τ , follows from [42]. For τ discontinuous, see [64,Cor. 4.4].
Theorem 10. 5. Let τ : I → I be piecewise monotonic, not essentially injective, and transitive, and let F τ = ⊕A i be the decomposition as a sum of simple AFalgebras given in Corollary 6. 13. Then each A i has a unique tracial state τ i . There is a tracial state on F τ scaled by Φ, and such a tracial state is unique. The scaling factor is e −hτ .
Proof. Let σ : X → X be the associated local homeomorphism. Let X 1 , . . . , X n be a partition of X into clopen sets, cyclically permuted by σ, such that σ n restricted to each X i is topologically exact, cf. Theorem 6. 12. There is a unique state on DG(τ ) ∼ = K 0 (F τ ) scaled by L * by the factor s = exp(h τ ) ([64, Thm. 5.3]). Since there is a bijection of tracial states on a unital AFalgebra and states on the associated dimension group, cf., e.g., [61, Prop. 1. 5.5], and since Φ * = L −1 * , it follows that there is a unique tracial state on F τ scaled by Φ by the factor 1/s = exp(−h τ ). Furthermore, by Theorem 7.9, K 0 (A i ) = K 0 (C * (R(X i , σ n ))) is isomorphic to G σ n |X i , which has a unique state ([64, Thm. 5.3]). It follows that A i has a unique tracial state.
The uniqueness statement regarding the scaled trace in Theorem 10.5 can fail if τ is essentially injective. For example, there are minimal interval exchange maps with more than one invariant measure, cf. [31] or [32]. Each invariant measure induces a trace on F τ scaled by Φ by the factor s = 1.
Corollary 10. 6. If τ : [0, 1] → [0, 1] is piecewise monotonic and is topologically exact, then F τ is simple and has a unique tracial state. The canonical endomorphism Φ of F τ scales the trace by a factor e −hτ , where h τ is the topological entropy of τ .
Proof. Topological exactness of τ implies topological exactness of σ, so there is just one summand in the decomposition F τ = ⊕A i in Theorem 10.5. We turn to the question of KMS-states on the algebra O τ . Suppose that τ : I → I is surjective, so that O τ is isomorphic to F τ × Φ N. Then O τ is the universal C*algebra generated by F τ and an isometry v such that vav * = Φ(a) for a ∈ F τ . For each scalar λ with |λ| = 1, there is a unique *-automorphism α λ of O τ which fixes F τ and takes v to λv. This gives an action of T on O τ , or alternatively, an action of R (by taking λ = e it ). With respect to this action, the set of fixed points is F τ , and there is a faithful conditional expectation from O τ onto F τ given by [48,49].
Theorem 10.7. If τ : I → I is transitive, and is not essentially injective, then O τ has a unique β-KMS state for β = h τ , and there is no other β-KMS state for 0 ≤ β < ∞.

Proof.
In the remarks at the end of [48], it is observed that β-KMS states for 0 < β < ∞ are precisely those of the form tr •E, where tr is a trace scaled by Φ by the factor e −β , and E is the conditional expectation from O τ onto F τ described above. We know F τ has a unique trace scaled by Φ, and that the scaling factor is exp(−h τ ), so the desired result follows for 0 < β. For the case β = 0, a KMS-state is a tracial state on O τ . Since the latter is simple and purely infinite (Theorem 9.5), no such tracial state exists, so no β-KMS state exists for β = 0.
If σ is exact and positively expansive, the results on KMS-states in Theorem 10.7 follow from [39, Thm. 3.5]. However, the map σ will not be positively expansive unless the forward orbit under τ of the set C of endpoints of intervals of monotonicity is finite, i.e., τ is Markov (as defined below). (For details, see [64, remark after Thm. 4.5]).

Markov maps
In the remaining sections, we will apply the results developed so far to particular families of interval maps. In Figure 1, some of the examples that will be discussed are portrayed.
Recall that if τ : I → I is piecewise monotonic, with associated partition C, and is discontinuous at c ∈ C, we view τ as multivalued at c, with the values given by the left and right limits of τ . For A ⊂ [0, 1], we write τ (A) to denote the set of values of τ at points in A, including the possible multiple values at points in C ∩ A.
Definition 11.1. Let τ : I → I be piecewise monotonic. A Markov partition for τ is a partition 0 = b 0 < b 1 < . . . < b n = 1, with each b i being in I 1 , such that for each i, τ is monotonic on , and such that for some n ≥ 0, τ n (C) ⊂ {b 0 , b 1 , . . . , b n }. We say τ is Markov if it admits a Markov partition, The incidence matrix for τ with respect to this Markov partition is the zero-one n × n matrix A where A ij = 1 iff If τ is piecewise monotonic with associated partition C, and is Markov, then one possible Markov partition is the set of points in the forward orbit of C. This will be the most common kind of Markov partition that we use, but the slightly greater generality allowed in the definition above will be useful in showing that all Cuntz-Krieger algebras O A arise as O τ for some piecewise monotonic map τ , cf. Corollary 11. 9. If σ : X → X is the associated local homeomorphism, then the order intervals I(b i−1 , b i ) will form a partition of X such that each σ(I(b i−1 , b i )) is a union of some order intervals of the form I(b j−1 , b j ). Since the range of τ or σ will be a union of partition intervals, it follows that both τ and σ will be eventually surjective.
Definition 11.2. If A is an n × n zero-one matrix, G A denotes the stationary inductive limit Z n A −→Z n in the category of ordered abelian groups. Here A acts by right multiplication, and G A will be a dimension group. The action of A induces an automorphism of G A denoted A * . Proposition 11.3. If τ : I → I is piecewise monotonic and Markov, with incidence matrix A, then K 0 (F τ ) ∼ = G A . If Φ is the canonical *-automorphism of F τ , then the induced automorphism L * = (Φ * ) −1 on K 0 (F τ ) is carried to A * .
Corollary 11.4. If τ : I → I is piecewise monotonic, surjective, and Markov, with incidence matrix A, then Proof. By Propositions 9.4 and 11.3, which is isomorphic to coker(id −A) by Proposition 9. 3. The second isomorphism in the statement of the corollary follows in a similar fashion.
If a piecewise monotonic Markov map τ is not surjective, then the eventual range Y of σ will be a union of some of the order intervals in the Markov partition. Restricting σ to its eventual range gives a Markov map whose incidence matrix A Y will be the restriction of the original incidence matrix to the rows and columns corresponding to intervals in the eventual range. The corresponding algebras C * (R(Y, σ| Y )) and C * (Y, σ| Y ) will be strongly Morita equivalent to F τ = C * (R(X, σ)) and O τ = C * (X, σ) respectively, as remarked at the end of Section 3. Morita equivalent C*-algebras have the same K-groups, so the isomorphisms in Corollary 11.4 will hold with A Y in place of A.
Let X A be the set of all sequences in {1, 2, . . . , n} N such that i is followed by j only if A ij = 1, and let σ A be the one-sided shift map on X A . We are going to characterize when (X, σ) is conjugate to (X A , σ A ).
For each n × n zero-one matrix A satisfying a certain "Condition I", Cuntz and Krieger [CK] defined an AF-algebra F A and an algebra O A . For finite directed graphs, Condition I is equivalent to the following condition of Kumjian-Pask-Raeburn [38,Lemma 3.3]. (For a directed graph we will always require that there are no multiple edges between vertices).
there is an index i and an edge from v i to a vertex other than v i+1 mod n . If A is a zero-one n × n matrix, A satisfies Condition L if the associated directed graph satisfies Condition L.
Definition 11. 6. Let τ : I → I be piecewise monotonic and Markov, with associated local homeomorphism σ : X → X. Let E 1 , . . . , E n be the associated Markov partition for σ : X → X, with incidence matrix A. The itinerary map S : X → X A is given by S(x) = s 0 s 1 s 2 . . ., where σ k (x) ∈ E s k . We say itineraries separate points of X if the itinerary map is 1-1. (This is independent of the choice of Markov partition.) We now describe when itineraries separate points. Recall that an interval J is a homterval for a piecewise monotonic map τ : I → I if τ n is a homeomorphism on J for all n. A piecewise monotonic Markov map τ : I → I is piecewise linear if for each interval J of the Markov partition, there is a partition of J into subintervals on whose interior τ is linear. We allow τ to be discontinuous at finitely many points in each Markov partition interval, but require that its slope be constant within each Markov partition interval. (See the Markov map in Figure 1 on page 23 for an example.) Proposition 11.7. Let τ : I → I be piecewise monotonic and Markov, with incidence matrix A. These are equivalent.
(3) (X, σ) is conjugate to the one-sided shift of finite type (X A , σ A ). If τ is also piecewise linear, these conditions are equivalent to A satisfying Condition L.
Proof. [63, Props. 8.5 and 8.8] Proposition 11.8. Let τ : I → I be piecewise monotonic and Markov, and let σ : X → X be the associated local homeomorphism. If A is the associated incidence matrix, and if itineraries separate points of X, then F τ ∼ = F A and O τ ∼ = O A . Proof. By Proposition 11.7, (X, σ) is conjugate to (X A , σ A ). Thus C * (R(X, σ)) ∼ = C * (R(X A , σ A )), and C * (R(X A , σ A )) ∼ = F A by ([12, Example 2]). Hence F τ ∼ = F A . In a similar fashion, since Corollary 11. 9. For each n × n zero-one matrix A satisfying Cuntz-Krieger's Condition I, there is a piecewise linear Markov map τ such that F τ ∼ = F A , and Proof. Let τ be any piecewise linear Markov map, with a Markov partition whose associated incidence matrix is A. Since Condition I is equivalent to Condition L [38, Lemma 3.3], the corollary follows from Propositions 11.7 and 11.8.
If τ is piecewise linear, Markov, and surjective, with incidence matrix A, we can determine simplicity of F τ and O τ from the matrix A. Indeed, τ is topologically exact iff A is primitive ([63, Cor. 8.9]), and this is equivalent to F τ being simple (Proposition 5.2). Similarly, τ is transitive iff A is irreducible and is not a permutation matrix, and this is equivalent to O τ being simple (Proposition 5.3).
The next two results will be used later for Markov maps that are unimodal or are β-transformations. We write Z[t]I(0, 1) for the set of elements in C(X, Z) of the form p( L)I(0, 1), with p a polynomial with integral coefficients.
Lemma 11. 10. Let L be an endomorphism of an abelian group G, and M a subgroup of G invariant under L, such that for each g ∈ G there exists k ∈ N such that L k g ∈ M . Then If m ∈ (id −L)M , then m ∈ (id −L)G, so φ is well defined. If m ∈ M and φ(m) = 0, choose g ∈ G so that m = (id −L)g. Now choose k ∈ N so that L k g ∈ M . Then To see that φ is surjective, let g ∈ G, and again choose k ∈ N such that L k g ∈ M . Then Thus φ is surjective, which completes the proof.
Recall that a module is cyclic if it is generated by a single element, called a cyclic element. If M is a subgroup of C(X, Z), invariant under the transfer map L, and p ∈ Z[t] is a monic polynomial such that p( L)M = 0, we say p is the minimal polynomial for L on M if no polynomial of lower degree annihilates L| M . Note that it may happen that no polynomial annihilates L| M , so that there is no minimal polynomial for L on M .
Proposition 11.11. Let τ : I → I be piecewise monotonic and surjective. Assume that I(0, 1) is a cyclic element for the module DG(τ ), and that the minimal polynomial for L on M = Z[t]I(0, 1) ⊂ C(X, Z) is and if in addition τ is transitive, Proof. We first show that (12) f ∈ C(X, Z) =⇒ ∃k ∈ N such that L k f ∈ M.
Since I(0, 1) is cyclic for L * , there exists a Laurent polynomial q with integer coefficients such that [f ] = q( L * )I(0, 1). Choose j ∈ N such that t j q(t) ∈ Z[t].
Then 1 0 · · · 0 0 0 1 · · · 0 0 0 0 · · · 0 · · · · · · · · · · · · · · · 0 0 0 · · · where B acts by right multiplication on Q p . Now by elementary operations over Z, (i.e., operations that exchange rows, multiply a row by -1, or add an integral multiple of one row to another, or the analogous column operations), the matrix (id −B) can be transformed to a diagonal matrix D with entries (1, 1, . . . , 1, n), Hence the image of v 0 = I(0, 1) generates the quotient. The element of K 0 (F τ ) corresponding to the identity in F τ is I(0, 1), and the inclusion of Finally, suppose that n = 0 and that τ is transitive. We would like to apply Theorem 9.5; for that purpose, we need to know that τ is not essentially injective. If τ were essentially injective, then σ would be injective ( [63,Lemma 11.3]). Since τ (and then σ) are surjective, then σ would be a homeomorphism, so LI(0, 1) = I(0, 1). Thus the minimal polynomial of L on M = Z[t]I(0, 1) would be m(t) = t−1, so m(1) = 0, contrary to assumption. Thus τ is not essentially injective, so by Theorem 9.5, O τ is simple, separable, nuclear, purely infinite, and in the UCT class N , so by the results of Kirchberg [33] and Phillips [50] (Theorem 9.6), is determined by its K-groups. By the isomorphisms in (11), the K-groups of O τ coincide with those of O n+1 , so these algebras are isomorphic.
If m(t) is the minimal polynomial for L on M = Z[t]I(0, 1), and p(t) is the minimal polynomial for the incidence matrix A, then m and p are closely related. In Proposition 11.11, let E 1 , . . . , E n be the intervals for a Markov partition of σ, and A the incidence matrix. The action of L on i z i E i for z i ∈ Z is given by right multiplication by A on (z 1 , z 2 , . . . , z n ). Since I(0, 1) = E 1 + · · · + E n , then Example 11. 12. Let τ be the tent map, i.e., τ (x) = 2x if 0 ≤ x ≤ 1/2, and τ (x) = 2−2x if 1/2 ≤ x ≤ 1. Then τ is topologically exact, so F τ is a simple AF-algebra and O τ is simple and purely infinite. Furthermore, τ is Markov, with incidence matrix 1 1 1 1 , and the inductive limit of Z 2 with respect to this matrix is the dyadic rationals, so by Proposition 11.3, K 0 (F τ ) is isomorphic to the dyadic rationals, with order unit 1, and with the canonical automorphism being multiplication by 2. Thus K 0 (F τ ) is unitally isomorphic to K 0 (M 2 ∞ ), and F τ and M 2 ∞ are unital AF-algebras, so Example 11. 13. Let τ be the Markov map in Figure 1 on page 23. Then the associated local homeomorphism σ : X → X is Markov with respect to the partition Since K 0 (F τ ) ∼ = DG(τ ), and DG(τ ) is described for most of the following examples in [63] and [64], in the rest of this paper we will concentrate on describing the algebras O τ and their K-groups. We first analyze the case where the orbit of 0 (or equivalently, of the critical point c) is periodic. Let J 1 = [0, c] and J −1 = (c, 1], and let the itinerary of 0 be the sequence n 0 , n 1 , n 2 , . . .. (In other words, τ k 0 ∈ J n k for k ≥ 0.) Define a k to be the product n 0 n 1 . . . n k for k ≥ 0, and a −1 = 1. With latter convention, note that a k a k−1 = n k for all k ≥ 0. (1) If 0 is periodic with period p ≥ 3,
(4) Assume that 0 is eventually periodic, with k > 1. Then applying (15) with j = p and j = k, we conclude that m( L)I(0, 1) = 0, where m is as defined in (4). Since I(0, 1) is a cyclic element for L on M , then m( L) = 0 on M . We show that no polynomial of lower degree annihilates L on M . The points {τ j 0 | 0 ≤ j ≤ p − 1} are distinct, and none equals 1, so the characteristic functions for the p intervals {I(τ j 0, 1) | 0 ≤ j ≤ p − 1} are linearly independent in C(X, R). Thus the linear span V of this set is p dimensional. As remarked above, the linear span of the p vectors { L k I(0, 1) | 0 ≤ k ≤ p− 1} equals V , and thus these vectors also are linearly independent. Hence, no linear combination of 1, L, L 2 , . . . , L p−1 annihilates I(0, 1). It follows that m is the minimal polynomial of L.
(5) Assume that 0 is eventually periodic, with k = 1. Then combining (15) for j = k = 1 and (15) for j = p gives the desired formula for m(t). That m is the minimal polynomial of L follows in a similar way to (4).
In the statement of the following theorem, periodic points are also viewed as eventually periodic. Given 1 < s < 2, we define the restricted tent map T s by where c = 1 − 1/s. (This is the usual symmetric tent map τ on [0, 1] with slopes ±s, restricted to the interval [τ 2 (1/2), τ (1/2)], which is the interval of most interest for the dynamics. Then the map has been rescaled so that its domain is [0,1]. See Figure 1 on page 23.) Note that T s (c) = 1 and T s (1) = 0.
Example 12. 3. Let τ = T s be the restricted tent map, with √ 2 < s < 2, and assume that the orbit of the critical point is not eventually periodic. (For example, this occurs if s = 3/2, or if s is transcendental.) By [25] or [64,Lemma 8.1], τ is topologically exact. By Theorem 12.2, O τ ∼ = O ∞ . Thus all such tent maps give isomorphic C*-algebras O τ . However, O τ comes equipped with an action of T (or of R), (see the discussion preceding Theorem 10.7) and these actions give different KMS-states. In fact, by Theorem 10.7, since ln s is the entropy of τ = T s , there will be a unique β-KMS state for β = ln s. Furthermore, the range of the unique state on the dimension group of the AF-algebras F τ will be Z[s, s −1 ] ([64, Prop. 8.2]), as s varies, many of the algebras F τ for τ = T s will be non-isomorphic, even though the associated algebras O τ will all be isomorphic to O ∞ .

Multimodal maps
By a multimodal map we mean a continuous piecewise monotonic map. In this section, we compute the K-groups of O τ for some multimodal maps.
Observe that for the maps in Examples 13.1 and 13.2, K 1 (O τ ) is not the torsion free part of K 0 (O τ ), so O τ won't be a Cuntz-Krieger algebra. 14. Interval exchange maps Definition 14.1. A piecewise monotonic map τ : [0, 1) → [0, 1) is an generalized interval exchange map if τ is increasing on each interval of monotonicity, is bijective, and is right continuous at all points. We will usually identify τ with its extension to a map from [0, 1] into [0, 1], defined to be left continuous at 1.
Note that if 0 = a 0 < a 1 < . . . < a n = 1 is the partition associated with a generalized interval exchange map τ , then τ maps the intervals [a i−1 , a i ) to new intervals which partition [0, 1). If a generalized interval exchange map is linear with slope 1 on each interval of monotonicity, it is called an interval exchange map; in this case τ maps each interval [a i−1 , a i ) to an interval of the same length.
For a generalized interval exchange map, since τ : [0, 1) → [0, 1) is bijective, the associated local homeomorphism σ : X → X will be a homeomorphism. Then, as we observed when C * (X, σ) was defined, C * (X, σ) will be isomorphic to the transformation group C*-algebra C(X) × ψ Z, where the action of Z is given by the *-automorphism ψ defined by ψ(f ) = f • σ −1 . If τ is an interval exchange map, then our construction of the homeomorphism σ : X → X is the same as that of Putnam [55]. Thus we have the following. When we have referred to the generalized orbit of a point with respect to a piecewise monotonic map with some discontinuities, we have previously used the orbit with respect to the multivalued map τ . In the next proposition, we need to refer to the orbit with respect to τ itself, and we will call this the τ -orbit to avoid confusion with our earlier usage. If a set B has the property that its points have orbits that are infinite and disjoint, we refer to this property as the IDOC for B.
Since τ is exact, then (iii) follows from Corollary 10. 6. The fact that the entropy of τ is ln β can be found in [64,Prop. 3.7]. The statement about KMS-states follows from Theorem 10. 7.
If x ∈ [0, 1], then the β-expansion of x is x = n0 β + n1 β 2 +· · · , where n 0 is the greatest integer in βx, and for each k ≥ 1, n k is the greatest integer in β k+1 (x− k−1 i=0 ni β i+1 ). If β / ∈ N, the itinerary of 1 will be the sequence of coefficients (n 1 n 2 · · · ) in the β-expansion of 1. (If β = n ∈ N, the itinerary of 1 will be (n − 1)(n − 1) . . ., while the β-expansion of 1 will be n000 · · · .) The β-expansion of 1 will be finite iff either τ 1 = 1 or the orbit of 1 lands on 0. Since τ is topologically exact, then itineraries separate points, so the β-expansion of 1 will be eventually periodic iff 1 is eventually periodic.  4.13], it also is shown that O β ∼ = O n+1 if the β-expansion of 1 is eventually periodic, and O β ∼ = O ∞ otherwise. A formula for n is given in terms of the β-expansion of 1; this formula gives the same value for n as in Proposition 15.2. Thus O τ ∼ = O β in either case.