𝐶^∗-Algebras from Groupoids on Self-Similar Groups

Abstract

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 (𝑆_𝐺,𝜎) of a self-similar group (𝐺,𝑋). Instead of using the groupoids 𝐺_𝑠 and 𝐺_𝑠⋊ℤ on the Smale space (𝑆_𝐺,𝜎) as Putnam [3, 4] and Nekrashevych [2] did, we consider the essentially principal groupoids 𝑅_∞ and Γ(𝐽_𝐺,𝜎) of Anantharaman-Delaroche [8] and Deaconu [9] on a presentation (𝐽_𝐺,𝜎) of (𝑆_𝐺,𝜎). While 𝐺_𝑠 and 𝐺_𝑠⋊ℤ are not 𝑟-discrete groupoids, 𝑅_∞ and Γ(𝐽_𝐺,𝜎) are 𝑟-discrete. And 𝑅_∞ and Γ(𝐽_𝐺,𝜎) are defined on (𝐽_𝐺,𝜎) so that we do not need to entail the inverse limit structure of (𝑆_𝐺,𝜎). Thus 𝑅_∞ and Γ(𝐽_𝐺,𝜎) are more manageable than 𝐺_𝑠 and 𝐺_𝑠⋊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 (𝐽_𝐺,𝜎) is topologically mixing so that (𝑆_𝐺,𝜎) is an irreducible Smale space. And we show that 𝑅_∞ is equivalent to 𝐺_𝑠 and Γ(𝐽_𝐺,𝑠) is equivalent to 𝐺_𝑠⋊ℤ in the sense of Muhly et al. [10]. Consequently, the groupoid 𝐶^∗-algebras 𝐶^∗(𝑅_∞) and 𝐶^∗(Γ(𝐽_𝐺,𝜎)) are strongly Morita equivalent to the stable algebra 𝑆 and the stable Ruelle algebra 𝑅_𝑠, respectively, of (𝑆_𝐺,𝜎). Then we use 𝑅_∞ and Γ(𝐽_𝐺,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 𝐺_𝑠 and Γ(𝐽_𝐺,𝑠) is equivalent to 𝐺_𝑠⋊ℤ. In Section 4, we give a proof that its groupoid algebra 𝐶^∗(Γ(𝐽_𝐺,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 𝑋^𝑛 the set of words of length 𝑛 in 𝑋 with 𝑋⁰={∅}, and define 𝑋^∗=∪^∞_𝑛=0𝑋^𝑛. 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 𝑔(𝑥𝑤)=𝑦ℎ(𝑤)forevery𝑤∈𝑋^∗. (1) The above equality is written formally as 𝑔⋅𝑥=𝑦⋅ℎ. (2)

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 𝑔⋅𝑣=𝑢⋅ℎ. (3)

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 𝑋^𝜔 of right-infinite paths of the form 𝑥₁𝑥₂⋯, where 𝑥_𝑖∈𝑋. The product topology of discrete set 𝑋 is given on 𝑋^𝜔.

We say that a self-similar group (𝐺,𝑋) is regular if, for every 𝑔∈𝐺 and every 𝑤∈𝑋^𝜔, 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 𝑔_𝑛∈𝐼 such that 𝑔_𝑛𝑥_−𝑛⋯𝑥₋₁=𝑦_−𝑛⋯𝑦₋₁, (4) 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 𝐽_𝐺. 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∶𝐽_𝐺→𝐽_𝐺. We call the induced dynamical system (𝐽_𝐺,𝜎) the limit dynamical system of (𝐺,𝑋) (see [1, 2] for details).

Remark 1. Recurrent and finitely generated conditions imply that 𝐽_𝐺 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≥0𝑋^𝑛 is defined as follows: 𝑍(𝑢)=𝜉∈𝑋^−𝜔∶𝜉=⋯𝑥_−𝑛−1𝑥_−𝑛⋯𝑥₋₁suchthat𝑥_−𝑛⋯𝑥₋₁=𝑢. (5) 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 𝑓^𝑘(𝐴)∩𝐵≠∅ for every 𝑘≥𝑛.

Theorem 2. (𝐽_𝐺,𝜎) is a topologically mixing system.

Proof. As 𝑋^−𝜔 has the product topology and 𝐽_𝐺 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∈𝑋^𝑛 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∈𝑋^𝑚−𝑛 so that 𝑎𝑢 and 𝑏𝑣 are elements of 𝑋^𝑚. Then by recurrent condition and [1, Corollary  2.8.5], for 𝑎𝑢 and 𝑏𝑣 in 𝑋^𝑚, 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 𝑁=∪_𝑔∈𝐺∩_𝑛≥0𝑔|_𝑣∶𝑣∈𝑋^∗,|𝑣|≥𝑛. (6) 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 𝑔_−𝑛−1𝑥_−𝑛−1𝑢=𝑦_−𝑛−1𝑣. (7) So by induction there are a letter 𝑥_−𝑚 and a 𝑔_−𝑚∈𝑁 for every 𝑚≥𝑛 such that 𝑔_−𝑚𝑥_−𝑚⋯𝑥_−𝑛−1𝑢=𝑦_−𝑚⋯𝑦_−𝑛−1𝑣. (8) 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 (𝐽_𝐺,𝜎) is topologically mixing.

Let 𝑋^ℤ 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 𝑋^ℤ. We say that two paths ⋯𝑥₋₁𝑥₀⋅𝑥₁𝑥₂⋯ and ⋯𝑦₋₁𝑦₀⋯𝑦₁𝑦₂⋯ in 𝑋^ℤ are asymptotically equivalent if there is a finite set 𝐼⊂𝐺 and a sequence 𝑔_𝑛∈𝐼 such that 𝑔_𝑛𝑥_𝑛𝑥_𝑛+1⋯=𝑦_𝑛𝑦_𝑛+1…, (9) for every 𝑛∈ℤ. The quotient of 𝑋^ℤ by the asymptotic equivalence relation is called the limit solenoid of (𝐺,𝑋) and is denoted by 𝑆_𝐺. As in the case of 𝐽_𝐺, the shift map on 𝑋^ℤ is transferred to an induced homeomorphism on 𝑆_𝐺, which we will denote by 𝜎.

Theorem 3 (see [1, 2]). The limit solenoid 𝑆_𝐺 is homeomorphic to the inverse limit space of (𝐽_𝐺,𝜎)𝐽←_𝐺^𝜎𝐽←_𝐺^𝜎⋯=𝜉₀,𝜉₁,𝜉₂,…∈^∞_𝑛≥0𝐽_𝐺∶𝜎𝜉_𝑛+1=𝜉_𝑛forevery𝑛≥0, (10) and 𝜎∶𝑆_𝐺→𝑆_𝐺 is the induced homeomorphism defined by 𝜉₀,𝜉₁,𝜉₂,…⟼𝜎𝜉₀,𝜎𝜉₁,𝜎𝜉₂=𝜎𝜉,…₀,𝜉₀,𝜉₁,…. (11) Moreover, the limit solenoid system (𝑆_𝐺,𝜎) is a Smale space.

Then we have the following from Theorem 2.

Corollary 4. (𝑆_𝐺,𝜎) is topologically mixing.

We have a natural projection 𝜋∶𝑆_𝐺→𝐽_𝐺 induced from the map ⋯𝑥₋₁𝑥₀⋅𝑥₁𝑥₂⋯⟼⋯𝑥₋₁𝑥₀, (12) and the relation that ⋯𝑥_𝑛−1𝑥_𝑛∈𝑋^−𝜔 represents 𝜉_𝑛∈𝐽_𝐺. Then it is easy to check 𝜋∘𝜎=𝜎∘𝜋. The stable equivalence relation on (𝑆_𝐺,𝜎) is defined as follows [2, Proposition 6.8]:

Definition 5. One says that two elements 𝛼 and 𝛽 in 𝑆_𝐺 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 (𝑥_𝑛)_𝑛∈ℤ and (𝑦_𝑛)_𝑛∈ℤ in 𝑋^ℤ, 𝛼∼_𝑠𝛽 if and only if the corresponding left-infinite paths ⋯𝑥_𝑘−1𝑥_𝑘 and ⋯𝑦_𝑘−1𝑦_𝑘 in 𝑋^−𝜔 are asymptotically equivalent for some 𝑘∈ℤ.

Groupoids on (𝐽_𝐺,𝜎) and (𝑆_𝐺,𝜎).Suppose that (𝐺,𝑋) is a self-similar group and (𝑆_𝐺,𝜎) is its corresponding limit solenoid. We recall from [3] that the stable equivalence groupoid 𝐺_𝑠 on 𝑆_𝐺 and its semidirect product by ℤ are defined to be 𝐺_𝑠=𝛼,𝛽∈𝑆_𝐺×𝑆_𝐺∶𝛼∼_𝑠𝛽,𝐺_𝑠⋊ℤ=𝛼,𝑛,𝛽∈𝑆_𝐺×ℤ×𝑆_𝐺∶𝑛∈ℤ,𝜎^𝑛(𝛼),𝛽∈𝐺_𝑠. (13) Then 𝐺_𝑠 and 𝐺_𝑠⋊ℤ are groupoids with the natural structure maps. The unit spaces of 𝐺_𝑠 and 𝐺_𝑠⋊ℤ are identified with 𝑆_𝐺 via the maps 𝛼∈𝑆_𝐺↦(𝛼,𝛼)∈𝐺_𝑠 and 𝛼↦(𝛼,0,𝛼)∈𝐺_𝑠⋊ℤ, respectively.

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

For the limit dynamical system (𝐽_𝐺⋅𝜎) of a self-similar group (𝐺,𝑋), we construct groupoids 𝑅_∞ and Γ(𝐽_𝐺,𝜎) of Anantharaman-Delaroche [8] and Deaconu [9]. Let 𝑅_𝑛={(𝜉,𝜂)∈𝐽_𝐺×𝐽_𝐺∶𝜎^𝑛(𝜉)=𝜎^𝑛(𝜂)} for 𝑛≥0 and define 𝑅_∞=_𝑛≥0𝑅_𝑛,Γ𝐽_𝐺,𝜎=𝜉,𝑛,𝜂∈𝐽_𝐺×ℤ×𝐽_𝐺∶∃𝑘,𝑙≥0,𝑛=𝑘−𝑙,𝜎^𝑘(𝜉)=𝜎^𝑙𝜂 (17) with the natural structure maps. The unit spaces of 𝑅_∞ and Γ(𝐽_𝐺,𝜎) are identified with 𝐽_𝐺 via 𝜉↦(𝜉,𝜉) and 𝜉↦(𝜉,0,𝜉).

We give the relative topology from 𝐽_𝐺×𝐽_𝐺 on 𝑅_𝑛 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(𝐽_𝐺,𝜎) is given by basis of the form Λ(𝑈,𝑉,𝑘⋅𝑙)=𝜉,𝑘−𝑙,𝜎^𝑙∣_𝑉⁻¹∘𝜎^𝑘(𝜉)∶𝜉∈𝑈, (18) where 𝑈 and 𝑉 are open sets in 𝐽_𝐺 and 𝑘,𝑙≥0 such that 𝜎^𝑘|_𝑈 and 𝜎^𝑙|_𝑉 are homeomorphisms with the same range. Then Γ(𝐽_𝐺,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(𝐽_𝐺,𝜎) is explained in Proposition 2.4 of [12]. We denote the groupoid 𝐶^∗-algebras of 𝑅_∞ and Γ(𝐽_𝐺,𝜎) by 𝐶^∗(𝑅_∞) and 𝐶^∗(Γ(𝐽_𝐺,𝜎)), respectively.

3. Groupoid Equivalence

We follow Kumjian and Pask [13, Section 5] to obtain equivalence of groupoids between 𝐺_𝑠 and 𝑅_∞ and between 𝐺_𝑠⋊ℤ and Γ(𝐽_𝐺,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 𝑍=𝑌∗Γ=𝑦,𝛾∶𝑦∈𝑌,𝛾∈Γ,𝜙𝑦=𝑠𝛾 (19) with the relative topology in 𝑌×Γ and a locally compact Hausdorff groupoid Γ^𝜙=𝑦₁,𝛾,𝑦₂∶𝑦₁,𝑦₂𝜙𝑦∈𝑌,𝛾∈Γ,₁=𝑠𝛾,𝑟𝛾=𝜙𝑦₂ (20) with the relative topology.

Theorem 6 (see [13, Lemma 5.1]). Suppose that 𝑌, Γ, 𝜙, 𝑍, and Γ^𝜙 are as previous. Then 𝑍 implements an equivalence between Γ and Γ^𝜙 in the sense of Muhly-Renault-Williams.

Now we consider 𝜙∶𝑆_𝐺→𝑅⁰_∞ defined by 𝛼↦(𝜋(𝛼),𝜋(𝛼)). Since 𝜙 is the composition of the projection map 𝜋∶𝑆_𝐺→𝐽_𝐺 and the identity map from 𝐽_𝐺 to 𝑅⁰_∞, 𝜙 is a continuous open surjection. Then we have 𝑅^𝜙_∞=𝛼,𝜋(𝛼),𝜋𝛽,𝛽∶𝛼,𝛽∈𝑆_𝐺,𝜋(𝛼),𝜋𝛽∈𝑅_∞. (21) It is not difficult to check that 𝑅^𝜙_∞=∪_𝑛≥0𝑅^𝜙_𝑛, where 𝑅^𝜙_𝑛=𝛼,𝜋(𝛼),𝜋𝛽,𝛽∶𝛼,𝛽∈𝑆_𝐺,𝜎^𝑛(𝜋(𝛼))=𝜎^𝑛𝜋𝛽, (22) and that the relative topology on 𝑅^𝜙_∞ is equivalent to the inductive limit topology.

Lemma 7. Suppose that (𝑆_𝐺,𝜎) is the limit solenoid system induced from a self-similar group (𝐺,𝑋) and that 𝐺_𝑠 is the stable equivalence groupoid associated with (𝑆_𝐺,𝜎). Then 𝜙∶𝐺_𝑠→𝑅^𝜙_∞ defined by (𝛼,𝛽)↦(𝛼,(𝜋(𝛼),𝜋(𝛽)),𝛽) is a groupoid isomorphism.

Proof. Remember that 𝐺_𝑠=∪_𝑛≥0𝐺_𝑠,𝑛 and 𝑅^𝛼_∞=∪_𝑛≥0𝑅^𝛼_𝑛. From the commutative relation 𝜎𝜋=𝜋𝜎, we observe 𝛼,𝛽∈𝐺_𝑠,𝑛⟺𝜋𝜎^𝑛(𝛼)=𝜋𝜎^𝑛𝛽⟺𝜎^𝑛𝜋(𝛼)=𝜎^𝑛𝜋𝛽. (23) Hence 𝜙|_{𝐺𝑠,𝑛} is a well-defined bijective map between 𝐺_𝑠,𝑛 and 𝑅^𝜙_𝑛.
Since topologies on 𝐺_𝑠,𝑛 and 𝑅^𝜙_𝑛 are relative topologies from 𝑆_𝐺×𝑆_𝐺, 𝜙|_{𝐺𝑠,𝑛} is a homeomorphism. Then 𝜙 is a homeomorphism as the inductive limit topologies are given on 𝐺_𝑠 and 𝑅^𝜙_∞. It is routine to check that 𝜙 is a groupoid morphism.

The groupoid equivalence between 𝑅_∞ and 𝐺_𝑠 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 (𝐽_𝐺,𝜎), and that 𝐺_𝑠 is the stable equivalence groupoid associated with (𝑆_𝐺,𝜎). Then 𝑅_∞ and 𝐺_𝑠 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 (𝑆_𝐺,𝜎).

Analogous assertions hold for Γ(𝐽_𝐺,𝜎) and 𝐺_𝑠⋊ℤ. For 𝜓∶𝑆_𝐺→Γ(𝐽_𝐺,𝜎)⁰ defined by 𝛼↦(𝜋(𝛼),0,𝜋(𝛼)), we observe Γ𝐽_𝐺,𝜎^𝜓=𝛼,𝜋(𝛼),𝑛,𝜋𝛽,𝛽∶𝛼,𝛽∈𝑆_𝐺,𝜋(𝛼),𝑛,𝜋𝛽∈Γ𝐽_𝐺,𝜎. (24)

Lemma 9. Suppose that 𝐺_𝑠 is the stable equivalence groupoid of (𝑆_𝐺,𝜎) and that 𝐺_𝑠⋊ℤ is the semidirect product groupoid. Then 𝜓∶𝐺_𝑠⋊ℤ→Γ(𝐽_𝐺,𝜎)^𝜓 defined by (𝛼,𝑛,𝛽)↦(𝛼,(𝜋(𝛼),𝑛,𝜋(𝛽)),𝛽) is a groupoid isomorphism.

Proof. Recall that (𝛼,𝑛,𝛽)∈𝐺_𝑠⋊ℤ⇔(𝜎^𝑛(𝛼),𝛽)∈𝐺_𝑠. Then 𝐺_𝑠=∪_𝑛≥0𝐺_𝑠,𝑛 implies that (𝜎^𝑛(𝛼),𝛽)∈𝐺_𝑠,𝑙 for some 𝑙≥0. So from the proof of Lemma 7, we obtain that 𝜎^𝑛(𝛼),𝛽∈𝐺_𝑠,𝑙⟺𝜎^𝑛+𝑙(𝜋(𝛼))=𝜎^𝑙𝜋𝛽⟺𝜋(𝛼),𝑛,𝜋𝛽∈Γ𝐽_𝐺,𝜎. (25) Thus 𝜓 is a well-defined bijective map. As 𝐺_𝑠⋊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 𝛼,𝜋(𝛼),𝑛,𝜋𝛽,𝛽∶𝛼,𝛽∈𝑆_𝐺,𝜎^𝑛+𝑙(𝜋(𝛼))=𝜎^𝑙𝜋𝛽. (26) It is trivial that 𝜓 is a groupoid morphism.

Theorem 10. Suppose that (𝐺,𝑋) is a self-similar group. Then Γ(𝐽_𝐺,𝜎) and 𝐺_𝑠⋊ℤ are equivalent in the sense of Muhly-Renault-Williams. Therefore 𝐶^∗(Γ(𝐽_𝐺,𝜎)) is strongly Morita equivalent to the stable Ruelle algebra 𝑅_𝑠 on (𝑆_𝐺,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(𝐽_𝐺,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(𝐽_𝐺,𝜎) is essentially principal.

Proof. Let 𝐴=𝜉∈𝐽_𝐺∶for𝑘,𝑙≥0,𝜎^𝑘(𝜉)=𝜎^𝑙(𝜉)implies,𝐽𝑘=𝑙𝐵=𝑏∈Γ_𝑓,𝑓⁰∶𝑟⁻¹(𝑏)∩𝑠⁻¹(𝑏)={𝑏}. (27) Then we observe 𝜉∈𝐴⇔(𝜉,0,𝜉)∈𝐵. Hence 𝐴 is dense in 𝑋 implying that 𝐵 is dense in Γ(𝐽_𝑓,𝑓)⁰ so that Γ(𝐽_𝐺,𝜎) is essentially principal.
To show that 𝐴 is dense in 𝐽_𝐺, we assume 𝐴 is not dense in 𝐽_𝐺. Then we can find an open set 𝑈⊂𝐽_𝐺 such that 𝑈∩𝐴=∅ as 𝐽_𝐺 is a compact Hausdorff space. Since 𝐽_𝐺−𝐴=^∞∞_𝑛=1𝑘=0𝜎^−𝑘Per_𝑛, (28) where Per_𝑛={𝜉∈𝐽_𝐺∶𝜎^𝑛(𝜉)=𝜉}, we have 𝑈=𝑈∩𝐽_𝑓−𝐴=^∞∞_𝑛=1𝑘=0𝑈∩𝜎^−𝑘Per_𝑛. (29) Then by Baire category theorem, there exist some integers 𝑛≥1 and 𝑘≥0 such that 𝑈∩𝜎^−𝑘(Per_𝑛) has nonempty interior. But Per_𝑛={𝜉∈𝐽_𝐺∶𝜎^𝑛(𝜉)=𝜉} is a finite set because 𝑋 is a finite set, and 𝜎^−𝑘(Per_𝑛) is a finite set as 𝜎 is an |𝑋|-fold covering map, a contradiction. Therefore 𝐴 is dense in 𝐽_𝐺, and Γ(𝐽_𝐺,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 [12]). 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.(1)Assume that for every nonempty open set 𝑈⊂𝑋 and every 𝑥∈𝑋, there exist 𝑚,𝑛∈ℕ such that 𝑇^𝑛(𝑥)∈𝑇^𝑚(𝑈). Then 𝐶^∗(𝑋,𝑇) is simple.(2)Assume that for every nonempty open set 𝑈⊂𝑋, there exist an open set 𝑉⊂𝑈 and 𝑚,𝑛∈ℕ such that 𝑇^𝑚(𝑉) is strictly contained in 𝑇^𝑛(𝑉). Then 𝐶^∗(𝑋,𝑇) is purely infinite.

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

Theorem 14. The algebra 𝐶^∗(Γ(𝐽_𝐺,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 𝐽_𝐺. 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∈𝑋^𝑛 such that 𝑍(𝑢)⊂𝑈^′. By definition of cylinder sets, we have 𝜎^𝑛(𝑍(𝑢))=𝑋^−𝜔⊆𝜎^𝑛(𝑈^′), which implies that 𝜎^𝑛(𝑈)=𝐽_𝐺 on the quotient space. Thus for every 𝜉∈𝐽_𝐺, 𝜉∈𝜎^𝑛(𝑈) and 𝐶^∗(Γ(𝐽_𝐺,𝜎)) is simple.
For an open set 𝑈 of 𝐽_𝐺, 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∈𝑋^𝑛 for some 𝑛≥2. Then we obtain 𝜎^𝑛(𝑉)=𝐽_𝐺 as in the previous, and 𝜎^𝑚(𝑉) is a proper subset of 𝜎^𝑛(𝑉) for every 1≤𝑚⪇𝑛. Hence 𝐶^∗(Γ(𝐽_𝐺,𝜎)) is purely infinite.
Since Γ(𝐽_𝐺,𝜎) is locally compact and second countable, 𝐶^∗(Γ(𝐽_𝐺,𝜎)) is 𝜎-unital, nonunital, and separable. So Zhang's dichotomy [16, Theorem 1.2] implies that 𝐶^∗(Γ(𝐽_𝐺,𝜎)) is stable. By Proposition 2.4 of [12], nuclear is an easy consequence from amenability of Γ(𝐽_𝐺,𝜎). Because Γ(𝐽_𝐺,𝜎) is a locally compact amenable groupoid with Haar system, 𝐶^∗(Γ(𝐽_𝐺,𝜎)) satisfied the Universal Coefficient Theorem by Lemma 3.5 and Proposition 10.7 of [17].

Corollary 15. 𝐶^∗(Γ(𝐽_𝐺,𝜎)) is ∗-isomorphic to the stable Ruelle algebra 𝑅_𝑠.

Proof. Because 𝐶^∗(Γ(𝐽_𝐺,𝜎)) and 𝑅_𝑠 are stable, this is trivial from Theorem 10.

For 𝐶^∗(𝑅_∞), we use the fact that 𝑅_∞=∪𝑅_𝑛 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 𝜎^𝑛(𝑍(𝑢))=𝑋^−𝜔. Since the inverse image of a nonempty open set 𝑈 in 𝐽_𝐺 contains a cylinder set 𝑍(𝑢), this observation induces that 𝜎^𝑛(𝑈)=𝐽_𝐺 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^∗(𝑅_𝑛). And each 𝐶^∗(𝑅_𝑛) is strongly Morita equivalent to 𝐶(𝑅⁰_𝑛/𝑅_𝑛)=𝐶(𝐽_𝐺/𝑅_𝑛).

Proof. Note that 𝑅_∞=∪_𝑛≥0𝑅_𝑛 is the groupoid representing an AP equivalence relation on stationary sequence 𝐽→−_𝐺^𝜎𝐽→−_𝐺^𝜎⋯. Thus it is easy to check that Corollary 2.2 of [18] implies the inductive limit structure.
Clearly 𝑅_𝑛={(𝑢,𝑣)∈𝐽_𝐺×𝐽_𝐺∶𝜎^𝑛(𝑢)=𝜎^𝑛(𝑣)} is the groupoid representing an equivalence relation on 𝐽_𝐺 defined by 𝑢∼_𝑛𝑣 if and only if 𝜎^𝑛(𝑢)=𝜎^𝑛(𝑣). And (𝑠×𝑟)(𝑅_𝑛)=(𝜎^−𝑛×𝜎^−𝑛)(Δ), where Δ={(𝑢,𝑢)∈𝐽_𝐺×𝐽_𝐺} implies that (𝑠×𝑟)(𝑅_𝑛) is a closed subset of 𝐽_𝐺×𝐽_𝐺. Thus we have strong Morita equivalence of 𝐶^∗(𝑅_𝑛) and 𝐶(𝐽_𝐺/𝑅_𝑛) by [20, Proposition 2.2].

Corollary 18. 𝐶^∗(𝑅_∞) is a nuclear algebra.

Proof. Since 𝐶(𝐽_𝐺/𝑅_𝑛) is nuclear, 𝐶^∗(𝑅_𝑛) 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 𝐽_𝑓, the group IMG(𝑓) is contracting, recurrent, regular, and finitely generated, and the limit dynamical system 𝜎∶𝐽_IMG(𝑓)→𝐽_IMG(𝑓) is topologically conjugate with the action of 𝑓 on its Julia set 𝐽_𝑓 (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∶𝐽_𝑓→𝐽_𝑓 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.