2.1 Directed graphs and their path spaces

We quickly introduce directed graphs; for a detailed treatment, see Raeburn’s seminal book [Reference Raeburn14].

A directed graph E is a quadruple $E = (E^0, E^1, r, s)$ consisting of two sets $E^0$ and $E^1$ along with two functions $r,s : E^1 \to E^0$ called the range and source maps, respectively. Elements in $E^0$ are vertices and elements in $E^1$ are edges. We think of an edge e as a directed arrow from its source vertex $s(e)$ to its range $r(e)$ .

Perhaps the most important aspect of a directed graph is its path space. A finite path $\mu $ in a directed graph E is either a vertex $\mu = v$ , or a finite sequence of edges $\mu = e_1 \cdots e_n$ such that $s(e_i) = r(e_{i+1})$ for all $i\leq n-1$ . Let the paths of length n in E be denoted by $E^n = \{e_1 \cdots e_n : e_i \in E^1, s(e_i) = r(e_{i+1})\}$ . We then let denote the set of all finite paths in E. For a path $\mu =\mu _1 \cdots \mu _n$ in $E^n$ , let $r(\mu ) =r(\mu _1)$ and $s(\mu )=s(\mu _n)$ . For $\mu \in E^*$ and $X \subseteq E^*$ , we define

$$ \begin{align*} \mu X = \{\mu \nu : \nu \in X, s(\mu) = r(\nu)\}\quad\text{and}\quad X \mu = \{\nu\mu : \nu \in X, r(\mu) = s(\nu)\}. \end{align*} $$

We then have $\mu X \nu = \mu X \cap X \nu $ .

A graph is finite if both $E^0$ and $E^1$ are finite. A graph is strongly connected if, for all $v,w \in E^0$ , the set $v E^* w$ is nonempty. Notice that if E is strongly connected, then $vE^1$ and $E^1v$ are nonempty for all $v \in E^0$ , unless E is the graph with one vertex and no edges. We say a vertex v in a graph E is a source if $vE^1=\emptyset $ and a sink if $E^1v=\emptyset $ .

In this paper, we need to work with both left-, right- and bi-infinite paths in a graph E. Thus, we define:

• • ;

• • ; and

• • .

As usual, we endow these spaces with the product topology, with a basis of cylinder sets. These are indexed by finite paths in each of the three spaces, so we distinguish them as follows. For

• • $E^{+\infty }$ : when $\mu \in E^n$ , let ; for

• • $E^{-\infty }$ : when $\mu \in E^n$ , let ; and for

• • $E^{\mathbb {Z}}$ : when $n \ge 0$ and $\mu \in E^{2n+1}$ , let .

If x is an element in any of the spaces above and $m < n \in \mathbb {Z}$ appropriately chosen for the space in question, we define

2.2 Self-similar actions of groupoids on graphs

Katsura–Exel–Pardo actions are a type of self-similar groupoid action on a graph, so we take a few paragraphs to introduce them. For further details, see [Reference Laca, Raeburn, Ramagge and Whittaker10].

Suppose E is a directed graph. Given $v,w \in E^0$ , a partial isomorphism of $E^*$ is a bijection $g : vE^* \to wE^*$ that is length and path preserving in the sense that $|g(\mu )|=|\mu |$ and $g(\mu e) \in g(\mu )E^1$ for all $\mu \in E^*$ and $e \in E^1$ satisfying $s(\mu ) = r(e)$ . We use the notation to reduce the number of parentheses. These two conditions are equivalent to the following property: g is length preserving, and for every $\mu \in vE^{*}$ , there is a partial isomorphism $h:s(\mu )E^{*}\to s(g \cdot \mu )E^{*}$ such that

(2-1) $$ \begin{align} g \cdot(\mu\nu) = (g\cdot \mu)(h \cdot \nu)\quad\text{for all } \nu \in s(\mu)E^{*}. \end{align} $$

We write $h = g|_{\mu }$ , as it is uniquely defined by the above property, and call it the restriction of g to $\mu $ .

Let $\operatorname {PIso}(E^*)$ denote the set of all partial isomorphisms of $E^*$ , which is itself a groupoid with units $\operatorname {id}_v: vE^* \to vE^*$ defined by $\operatorname {id}_v(\mu )=\mu $ for all $\mu \in vE^*$ and multiplication given by composition of maps. Since units are associated with vertices, we go ahead and identify the unit space of $\operatorname {PIso}(E^*)$ with $E^0$ . The isotropy group of a unit $v \in E^0$ is the set of partial isomorphisms from $vE^*$ to $vE^*$ .

Given a partial isomorphism $g : vE^* \to wE^*$ , define its domain to be $d(g) = v$ and its codomain to be $c(g) = w$ . That is, we are renaming the range and source maps in the groupoid $\operatorname {PIso}(E^*)$ by the terms codomain and domain since the symbols s and r are already in use.

Restriction and multiplication of elements satisfy several relations, which we record in the following lemma.

A groupoid G with unit space $E^0$ acts on $E^*$ if there is a groupoid homomorphism $\phi : G \to \operatorname {PIso}(E^*)$ that restricts to the identity map on $E^0$ . Define $\operatorname {Ker}(\phi ) = \phi ^{-1}(E^{0})$ , which is a normal sub-groupoid of G. We say G acts faithfully on $E^*$ if $\phi $ is injective or, in other words, $\operatorname {Ker}(\phi ) = E^0$ . We write $g \cdot \mu $ in place of $\phi (g)(\mu )$ .

Definition 2.2. Suppose $E = (E^0, E^1, r, s)$ is a directed graph, and G is a groupoid with unit space $E^0$ and a faithful action $\phi :G\to \operatorname {PIso}(E^*)$ . We say $(G, E)$ is a self-similar groupoid action if for every $g\in G$ and $e\in d(g)E^*$ , there is $h\in G$ such that $\phi (g)|_{e} = \phi (h)$ . We write and call it the restriction of g to e. Using Lemma 2.1, $g|_{\mu }\in G$ for $\mu \in d(g)E^{n}$ is well defined and satisfies

(2-2) $$ \begin{align} g \cdot(\mu\nu) = (g\cdot \mu)(g|_{\mu} \cdot \nu)\quad\text{for all } \nu \in s(\mu)E^{*}. \end{align} $$

Moreover, all the conclusions in Lemma 2.1 hold for the restriction and multiplication when $\operatorname {PIso}(E^*)$ is replaced with G.

If the action of G on $E^*$ is range preserving, then is a group and, hence, G is a group bundle. In that case, we say $(G,E)$ is a self-similar group bundle on E. As we see later, this holds for all Katsura–Exel–Pardo actions by definition.

It is useful in this paper to work with nonfaithful actions of groupoids by partial isomorphisms whose faithful quotient is self-similar. These should be considered as nonfaithful self-similar groupoids. However, such a term is an oxymoron, so we name them as below.

Definition 2.3. Let E be a directed graph and G a groupoid with unit space $E^{0}$ . An action-restriction pair for $(G,E)$ is a map

(2-3)

such that:

1. (A0) $r(g\cdot e) = c(g)$ and $d(g|_{e}) = s(e)$ for every ;

2. (A1) $r(e)\cdot e = e$ and $r(e)|_{e} = s(e)$ for every $e\in vE^{1}$ ;

3. (A2) $gh\cdot e = g \cdot (h\cdot e)$ for every $(g,h)\in G^{(2)}$ and $e\in d(g)E^1$ ;

4. (A3) $g^{-1}|_{e} = (g|_{g^{-1}\cdot e})^{-1}$ for every $g\in G$ and $e\in c(g)E^1$ .

We usually denote an action-restriction pair by $(G,E)$ when there is no ambiguity.

If we replace G in the definition above with a finite set A with $E^{0}\subseteq A$ and retracts $c,d:A\to E^{0}$ , then this is the notion of an automaton defined in [Reference Laca, Raeburn, Ramagge and Whittaker10], and one checks that such a pair extends to an action-restriction pair of the free groupoid associated to $(A,c,d)$ . Katsura–Exel–Pardo actions are, in general, not generated from an automaton but an action-restriction pair as defined above.

If we consider a group G, a finite directed graph E, an action $\sigma :G\times E^{1}\to E^{1}$ and a one-cocycle $\varphi : G\times E^{1}\to E^{1}$ satisfying Exel and Pardo’s conditions in [Reference Exel and Pardo4, Section 2.3], then such a pair defines an action-restriction pair on the group bundle $G\times E^{0} =\{g_{v} : g\in G, v\in E^{0}\}$ by

$$ \begin{align*}(g_{v},e)\to (\sigma(g,e), (\varphi(g,e)_{s(e)})).\end{align*} $$

An action-restriction pair for $(G,E)$ defines a (not-necessarily faithful) action ${\phi :G\to \operatorname {PIso}(E^{*})}$ : for $g\in G$ and $\mu = e\nu \in d(g)E^{n}$ , we inductively define

(2-4) $$ \begin{align} g \cdot \mu = (g\cdot e)(g|_{e} \cdot \nu). \end{align} $$

Note that if $g\in \operatorname {Ker}(\phi )$ and $e\in d(g)E^{1}$ , then for all $\nu \in d(g|_{e})E^{*} = s(e)E^{*}$ , we have $e g|_{e}\cdot \nu = g\cdot (e\nu ) = e\nu $ and, hence, $g|_{e}\cdot \nu = \nu $ . Therefore, $g|_{e}\in \operatorname {Ker}(\phi )$ .

It follows that if is the quotient map, then $q(g|_{e}) = q(g)|_{e}$ for all $g\in G$ and $e\in d(g)E^{1} = d(q(g))E^{1}$ . Hence, the induced action $G_{\phi }\to \operatorname {PIso}(E^*)$ is self-similar.

The case that the self-similar action comes from an action-restriction pair induced by an Exel–Pardo action as above is covered in more detail in [Reference Laca, Raeburn, Ramagge and Whittaker10, Appendix A].

Every self-similar groupoid is an example of an action-restriction pair. Moreover, self-similar groupoids are in one–one correspondence with action-restriction pairs whose induced actions as partial isomorphisms are faithful.

2.3 Properties and limit spaces of self-similar groupoid actions on graphs

In this section, we recall standard properties and constructions associated to action-restriction pairs and self-similar groupoids acting on the path space of a graph.

Definition 2.4. Let G be a groupoid and E a finite directed graph. An action-restriction pair $(G, E)$ is contracting if there exists a finite subset $F \subseteq G$ so that, for every $g \in G$ , there is an $n \ge 0$ such that $g|_{\mu }\in F$ for every path $\mu \in E^{k}$ , $k\geq n$ . Such a subset F is called a contracting core of $(G,E)$ . The nucleus of $(G,E)$ is the set

Let us see that this notion of regularity is equivalent to that of [Reference Brownlowe, Buss, Gonçalves, Hume, Sims and Whittaker3, Definition 4.1] for self-similar groupoids.

We recall from [Reference Brownlowe, Buss, Gonçalves, Hume, Sims and Whittaker3, Section 3] (see [Reference Nekrashevych11, Ch. 3]) the construction of the limit space from a self-similar groupoid. Let $(G, E)$ be a self-similar groupoid. For $\mu ,\nu \in E^{-\infty }$ , we say $\mu $ is asymptotically equivalent to $\nu $ if there is a finite set $F\subseteq G$ and a sequence $(g_{n})_{n<0}\subseteq F$ such that $d(g_{n}) = r(\mu _{n})$ and $g_{n}\cdot \mu _{n}\cdots \mu _{-1} = \nu _{n}\cdots \nu _{-1}$ for all $n < 0$ . We write $\mu \sim _{ae} \nu $ . It is shown in [Reference Brownlowe, Buss, Gonçalves, Hume, Sims and Whittaker3, Section 3] that $\sim _{ae}$ is an equivalence relation and $\mu \sim _{ae} \nu $ implies $\sigma (\mu )\sim _{ae}\sigma (\nu )$ . The quotient space $E^{-\infty }/\sim _{ae}$ is called the limit space of $(G,E)$ and is denoted $\mathcal {J}_{G, E}$ . The induced continuous mapping from $(\sigma , E^{-\infty })$ is called the shift on $\mathcal {J}_{G, E}$ and is denoted $\tilde {\sigma }$ .

It is shown in [Reference Brownlowe, Buss, Gonçalves, Hume, Sims and Whittaker3, Theorem 4.3] that if $(G, E)$ is a contracting and regular self-similar groupoid such that E has no sources, then $(\tilde {\sigma }, \mathcal {J}_{G,E})$ is an open, surjective and positively expansive local homeomorphism.

In many ways, the remainder of this paper is dedicated to understanding, in various contexts, this limit space dynamical system $(\tilde {\sigma }, \mathcal {J}_{G,E})$ and the conditions above on $(G,E)$ that give rise to its regularity properties.

4.1 Putnam’s construction

Let E and H be finite directed graphs, and let $\xi = \xi ^{0},\xi ^{1}:H \to E$ be two injective graph homomorphisms (embeddings) satisfying $\xi ^{0}|_{H^{0}} = \xi ^{1}|_{H^{0}}$ and $\xi ^{0}(H^{1})\cap \xi ^{1}(H^{1})=\emptyset $ . We refer to $\xi $ satisfying the properties above as an embedding pair.

For $\mu ,\nu \in E^{-\infty }$ with $\mu = \cdots \mu _{-2}\mu _{-1}$ and $\nu = \cdots \nu _{-2}\nu _{-1}$ , we say $\mu \sim _{\xi } \nu $ if $\mu = \nu $ , or there is $n<0$ , $i\in \{0,1\}$ and $(y_{k})_{k\leq n}\subseteq H^{1}$ such that $\mu _{k} = \xi ^{i}(y_{k})$ and $\nu _{k} = \xi ^{1-i}(y_{k})$ for all $k\leq n$ , and one of the following holds:

1. (1) $n=-1$ ;

2. (2) $n < 1$ , $\mu _{j} = \nu _{j}$ for all $j> n+1$ , and there is $y_{n+1}\in H^{1}$ such that $\mu _{n+1} = \xi ^{1-i}(y_{n+1})$ and $\nu _{n+1} = \xi ^{i}(y_{n+1})$ ; or

3. (3) $n<1$ , $\mu _{j} = \nu _{j}$ for all $j\geq n+1$ , and $\mu _{n+1}= \nu _{n+1}\notin H_{\xi }^{1}$ .

By [Reference Putnam13, Proposition 3.7], $\sim _{\xi }$ is an equivalence relation and $\mu \sim _{\xi } \nu $ for $\mu ,\nu \in E^{-\infty }$ implies $\sigma (\mu )\sim _{\xi }\sigma (\nu )$ . Therefore, if we denote by $\mathcal {J}_{\xi }$ the quotient space $E^{-\infty }/ \sim _{\xi }$ , the shift $\sigma $ descends to a continuous mapping $\sigma _{\xi }:\mathcal {J}_{\xi } \to \mathcal {J}_{\xi }$ .

Putnam shows in [Reference Putnam13, Section 3] that if E is assumed primitive and $\xi $ satisfies an extra hypothesis (H3), then $\sigma _{\xi }$ is an expanding surjective local homeomorphism. In addition, he describes the expanding metric in great detail. We show the same, but with hypothesis (H3) removed and primitive weakened to no sources by proving that $(\sigma _{\xi }, \mathcal {J}_{\xi })$ is isomorphic to the limit space dynamical system of a contracting and regular self-similar groupoid acting on E. We then apply the recent work on these dynamical systems in [Reference Brownlowe, Buss, Gonçalves, Hume, Sims and Whittaker3]. We do not, however, extend Putnam’s metric results.

4.2 Putnam’s binary factor maps as self-similar groupoid actions on graphs

In this section, we show that Putnam’s construction gives rise to a self-similar groupoid on a graph.

Let $\xi = \xi ^{0},\xi ^{1}:H \to E$ be an embedding pair. Denote and . Let $\tilde {G}_{\xi } = \mathbb {Z}\times E^{0}$ , with groupoid structure determined by the projection $\pi :\tilde {G}_{\xi } \to E^{0}$ . This means that $(m,v),(n,w)\in \tilde {G}_{\xi }$ are composable if and only if $v = w$ , in which case, $(m,v)(n,v) = (m+n, v)$ . Hence, $d(m,v)=c(m,v)= \pi (m,v)$ , so that $\tilde {G}_{\xi }$ is a group bundle in the sense that $\pi ^{-1}(v)=\mathbb {Z}$ for all $v \in E^0$ .

For $v\in H^{0}$ , let $\ell (v)$ be the maximum length of a path y in H satisfying ${r(y) = v}$ . Consider the quotient bundle $G_{\xi } = (\bigcup _{v\in H^{0}} \mathbb {Z}/2^{\ell (v)}\mathbb {Z}\times \{\xi ^{0}(v)\}) \cup (\{0\}\times (E^{0}\setminus H_{\xi }^{0}))$ , where we make the convention that if $\ell (v) = \infty $ , then $\mathbb {Z}/2^{\ell (v)}\mathbb {Z} = \mathbb {Z}$ . We aim to define a (faithful) self-similar groupoid action of $G_{\xi }$ on E.

For $(m,v) \in \tilde {G}_{\xi }=\mathbb {Z}\times E^{0}$ and $e\in vE^{1}$ , we define

(4-1) $$ \begin{align} (m,v) \cdot e &= \begin{cases} e & \text{ if } e \notin H^{1}_{\xi},\\ \xi^{j}(h)&\text{ if } e =\xi^{i}(h)\ \text{for } h\in H^{1}, i,j,n \in\{0,1\}\ \text{such that } m+i = 2n+j, \end{cases} \notag\\ (m,v)|_{e} &= \begin{cases} (0,s(e)) & \text{ if } e \notin H^{1}_{\xi},\\ (n, s(e)) & \text{ if } e =\xi^{i}(h)\ \text{for }h\in H^{1}, i,j,n\in\{0,1\}\ \text{such that } m+i = 2n+j. \end{cases} \end{align} $$

This defines an action-restriction pair in the sense of Definition 2.3. To find the kernel of the induced action $\phi :\tilde {G}_{\xi }\to \operatorname {PIso}(E^{*})$ , let us describe the action and restriction in terms of the binary odometer action. Let $\alpha :\mathbb {Z} \curvearrowright \{0,1\}^{*}$ be the self-similar group representation of $\mathbb {Z}$ by the $2$ -odometer action [Reference Nekrashevych11, Section 1.7.1]. In our language, this is the self-similar group defined by the Katsura pair $A = (2)$ and $B = (1)$ . More explicitly, for $m\in \mathbb {Z}$ and $i\in \{0,1\}$ , we have

$$ \begin{align*} a^{m}\cdot i = j\quad\text{and}\quad a^{m}|_{i} = a^{n}\quad \text{where }m + i = 2n + j. \end{align*} $$

For example, we have $a\cdot 1^{k} = 0^{k}$ , $a|_{1^{k}} = a$ and $a\cdot 0 = 1$ , $a|_{0} = \text {id}$ . This completely describes a as an automorphism of $\{0,1\}^{*}$ .

For $n\geq 0$ , $i = i_{1}i_{2}\cdots i_{n}\in \{0,1\}^{n}$ and $h = h_{1}h_{2}\cdots h_{n}\in H^{n}$ , let

(4-2)

Notice that the embedding $\xi :H^1 \to E^1$ extends to an embedding $\xi ^i:H^n \to E^n$ via Equation (4-2). If $\mu \in E^n$ satisfies $\mu = \xi ^{i}(h)$ for some $i\in \{0,1\}^{n}$ and $h\in vH^{n}$ , then for any $\nu \in s(\mu )E^*$ , we have

(4-3) $$ \begin{align} (m,r(\mu)) \cdot \mu\nu = \xi^{(a^m \cdot i)}(h)\, (a^m|_i, s(\mu)) \cdot \nu. \end{align} $$

If $e\notin H_{\xi }^{1}$ , then for any $\nu \in s(e)H^{*}$ , we have $(m,r(e)) \cdot e\nu = e\nu $ . Thus, the action ${\phi :\tilde {G}_{\xi }\to \operatorname {PIso}(E^{*})}$ is completely described by the $2$ -odometer action and the trivial action.

Since the kernel of $\alpha :\mathbb {Z}\curvearrowright \{0,1\}^{n}$ is $2^{n}\mathbb {Z}$ , the kernel of the $\tilde {G}_{\xi }$ -action on $E^{*}$ is given by $(\bigcup _{v \in H^{0}}2^{\ell (v)}\mathbb {Z}\times \{\xi ^{0}(v)\}) \cup (\mathbb {Z} \times (E^{0}\setminus H_{\xi }^{0}))$ . Hence, the quotient of the $\tilde {G}_{\xi }$ -action is $G_{\xi }$ . The quotient action $\phi :G_{\xi }\to \operatorname {PIso}(E^*)$ is then self-similar by the results in Section 2.2.

Example 4.1. Consider the graphs E and H in Figure 2 along with the embedding pair $\xi :H \to E$ defined by $\xi ^{0}(e)=e_{0}, \, \xi ^{1}(e)=e_{1}.$ Then, Equation (4-1) gives partial isomorphisms generating a self-similar groupoid $(G_\xi ,E)$ via:

$$ \begin{align*} &(1,v) \cdot e_{0} =e_{1} \quad (1,v)|_{e_{0}}=(0,v), & &(1,v) \cdot e_{1} =e_{0} \quad (1,v)|_{e_{1}}=(1,v),\\ &(1,v) \cdot f =f \quad (1,v)|_f=(0,v), \end{align*} $$

where $(G_{\xi }, E)$ is not a self-similar groupoid action arising from a KEP-action. Indeed, if $(G_{\xi }, E)$ was isomorphic to $(G_{B}, E_{A})$ for some Katsura pair, then $A = (3)$ and $B = (n)$ for some $n\in \mathbb {Z}$ . The action of $G_{\xi } = \mathbb {Z}$ on E is nontrivial and contracting, so its contracting coefficient satisfies $0 <\rho = {|n|}/{3} < 1$ . So either $|n| = 1$ or $|n| = 2$ , but neither of these cases allow for $\mathbb {Z}\cdot f = f$ , $\mathbb {Z}|_{f} = 0$ . Thus, $(G_{\xi }, E)$ is not isomorphic to a KEP-action.

Figure 2

The embedding pair $\xi : H \to E$ for Example 4.1.

However, we show in Section 5 that every self-similar groupoid from an embedding pair is an out-splitting of a KEP-action.

4.3 Properties of $(G_{\xi }, E)$

In this section, we prove that the self-similar groupoid actions associated with a binary factor are contracting and regular. Moreover, we prove that the shift map on the limit space of the self-similar groupoid is conjugate to Putnam’s expanding local homeomorphism on the quotient space $\mathcal {J}_{\xi }=E^{-\infty }/\sim _{\xi }$ described in Section 4.

Proof. We first show $(G_{\xi },E)$ is contracting. We show the nucleus is contained in $\mathcal {N} = (\{-1,0,1\}\times H_{\xi }^{0})\cup \{0\}\times (E^{0}\setminus H_{\xi }^{0}).$ Let $g = (m,w)\in G_{\xi }$ . If $w\notin H^{0}_{\xi }$ , then $m = 0$ and, hence, $g\in \mathcal {N}$ . So suppose $w = \xi ^{0}(v)$ . First, assume $\ell (v) < \infty $ . Then, if there is a path $\mu \in wE^{*}$ such that $|\mu | = n\geq \ell (v) +1$ , at least one of its edges $\mu _{k}\notin H_{\xi }^{1}$ . So, we have by Equation (4-1) that

(4-4) $$ \begin{align} g|_{\mu} = (g|_{\mu_{1}\cdots \mu_{k}})|_{\mu_{k+1}\cdots \mu_{n}} = (0,s(\mu_{k}))|_{{\mu_{k+1}\cdots \mu_{n}}} = (0, s(\mu_{n})) \in \mathcal{N}. \end{align} $$

If there are no paths $\mu $ of length $|\mu |\geq \ell (v)+1$ in $E^{*}$ ending at $\xi ^{0}(v)$ , then the contracting condition is satisfied for g vacuously.

Now, suppose $w =\xi ^{0}(v)$ and $\ell (v) = \infty $ . The 2-odometer action of $\mathbb {Z}$ is contracting, with $\mathcal {N}=\{-1,0,1\}$ , see [Reference Nekrashevych11, Section 1.7.1]. So, let $K\in \mathbb {N}$ be the number such that $a^m|_{i}\in \{-1,0,1\}$ for all $|i|\geq K$ . Let $\mu \in wE^{*}$ have length $|\mu | = n\geq K$ . If there is ${k\leq n}$ such that $\mu _{k}\notin \xi ^{0}(H^{1})\cup \xi ^{1}(H^{1})$ , then Equation (4-4) implies $g|_{\mu } = (0, s(\mu )) \in \mathcal {N}$ . Otherwise, $\mu _{1}\cdots \mu _{n} = \xi ^{i}(h)$ for some $i\in \{0,1\}^{n}$ and $h\in H^{n}$ . By Equation (4-3), letting $a^{l} = a^{m}|_{i}$ , we have

$$ \begin{align*} g|_{\mu} = (l, s(\mu))\in\mathcal{N}. \end{align*} $$

Thus, $(G_{\xi }, E)$ is contracting.

We now show $(G_{\xi }, E)$ is regular. Let $g = (m,w)$ and $\mu \in wE^{*}$ satisfy $g\cdot \mu = \mu $ and $g|_{\mu }\neq s(\mu )$ . Then, Equation (4-4) implies $\mu = \xi ^{i}(h)$ for some $i\in \{0,1\}^{*}$ and $h\in H^{*}$ . By Equation (4-3), we have

$$ \begin{align*}\xi^{a^{m}\cdot i}(h) = g\cdot \mu = \mu = \xi^{i}(h).\end{align*} $$

The $2$ -odometer action is regular, so let $M\in \mathbb {N}$ be such that if $a^{m}\cdot \tilde {i} = \tilde {i}$ and $|\tilde {i}|> M$ , then $a^{m}|_{\tilde {i}} = e$ . Hence, we have $|\mu | = |i| \leq M$ . Therefore, $(G_{\xi }, E)$ is regular.

4.4 Equality of the dynamical systems $(\sigma _{\xi }, \mathcal {J}_{\xi })$ and $({\tilde \sigma }, \mathcal {J}_{G_{\xi },E})$

In this section we prove the following theorem.

We begin by first proving a couple of lemmas.

Proof. We prove the forward direction first. Suppose that $\mu \sim _{ae} \nu $ and $\mu \neq \nu $ . From Lemma 4.4, we know that there is $n<0$ such that for all $k \leq n$ , there is $i_{k}, i^{\prime }_{k} \in \{0,1\}$ and $y_{k}, y^{\prime }_{k}\in H^{1}$ with $\mu _{k} = \xi ^{i_{k}}(y_{k})$ and $\nu _{k} = \xi ^{i^{\prime }_{k}}(y^{\prime }_{k})$ . Let n be the largest such n for which this is true, and let $(g_{k})_{k<0}$ be a sequence contained in some finite set F of $G_{\xi }$ satisfying $d(g_{k}) = r(\mu _{k})$ and $g_{k}\cdot \mu _{k}\cdots \mu _{-1} = \nu _{k}\cdots \mu _{-1}$ for all $k<0$ .

Let $\tilde {F}$ be a finite set in $\mathbb {Z}$ such that $(g_{k})_{k\leq n}\subseteq \pi (\tilde {F}\times H_{\xi }^{0})$ , where $\pi :\tilde {G}_{\xi }\to G_{\xi }$ is the quotient map. Then, there is $(n_{k})_{k\leq n}\subseteq \tilde {F}$ such that

$$ \begin{align*} \xi^{a^{n_{k}}\cdot (i_{k}\cdots i_{n})}(y_{k}\cdots y_{n}) = g_{k}\cdot \mu_{k}\cdots \mu_{n} = \nu_{k}\cdots \nu_{n} = \xi^{i^{\prime}_{k}\cdots i^{\prime}_{n}}(y^{\prime}_{k}\cdots y^{\prime}_{n}). \end{align*} $$

It follows that $y_{k} = y^{\prime }_{k}$ for all $k\leq n$ and $(\cdots i_{n-1}i_{n})\sim _{ae} (\cdots i^{\prime }_{n-1}i^{\prime }_{n})$ .

Now, if $n =-1$ , then we are done, so assume $n<1$ and, without loss of generality, that $\mu _{n+1}\notin H^{1}_{\xi }$ . Then $\nu _{n+1} = g_{n+1}\cdot \mu _{n+1} = \mu _{n+1}$ and $g_{n+1}|_{\mu _{n+1}} = (0, s(\mu _{n+1}))$ , so that $\nu _{n+1}\cdots \nu _{-1} = g_{n+1}\cdot \mu _{n+1}\cdots \mu _{-1} = \mu _{n+1}\cdots \mu _{-1}$ .

Now, we prove the reverse direction. Let $\tilde {F}$ be a finite set in $\mathbb {Z}$ and $(n_{k})_{k\leq n}\subseteq \tilde {F}$ a sequence such that $\alpha ^{n_{k}}\cdot i_{k}\cdots i_{n} = i^{\prime }_{k}\cdots i^{\prime }_{n}$ for all $k\leq n$ . It follows from the definition of the action of $G_{\xi }$ that if we let $g_{k} = \pi ((n_{k}),s(\mu _{k}))$ , then $g_{k}\cdot \mu _{k}\cdots \mu _{n} = \nu _{k}\cdots \nu _{n}$ for all $k\leq n$ . If $n=-1$ , then we are done, so assume $n < -1$ . Since $\mu _{n+1} = \nu _{n+1}\notin H^{1}_{\xi }$ , it and its extension to the path $\mu _{n+1}\cdots \mu _{-1} = \nu _{n+1}\cdots \nu _{-1}$ are fixed by any element in $G_{\xi }\cap d^{-1}(s(\mu _{n+1}))$ . Then,

$$ \begin{align*} g_{k}\cdot \mu_{k}\cdots \mu_{-1} &= \nu_{k}\cdots \nu_{n}(g|_{\mu_{n}}\cdot \mu_{n+1}\cdots \mu_{-1})\\& = \nu_{k}\cdots \nu_{n} \nu_{n+1}((0,s(\mu_{n+1})) \cdot \mu_{n+2}\cdots \mu_{-1}) = \nu_{k}\cdots \nu_{-1}. \end{align*} $$

So, if we let $g_{j} = (0, s(\mu _{j}))$ for $j\geq n+1$ , then $(g_{k})_{k<0}$ is a sequence contained in $\pi (\tilde {F}\times H^{0}_{\xi })\cup \{0\}\times (E^{0}\setminus H^{0}_{\xi })$ implementing $\mu \sim _{ae} \nu $ .

We now prove the main theorem.

5.1 Out-splits of self-similar group bundles

We take the approach on out-splits found in [Reference Brix, Mundey and Rennie2]. For the ‘classical’ approach, see [Reference Kitchens9]. Let E be a directed graph and let $OS = (\pi , \beta )$ be a tuple, where $\pi : E^{1}\to E^{0}_{OS}$ and $\beta :E_{OS}^{0}\to E^{0}$ are maps such that ${s = \beta \circ \pi} $ . The out-split of E by $OS$ is the graph $E_{OS} = (E^{0}_{OS}, E^{1}_{OS}, r_{OS}, s_{OS})$ , where and $r_{OS},s_{OS}:E^{1}_{OS}\to E_{OS}^{0}$ are defined for $(v,e)\in E_{OS}^{1}$ as ${r_{OS}(v,e) = v}$ and $s_{OS}(v,e) = \pi (e)$ .

Example 5.1. Let E be the graph on the left of Figure 3. Let and $\pi : E^{1}\to E^{0}_{OS}$ be defined by

$$ \begin{align*} \pi(1)=v_1,\quad \pi(2)=v_2,\quad \pi(3)=v_3\quad \text{and}\quad \pi(4)=v_3. \end{align*} $$

Since $s = \beta \circ \pi $ , the map $\beta :E_{OS}^{0}\to E^{0}$ is given by

$$ \begin{align*} \beta(v_1)= \beta(\pi(1))= s(1)=x,\quad \beta(v_2)=x\quad\text{and}\quad \beta(v_3)=y. \end{align*} $$

Then,

with the out-split graph $E_{OS}$ depicted on the right of Figure 3.

Figure 3

An out-split of the graph on the left appears on the right.

It is routine to check that the dynamical systems $(\sigma , E^{-\infty })$ and $(\sigma , E_{OS}^{-\infty })$ are topologically conjugate via the map $I: E^{-\infty } \to E_{OS}^{-\infty }$ given by

$$ \begin{align*} I(\ldots ,e_{-2}, e_{-1})= (\ldots ,(\pi(e_{-3}), e_{-2}), (\pi(e_{-2}), e_{-1})). \end{align*} $$

More generally, for every $n\in \mathbb {N}$ , there is a bijection defined for with $\mu =e_{-n} \cdots e_{-1}$ by

$$ \begin{align*} I_{n}(v,\mu) = (v,e_{-n})(\pi(e_{-n}), e_{-n+1}) \cdots (\pi(e_{-2}), e_{-1}). \end{align*} $$

Now, suppose $(G,E)$ is a self-similar group bundle; that is, the action of G on $E^*$ is range preserving. We can define a new group bundle $(G_{OS},E_{OS})$ with

$$ \begin{align*} G_{OS} = \{(g,v)\in G \times E_{OS}^{0} : d(g)=c(g)=\beta(v)\}, \end{align*} $$

where $(g,v)$ and $(g, v')$ are composable if and only if $v = v'$ , in which case

$$ \begin{align*} (g,v)\cdot (g',v) = (gg', v). \end{align*} $$

The action-restriction pair $(G_{OS},E_{OS})$ is defined for $(g,v)\in G_{OS}$ and $(v, e)\in E_{OS}^{1}$ , as

(5-1) $$ \begin{align} (g,v) \cdot (v, e) = (v,g\cdot e) \quad \text{and} \quad (g, v)|_{(v, e)} = (\pi(e), g|_{e}). \end{align} $$

The induced action of $(g,v)$ on a path $I_{n}(v,e)$ , for , is

(5-2) $$ \begin{align} (g,v)\cdot I_{n}(v,e) = I_{n}(v, g\cdot e), \end{align} $$

making it clear that the homomorphism $\phi :G_{OS}\to \operatorname {PIso}(E^{*}_{OS})$ is faithful. Therefore, $(G_{OS}, E_{OS})$ is a self-similar groupoid action on a graph. We call $(G_{OS}, E_{OS})$ the out-split of $(G,E)$ by $OS$ .

5.2 KEP-action models for embedding pairs

Suppose $\xi = \xi ^{0},\xi ^{1}:H\to E$ is an embedding pair and $(G_{\xi }, E)$ its associated self-similar groupoid action. We show there is a Katsura pair $(A,B)$ such that $(G_{A},E_{B})$ is the out-split of $(G_{\xi }, E)$ .

Let $E_{OS}^{0}$ be the set obtained from $E^{1}$ by identifying the edges $\xi ^{0}(h)$ and $\xi ^{1}(h)$ for all $h\in H^{1}$ . Let $\pi :E^{1}\to E_{OS}^{0}$ be the quotient map. Since the edges being identified share the same source, there is a unique map $\beta :E_{OS}^{0}\to E^0$ satisfying $\beta \circ \pi = s$ . Therefore, the tuple $OS = (\pi , \beta )$ determines an out-split $(G_{OS}, E_{OS})$ of the self-similar group bundle $(G_{\xi }, E)$ . We show $(G_{OS},E_{OS})$ is isomorphic to a KEP-action.

If $w\in \pi (H_{\xi }^{1})$ , there is a unique $h\in H^{1}$ such that $\pi ^{-1}(w) = \{\xi ^{0}(h),\xi ^{1}(h)\}$ . Otherwise, $\pi ^{-1}(w) = \{w\}\subseteq E^{1}\setminus H_{\xi }^{1}$ .

Therefore, for $v,w\in E_{OS}^0$ , if , we have

(5-3) $$ \begin{align} v(E_{OS}^{1})w &= \begin{cases} \{(v, \xi^{0}(h)), (v,\xi^{1}(h))\} \text{ for some }h\in H^{1}& \text{ if } w\in\pi(H_{\xi}^{1}),\\ \{(v, w)\} &\text{ otherwise.} \end{cases} \end{align} $$

In the first case, we denote for $m\in \{0,1\}$ and in the second case, denote . Replacing the notation $(k,v)\in G_{OS}$ with $a_{v}^{k}$ , we see that when $A_{v,w}\neq 0$ , the action and restriction $(G_{OS})_{v}\times v(E_{OS}^{1})w\to v(E_{OS}^{1})w\times (G_{OS})_{w}$ is given by $(a_{v}^{k}, e_{v,w,m})\to (e_{v,w, \hat {m}}, a_{w}^{\hat {k}})$ , where

(5-4) $$ \begin{align} k(A_{v,w}-1) + m = \hat{k}(A_{v,w}) + \hat{m}\quad \text{and}\quad 0\leq m,\quad \hat{m} < A_{v,w} - 1. \end{align} $$

Comparing Equation (5-4) with Equations (3-1) and (3-2), we see that $(G_{OS}, E_{OS})$ is canonically isomorphic to $(G_{B}, E_{A})$ , where $B = (\max \{0, A_{v,w} - 1\})_{v,w\in E_{OS}^{0}}$ . We record this formally as a corollary to Theorem 5.2.

Example 5.5. Consider again Example 4.1. We have $E^{0}_{OS} = \{e,f\}$ and the quotient map $\pi :E^{1}\to E^{0}_{OS}$ satisfies

$$ \begin{align*} \pi(e_{0})=\pi(e_{1}) = e, \quad \pi(f)=f \end{align*} $$

and

$$ \begin{align*} E_{OS}^1=\{(f,f), (e,f), (e,e_{i}),(f,e_{j}) : i,j\in\{0,1\}\}. \end{align*} $$

The graph E and its out-split $OS$ is recorded in Figure 4. If we order $e< f$ , then the KEP-action is defined by the matrices

$$ \begin{align*} A = \left( \begin{matrix} 2 & 1 \\ 2 & 1 \\ \end{matrix} \right) \quad \text{and} \quad B = \left( \begin{matrix} 1 & 0\\ 1 & 0 \\ \end{matrix} \right). \end{align*} $$

Figure 4

The out-split associated with Examples 4.1 and 5.5.