Groupoid Actions on Fractafolds

We define a bundle over a totally disconnected set such that each fiber is homeomorphic to a fractal blowup. We prove that there is a natural action of a Renault-Deaconu groupoid on our fractafold bundle and that the resulting action groupoid is a Renault-Deaconu groupoid itself. We also show that when the bundle is locally compact the associated $C^*$-algebra is primitive and has a densely defined lower-semicontinuous trace.


Introduction
The goal of this paper is to find and analyze symmetries of fractals associated to iterated function systems (F 1 , . . . , F N ) and to study the associated C * -algebras that might arise from the dynamics. In [19] Stricharz constructed a family of fractafold blowups of the invariant set of an iterated function system which is parameterized by infinite words in the alphabet {1, . . . , N } and observed that two such blowups are naturally homeomorphic if the parametrizing words are eventually the same. We endow these fractafold blowups with the inductive limit topology and assemble them into a fractafold bundle L.
In general there do not appear to be any natural nontrivial symmetries of a generic blowup but Stricharz's observation suggests that we look for symmetries of the bundle instead. Indeed we show that the homeomorphisms between fibers observed by Stricharz give rise to a natural groupoid action on L, the fractafold bundle. This groupoid action and the associated action groupoidG constitute the main focus for this work.
We prove that the there is a local homeomorphismσ on L such thatG is isomorphic to the Renault-Deaconu groupoid associated toσ and, in particular,G isétale. We also prove thatG is topologically principal and has a dense orbit. If the iterated function system satisfies the open set condition then we construct aG-invariant measure on the unit spaceG 0 . Now suppose that L is locally compact. Then the associated C * -algebra, C * (G), is primitive. If, in addition the iterated function system satisfies the open set condition, then the associated C * -algebra has a densely defined lower semi-continuous trace.
We begin by reviewing some of the background material and by proving some general results about Renault-Deaconu groupoids in Section 2. These results will be useful in our analysis of the fractafold bundle L and the associated action groupoidG in Section 3. In the final section we consider some examples to illustrate the theory. We show that the action groupoid is not in general minimal. We also point out cases when the fractafold bundle L fails to be locally compact.

Renault-Deaconu groupoid and groupoid actions
In this section we prove that if G is the Renault-Deaconu groupoid associated to a local homeomorphism σ on a topological space X such that G acts on a topological space Z, then σ lifts to a natural local homeomorphismσ on Z. Moreover, we show that the resulting action groupoid is isomorphic to the Renault-Deaconu groupoid associated toσ. We begin by reviewing some of the background material that we need. While some of the motivation for this work came from the theory of C * -algebras associated to groupoids, many of our results hold for topological groupoids that are not necessarily locally compact. Since some of the groupoids in our examples are not locally compact, the only generic assumption on the topological spaces and topological groupoids that we make is that they are Hausdorff.
Let G be a Hausdorff topological groupoid (see [18], cf. also [16]). Then the structure maps are continuous and, in addition, both the range map r (where r(x) = xx −1 ) and the source map s (where s(x) = x −1 x) are open. We write G 0 for the unit space of G. The groupoid G is said to beétale if s is a local homeomorphism (or equivalently, r is a local homeomorphism). A subset S ⊂ G is called a G-set or a bisection if the restrictions, r| S , s| S , are both injective. If G iś etale, then it has a cover of open G-sets and the restriction of either r or s to an open G-set is a homeomorphism onto an open subset of G 0 .
Let G be topological groupoid. If the set of points in G 0 with trivial isotropy, is dense in G 0 , we say that G is topologically principal (see [18,Definition 3.5(ii)]). If G is a locally compactétale groupoid we let C * (G) denote the full C * -algebra of G and C * r (G) denote the reduced C * -algebra of G. If, in addition, G is amenable 1 the canonical quotient map C * (G) → C * r (G) is an isomorphism. We regard C 0 (G 0 ) as an abelian C * -subalgebra of both. Let X be a topological space and let σ : X → X be a local homeomorphism on X. The Renault-Deaconu groupoid associated to σ [1,7,8,16,17] is Two elements (x, m, y) and (z, n, w) in G are composable if and only if y = z and, in this case, their product is (x, m, y)(y, n, w) = (x, m + n, w).
The inverse of an element in G is defined by (x, n, y) −1 = (y, −n, x).
Thus the range and source maps are given by r(x, n, y) = (x, 0, x) and s(x, n, y) = (y, 0, y). Hence G 0 , the unit space of G, may be identified with X via the map (x, 0, x) → x (in the sequel we will often make this identification without comment). A basis for the topology consists of sets of the form G(U, m, n, V ) = (x, m − n, y) : σ m (x) = σ n (y), x ∈ U, y ∈ V , where m and n ∈ N, U and V are open subsets of X such that both σ m | U and σ n | V are injective and σ m (U ) = σ n (V ). Note that the range map r induces a homeomorphism G(U, m, n, V ) ∼ = U . Hence, with this topology G is anétale groupoid. If X is locally compact then G = G(X, σ) is a locally compact groupoid.
Therefore s 1 = s 2 , ρ(t 1 ) = ρ(t 2 ), and n 1 = n 2 . Moreover Hence Ψ is one-to-one. To see that Ψ is onto, let ((x, n, y), s) ∈ G * Z, and recall that y = ρ(s). Let t = (x, n, y) · s. Then ρ(t) = x and One can repeat the argument that we made when showing thatσ is a local homeomorphism and prove thatσ m (ρ is an open cylinder inG and Ψ is continuous. Finally, Ψ −1 is continuous since it is a composition of continuous maps.
For the rest of this section we assume that both X and Z are locally compact spaces. This assumption implies that both G andG are locally compact groupoids.
Assume that σ is a local homeomorphism on a locally compact space X and assume that the Renault-Deaconu groupoid G = G(X, σ) acts on the locally compact space Z. The groupoidG ∼ = G * Z is amenable and the full and reduced C * -algebra of G * Z coincide.
As another application of Theorem 2.2, we can identify C * (G * Z) as a Cuntz-Pimsner algebra [15]. The closure of C c (Z) under a suitable norm may be viewed as a C * -correspondence Z over the C * -algebra C 0 (Z). The inner product is defined for ξ, η ∈ C c (Z) by ξ, η (z) = σ(y)=z ξ(y)η(y); the left and right actions of C 0 (Z) are given by (a · ξ · b)(z) = a(z)ξ(z)b(σ(z)) where a, b ∈ C 0 (Z) and ξ ∈ C c (Z). The following corollary follows from [8,Theorem 7].

Groupoid actions on fractafolds
In this section we build a fractafold bundle L such that the Renault-Deaconu G groupoid associated to a one-sided shift map σ acts naturally on L. The action groupoid G * L encapsulates the symmetries of the fractafold bundle. Using the results of the previous section, σ extends to a local homeomorphismσ on L. We prove thatσ is essentially free and, hence, G * L is topologically principal. Moreover, we show that the action groupoid G * L contains a dense orbit. If L is locally compact, it follows that C * (G * L) is primitive. If the iterated function system defining L satisfies the open set condition, then we build a G * L-invariant measure on L. Thus if L is, in addition, locally compact, then C * (G * L) has a densely defined lower semicontinuous trace.
Let (T, d) be a complete metric space and let (F 1 , . . . , F N ) be an iterated function system on T [3,9,11]. That is, each F i is a strict contraction on T . We assume that the iterated function system is non-degenerate, meaning that there are constants 0 < r i ≤ R i < 1, i = 1, . . . , N , such that We further assume map F i is surjective for all i = 1, . . . , N ; so the F i are homeomorphisms. If we can chose r i = R i in (3.1), then F i is a similarity (or a similitude). For an iterated function system there is a unique compact invariant set K [11, Theorem 3.1.3] such that . . , N } and define W * = n∈N W n to be the set of finite words over the alphabet W , and X = W ∞ to be the set of infinite words (sequences) with elements in W . If ω ∈ W n we say that the length of ω, denoted by |ω|, is n.
, r ω := r ω 1 · · · r ωn and similarly for R ω . If ω ∈ W n or x ∈ X and if k ∈ N such that k ≤ n (if applicable), then we write Given a finite sequence ω ∈ W n or an infinite sequence x ∈ X set Then, if x ∈ X, L n (x) ⊂ L n+1 (x) and we define L(x) := n∈N L n (x) endowed with the inductive limit topology. We refer to L(x) as the infinite blow-up of K at x. Our definition is not quite the same as the one used by Stricharz since he uses the relative topology (see [19] and [20,Chapter 5.4]); but the two seem to agree in some cases, for example, when the blowups are zero dimensional. Two blow-ups L(x) and L(y) are homeomorphic if the infinite sequences x and y differ in a finite number of indices. For example, if y = x 2 x 3 · · · , F x 1 extends to a homeomorphism from L(x) to L(y).
If U is a subset of K and ω ∈ W n is a finite sequence or x ∈ X is an infinite word, then we write L ω n (U ) for F −1 ω(n) (U ) and L x n (U ) for L x(n) n (U ). If ω ∈ W * , then the clopen cylinder Z(ω) ⊂ X = W ∞ is defined via The collection {Z(ω)} ω∈W * forms a basis of a topology on X, and, endowed with this topology, X is a totally disconnected compact space. Moreover, the shift map σ : X → X defined by σ(x 1 x 2 · · · ) = (x 2 x 3 · · · ) is a local homeomorphism on X. We write G for the Renault-Deaconu groupoid associated to σ as in Section 2. Recall that the unit space G 0 is homeomorphic to X.
Next we build a fractafold bundle on which the groupoid G acts. For n ≥ 0 define Then each L n is a compact space and L n ⊂ L n+1 . Observe that that L 0 = X ×K. We define the fractafold bundle L to be the increasing union of L n , L = n≥0 L n , endowed with the inductive limit topology. That is, a set U ⊂ L is open if and only if U ∩ L n is open for all n ≥ 0. We will show below that the base space is X. The following characterization of open sets is used later in this section.
Proof . Recall that the maps F i , i = 1, . . . , N , are homeomorphisms. Therefore the maps F ω and F −1 ω are homeomorphisms for all ω ∈ W * . The conclusion of the lemma follows immediately.
Recall that, in general, the inductive limit of an increasing sequence of Hausdorff spaces might fail to be Hausdorff. However, as we prove in the following lemma, the bundle L is Hausdorff.
Lemma 3.2. The bundle L endowed with the inductive limit topology as above is a Hausdorff space.
Proof . Let ι : L → X ×T be the inclusion map. If U is an open set in X ×T , then ι −1 (U )∩L n = U ∩ L n is open in L n for all n ≥ 1. Therefore ι is a continuous one-to-one map. Since X × T is Hausdorff, it follows that L is Hausdorff as well.
Before we define the action of the Renault-Deaconu groupoid on L we show that the natural projection from L into X is an open map. Proof . Since the restriction of π to L n is continuous for each n and L is endowed with the inductive limit topology, π is also continuous. Let U be an open set in L. Therefore U n := U ∩L n is open for all n ≥ 0. Hence U n ∩ (Z(ω) × L n (ω)) is open for all ω ∈ W n . For ω ∈ W n , the map π| Z(ω)×Ln(ω) : Z(ω) × L n (ω) → Z(ω) is open because it is just the projection onto the first coordinate. Since Z(ω) is an open subset of X it follows that π(U n ∩ (Z(ω) × L n (ω))) is open in X for all ω ∈ W n and n ≥ 0. Therefore, π(U n ) is open in X for all n ≥ 0, and hence π(U ) = n π(U n ) is open. Thus π is an open map. The final assertion is obvious.
The next results shows that the Renault-Deaconu groupoid G associated to the shift map on X acts on the left on the space L. Note that if γ = (x, m − n, y) ∈ G and (z, t) ∈ L, then s(γ) = π(z, t) if and only if y = z. Let G * L = {((x, m − n, y), (y, t)) : (x, m − n, y) ∈ G, (y, t) ∈ L}.
(3.2) Theorem 3.4. With the notation as above, the Renault-Deaconu groupoid G associated to the shift map σ on X acts on the fractafold bundle L via the map ((x, m − n, y), (y, t)) → (x, m − n, y) · (y, t) defined on G * L, where

Moreover, the action map is open.
Proof . We need to prove that the above map is well defined; that is, we need to show that the range of the map is L. Let (x, m−n, y) ∈ G and (y, t) ∈ L. Let k ≥ 0 such that (y, t) ∈ L k . Then there is ω ∈ W k such that y ∈ Z(ω) and t ∈ L k (ω). Therefore y i = ω i for i = 1, . . . , k. Notice that it suffices to assume that n ≥ k. Indeed, if m −n = m−n and σ m (x) = σ n (y) with m > m and n > n, then x m+i = y n+i for all i = 1, . . . , m −m and so F −1

t). Then
F y(n) (t) ∈ K and, thus, So the action is well defined.
The left action is continuous since for fixed m and n in N and fixed words α ∈ W m and β ∈ W n , we have that the map F −1 α • F β is continuous. Thus the Renault-Deaconu groupoid G acts on the left on L. The last part of the theorem is an immediate consequence of Lemma 2.1.
Corollary 3.6. Letσ be the local homeomorphism on L provided by Corollary 3.5. Then the left action groupoid G * L is homeomorphic to the Renault-Deaconu groupoidG associated toσ.
Proof . This is an immediate consequence of the second part of Theorem 2.2.

Recall from [7, Definition on p. 1781] that a local homeomorphism τ on a topological space
If τ is essentially free then the Renault-Deaconu groupoid G(Z, τ ) is topologically principal because, for z ∈ Z, the isotropy group G(z) is nontrivial if and only if there are k, l ≥ 0 with k = l such that τ k (z) = τ l (z) [1, Example 1.2c]. It is easy to see that the shift map σ on X is essentially free [7, Example 2].
Proposition 3.7. The local homeomorphismσ on L defined in Corollary 3.5 is essentially free. Hence, the Renault-Deaconu groupoidG associated toσ is topologically principal.
Proof . Let U be a nonempty open subset of L. We need to find (x, t) ∈ U such that for all k, l ≥ 0 ifσ k (x, t) =σ l (x, t) then k = l. Corollary 3.5 implies that ifσ k (x, t) =σ l (x, t) then σ k (x) = σ l (x). Since π(U ) is open in X there is x ∈ π(U ) such that for all k, l ≥ 0 if σ k (x) = σ l (x) then k = l. It follows that if we pick t ∈ L x such that (x, t) ∈ U then (x, t) satisfies the desired property. Henceσ is essentially free andG is topologically principal. Proposition 3.8. Let G be the Renault-Deaconu groupoid associated to the shift map σ on X and let L be the fractafold bundle associated to a non-degenerate iterated function system (F 1 , . . . , F N ) on a complete metric space Y . Let G * L be the left action groupoid defined via (3.2) and (3.3). Let x ∈ X be an infinite word obtained by concatenating all the finite words in W * . Then for all t ∈ K ⊂ L x , the orbit of (x, t) ∈ L (G * L) 0 is dense.
Proof . Recall (see, for example, [3, Theorem 4.2.1]) that every point v in K has at least one address y ∈ X, that is, {v} = n≥0 F y 1 · · · F yn (K). Notice that the diameter of F y 1 · · · F yn (K) is less than R y(n) · diam K and recall that each R i is strictly smaller than 1. We claim that the sequence {F xn · · · F x 1 (t)} n∈N is dense in K for all t ∈ K. To prove the claim, let v ∈ K and ε > 0. There is k ∈ N and ω ∈ W k such that d(v, F ω 1 · · · F ω k (u)) < ε for all u ∈ K. By hypothesis, x contains the word ω k · · · ω 1 . That is, there is l ≥ 1 such that x l = ω k , . . . , x l+k = ω 1 . Then d(v, F x l+k · · · F x l (F x l−1 · · · F x 1 (t))) = d(v, F ω 1 · · · F ω k (F x l−1 · · · F x 1 (t))) < ε for all t ∈ K. The claim follows.
We prove that the orbit of (x, t) ∈ L 0 is dense, where t is an arbitrary point in K. Let (y, v) ∈ L and let U be a neighborhood of (y, v) in L. Let m ∈ N and α ∈ W m such that (y, v) ∈ U m := U ∩(Z(α)×L m (α)). Then there is V open in K such that (y, v) ∈ Z(α)×L α m (V ). We need to find γ ∈ G such that γ · (x, t) ∈ U . Let n ∈ N be such that F x(n) (t) ∈ V . Then . Define y ∈ X such that y i = α i , i = 1, . . . , m and y m+i = x n+i for all i ≥ 1. Then (y, m − n, x) ∈ G and (y, m − n, x) · (x, t) ∈ U .
In general L may fail to be locally compact (see Section 4 for specific examples of when this property fails). However, if F i (K) is open in K for all i = 1, . . . , N , then L is locally compact. In this case K is a totally disconnected set. One can easily check that this condition is satisfied if the iterated function system is totally disconnected.
Suppose that L is locally compact, then as observed at the end of Section 2, C * (G * L) is nuclear and a Cuntz-Pimsner algebra (where the correspondence is defined over C 0 (L)). Proposition 3.8 allows us to conclude a bit more about C * (G * L).
A C * -algebra is defined to be primitive (see [14, § 3.13.7]) if it has a faithful irreducible representation. Proof . Let z = (x, t) ∈ L (G * L) 0 be a point with dense orbit as guaranteed by Proposition 3.8. Note that point evaluation at z defines a pure state on C 0 (L), the canonical masa (maximal abelian subalgebra) in C * (G * L). Since it extends to a pure state on C * (G * L), the GNS construction provides an irreducible representation π z of C * (G * L) and a unit vector ξ z in the associated Hilbert space H πz such that π z (f )ξ z , ξ z = f (z) for all f ∈ C 0 (L). Since ker π z ∩ C 0 (L) is supported on an open set which does not contain any points in the orbit of z, ker π z ∩ C 0 (L) = {0}. By Corollary 2.3 and Proposition 3.7, G * L =G is both amenable and topologically principal. Hence, every nonzero ideal in C * (G * L) must have a nontrivial intersection with C 0 (L) by [10,Theorem 4.4] (see also [5,Proposition 5.5]). Hence, ker π z = {0} and thus C * (G * L) has a faithful irreducible representation.
If G is anétale groupoid, a G-invariant measure µ on G 0 is a measure such that for any open G-set U , µ(r(U )) = µ(s(U )) (it suffices to prove this for basis of open G-sets). We show below that there is a G * L-invariant measure µ ∞ on L provided that the iterated function system satisfies one additional hypothesis. We will assume that the iterated function system ( We will need to invoke a measure theoretic extension theorem (see [12,Theorem 6.2]) to extend a measure on a semialgebra of subsets of L to the σ-algebra of Borel sets on L. Recall that a collection C of subsets of a set Ω is called a semialgebra if it is closed under finite intersections and if the complement of B ∈ C is expressible as a finite disjoint union of elements of C. For each n ∈ N, let C n be the collection of Borel subsets of L n . Then C = {L c n : n ≥ 1} ∪ n∈N C n is a semialgebra.
By [11,Theorem 4.4.1], there is a unique Borel probability measure µ on K such that for all Borel subsets of K. Then, for each n ∈ N and ω ∈ W n we can define a measure µ ω on L n (ω) via µ ω (A) = N n µ(F ω (A)). We let ν be the product measure on X generated by the weights {1/N, . . . , 1/N } on the set {1, . . . , N }. Therefore, if ω ∈ W n then ν(Z(ω)) = (1/N ) n . Then one can define a Borel measure µ n on L n such that µ n (Z(ω) × A) = µ(F ω (A)), and, more generally, µ n (U × A) = ν(σ n (U )) · µ(F ω (A)) if U × A ⊂ Z(ω) × L n (ω). Notice that if m < n and B is a Borel subset of L m then B is a Borel subset of L n . However, more is true if the iterated function system satisfies the open set condition. Proof . The first part is an immediate consequence of [11,Theorem 5.3.1]. The second part follows from the first part.
Proof . As noted above C = {L c n : n ≥ 1} ∪ n∈N C n is a semialgebra. By [12, Theorem 6.2], µ ∞ extends uniquely to a measure on the σ-algebra generated by C if the following conditions hold i. If ∅ ∈ C, µ ∞ (∅) = 0.
Conditions (i) and (ii) are easy to check. The only non-trivial case for condition (iii) is when C = L c k for some k ∈ N, and for each n ∈ N there is k n ∈ N such that C n ∈ C kn . Then µ ∞ (C) = ∞ and lim k→∞ µ ∞ (L k C) = ∞. Let k ∈ N be fixed. Then the sets L k ∩ C n are Borel subsets of L k that cover L k ∩ C. Hence, using Lemma 3.10, we have that Since the left hand side of the inequality goes to ∞, it follows that To see that condition (iv) holds we first note that for each α ∈ W n , C α = Z(α) × L n (α) ∈ C, µ(C α ) = µ(K) = 1 and L = α∈W * C α . It is straightforward to show that the σ-algebra generated by C is the Borels.
To prove that µ ∞ isG-invariant we use the following fact that is probably known to specialists: if σ is a local homeomorphism on a Hausdorff topological space X, then G(U, m, n, V ) is a G-section (where G = G(X, σ) is the associated Renault-Deaconu groupoid), if and only if σ m | U is a homeomorphism onto σ m (U ) and σ n | V is a homeomorphism onto σ n (V ). Moreover, r(G(U, m, n, V )) = U and s(G(U, m, n, V )) = V .
Let V be open in L such thatσ| V is a homeomorphism ontoσ(V ). It follows from the above remark that in order to show that µ ∞ isG-invariant it is enough to prove that µ ∞ (V ) = µ ∞ (σ(V )). We have that Hence, it suffices to prove that if ω ∈ W n and U × A ⊂ Z(ω) × L n (ω) is open such thatσ| U ×A is a homeomorphism ontoσ(U × A), then µ(σ(U × A)) = µ(U × A). By the definition ofσ we have thatσ(U × A) = σ(U ) × F ω 1 (A). Hence, using Lemma 3.10, Thus µ ∞ isG-invariant.
extends to a densely defined lower semi-continuous trace on C * (G) C * (G * L).
The proof of the corollary follows from the following lemma which is known to specialists. We were unable, however, to find a specific reference in the literature and we include a short proof for completeness. Lemma 3.13. Suppose that G is anétale locally compact groupoid and that µ is a (Radon) Ginvariant measure on G 0 . Then the map extends to a densely defined lower semi-continuous trace on C * (G).
Proof . Recall that for anétale locally compact groupoid G the restriction map from C c (G) to C c (G 0 ) extends to a continuous expectation from C * (G) to C * (G 0 ) [

Examples
In this section we provide detailed descriptions of the fractafold bundle defined in Section 3 for some specific iterated function systems. We point out some cases when the bundle L is not locally compact. We show in the second example that the action groupoid G * L is not, in general, minimal.
Example 4.1. Let F 0 , F 1 : R → R be the maps F 0 (x) = 1 2 x and F 1 (x) = 1 2 x + 1 2 . Then {F 0 , F 1 } is an iterated function system whose invariant set is [0, 1]. In this example, W = {0, 1}, X = W ∞ = {0, 1} ∞ and the fractafolds defined in Section 3 have an easy description. Notice that, unlike in the previous section, we index our maps using 0 and 1; the index of an element in X starts at 0 as well. This makes the formulas that we describe next more tractable. For x ∈ X define a n (x) = F −1 x(n) (0) and b n (x) = F −1 x(n) (1) for n ≥ 1. Then and L(x) is either the real line, a left half-closed infinite interval, or a right half-closed infinite interval. To prove this claim notice that {a n (x)} n is a decreasing sequence and {b n (x)} n is an increasing sequence. Indeed, if x n = 0 then a n+1 (x) = a n (x) and b n (x) < b n+1 (x), and if x n = 1 then a n+1 (x) < a n (x) and b n (x) = b n+1 (x). It follows that if there is n ∈ N such that x i = 0 for all i ≥ n, then L(x) = [a n (x), ∞) and, if We claim that for any x ∈ X we have that a n (x) = − x j 2 j = 2 n+1 − n j=1 x j 2 j .
Thus the induction holds and our claim is proved.
Note that L is not locally compact. For example, the point (x, 0) ∈ L does not have a compact neighborhood, where x is the sequence for which x i = 0 for all i ∈ N. To see this, recall that if C ⊂ L is compact then there is n ≥ 0 such that C ⊂ L n (see [6, p. 2]). However, one can check that there is no open set in L containing (x, 0) that is a subset of any of the L n 's. Example 4.2. Fix N > 1, r ∈ (0, 1) and let e 1 , . . . , e N ∈ R N be the standard basis elements. For j = 1, . . . , N , define F j : R N → R N by F j (x) = rx + (1 − r)e j . Then (F 1 , . . . , F N ) forms an iterated function system on Y = R N (endowed with the usual metric). If N = 2 and r = 1/2, then the invariant set K is homeomorphic to the unit interval and (F 1 , F 2 ) is conjugate to the iterated function system of the previous example. If N = 3 and r = 1/2 then K is homeomorphic to the Sierpinski gasket.
We have W = {1, . . . , N }, X = W ∞ = {1, . . . , N } ∞ . Note that for x ∈ X we have Hence, the invariant set is given by K = (1 − r) ∞ j=1 r j−1 e x j : x ∈ X . It is straightforward to check that the iterated function system is totally disconnected if r < 1/2. Observe that F −1 j (x) = 1 r (x − (1 − r)e j ). For α ∈ W n , L n (α) consists of all points of the form And, of course, for each x ∈ X, we have L(x) = n L n (x) (with the inductive limit topology). We will show that the groupoid G * L is not minimal. Let y ∈ X be the sequence for which y i = 1 for all i. We claim that the orbit of (y, e 1 ) ∈ L is not dense. In particular we show that (y, e 2 ) is not in the closure of {γ · (y, e 1 ) : s(γ) = y}. If s(γ) = y, then γ = (x, m − n, y) (note x i = 1 for i ≥ m). Since e 1 is a fixed point for F 1 γ · (y, e 1 ) = x, F −1 x(m) • F y(n) (e 1 ) = x, F −1 x(m) (e 1 ) Define f : R N → R by f (t 1 , . . . , t N ) = t 2 . Note that F −1 j (t) = 1 r (t − (1 − r)e j ), so if f (t) ≤ 0, then f (F −1 j (t)) ≤ 0. Observe that projection π 2 : L → R N is continuous. An easy induction argument now shows that (f • π 2 )(γ · (y, e 1 )) = f F −1 x(m) (e 1 ) ≤ 0.
Since (f • π 2 )(y, e 2 ) = 1 (and f • π 2 is continuous), (y, e 2 ) is not in the closure of the orbit of (y, e 1 ). If r < 1/2 the iterated function system is totally disconnected, and, hence, F i (K) is open in K. Thus, if r < 1/2, L is locally compact. However, if r ≥ 1/2 then L is not locally compact. An argument similar with the one at the end of Example 4.1 shows that, if r ≥ 1/2, the point (y, e 1 ) ∈ L does not have a compact neighborhood in L, where y is the sequence for which y i = 1 for all i ∈ N.