We show that the Smale spaces from self-similar groups are topologically mixing and their stable algebra and stable Ruelle algebra are strongly Morita equivalent to groupoid algebras of Anantharaman-Delaroche and Deaconu. And we show that C∗(R∞) associated to a postcritically finite hyperbolic rational function is an AT-algebra of real-rank zero with a unique trace state.

1. Introduction

Nekrashevych has developed a theory of dynamical systems and C*-algebras for self-similar groups in [1, 2]. These groups include groups acting on rooted trees and finite automata and iterated monodromy groups of self-covering on topological spaces. From self-similar groups, Nekrashevych constructed Smale spaces of Ruelle and Putnam with their corresponding stable and unstable algebras and those of Ruelle algebras for various equivalence relations on the Smale spaces [3–7].

Main approach to C*-algebras structures in [2] is based on Cuntz-Pimsner algebras generated by self-similar groups. However Smale spaces and their corresponding C*-algebras have rich dynamical structures, and it is conceivable that dynamical systems associated with self-similar groups may give another way to study C*-algebras from self-similar groups. Our intention is to elucidate self-similar groups from the perspective of dynamical systems.

This paper is concerned with groupoids and their groupoid C*-algebras from the stable equivalence relation on the limit solenoid (SG,σ¯) of a self-similar group (G,X). Instead of using the groupoids Gs and Gs⋊ℤ on the Smale space (SG,σ¯) as Putnam [3, 4] and Nekrashevych [2] did, we consider the essentially principal groupoids R∞ and Γ(JG,σ) of Anantharaman-Delaroche [8] and Deaconu [9] on a presentation (JG,σ) of (SG,σ¯). While Gs and Gs⋊ℤ are not r-discrete groupoids, R∞ and Γ(JG,σ) are r-discrete. And R∞ and Γ(JG,σ) are defined on (JG,σ) so that we do not need to entail the inverse limit structure of (SG,σ¯). Thus R∞ and Γ(JG,σ) are more manageable than Gs and Gs⋊ℤ for the structures of their C*-algebras.

In this paper, we prove that, for a self-similar group (G,X), its limit dynamical system (JG,σ) is topologically mixing so that (SG,σ¯) is an irreducible Smale space. And we show that R∞ is equivalent to Gs and Γ(JG,s) is equivalent to Gs⋊ℤ in the sense of Muhly et al. [10]. Consequently, the groupoid C*-algebras C*(R∞) and C*(Γ(JG,σ)) are strongly Morita equivalent to the stable algebra S and the stable Ruelle algebra Rs, respectively, of (SG,σ¯). Then we use R∞ and Γ(JG,σ) to study structures of C*-algebras from a self-similar group (G,X). Finally we show that groupoid algebras of R∞ from postcritically finite hyperbolic rational functions are AT-algebras of real-rank zero.

The outline of the paper is as follows. In Section 2, we review the notions of self-similar groups and their groupoids and show that the induced limit dynamical system and the limit solenoid of a self-similar group are topologically mixing. In Section 3, we observe that R∞ is equivalent to Gs and Γ(JG,s) is equivalent to Gs⋊ℤ. In Section 4, we give a proof that its groupoid algebra C*(Γ(JG,σ)) is simple, purely infinite, separable, stable, and nuclear and satisfies the Universal Coefficient Theorem. For R∞, we show that C*(R∞) is simple and nuclear. And, when self-similar group is defined by a postcritically finite hyperbolic rational function and its Julia set, we show that C*(R∞) is an AT-algebra.

2. Self-Similar Groups

We review the properties of self-similar groups. As for general references for the notions of self-similar groups, we refer to [1, 2].

Suppose that X is a finite set. We denote by Xn the set of words of length n in X with X0={∅}, and define X*=∪n=0∞Xn. A self-similar group (G,X) consists of an X and a faithful action of a group G on X such that, for all g∈G and x∈X, there exist unique y∈X and h∈G such that
(1)g(xw)=yh(w)foreveryw∈X*.
The above equality is written formally as
(2)g·x=y·h.

We observe that for any g∈G and v∈X*, there exists a unique element h∈G such that g(vw)=g(v)h(w) for every w∈X*. The unique element h is called the restriction of g at v and is denoted by g|v. For u=g(v) and h=g|v, we write
(3)g·v=u·h.

A self-similar group (G,X) is called recurrent if, for all x,y∈X, there is a g∈G such that g·x=y·1; that is, g(xw)=yw for every w∈X*. We say that (G,X) is contracting if there is a finite subset N of G satisfying the following: for every g∈G, there is n≥0 such that g|v∈N for every v∈X* of length |v|≥n. If the group is contracting, the smallest set N satisfying this condition is called the nucleus of the group.

Standing Assumption. We assume that our self-similar group (G,X) is a contracting, recurrent, and regular group and that the group G is finitely generated.

Path Spaces. For a self-similar group (G,X), the set X* has a natural structure of a rooted tree: the root is ∅, the vertices are words in X*, and the edges are of the form v to vx, where v∈X* and x∈X. Then the boundary of the tree X* is identified with the space Xω of right-infinite paths of the form x1x2⋯, where xi∈X. The product topology of discrete set X is given on Xω.

We say that a self-similar group (G,X) is regular if, for every g∈G and every w∈Xω, either g(w)≠w or there is a neighborhood of w such that every point in the neighborhood is fixed by g.

We also consider the space X-ω of left-infinite paths ⋯x-2x-1 over X with the product topology. Two paths ⋯x-2x-1 and ⋯y-2y-1 in X-ω are said to be asymptotically equivalent if there is a finite set I⊂G and a sequence gn∈I such that
(4)gn(x-n⋯x-1)=y-n⋯y-1,
for every n∈ℕ. The quotient of the space X-ω by the asymptotic equivalence relation is called the limit space of (G,X) and is denoted by JG. Since the asymptotic equivalence relation is invariant under the shift map ⋯x-2x-1↦⋯x-3x-2, the shift map induces a continuous map σ:JG→JG. We call the induced dynamical system (JG,σ) the limit dynamical system of (G,X) (see [1, 2] for details).

Remark 1.

Recurrent and finitely generated conditions imply that JG is a compact, connected, locally connected, metrizable space of a finite dimension by Corollary 2.8.5 and Theorem 3.6.4 of [1]. And regular condition implies that σ is an |X|-fold self-covering map by Proposition 6.1 of [2].

A cylinder set Z(u) for each u∈X*=∪n≥0Xn is defined as follows:
(5)Z(u)={ξ∈X-ω:ξ=⋯x-n-1x-n⋯x-1Z(u)1suchthatx-n⋯x-1=u}.
Then the collection of all such cylinder sets forms a basis for the product topology on X-ω. And we recall that a dynamical system (Y,f) is called topologically mixing if, for every pair of nonempty open sets A,B in Y, there is an n∈ℕ such that fk(A)∩B≠∅ for every k≥n.

Theorem 2.

(JG,σ) is a topologically mixing system.

Proof.

As X-ω has the product topology and JG has the quotient topology induced from asymptotic equivalence relation, it is sufficient to show that, for arbitrary cylinder sets Z(u) and Z(v) of X-ω, there are infinite paths ξ=⋯x-2x-1∈Z(u) and η=⋯y-2y-1∈Z(v) such that ξ is asymptotically equivalent to η. Moreover we can assume that u,v∈Xn for some n∈ℕ so that u=x-n⋯x-1 and v=y-n⋯y-1.

We choose sufficiently large m and let a,b∈Xm-n so that au and bv are elements of Xm. Then by recurrent condition and [1, Corollary 2.8.5], for au and bv in Xm, there is a g∈G such that g(au)=g(a)g|a(u)=bv. Since we chose large m, by contracting condition, g|a is an element of the nucleus of (G,X).

We remind that the nucleus of (G,X) is a finite set and equal to
(6)N=∪g∈G∩n≥0{g|v:v∈X*,|v|≥n}.
So an element of the nucleus is a restriction of another element of the nucleus. Hence g|a∈N implies that there exist a letter x-n-1 and a g-n-1∈N such that g-n-1|x-n-1=g|a. Then, for g-n-1(x-n-1)=y-n-1, we have
(7)g-n-1(x-n-1u)=y-n-1v.
So by induction there are a letter x-m and a g-m∈N for every m≥n such that
(8)g-m(x-m⋯x-n-1u)=y-m⋯y-n-1v.
Let ξ=⋯x-2x-1 and let η=⋯y-2y-1. Then it is trivial that ξ∈Z(u) and η∈Z(v). And ξ is asymptotically equivalent to η. Therefore the limit dynamical system (JG,σ) is topologically mixing.

Let Xℤ be the space of bi-infinite paths ⋯x-1x0·x1x2⋯ over the alphabet X. The direct product topology of the discrete set X is given on Xℤ. We say that two paths ⋯x-1x0·x1x2⋯ and ⋯y-1y0⋯y1y2⋯ in Xℤ are asymptotically equivalent if there is a finite set I⊂G and a sequence gn∈I such that
(9)gn(xnxn+1⋯)=ynyn+1…,
for every n∈ℤ. The quotient of Xℤ by the asymptotic equivalence relation is called the limit solenoid of (G,X) and is denoted by SG. As in the case of JG, the shift map on Xℤ is transferred to an induced homeomorphism on SG, which we will denote by σ¯.

Theorem 3 (see [<xref ref-type="bibr" rid="B12">1</xref>, <xref ref-type="bibr" rid="B13">2</xref>]).

The limit solenoid SG is homeomorphic to the inverse limit space of (JG,σ)(10)JG←σJG←σ⋯={(ξ0,ξ1,ξ2,…)∈∏n≥0∞JG:σ(ξn+1)JG←σJG←σ⋯=∏n≥0∞JG=ξnforeveryn≥0},
and σ¯:SG→SG is the induced homeomorphism defined by
(11)(ξ0,ξ1,ξ2,…)⟼(σ(ξ0),σ(ξ1),σ(ξ2),…)(ξ0,ξ1,ξ2,…)=(σ(ξ0),ξ0,ξ1,…).
Moreover, the limit solenoid system (SG,σ¯) is a Smale space.

Then we have the following from Theorem 2.

Corollary 4.

(SG,σ¯) is topologically mixing.

We have a natural projection π:SG→JG induced from the map
(12)⋯x-1x0·x1x2⋯⟼⋯x-1x0,
and the relation that ⋯xn-1xn∈X-ω represents ξn∈JG. Then it is easy to check π∘σ¯=σ∘π. The stable equivalence relation on (SG,σ¯) is defined as follows [2, Proposition 6.8]:

Definition 5.

One says that two elements α and β in SG are stably equivalent and write α~sβ if there is a k∈ℤ such that πσ¯k(α)=πσ¯k(β).

In other words, when α and β are represented by infinite paths (xn)n∈ℤ and (yn)n∈ℤ in Xℤ, α~sβ if and only if the corresponding left-infinite paths ⋯xk-1xk and ⋯yk-1yk in X-ω are asymptotically equivalent for some k∈ℤ.

Groupoids on (JG,σ) and (SG,σ¯).Suppose that (G,X) is a self-similar group and (SG,σ¯) is its corresponding limit solenoid. We recall from [3] that the stable equivalence groupoid Gs on SG and its semidirect product by ℤ are defined to be
(13)Gs={(α,β)∈SG×SG:α~sβ},Gs⋊ℤ={(α,n,β)∈SG×ℤ×SG:n∈ℤ,(σ¯n(α),β)Gs⋊ℤ=(σ¯n(α),β)∈Gs}.
Then Gs and Gs⋊ℤ are groupoids with the natural structure maps. The unit spaces of Gs and Gs⋊ℤ are identified with SG via the maps α∈SG↦(α,α)∈Gs and α↦(α,0,α)∈Gs⋊ℤ, respectively.

To give topologies on these groupoids, we consider subgroupoids of Gs. For each n≥0, set
(14)Gs,n={(α,β)∈SG×SG:πσ¯n(α)=πσ¯n(β)}.
Then Gs,n is a subgroupoid of Gs. Note that if μ and ν in SG are stably equivalent with πσ¯l(μ)=πσ¯l(ν) for some negative integer l, then
(15)π(μ)=σ-lπσ¯l(μ)=σ-lπσ¯l(ν)=π(ν)
implies that (μ,ν)∈Gs,0. So we obtain the stable equivalence groupoid
(16)Gs=⋃n≥0Gs,n.
Each Gs,n is given the relative topology from SG×SG, and Gs is given the inductive limit topology. Under this topology, it is not difficult to check that Gs is a locally compact Hausdorff principal groupoid with the natural structure maps. For Gs⋊ℤ, we transfer the product topology of Gs×ℤ to Gs⋊ℤ via the map ((α,β),n)↦(α,n,σ¯(β)). Amenability and Haar systems on Gs and Gs×ℤ are explained in [2–4]. We denote the groupoid C*-algebra of Gs by S and that of Gs⋊ℤ by Rs and call it stable Ruelle algebra on (SG,σ¯).

For the limit dynamical system (JG·σ) of a self-similar group (G,X), we construct groupoids R∞ and Γ(JG,σ) of Anantharaman-Delaroche [8] and Deaconu [9]. Let Rn={(ξ,η)∈JG×JG:σn(ξ)=σn(η)} for n≥0 and define
(17)R∞=⋃n≥0Rn,Γ(JG,σ)={σk(ξ)(ξ,n,η)∈JG×ℤ×JG:∃k,l≥0,Γ(JG,σ)=n=k-l,σk(ξ)=σl(η)}
with the natural structure maps. The unit spaces of R∞ and Γ(JG,σ) are identified with JG via ξ↦(ξ,ξ) and ξ↦(ξ,0,ξ).

We give the relative topology from JG×JG on Rn and the inductive limit topology on R∞. Then R∞ is a second countable, locally compact, Hausdorff, r-discrete groupoid with the Haar system given by the counting measures. A topology on Γ(JG,σ) is given by basis of the form
(18)Λ(U,V,k·l)={(ξ,k-l,(σl∣V)-1∘σk(ξ)):ξ∈U},
where U and V are open sets in JG and k,l≥0 such that σk|U and σl|V are homeomorphisms with the same range. Then Γ(JG,σ) is a second countable, locally compact, Hausdorff, r-discrete groupoid, and the counting measure is a Haar system [9, 11]. Amenability of R∞ and Γ(JG,σ) is explained in Proposition 2.4 of [12]. We denote the groupoid C*-algebras of R∞ and Γ(JG,σ) by C*(R∞) and C*(Γ(JG,σ)), respectively.

3. Groupoid Equivalence

We follow Kumjian and Pask [13, Section 5] to obtain equivalence of groupoids between Gs and R∞ and between Gs⋊ℤ and Γ(JG,σ), respectively, in the sense of Muhly et al. [10].

We repeat Kumjian and Pask's observation [13]. Suppose that Y is a locally compact Hausdorff space and that Γ is a locally compact Hausdorff groupoid. For a continuous open surjection ϕ:Y→Γ0, we set a topological space
(19)Z=Y*Γ={(y,γ):y∈Y,γ∈Γ,ϕ(y)=s(γ)}
with the relative topology in Y×Γ and a locally compact Hausdorff groupoid
(20)Γϕ={(y1,γ,y2):y1,y2∈Y,γ∈Γ,Γϕ=1ϕ(y1)=s(γ),r(γ)=ϕ(y2)}
with the relative topology.

Theorem 6 (see [<xref ref-type="bibr" rid="B8">13</xref>, Lemma 5.1]).

Suppose that Y, Γ, ϕ, Z, and Γϕ are as previous. Then Z implements an equivalence between Γ and Γϕ in the sense of Muhly-Renault-Williams.

Now we consider ϕ:SG→R∞0 defined by α↦(π(α),π(α)). Since ϕ is the composition of the projection map π:SG→JG and the identity map from JG to R∞0, ϕ is a continuous open surjection. Then we have
(21)R∞ϕ={(α,(π(α),π(β)),β):α,β∈SG,R∞ϕ=1(π(α),π(β))∈R∞}.
It is not difficult to check that R∞ϕ=∪n≥0Rnϕ, where
(22)Rnϕ={(α,(π(α),π(β)),β):α,β∈SG,Rnϕ=1σn(π(α))=σn(π(β))},
and that the relative topology on R∞ϕ is equivalent to the inductive limit topology.

Lemma 7.

Suppose that (SG,σ¯) is the limit solenoid system induced from a self-similar group (G,X) and that Gs is the stable equivalence groupoid associated with (SG,σ¯). Then ϕ~:Gs→R∞ϕ defined by (α,β)↦(α,(π(α),π(β)),β) is a groupoid isomorphism.

Proof.

Remember that Gs=∪n≥0Gs,n and R∞α=∪n≥0Rnα. From the commutative relation σπ=πσ¯, we observe
(23)(α,β)∈Gs,n⟺πσ¯n(α)=πσ¯n(β)⟺σnπ(α)(α,β)∈Gs,n⟺πσ¯n(α)=σnπ(β).
Hence ϕ~|Gs,n is a well-defined bijective map between Gs,n and Rnϕ.

Since topologies on Gs,n and Rnϕ are relative topologies from SG×SG, ϕ~|Gs,n is a homeomorphism. Then ϕ~ is a homeomorphism as the inductive limit topologies are given on Gs and R∞ϕ. It is routine to check that ϕ~ is a groupoid morphism.

The groupoid equivalence between R∞ and Gs follows from Theorem 6 and Lemma 7. Strong Morita equivalence is from [10, Proposition 2.8] as both groupoids have Haar systems.

Theorem 8.

Suppose that (G,X) is a self-similar group, that R∞ is the groupoid associated with (JG,σ), and that Gs is the stable equivalence groupoid associated with (SG,σ¯). Then R∞ and Gs are equivalent in the sense of Muhly-Renault-Williams. Therefore C*(R∞) is strongly Morita equivalent to the stable algebra S on the limit solenoid system (SG,σ¯).

Analogous assertions hold for Γ(JG,σ) and Gs⋊ℤ. For ψ:SG→Γ(JG,σ)0 defined by α↦(π(α),0,π(α)), we observe
(24)Γ(JG,σ)ψ={(α,(π(α),n,π(β)),β):α,β∈SG,Γ(JG,σ)ψ=1(π(α),n,π(β))∈Γ(JG,σ)}.

Lemma 9.

Suppose that Gs is the stable equivalence groupoid of (SG,σ¯) and that Gs⋊ℤ is the semidirect product groupoid. Then ψ~:Gs⋊ℤ→Γ(JG,σ)ψ defined by (α,n,β)↦(α,(π(α),n,π(β)),β) is a groupoid isomorphism.

Proof.

Recall that (α,n,β)∈Gs⋊ℤ⇔(σ¯n(α),β)∈Gs. Then Gs=∪n≥0Gs,n implies that (σ¯n(α),β)∈Gs,l for some l≥0. So from the proof of Lemma 7, we obtain that
(25)(σ¯n(α),β)∈Gs,l⟺σn+l(π(α))=σl(π(β))⟺(π(α),n,π(β))∈Γ(JG,σ).
Thus ψ~ is a well-defined bijective map. As Gs⋊ℤ has the product topology, we notice that ψ~|Gs⋊{0} is the homeomorphism ϕ~ defined in Lemma 7 and that ψ~|Gs⋊{n} is homeomorphism onto
(26){σn+l(α,(π(α),n,π(β)),β):α,β∈SG,1σn+l(π(α))=σl(π(β))}.
It is trivial that ψ~ is a groupoid morphism.

Theorem 10.

Suppose that (G,X) is a self-similar group. Then Γ(JG,σ) and Gs⋊ℤ are equivalent in the sense of Muhly-Renault-Williams. Therefore C*(Γ(JG,σ)) is strongly Morita equivalent to the stable Ruelle algebra Rs on (SG,σ¯).

Remark 11.

In [11], Chen and Hou showed similar result under an extra condition that a Smale space is the inverse limit of an expanding surjection on a compact metric space.

4. Groupoid Algebras

Suppose that (G,X) is a self-similar group. We use its corresponding R∞ and Γ(JG,σ) to study C*-algebraic structures of stable algebra and stable Ruelle algebra from (G,X).

Following Renault [15], we say that a topological groupoid Γ with an open range map is essentially principal if Γ is locally compact and, for every closed invariant subset E of its unit space Γ0, {u∈E:r-1(u)∩s-1(u)={u}} is dense in E. A subset E of Γ0 is called invariant if r∘s-1(E)=E. And Γ is called minimal if the only open invariant subsets of Γ0 are the empty set ∅ and Γ0 itself. We refer [15] for details.

Proposition 12.

The groupoid Γ(JG,σ) is essentially principal.

Proof.

Let
(27)A={ξ∈JG:fork,l≥0,σk(ξ)=σl(ξ)impliesk=l},B={b∈Γ(Jf,f)0:r-1(b)∩s-1(b)={b}}.
Then we observe ξ∈A⇔(ξ,0,ξ)∈B. Hence A is dense in X implying that B is dense in Γ(Jf,f)0 so that Γ(JG,σ) is essentially principal.

To show that A is dense in JG, we assume A is not dense in JG. Then we can find an open set U⊂JG such that U¯∩A=∅ as JG is a compact Hausdorff space. Since
(28)JG-A=⋃n=1∞⋃k=0∞σ-k(Pern),
where Pern={ξ∈JG:σn(ξ)=ξ}, we have
(29)U¯=U¯∩(Jf-A)=⋃n=1∞⋃k=0∞U¯∩σ-k(Pern).
Then by Baire category theorem, there exist some integers n≥1 and k≥0 such that U¯∩σ-k(Pern) has nonempty interior. But Pern={ξ∈JG:σn(ξ)=ξ} is a finite set because X is a finite set, and σ-k(Pern) is a finite set as σ is an |X|-fold covering map, a contradiction. Therefore A is dense in JG, and Γ(JG,σ) is an essentially principal groupoid.

There are excellent criteria for groupoid algebras from dynamical systems to be simple and purely infinite developed by Renault [12].

Lemma 13 (see [<xref ref-type="bibr" rid="B18">12</xref>]).

For a topological space X and a local homeomorphism T:X→X, let Γ(X,T) be the groupoid of Anantharaman-Delaroche and Deaconu. Suppose that Γ(X,T) is an essentially principal groupoid and C*(X,T) is its groupoid algebra.

Assume that for every nonempty open set U⊂X and every x∈X, there exist m,n∈ℕ such that Tn(x)∈Tm(U). Then C*(X,T) is simple.

Assume that for every nonempty open set U⊂X, there exist an open set V⊂U and m,n∈ℕ such that Tm(V) is strictly contained in Tn(V). Then C*(X,T) is purely infinite.

As Γ(JG,σ) is an essentially principal groupoid, we have an alternative proof for Theorem 6.5 of [2].

Theorem 14.

The algebra C*(Γ(JG,σ)) is simple, purely infinite, separable, stable, and nuclear and satisfies the Universal Coefficient Theorem of Rosenberg-Schochet.

Proof.

Suppose that U is an open set in JG. Then the inverse image of U in X-ω, say U′, is open, and there is a cylinder set Z(u) defined by some u∈Xn such that Z(u)⊂U′. By definition of cylinder sets, we have σn(Z(u))=X-ω⊆σn(U′), which implies that σn(U)=JG on the quotient space. Thus for every ξ∈JG, ξ∈σn(U) and C*(Γ(JG,σ)) is simple.

For an open set U of JG, let V be an open subset of U such that the inverse image of V in X-ω is equal to the cylinder set Z(v), where v∈Xn for some n≥2. Then we obtain σn(V)=JG as in the previous, and σm(V) is a proper subset of σn(V) for every 1≤m⪇n. Hence C*(Γ(JG,σ)) is purely infinite.

Since Γ(JG,σ) is locally compact and second countable, C*(Γ(JG,σ)) is σ-unital, nonunital, and separable. So Zhang's dichotomy [16, Theorem 1.2] implies that C*(Γ(JG,σ)) is stable. By Proposition 2.4 of [12], nuclear is an easy consequence from amenability of Γ(JG,σ). Because Γ(JG,σ) is a locally compact amenable groupoid with Haar system, C*(Γ(JG,σ)) satisfied the Universal Coefficient Theorem by Lemma 3.5 and Proposition 10.7 of [17].

Corollary 15.

C*(Γ(JG,σ)) is *-isomorphic to the stable Ruelle algebra Rs.

Proof.

Because C*(Γ(JG,σ)) and Rs are stable, this is trivial from Theorem 10.

For C*(R∞), we use the fact that R∞=∪Rn is a principal groupoid representing an AP equivalence relation [18].

Proposition 16.

The groupoid R∞ is minimal, and its groupoid algebra C*(R∞) is simple.

Proof.

In the proof of Theorem 14, we observed that for every cylinder set Z(u) of X-ω, there is an n>0 such that σn(Z(u))=X-ω. Since the inverse image of a nonempty open set U in JG contains a cylinder set Z(u), this observation induces that σn(U)=JG on the quotient space. Then R∞ is a minimal groupoid by [19, Proposition 2.1]. And simplicity of C*(R∞) follows from [15, Proposition II.4.6] as R∞ is an r-discrete principal groupoid.

Proposition 17.

C*(R∞) is the inductive limit of C*(Rn). And each C*(Rn) is strongly Morita equivalent to C(Rn0/Rn)=C(JG/Rn).

Proof.

Note that R∞=∪n≥0Rn is the groupoid representing an AP equivalence relation on stationary sequence JG→σJG→σ⋯. Thus it is easy to check that Corollary 2.2 of [18] implies the inductive limit structure.

Clearly Rn={(u,v)∈JG×JG:σn(u)=σn(v)} is the groupoid representing an equivalence relation on JG defined by u~nv if and only if σn(u)=σn(v). And (s×r)(Rn)=(σ-n×σ-n)(Δ), where Δ={(u,u)∈JG×JG} implies that (s×r)(Rn) is a closed subset of JG×JG. Thus we have strong Morita equivalence of C*(Rn) and C(JG/Rn) by [20, Proposition 2.2].

Corollary 18.

C*(R∞) is a nuclear algebra.

Proof.

Since C(JG/Rn) is nuclear, C*(Rn) is also nuclear by [21, Theorem 15]. And it is a well-known fact that the class of nuclear C*-algebras is closed under inductive limit. So C*(R∞) is nuclear.

Postcritically Finite Rational Maps. Suppose that f:ℂ→ℂ is a postcritically finite hyperbolic rational function of degree more than one, that is, a rational function of degree more than one such that the orbit of every critical point of f eventually belongs to a cycle containing a critical point. Then f is expanding on a neighborhood of its Julia set Jf, the group IMG(f) is contracting, recurrent, regular, and finitely generated, and the limit dynamical system σ:JIMG(f)→JIMG(f) is topologically conjugate with the action of f on its Julia set Jf (see [2, Sections 2 and 6] for details).

We borrowed the following theorem from Theorem 3.16 and Remark 4.23 of [22].

Theorem 19.

Let f:ℂ→ℂ be a postcritically finite hyperbolic rational function of degree more than one and let R∞ be the groupoid on its limit dynamical system as in Section 2. Then C*(R∞) is an AT-algebra of real-rank zero with a unique trace state.

Proof.

To show that C*(R∞) is an AT-algebra, we use the work of Gong [23, Corollary 6.7]. By Propositions 16 and 17, C*(R∞) is a simple algebra which is an inductive limit of an AH system with uniformly bounded dimensions of local spectra. And Nekrashevych showed that K-groups of C*(R∞) for postcritically finite hyperbolic rational functions are torsion free in [2, Theorem 6.6]. Hence C*(R∞) is an AT-algebra.

As f:Jf→Jf is an expanding local homeomorphism (see [2, Section 6.4]) and exact by Proposition 16 and [19, Proposition 2.1], C*(R∞) has a unique trace state by Remark 3.6 of [19]. Simplicity and uniformly bounded dimension conditions imply that C*(R∞) is approximately divisible in the sense of Blackadar et al. [24] as shown by Elliot et al. [14]. Therefore C*(R∞) has real-rank zero by Theorem 1.4 of [24].

Corollary 20.

C*(R∞) associated with postcritically finite hyperbolic rational functions of degree more than one belongs to the class of C*-algebras covered by Elliot classification program.

Acknowledgment

The author would like to express gratitude to the referees for their kind suggestions.

NekrashevychV.NekrashevychV.C∗-algebras and self-similar groupsPutnamI. F.C∗-algebras from Smale spacesPutnamI.PutnamI. F.SpielbergJ.The structure of C∗-algebras associated with hyperbolic dynamical systemsRuelleD.RuelleD.Noncommutative algebras for hyperbolic diffeomorphismsAnantharaman-DelarocheC.Purely infinite C∗-algebras arising from dynamical systemsDeaconuV.Groupoids associated with endomorphismsMuhlyP. S.RenaultJ. N.WilliamsD. P.Equivalence and isomorphism for groupoid C∗-algebrasChenX.HouC.Morita equivalence of groupoid C∗-algebras arising from dynamical systemsRenaultJ.Cuntz-like algebrasKumjianA.PaskD.Actions of ℤk associated to higher rank graphsElliottG. A.GongG.LiL.Approximate divisibility of simple inductive limit C∗-algebrasRenaultJ.ZhangS.Certain C∗-algebras with real rank zero and their corona and multiplier algebras. ITuJ.-L.La conjecture de Baum-Connes pour les feuilletages moyennablesRenaultJ.The Radon-Nikodym problem for appoximately proper equivalence relationsKumjianA.RenaultJ.KMS states on C∗-algebras associated to expansive mapsMuhlyP. S.WilliamsD. P.Continuous trace groupoid C∗-algebrasAn HuefA.RaeburnI.WilliamsD. P.Properties preserved under Morita equivalence of C∗-algebrasThomsenK.C∗-algebras of homoclinic and heteroclinic structure in expansive dynamicsGongG.On the classification of simple inductive limit C∗-algebras—I. The reduction theoremBlackadarB.KumjianA.RørdamM.Approximately central matrix units and the structure of noncommutative tori