2 Preliminaries

In this section, we set out some well-known preliminaries on shift spaces. An excellent reference is the book of Lind and Marcus [Reference Lind and Marcus18]. As discussed in the introduction, we are considering shifts of finite type. However, we will deal exclusively with edge shifts of finite directed graphs. These are dynamically equivalent, as a consequence of [Reference Lind and Marcus18, Example 1.5.10 and Theorem 2.3.2].

By a finite directed graph, G, we mean two finite sets $G^{0}$ (the vertex set) and $G^{1}$ (the edge set) along with maps $i,t: G^{1} \rightarrow G^{0}$ . A path of length n in G is a finite sequence $p = (p_{1}, p_{2}, \ldots , p_{n})$ in $G^{1}$ such that $t(p_{i}) = i(p_{i+1})$ , for all $ 1 \leq i < n$ . We define $i(p) = i(p_{1})$ and $t(p)=t(p_{n})$ .

The adjacency matrix for the graph is denoted $A_{G}$ and we regard it as the function on $G^{0} \times G^{0}$ whose value at $(v,w)$ is the number of edges e with $i(e) =w$ and $t(e) = v$ . We also regard it as an endomorphism of $\mathbb {Z}^{G^{0}}$ by matrix multiplication.

We say G is irreducible if, for every ordered pair of vertices $v, w$ , there is a path p with $i(p)=v, t(p)=w$ . Equivalently, G is irreducible if, for every $(v,w)$ , there is a positive integer k with $A_{G}^{k}(w,v)$ positive. We say G is primitive if there is a positive integer k such that $A_{G}^{k}(w,v)$ is positive, for all $v,w$ .

We let $X_{G}^{+}$ be the one-sided infinite path space: an element is a sequence $(x_{1}, x_{2}, \ldots )$ in $G^{1}$ with $t(x_{n})= i(x_{n+1})$ for all $n \geq 1$ . The bi-infinite path space $X_{G}$ is defined analogously with sequences indexed by the integers. We let $\chi _{G}^{+}$ denote the obvious map from $X_{G}$ to $X_{G}^{+}$ , which simply restricts the domain of the sequence.

The spaces $X^{+}_{G}$ and $X_{G}$ both have canonical metrics. For $x,y$ in $X_{G}^{+}$ , we define

$$ \begin{align*} d_{G}(x,y) = \inf \{ 2^{-n} \mid n \geq 0, x_{i} = y_{i}, 1 \leq i \leq n \} \end{align*} $$

and for $x,y$ in $X_{G}$ , we define

$$ \begin{align*} d_{G}(x,y) = \inf \{ 2^{-n} \mid n \geq 0, x_{i} = y_{i}, 1-n \leq i \leq n \}. \end{align*} $$

Using the same notation should cause no confusion. Observe that $\chi _{G}^{+}$ is a contraction.

The spaces $X_{G}^{+}$ and $X_{G}$ both carry dynamical systems. We define $\sigma $ by $\sigma (x)_{n} = x_{n+1}$ . Here, either x is in $X_{G}^{+}$ and $n \geq 1$ or x is in $X_{G}$ and n is an integer, making $\sigma $ a self-map of either space. Both are continuous and surjective and the latter is a homeomorphism. We make the easy observation that $d_{G}(\sigma ^{n}(x), \sigma ^{n}(y)) \leq 2^{n} d_{G}(x,y)$ , for $x, y$ in $X_{G}^{+}$ .

Let $G, H$ be two finite directed graphs. By a graph homomorphism from H to G, we mean a function, $\xi : H^{0} \cup H^{1} \rightarrow G^{0} \cup G^{1}$ such that $\xi (H^{0}) \subseteq G^{0}, \xi (H^{1}) \subseteq G^{1}$ and satisfying $t_{G} \circ \xi |_{H^{0}} = \xi \circ t_{H}|_{H^{0}}, i_{G} \circ \xi |_{H^{0}} = \xi \circ i_{H}|_{H^{0}}$ . It will usually not cause any confusion to drop the subscripts on $i, t$ . We usually write such a function as $\xi : H \rightarrow G$ . A graph embedding is a graph homomorphism which is injective.

If $(X, d)$ is a metric space, x is in X and $\epsilon $ is positive, then we let $X(x,\epsilon )$ denote the ball centred at x of radius $\epsilon $ . If A is any subset of X, we let $\mathrm{Cl}(A)$ denote its closure. If $(X_{1}, d_{1})$ and $(X_{2}, d_{2})$ are metric spaces, we let

$$ \begin{align*} d_{1} \times d_{2}((x_{1}, x_{2}), (y_{1}, y_{2})) = d_{1}(x_{1}, y_{1}) +d_{2}(x_{2}, y_{2}), \end{align*} $$

for all $x_{1}, y_{1}$ in $X_{1}$ and $x_{2}, y_{2}$ in $X_{2}$ , which is a metric for the product space $X_{1} \times X_{2}$ .

3 Construction

In this section, we describe our basic construction.

The first two conditions will be essential for our construction. The third is a convenience and is not always needed. It also seems likely that the hypothesis that G is primitive can be weakened.

We observe a simple consequence of hypotheses (H0) and (H1): if y is any edge in $H^{1}$ , then $\xi ^{0}(y)$ and $\xi ^{1}(y)$ are distinct, but have the same initial and terminal vertices in $G^{0}$ .

The following result is worth noting but its proof is trivial and we omit it.

The following is a summary of some convenient topological properties of the sets $X_{k}^{+}, k \geq 0$ . Essentially, these are consequences of our property (H2) and G being primitive.

We now give our two main definitions. The influence of binary expansion should be clear in the first.

It is worth noting that the n in parts (1) and (2) is unique (if it exists). The proofs of the following easy facts are left to the interested reader.

We can now define our main item of interest.

As indicated above, the space $X_{\xi }^{+}$ is given the quotient topology. It is not hard to use general topological results to show that $X_{\xi }^{+}$ is metrizable. That is not sufficient for our goals as we will need a metric with specific properties. The construction is rather subtle.

We begin the construction of a pseudo-metric, d, on $X^{+}_{G}$ . We will show that $d(x,y) \leq 3 d_{G}(x,y)$ , for all $x, y$ in $X^{+}_{G}$ . In addition, we will prove $d(x,y) = 0$ if and only if $x \sim _{\xi } y$ , for any $x,y$ in $X^{+}_{G}$ . The immediate consequence is that d induces a metric on the quotient space $X^{+}_{\xi }$ , denoted by $d_{\xi }$ , whose topology is the the quotient topology. This means that an alternate definition of our space $X_{\xi }^{+}$ is as the metric space naturally induced by the pseudo-metric d on $X_{G}^{+}$ . The definition given in Definitions 3.6 and 3.8 seems more intuitive.

For x in $X^{+}_{G}$ , if $\kappa (x)=0$ , we define

$$ \begin{align*} \theta(x) = \exp \bigg(2 \pi i \sum_{j=1}^{\infty} \varepsilon(x_{j}) 2^{-j} \bigg). \end{align*} $$

If $\kappa (x)> 0$ , we define

$$ \begin{align*} \theta(x) = \exp \bigg(2 \pi i \sum_{j=1}^{n(x)-1} \varepsilon(x_{j}) 2^{-j} \bigg). \end{align*} $$

Obviously, $\theta (x)$ lies on the unit circle, $\mathbb {T}$ , in either case and is a $2^{n(x)-1}$ th root of unity in the latter.

For $k \geq 0$ , we define $\unicode{x3bb} _{k}: X^{+}_{k} \times X^{+}_{k} \rightarrow [0, \infty )$ inductively as follows. First, we set

for $x,y$ in $X^{+}_{0}$ .

For $x,y$ in $X^{+}_{k}$ with $k> 0$ , we define

where we use $\delta ( \cdot , \cdot )$ for the Kronecker delta function.

Let us make a couple of comments on this definition. First, the value does not really depend on the actual edges $x_{j}$ but on $\varepsilon (x_{j})$ , that is, which of the two embeddings is used. Second, the value of $\unicode{x3bb} _{k}(x,y)$ depends only on $\varepsilon (x_{j}), 1 \leq j \leq n(x)$ and $\varepsilon (y_{j}), 1 \leq j \leq n(y)$ unless those two finite sequences are equal. In that case, they are deleted and the computation is done using $\unicode{x3bb} _{k-1}$ on the rest of the sequence (times a factor of $2^{-n(x)}$ ). Roughly speaking, this is an attempt to compute the distance between the binary factor parts of x and y.

Lemma 3.9. For $k \geq 0$ , we have

$$ \begin{align*} d_{\mathbb{T}}(\theta(x), \theta(y)) \leq d_{G}(x,y), \end{align*} $$

and

for all $x,y$ in $X^{+}_{k}$ .

Proof. We proceed by induction on k. For $k =0$ , suppose $d_{G}(x,y) = 2^{-m}$ , for some $m \geq 0$ , which implies $x_{j} = y_{j}$ for $1 \leq j \leq m$ . We have

$$ \begin{align*} d_{\mathbb{T}}(\theta(x), \theta(y)) & \leq \sum_{j=1}^{\infty} \vert \varepsilon(x_{j}) - \varepsilon(y_{j}) \vert 2^{-j} \\ & \leq \sum_{j=m+1}^{\infty} \vert \varepsilon(x_{j}) - \varepsilon(y_{j}) \vert 2^{-j} \\ & \leq 2^{-m} \\ & = d_{G}(x,y). \end{align*} $$

For $k=0$ , the second statement follows from the first as $\unicode{x3bb} _{0}(x,y) = d_{\mathbb {T}}(\theta (x), \theta (y))$ .

Now assume that $k>0$ and again $d_{G}(x, y) = 2^{-m}$ , so $x_{j} = y_{j}$ for $1 \leq j \leq m$ . If either $n(x) \leq m $ or $n(y) \leq m$ , then $n(x) = n(y)$ and $\vert 2^{-n(x)} - 2^{-n(y)} \vert =0$ . In addition, we have $\theta (x) = \theta (y)$ so $d_{\mathbb {T}}(\theta (x), \theta (y))=0$ .

Otherwise, we have $n(x), n(y)> m$ and

$$ \begin{align*} d_{\mathbb{T}}(\theta(x), \theta(y)) & \leq \bigg\vert\! \sum_{j=m+1}^{n(x)} \varepsilon(x_{j}) 2^{-j} - \sum_{j=m+1}^{n(y)} \varepsilon(y_{j})2^{-j} \bigg\vert \\ & \leq \sum_{j=m+1}^{\infty} 2^{-j} \\ & = 2^{-m} \\ & = d_{G}(x,y). \end{align*} $$

This establishes the first inequality. In addition, we note that

$$ \begin{align*} \vert 2^{-n(x)} - 2^{-n(y)} \vert \leq 2^{-m-1} < d_{G}(x,y). \end{align*} $$

Now, we consider $\unicode{x3bb} _{k}(x,y)$ . If either $n(x) \neq n(y)$ or $\theta (x) \neq \theta (y)$ , then we have

In the case $n(x) = n(y)=n$ and $\theta (x) = \theta (y)$ , from the definition and the induction hypothesis, we have

We can now begin our definition of our pseudo-metric on $X_{G}^{+}$ . Initially, we treat the spaces $X_{k}^{+}$ separately.

Proof. For the first part, the $d_{G_{\xi }}$ term is clearly a pseudometric, so it suffices to show that $\unicode{x3bb} _{k}$ is also, which we do by induction on k. This is clear for $k=0$ . For $k> 0$ , a simple induction argument shows that $\unicode{x3bb} _{k}$ is reflexive and symmetric. Now suppose that $x, y, z$ are in $X^{+}_{k}$ . We first consider the case $(n(x), \theta (x) ) \neq (n(y), \theta (y))$ , where we have

If the first two terms are both zero, then their sum is certainly less than or equal to $\unicode{x3bb} _{k}(x,z)$ . However, if either is non-zero, then their sum equals $\unicode{x3bb} _{k}(x,z)$ , by definition. The third and fourth terms are done similarly.

Now suppose that $(n(x), \theta (x) ) = (n(y), \theta (y))$ . If $(n(z), \theta (z) ) = (n(x), \theta (x))$ , then

follows from the induction hypothesis.

We next consider the case $n(z) \neq n(x)$ and so $n(z) \neq n(y)$ also. For any positive integers $m\neq n$ , we have

$$ \begin{align*} \vert 2^{-m} - 2^{-n} \vert = 2^{-\min\{ m,n \}} \vert 1 - 2^{-\vert m-n \vert} \vert \geq 2^{-\min\{ m,n \}-1} \geq 2^{-n-1} \end{align*} $$

and so

A similar estimate holds for $\unicode{x3bb} _{k}(z,y)$ .

However, we also have

Together this yields

The final case to consider is $n(x) = n(y) = n(z)$ and $\theta (z) \neq \theta (x)$ , so $\theta (z) \neq \theta (y)$ also. Then, as $\theta (x), \theta (y), \theta (z)$ are all $2^{n(x)-1}$ th roots of unity, we have

$$ \begin{align*} d_{\mathbb{T}}( \theta(x), \theta(z)), d_{\mathbb{T}}( \theta(z), \theta(y)) \geq 2^{-n(x)}. \end{align*} $$

It follows that

The second part is immediate from the second statement of Lemma 3.9.

For the third part, we consider the ‘if’ direction first. We proceed by induction on k, beginning with $k=0$ . Suppose $ x \sim _{\xi } y$ , where $x,y$ are in $X^{+}_{0}$ . From part (2) of Proposition 3.4, we know $\tau _{\xi }(x) = \tau _{\xi }(y)$ . In addition, $\varepsilon (x_{j}), j \geq 1$ and $\varepsilon (y_{j}), j \geq 1$ are two $0,1$ -sequences and the definition of $\sim _{\xi }$ implies that these sequences are dyadic representations of the same real number, modulo the integers. This implies $\theta (x) = \theta (y)$ and so

$$ \begin{align*} d_{0}(x,y) = d_{\mathbb{T}}(\theta(x), \theta(y)) + d_{G_{\xi}}(\tau_{\xi}(x), \tau_{\xi}(y)) = 0 +0=0. \end{align*} $$

Now assume the result is true for some k and let $x \sim _{\xi } y$ be in $X^{+}_{k+1}$ . From Proposition 3.4, we have $n(x) = n(y)$ and $x_{i} = y_{i}$ , for all $ 1 \leq i \leq n(x)$ . It follows that $\theta (x) = \theta (y)$ and

It follows from Proposition 3.4 that $\sigma ^{k}(x) \sim _{\xi } \sigma ^{k}(y)$ , so the conclusion follows from the induction hypothesis.

Now we consider the ‘only if’ direction. Suppose $d_{0}(x,y)=0 $ , where $x,y$ are in $X^{+}_{0}$ . This means that $x_{i}, y_{i}$ are in $\xi (H^{1})$ , for all $i \geq 1$ . The hypothesis obviously implies that $d_{G_{\xi }}(\tau _{\xi }(x), \tau _{\xi }(y)) =0$ so $\tau _{\xi }(x_{i}) = \tau _{\xi }(y_{i})$ , for all $i \geq 1$ . In other words, there is a path z in $X^{+}_{H}$ such that $\{ x_{i}, y_{i} \} \subseteq \{ \xi ^{0}(z_{i}), \xi ^{1}(z_{i}) \}$ for all i. The fact that $\unicode{x3bb} _{0}(x,y)=0$ implies that $\theta (x) = \theta (y)$ and this implies that the sequences $\varepsilon (x_{i}), \varepsilon (y_{i}), i \geq 1$ are the binary expansions of the same real number, modulo the integers. This implies $x \sim _{\xi } y$ .

We now consider $k> 0$ and $d_{k}(x,y)=0$ , where $x,y$ are in $X^{+}_{k}$ . The first consequence is that $\tau _{\xi }(x) = \tau _{\xi }(y)$ . This implies that $n(x) = n(y) = n$ and since $x_{n(x)}$ and $y_{n(y)}$ are not in $\xi ^{0}(H) \cup \xi ^{1}(H)$ , we have $x_{n(x)} = y_{n(y)}$ . In addition, there is a path $h_{1}, \ldots , h_{n-1}$ such that $x_{i}, y_{i}$ are in $\{ \xi ^{0}(h_{i}), \xi ^{1}(h_{i}) \}$ , for all $1 \leq i < n$ . The fact that $\unicode{x3bb} _{k}(x,y)=0$ implies that $\theta (x) = \theta (y)$ . The map which takes a $0,1$ sequence $\varepsilon _{1}, \ldots , \varepsilon _{n-1}$ to $\exp (2 \pi \sum _{j=1}^{n-1} \varepsilon _{j}2^{-j})$ is injective and so we conclude that $\varepsilon (x_{i}) = \varepsilon (y_{i})$ , for all $1 \leq i < n$ . It follows that $x_{i} = y_{i}$ , for $1 \leq i \leq n$ . The fact $\tau _{\xi }(x) = \tau _{\xi }(y)$ implies that $\tau _{\xi }(\sigma ^{n(x)}(x)) = \tau _{\xi }(\sigma ^{n(y)}(y))$ also and so we have

By our induction hypothesis, $\sigma ^{n}(x) \sim _{\xi } \sigma ^{n}(y) $ . Together with the fact that $x_{i} = y_{i}$ for $ 1 \leq i \leq n$ implies $x \sim _{\xi } y$ .

It will be useful to have a characterization of Cauchy sequences in our pseudo-metric.

Proof. The first part is clear from the definition of $d_{0}$ .

For the second part, we begin with the ‘only if’ direction. If $x^{l}, l \geq 1$ , is Cauchy in $d_{k}$ , then $\tau _{\xi }(x^{l}), l \geq 1,$ is clearly Cauchy and hence convergent in $X_{G_{\xi }}$ . In addition, both sequences $ 2^{-n(x^{l})}$ and $\theta (x^{l})$ are Cauchy. There are clearly two cases: either $n(x^{l}), l \geq 1$ , tends to infinity or it is eventually constant. In the first case, we have case (a). In the second case, if $n(x^{l})=n$ for all l sufficiently large, then for such l, $\theta (x^{l})$ lies in a finite set, so to be Cauchy, it must be eventually constant also, say with value $\theta $ . Then for sufficiently large values of $l, m$ , we have

It follows that $\sigma ^{n}(x^{l}), l \geq 1,$ is Cauchy in $d_{k-1}$ .

Conversely, if $x^{l}, l \geq 1,$ is any sequence in $X^{+}_{k}$ , then, for any $l,m$ , $d_{k}(x^{l}, x^{m})$ is equal to either

$$ \begin{align*} d_{G_{\xi}}( \tau_{\xi}(x^{l}), \tau_{\xi}(x^{m})) + \vert 2^{-n(x^{l})} - 2^{-n(x^{m})} \vert + d_{\mathbb{T}}(\theta(x^{l}), \theta(x^{m})) \end{align*} $$

or

It is then immediate that $\tau _{\xi }(x^{l}), l \geq 1,$ being convergent and either of the two conditions imply the sequence is Cauchy in $d_{k}$ .

At this point, we have defined a pseudo-metric on $X_{k}^{+}$ , for each value of k. Assuming condition (H2) and that G is primitive, we also know that $X_{k}^{+}$ will have limit points in $X_{j}^{+}$ , for every $ 0 \leq j < k$ . We now need to show these different pseudo-metrics are compatible, in an obvious sense. The following concerns a special case ( $j=0$ ) but will be used in the proof of the general statement as well.

Proof. For any $n \geq 1$ , we have $x_{n} = \lim _{l} x_{n}^{l}$ and $x_{n}^{l}$ is in $\xi (H^{1})$ , provided $n(x^{l})> n$ . It follows that $x_{n}$ is in $\xi (H^{1})$ , for every n, so x is in $X^{+}_{0}$ .

If we fix $m \geq 1$ , we may choose $l_{0}$ sufficiently large so that $n(x^{l})> m$ and $x_{i} = x_{i}^{l}$ , for all $1 \leq i \leq m$ and $l \geq l_{0}$ . For such l, we have

$$ \begin{align*} \vert \theta(x) - \theta(x^{l}) \vert & = \bigg\vert\! \exp\bigg( 2 \pi i \sum_{j=1}^{\infty} \varepsilon(x_{j}) 2^{-j}\bigg) - \exp\bigg( 2 \pi i \sum_{j=1}^{n(x^{l})-1} \varepsilon(x^{l}_{j}) 2^{-j}\bigg) \bigg\vert \\ & = \bigg\vert\! \exp\bigg( 2 \pi i \sum_{j=m+1}^{\infty} \varepsilon(x_{j}) 2^{-j}\bigg) - \exp\bigg( 2 \pi i \sum_{j=m+1}^{n(x^{l})-1} \varepsilon(x^{l}_{j}) 2^{-j}\bigg) \bigg\vert \\ & \leq 2 \pi \bigg\vert\! \sum_{j=m+1}^{\infty} \varepsilon(x_{j}) 2^{-j} - \sum_{j=m+1}^{n(x^{l})-1} \varepsilon(x^{l}_{j}) 2^{-j} \bigg\vert \\ & \leq 2 \pi 2^{-m}. \end{align*} $$

As m was arbitrary, this completes the proof.

The following summarizes the compatibility of the different pseudo-metrics.

Proposition 3.13. Suppose $ 0 \leq j < k$ , the sequences $x^{l}, l \geq 1,$ and $y^{l}, l \geq 1,$ are in $X^{+}_{k}$ , x and y are in $X^{+}_{j}$ , $x^{l}, l \geq 1,$ converges to x and $y^{l}, l \geq 1,$ converges to y in $X^{+}_{G}$ , then

$$ \begin{align*} d_{j}(x,y) = \lim_{l \rightarrow \infty} d_{k}(x^{l}, y^{l}). \end{align*} $$

Proof. We first consider the case $n(x^{l}), l \geq 1,$ tends to infinity. The fact that $x^{l},l \geq 1$ , converges in $X^{+}_{G}$ , along with part (2) of Proposition 3.10 shows that $x^{l}, l \geq 1$ , is Cauchy in $d_{k}$ . Proposition 3.11 shows that there are two cases to consider. We begin by assuming that part (a) holds so $n(x^{l})$ tends to infinity while $\theta (x^{l}), l \geq 1,$ converges.

Lemma 3.12 implies that x is in $X^{+}_{0}$ . Hence, we have $j=0$ . We now claim that the sequence $n(y^{l})$ must also tend to infinity. By the same argument as above, $y^{l}, l \geq 1,$ is Cauchy in $d_{k}$ . If $n(y^{l}), l \geq 1,$ is eventually constant, say n, then $y_{n}= \lim _{l} y_{n}^{l}$ is not in $\xi (H^{1})$ which means that y is not in $X^{+}_{0}$ , which is a contradiction.

Now, we consider the case $\theta (x) \neq \theta (y)$ . From Lemma 3.12, we see that $\theta (x^{l}) \neq \theta (y^{l})$ for l sufficiently large. In this case, we know that

$$ \begin{align*} d_{k}(x^{l}, y^{l}) = d_{G_{\xi}}(\tau_{\xi}(x^{l}), \tau_{\xi}(y^{l})) + \vert 2^{-n(x^{l})} - 2^{-n(y^{l})} \vert + d_{\mathbb{T}}( \theta(x^{l}), \theta(y^{l})). \end{align*} $$

The first term converges to $d_{G_{\xi }}(\tau _{\xi }(x), \tau _{\xi }(y))$ , the second to zero and the third to $d_{\mathbb {T}}( \theta (x), \theta (y))$ . The desired equality follows from the definition of $d_{0}(x,y)$ .

We next turn to the case $\theta (x)=\theta (y)$ . Here, we have either

$$ \begin{align*} d_{k}(x^{l}, y^{l}) = d_{G_{\xi}}(\tau_{\xi}(x^{l}), \tau_{\xi}(y^{l})) + \vert 2^{-n(x^{l})} - 2^{-n(y^{l})} \vert + d_{\mathbb{T}}( \theta(x^{l}), \theta(y^{l})) \end{align*} $$

or else

In either case, the limit is

$$ \begin{align*} d_{G_{\xi}}(\tau_{\xi}(x), \tau_{\xi}(y)) = d_{G_{\xi}}(\tau_{\xi}(x), \tau_{\xi}(y)) + d_{\mathbb{T}}(\theta(x), \theta(y)) = d_{0}(x,y). \end{align*} $$

We are left to consider the case that $n(x^{l})$ is eventually constant. It follows that $\theta (x^{l})$ is also. Suppose these values are n and $\theta $ , respectively. It follows that $n(x)=n, \theta (x) = \theta $ and $j>0$ . If the sequence $n(y^{l})$ is unbounded, then, arguing as before for $n(x^{l})$ , we have $j=0$ , which is a contradiction. Hence, the sequences $n(y^{l})$ and $\theta (y^{l})$ are also eventually constant, say with values $n'$ and $\theta '$ , respectively.

As $x^{l}_{i}$ ( $y^{l}_{i}$ ) converges to $x_{i}$ ( $y_{i}$ , respectively), for $1 \leq i \leq n$ ( $1 \leq i \leq n'$ , respectively), we know then that $n(x) = n$ , $\theta (x) = \theta $ , $n(y)=n'$ and $\theta (y) = \theta '$ . Let us first assume that $(n, \theta ) \neq (n', \theta ')$ . In this case, we have, for l sufficiently large,

$$ \begin{align*} d_{k}(x^{l}, y^{l})=d_{G_{\xi}}(\tau_{\xi}(x^{l}),\tau_{\xi}(y^{l}) ) + \vert 2^{-n} - 2^{-n'} \vert + d_{\mathbb{T}}(\theta, \theta'), \end{align*} $$

which clearly converges to

$$ \begin{align*} d_{G_{\xi}}(\tau_{\xi}(x),\tau_{\xi}(y) ) + \vert 2^{-n} - 2^{-n'} \vert + d_{\mathbb{T}}(\theta, \theta') = d_{j}(x, y). \end{align*} $$

The second case to consider is $(n, \theta ) = (n', \theta ')$ . In this case, we have, for l sufficiently large,

It is clear that the $\sigma ^{n}(x^{l}), \sigma ^{n}(y^{l}), l \geq 1,$ and points $\sigma ^{n}(x), \sigma ^{n}(y)$ satisfy the hypotheses for integer where $0 \leq j-1 < k-1$ , so by the induction hypothesis, the sequence above converges to

$$ \begin{align*} & d_{G_{\xi}}(\tau_{\xi}(x),\tau_{\xi}(y) ) - 2^{-2-n} d_{G_{\xi}}(\tau_{\xi}(\sigma^{n}(x)),\tau_{\xi}(\sigma^{n}(y)) ) & \\ &\qquad + 2^{-2-n} d_{j-1}(\sigma^{n}(x), \sigma^{n}(y)) & \\ &\quad= d_{j}(x,y). \\[-3.1pc] \end{align*} $$

We are now ready to unify our metrics $d_{k}$ into a single pseudo-metric.

Let us take a moment to recall (one approach to) taking the completion of a metric space $(X, d)$ (see [Reference Kelley17, p. 196]). A key step is showing that if $x^{l}, y^{l}, l \geq 1,$ are two Cauchy sequences in X with respect to d, then the sequence of real numbers, $d(x^{l}, y^{l}), l \geq 1,$ is also Cauchy and hence convergent. This proof only needs symmetry and the triangle inequality, and works equally well if $d $ is a pseudo-metric.

Let us describe how we will use this. For $x,y$ in $X_{k}^{+}$ , we have the inequality $d_{k}(x,y) \leq 3 d_{G}(x,y)$ . If we choose any sequences $x^{l}, y^{l}, l \geq 1,$ in $X_{k}^{+}$ , converging to $x,y$ in $X_{G}^{+}$ , respectively, then both are Cauchy in $d_{G}$ and hence also in $d_{k}$ . The argument outlined above then implies that $\lim _{l \rightarrow \infty } d_{k}(x^{l},y^{l})$ exists and it can be shown to be independent of the choice of sequences. If $x,y$ are actually in $X_{k}^{+}$ , then this limit agrees with $d_{k}(x,y)$ . We refer to this as an extension of $d_{k}$ . We use this repeatedly in the following theorem.

Proof. For $k \geq 0$ , the space $X_{k}^{+}$ carries two pseudo-metrics, $d_{G}$ and $d_{k}$ . The former dominates the latter (part (2) of Proposition 3.10), at least up to a factor of $3$ . In addition, it is clear that $d_{G_{\xi }}(\tau _{\xi }(x), \tau _{\xi }(y)) \leq d_{k}(x,y)$ , for all $x, y$ in $X_{k}^{+}$ . We also recall from Proposition 3.5 that the closure of $X_{k}^{+}$ in $X_{G}^{+}$ is $\mathrm{Cl}(X_{k}^{+}) = \bigcup _{j=0}^{k} X_{j}^{+}$ . It follows that $d_{k}$ extends to a pseudo-metric on $\mathrm{Cl}(X_{k}^{+})$ , which we denote $\bar {d}_{k}$ , as described above which is also dominated by $3d_{G}$ and bounded below by $d_{G_{\xi }}(\pi _{\xi }(\cdot ),\pi _{\xi }(\cdot ))$ . In addition, it follows from Proposition 3.13 that the restriction of $\bar {d}_{k}$ to $X_{j}^{+} \times X_{j}^{+}$ equals $d_{j}$ , for any $0 \leq j \leq k$ . It also follows that $\bar {d}_{k}$ equals $\bar {d}_{j}$ after extending the latter to $\bigcup _{i=0}^{j} X_{i}^{+}$ .

We may then define a pseudo-metric, $d_{\infty }$ , on $\bigcup _{j=0}^{\infty } X_{j}^{+}$ which is $\bar {d}_{k}$ on $\bigcup _{j=0}^{k} X_{j}^{+}$ . This is also bounded by $3d_{G}$ and so extends continuously to a pseudo-metric d on the closure of $\bigcup _{j=0}^{\infty } X_{j}^{+}$ which is $X_{G}^{+}$ (Proposition 3.5) and is bounded below by $d_{\xi }(\pi _{\xi }(\cdot ),\pi _{\xi }(\cdot ))$ and above by $3 d_{G}$ .

The first and third properties of d in the conclusion are immediate. For the second, we first suppose that $x \sim _{\xi } y$ . It follows from the definition of $\sim _{\xi }$ that $x,y$ lie in $X_{k}^{+}$ , for some $0 \leq k < \infty $ and hence $ d(x,y) = d_{k}(x,y) = 0$ , by Proposition 3.10.

Now assume that $x,y$ are in $X_{G}^{+}$ and $d(x,y) = 0$ . It follows that $\tau _{\xi }(x) = \tau _{\xi }(y)$ which, in turn, implies that $\kappa (x) = \kappa (y)$ by Proposition 3.4. If this is finite, say k, then $x,y$ are in $X_{k}^{+}$ and the conclusion follows from Proposition 3.10. We now assume $\kappa (x) = \infty $ and $\tau _{\xi }(x) = \tau _{\xi }(y)$ also implies that $n(x) = n(y)$ . From Proposition 3.5, we may find sequences, $x^{l}, y^{l} $ in $X_{l}^{+}$ converging to $x,y$ , respectively. This means that

$$ \begin{align*} 0 = d(x,y) = \lim_{l} d(x^{l}, y^{l}) = \lim_{l} d_{l}(x^{l}, y^{l}). \end{align*} $$

There exists $l_{0} \geq 1$ such that, for all $l \geq l_{0}$ , $x_{n} =x^{l}_{n}, y_{n} = y^{l}_{n}$ , for $1 \leq n \leq n(x)$ . From this, it follows that $n(x^{l}) = n(x) = n(y) = n(y^{l})$ and $\theta (x) = \theta (x^{l}), \theta (y) = \theta (y^{l})$ , for $l \geq l_{0}$ . If $\theta (x^{l})$ and $\theta (y^{l})$ are distinct for all l, then

$$ \begin{align*} d_{l}(x^{l}, y^{l}) \geq d_{\mathbb{T}}( \theta(x^{l}), \theta(y^{l})) \geq 2^{-n(x)}, \end{align*} $$

which is a contradiction. It follows that, for some l, we have $\theta (x) = \theta (x^{l}) = \theta (y^{l}) = \theta (y)$ . Together with the fact that $\tau _{\xi }(x_{n}) = \tau _{\xi }(y_{n})$ , this implies that $x_{n} = y_{n}$ , for $ 1 \leq n \leq n(x)$ .

We note that for $l \geq l_{0}$ , $\sigma ^{n(x)}(x^{l}),\sigma ^{n(x)}(y^{l})$ are in $X_{l-1}^{+}$ and we have

After re-indexing the sequence, we can apply the same argument to $\sigma ^{n(x)}(x)$ , $\sigma ^{n(x)}(y)$ , $\sigma ^{n(x)}(x^{l-1})$ , $\sigma ^{n(x)}(y^{l-1})$ , which shows that $x_{n} = y_{n}$ , for $ 1 \leq n \leq n(x) +n(\sigma ^{n(x)})$ . Continuing in this way, we see that $x=y$ , as desired.

Proof. It is a trivial consequence of the definition of $d_{k}$ that the inequality holds with $d_{k}$ on the right-hand side for $x,y$ in $X_{k}^{+}$ , for any $k \geq 1$ . The conclusion holds since $d_{\xi }$ is the common extension of all $d_{k}$ .

For any $k \geq 1$ and $x,y$ in $X_{k}^{+}$ , we have

Again, as $d_{\xi }$ is the common extension of all $d_{k}$ , this also holds for $d_{\xi }$ on $X_{\xi }$ .

As we mentioned earlier, it is an easy matter to see that $X_{\xi }^{+}$ is metrizable. We have gone to considerable trouble to actually produce a metric. The reason is that it satisfies some nice properties and the following is the most important. We remark that the same property holds for the map $\sigma $ on $(X_{G}^{+}, d_{G})$ .

Theorem 3.17. If $x,y$ in $X^{+}_{\xi }$ satisfy $d_{\xi }(x,y) \leq 2^{-2}$ , then

$$ \begin{align*} 2 d_{\xi}(x,y) \leq d_{\xi}(\sigma_{\xi}(x), \sigma_{\xi}(y)) \leq 8 d_{\xi}(x,y). \end{align*} $$

Proof. We will prove the analogous statement for $d_{k}, \sigma $ and $x,y$ in $X^{+}_{k}$ , for some fixed $k \geq 0$ .

We first consider $k=0$ and note that $\sigma (X_{0}^{+}) = X_{0}^{+}$ . For $x,y$ in $X_{0}^{+}$ , we have

$$ \begin{align*} d_{0}(x,y) = d_{G_{\xi}}(\tau_{\xi}(x), \tau_{\xi}(y)) + d_{\mathbb{T}}(\theta(x), \theta(y)). \end{align*} $$

Our hypothesis implies both terms on the right are at most $2^{-2}$ . This implies that $\tau _{\xi }(x_{1}) = \tau _{\xi }(y_{1})$ and hence

$$ \begin{align*} d_{G_{\xi}}(\tau_{\xi}(\sigma(x)), \tau_{\xi}(\sigma(y))) = d_{G_{\xi}}(\sigma\circ \tau_{\xi}(x), \sigma \circ \tau_{\xi}(y)) = 2 d_{G_{\xi}}(\tau_{\xi}(x), \tau_{\xi}(y)) \end{align*} $$

and $d_{\mathbb {T}}(\theta (x), \theta (y)) \leq 2^{-2}$ and hence

$$ \begin{align*} d_{\mathbb{T}}(\theta(\sigma(x)), \theta(\sigma(y))) = d_{\mathbb{T}}(\theta(x)^{2}, \theta(y)^{2}) = 2 d_{\mathbb{T}}(\theta(x), \theta(y)). \end{align*} $$

It follows that $d_{0}(\sigma (x), \sigma (y)) = 2 d_{0}(x,y)$ and we are done.

Now assume $k \geq 1$ . First, we note that $d_{k}(x,y) \leq 2^{-2}$ implies that $d_{G_{\xi }}(\tau _{\xi }(x), \tau _{\xi }(y)) \leq 2^{-2}$ so $\tau _{\xi }(x_{i}) = \tau _{\xi }(y_{i})$ , for $i = 1, 2$ . It follows that $n(x) = n(y)$ or both are at least $3$ .

Our first case is $n(x) =1 = n(y)$ . It follows that $\sigma (x), \sigma (y)$ are both in $X^{+}_{k-1}$ . We have $n(x) =1= n(y)$ and $\theta (x) = \theta (y)=1$ so

The right-hand side is clearly bounded above by

and below by

Next, we consider $n(x) = n(y) = 2$ . Also, $\theta (x), \theta (y)$ are both $ \pm 1 $ . If they are distinct, we have

which is a contradiction. Hence, we have $\theta (x)= \theta (y)$ . So we have $x, y$ are still in $X^{+}_{k}$ , $n(\sigma (x)) = n(x)-1 = n(\sigma (y))$ and $\theta (\sigma (x))= \theta (x)^{2} = \theta (y)^{2} = \theta (\sigma (y))$ and

Finally, we consider the case $n(x), n(y)> 2$ . Here again, we have $n(\sigma (x)) = n(x)-1$ , $n(\sigma (y)) = n(y)-1$ , $\theta (\sigma (x)) = \theta (x)^{2}$ and $\theta (\sigma (y)) = \theta (y)^{2}$ . Let us first assume that $(n(x), \theta (x)) \neq (n(y), \theta (y))$ . We claim that $(n(\sigma (x)), \theta (\sigma (x))) \neq (n(\sigma (y)), \theta (\sigma (y)))$ . If they are equal, then $n(x)-1 = n(y)-1$ so $n(x) = n(y)$ . It follows that $\theta (x) \neq \theta (y)$ , while $\theta (x)^{2} = \theta (y)^{2}$ , which implies $\theta (x) = - \theta (y)$ . This in turn implies that $d_{k}(x,y) \geq d_{\mathbb {T}}(\theta (x), \theta (y)) = 2^{-1}$ , which is a contradiction. From the fact that $2^{-2} \geq d_{k}(x,y) \geq d_{\mathbb {T}}(\theta (x), \theta (y)) $ , we see that $d_{\mathbb {T}}(\theta (x)^{2}, \theta (y)^{2}) = 2 d_{\mathbb {T}}(\theta (x), \theta (y)) $ .

Now we have

The final case is $(n(x), \theta (x)) =(n(y), \theta (y))$ . Here, we have $(n(\sigma (x)), \sigma (\theta (x))) =(n(x)-1, \theta (x)^{2})= (n(\sigma (y)), \sigma (\theta (y)))$ and

We finish this section with an alternate version of the local expanding property which will be useful in the later sections.

Proof. By Proposition 3.5, we may find sequences $x^{k}, k \geq 0,$ and $y^{k}, k \geq 0$ , converging to x and y, respectively, and $x^{k}, y^{k}$ are in $X_{k}^{+}$ . So it suffices to prove the statement for $x,y$ in $X_{k}^{+}$ , for $k \geq 2$ .

We consider three cases separately. The first is when $x_{1} = y_{1}$ is not in $\xi (H^{1})$ . This implies that $n(x) = n(y) = 1$ , $\theta (x) = \theta (y) =1$ and $n( \sigma (x)) = n(\sigma (y)) = n(x) -1$ . It also means that $\sigma (x), \sigma (y)$ are in $X_{k-1}^{+}$ . It follows that

The second case is $x_{1} = y_{1}$ is in $\xi (H^{1})$ , which implies that $n(x), n(y)> 1$ , and $(n(x),\theta (x)) \neq (n(y), \theta (y) )$ . We have $n( \sigma (x))= n(x) -1$ and $n(\sigma (y)) = n(y) -1$ , $\theta (\sigma (x))= \theta (x)^{2}, \theta (\sigma (y))= \theta (y)^{2}$ . It also means that $\sigma (x), \sigma (y)$ are in $X_{k}^{+}$ . We have

The hypothesis that $d_{\xi }(\sigma (x), \sigma (y)) \leq 2^{-1}$ implies that $d_{\mathbb {T}}(\theta (x)^{2}, \theta (y)^{2}) \leq 2^{-1}$ . The fact that $x_{1}=y_{1}$ implies $\varepsilon (x_{1}) = \varepsilon (y_{1})$ . Together, these imply that $ d_{\mathbb {T}}(\theta (x)^{2}, \theta (y)^{2}) = 2d_{\mathbb {T}}(\theta (x), \theta (y)) $ . Putting this in the last line above yields $ d_{\xi }(\sigma (x), \sigma (y)) = 2 d_{\xi }(x,y)$ .

The final case to consider is that $x_{1} = y_{1}$ is in $\xi (H^{1})$ , which implies that $n(x), n(y)> 1$ , and $(n(x),\theta (x)) =(n(y), \theta (y) )$ . Again, we know $\sigma (x), \sigma (y)$ are in $X_{k}^{+}$ . Here, we have $n( \sigma (x))= n(x) -1 = n(\sigma (y)) = n(y) -1$ and $\theta (\sigma (x))= \theta (x)^{2} = \theta (\sigma (y))= \theta (y)^{2}$ . We have

Theorem 3.19. If $x, y$ are in $X_{\xi }^{+}$ with $d_{\xi }(x, \sigma _{\xi }(y)) \leq 2^{-1}$ , then there is z in $X_{\xi }^{+}$ with $d_{\xi }(z,y) \leq 2^{-1} d_{\xi }(x, \sigma _{\xi }(y))$ and $\sigma _{\xi }(z) = x$ . In particular, we have

$$ \begin{align*} X_{\xi}^{+}(\sigma_{\xi}(x), \epsilon) \subseteq \sigma_{\xi}(X_{\xi}^{+}(x, 2^{-1} \epsilon)) \end{align*} $$

and the map $\sigma _{\xi }$ is open.

Proof. Write $x = \pi _{\xi }(x')$ , for some $x'$ in $X_{G}^{+}$ . Let $y'$ be in $X_{G}^{+}$ with $\pi _{\xi }(y') =y$ . As $\epsilon \leq 2^{-3}$ , we know that $d_{G_{\xi }}(\tau _{\xi }(x'), \tau _{\xi }(\sigma (y'))) \leq 2^{-3}$ also. This implies that $\tau _{\xi }(x^{\prime }_{1}) = \tau _{\xi }(y^{\prime }_{2})$ , which in turn implies that $i(x^{\prime }_{1}) = i(y_{2}')=t(y_{1}')$ . It follows that $z' = (y^{\prime }_{1}, x^{\prime }_{1}, x^{\prime }_{2}, \ldots )$ is in $X_{G}^{+}$ . Applying Lemma 3.18 to $z', y'$ gives

$$ \begin{align*} d_{\xi}(y,z) = d_{\xi}(y',z') \leq 2^{-1} d_{\xi}(\sigma(y'), \sigma(z')) = 2^{-1} d_{\xi}(\sigma(y), x).\\[-3.1pc] \end{align*} $$

4 Smale spaces

The results of §3 showed a construction of a compact metric space $(X_{\xi }^{+}, d_{\xi })$ along with a map $\sigma _{\xi }: X_{\xi }^{+} \rightarrow X_{\xi }^{+}$ which is continuous, surjective and open. It also satisfies a local expansiveness condition in Theorem 3.17. Our goal in this section is to replace this system with another where the dynamics is actually a homeomorphism. This is a standard construction via inverse limits. Without giving the precise definition for the moment, this introduces a new component to the space where the dynamics is contracting. In short, we are going to build a Smale space. Two references for Smale spaces are the works of Ruelle [Reference Ruelle30] and Putnam [Reference Putnam24].

Let us begin with recalling the definition of a Smale space. First of all, $(X, d)$ is a compact metric space and $\varphi $ is a homeomorphism of X. We assume that there are constants $\epsilon _{X}> 0, 0 < \unicode{x3bb} < 1$ and a continuous map

$$ \begin{align*} [ \cdot, \cdot] : \{ (x, y) \mid x,y \in X, d(x,y) \leq \epsilon_{X} \} \rightarrow X, \end{align*} $$

which satisfies the following statements:

1. B1 $[x,x] = x$ ;

2. B2 $ [x,[y,z]] = [x,z]$ ;

3. B3 $[[x,y],z] = [x,z]$ ;

4. B4 $[ \varphi (x), \varphi (y)] = \varphi [x,y]$ ,

for all $x, y, z$ in X, whenever both sides of an equation are defined, and finally:

1. C1 if $[x,y] = y$ , then $d(\varphi (x), \varphi (y)) \leq \unicode{x3bb} d(x,y)$ ;

2. C2 if $[x,y] = x$ , then $d(\varphi ^{-1}(x), \varphi ^{-1}(y)) \leq \unicode{x3bb} d(x,y)$ ,

for $x,y$ with $d(x,y) \leq \epsilon _{X}$ .

We say that $(X, d, \varphi )$ is a Smale space.

We now consider our specific case of interest. We will usually work under the standing hypotheses. It seems likely that it is sufficient for G to be irreducible. That is only used in an essential way in Proposition 3.5, but it seems possible that some other hypothesis on $\xi $ may be needed for the construction of the last section to work.

Definition 4.1. Let $G, H, \xi $ satisfy the standing hypotheses. We define $X_{\xi }$ to be the inverse limit of the system

That is,

$$ \begin{align*} X_{\xi} = \{ (x^{0}, x^{1}, \ldots ) \mid x^{n} \in X^{+}_{\xi}, x^{n} = \sigma_{\xi}(x^{n+1}), n \geq 0 \}. \end{align*} $$

We also define $\sigma _{\xi }: X_{\xi } \rightarrow X_{\xi }$ by $\sigma _{\xi }(x)^{n} = \sigma _{\xi }(x^{n})$ , for $n \geq 0$ and $x^{n}, n \geq 1$ in $X_{\xi }$ . It is a simple matter to check that the inverse is given by

$$ \begin{align*} \sigma_{\xi}^{-1}(x)^{n} = x^{n+1}, \quad n \geq 0, \end{align*} $$

for any x in $X_{\xi }$ .

Finally, we define a metric, also denoted $d_{\xi }$ , on $X_{\xi }$ by

$$ \begin{align*} d_{\xi}(x, y) = \sup\{ 2^{-n} d_{\xi}(x^{n}, y^{n}) \mid n \geq 0 \}, \end{align*} $$

for $x= (x^{n})_{ n \geq 0}, y = (y^{n})_{n \geq 0}$ in $X_{\xi }$ .

We remark that if we replace $X_{\xi }^{+}, \sigma _{\xi }$ by $X_{G}^{+}, \sigma _{G}$ , the result is $X_{G}, \sigma _{G}$ in a canonical way by identifying x in $X_{G}$ with the sequence $x^{n} = \chi _{G}^{+}(\sigma ^{-n}(x)) = (x_{1-n}, x_{2-n}, \ldots )$ , for $ n \geq 0$ . Pursuing this remark a little further, it is a fairly simple matter to check that the map which sends x in $X_{G}$ to the sequence $\pi _{\xi }(\chi _{G}^{+}(\sigma ^{-n}(x))), n \geq 0,$ in $X_{\xi }$ is well defined. We denote the map by $\pi _{\xi }$ . It is immediate that $\pi _{\xi } \circ \sigma = \sigma _{\xi } \circ \pi _{\xi } $ .

The system $(X_{G}, d_{G}, \sigma )$ is a Smale space with constants $\epsilon _{X_{G}} = \unicode{x3bb} = 2^{-1}$ and bracket defined as $[x,y] = ( \ldots y_{-1}, y_{0}, x_{1}, \ldots )$ , for $d_{G}(x,y) \leq 2^{-1}$ , which ensures $x_{0}=y_{0}, x_{1} = y_{1}$ .

To see that $(X_{\xi }, d_{\xi }, \sigma _{\xi })$ is a Smale space, we need to define the bracket map. Let $x= (x^{n})_{ n \geq 0}, y = (y^{n})_{n \geq 0}$ be in $X_{\xi }$ and assume that $d_{\xi }(x, y) \leq 2^{-1}$ .

We inductively define $z^{n}, n = 0, 1, 2, 3, \ldots $ in $X_{\xi }^{+}$ satisfying:

1. (1) $d_{\xi }(z^{n}, y^{n}) \leq 2^{-n-1};$

2. (2) $\sigma _{\xi }(z^{n+1}) = z^{n}$ .

Begin with $z^{0} = x^{0}$ . Observe the first condition holds for $n=0$ and

$$ \begin{align*} d_{\xi}(x^{0}, y^{0}) \leq d_{\xi}(x, y) \leq 2^{-1}. \end{align*} $$

To define $z^{n+1}$ , we have $d_{\xi }(z^{n}, \sigma _{\xi }(y^{n+1})) = d_{\xi }(z^{n},y^{n}) \leq 2^{-n-1}$ . By Theorem 3.19, we may find $z^{n+1}$ in $X_{\xi }^{+}$ with $\sigma _{\xi }(z^{n+1})=z^{n}$ and $d_{\xi }(z^{n+1}, y^{n+1}) \leq 2^{-1} d_{\xi }(z^{n},y^{n}) \leq 2^{-n-2}$ . So $z = (z^{0}, z^{1}, \ldots )$ lies in $X_{\xi }$ .

We use the constants $\epsilon _{X_{\xi }} = \unicode{x3bb} = 2^{-1}$ . Notice that $d_{\xi }(x,y) \leq 2^{-1}$ implies that $d_{\xi }(x^{0}, y^{0}) \leq 2^{-1}$ , which ensures that $[x,y]$ is defined.

We will not verify the axioms B1–B4. As for axiom C1, suppose that $z=[x,y]=y$ . It follows that $y^{0}=x^{0}$ and so

$$ \begin{align*} d(\sigma_{\xi}(x), \sigma_{\xi}(y)) & = \sup \{ 2^{-n}d_{\xi}(\sigma_{\xi}(x^{n}), \sigma_{\xi}(y^{n})) \mid n \geq 0 \} \\ & = \sup \{ 2^{-n}d_{\xi}(\sigma_{\xi}(x^{n}), \sigma_{\xi}(y^{n}) ) \mid n \geq 1 \} \\ & = \sup \{ 2^{-n}d_{\xi}(x^{n-1}, y^{n-1}) \mid n \geq 1 \} \\ & = 2^{-1} \sup \{ 2^{-m}d_{\xi}(x^{m}, y^{m}) \mid m \geq 0 \} \\ & = 2^{-1} d_{\xi}(x, y). \end{align*} $$

We also verify axiom C2: if $z = [x,y] = x$ , then we have

$$ \begin{align*} d(\sigma_{\xi}^{-1}(x), \sigma_{\xi}^{-1}(y)) & = \sup \{ 2^{-n}d_{\xi}(x^{n+1}, y^{n+1}) \mid n \geq 0 \} \\ & = \sup \{ 2^{-n} 2^{-1}d_{\xi}(x^{n}, y^{n}) \mid n \geq 0 \} \\ & = 2^{-1} d_{\xi}(x,y). \end{align*} $$

Proof. The first part is easy, as we have already noted.

For the second part, if $d_{G}(x,y) \leq 2^{-3}$ , then $[x,y]$ is defined. In addition, by Corollary 3.15, $d_{\xi }(\pi _{\xi }(x), \pi _{\xi }(y)) \leq 3 \cdot 2^{-3} < 2^{-1}$ , so $[\pi _{\xi }(x), \pi _{\xi }(y)]$ is also defined.

We define, for $n \geq 0$ , $z^{n} = \pi _{\xi }(y_{-n}, \ldots , y_{-1}, x_{0}, x_{1}, \ldots )$ in $X_{\xi }^{+}$ . It is clear that $z^{0} = \pi _{\xi }(\chi _{G}^{+}(x))$ and $\sigma _{\xi }(z^{n}) = z^{n-1}$ , for all $n \geq 1$ . We claim that $d_{\xi }(z^{n}, y^{n}) \leq 2^{-n-1}$ , which we show by induction. The case $n=0$ is simply the fact that $d_{\xi }(\pi _{\xi }(x), \pi _{\xi }(y)) \leq 2^{-1}$ , which we have already established. For any $n \geq 1$ , Lemma 3.18 and the induction hypothesis show that

$$ \begin{align*} d_{\xi}(z^{n}, y^{n}) \leq 2^{-1} d_{\xi}(z^{n-1}, y^{n-1}) \leq 2^{-n-1}. \end{align*} $$

It follows from the definition of the bracket on $X_{\xi }$ that $z = [\pi _{\xi }(x), \pi _{\xi }(y)]$ . However, we have

$$ \begin{align*} (\pi_{\xi}[x,y])^{n} = \pi_{\xi}([x,y]^{n})= \pi_{\xi}(\chi_{G}^{+}(\sigma^{-n}[x,y]) ) = z^{n}, \end{align*} $$

by definition.

For the third part, it is clear that if $x, y$ satisfy conditions (b) or (c), then $ \chi _{G}^{+}(\sigma ^{-n}(x)) \sim _{\xi } \chi _{G}^{+}(\sigma (y))$ , for all integers n, and it follows that $\pi _{\xi }(x)^{n} = \pi _{\xi }(y)^{n}$ , for all integers n.

Conversely, if $\pi _{\xi }(x) = \pi _{\xi }(y)$ , then $\chi _{G}^{+}(\sigma ^{-n}(x)) \sim _{\xi } \chi _{G}^{+}(\sigma ^{-n}(y))$ , for all integers n. According to the definition of $\sim _{\xi }$ , one possibility is that there is a unique path $z^{n}$ in $X_{H}^{+}$ and $i_{n}=0,1$ such that

$$ \begin{align*} \chi_{G}^{+}(\sigma^{-n}(x)) = \xi^{i_{n}}(z^{n}),\quad \chi_{G}^{+}(\sigma^{-n}(y)) = \xi^{1-i_{n}}(z^{n}). \end{align*} $$

Let us first suppose that this holds for all integers n. It follows $i_{n+1} = i_{n}$ and $\sigma (z^{n+1})= z^{n}$ . We then define z in $X_{H}$ by $z_{n}= z^{0}_{n}$ , for all $n \geq 1$ , and $z_{n} = z^{-n-1}_{1}$ , for all $n \leq 0$ , and part (b) holds with $i=i_{1}$ . Otherwise, the condition fails for some integer n. It is a simple matter to check that part (c) holds in this case.

For the last part, if $\pi _{\xi }(x) = \pi _{\xi }(y)$ , then by part (3), x and y are not stably equivalent (that is, right tail-equivalent) unless $x=y$ . This shows that $\pi _{\xi }| X^{s}_{G}(x)$ is injective. The surjectivity follows from the fact that G primitive implies $(X_{G}, \sigma )$ is non-wandering (see [Reference Lind and Marcus18, Proposition 2.2.14]), and [Reference Putnam24, Theorem 2.5.8].

5 Homology

In [Reference Putnam24], a homology theory for (non-wandering) Smale spaces is described. Specifically, given a non-wandering Smale space $(X, d, \varphi )$ and integer n, there are two countable abelian groups, $H^{s}_{n}(X, \varphi )$ and $H^{u}_{n}(X, \varphi )$ . The former invariant is covariant for s-bijective factor maps and contravariant for u-bijective factor maps, while the latter is covariant for u-bijective factor maps and contravariant for s-bijective factor maps. As such, the map $\varphi $ induces a pair of automorphisms of each. The aim of this section is to compute these invariants for the systems $(X_{\xi }, d_{\xi }, \sigma _{\xi })$ we have constructed.

Let us begin with some preliminary notions. If G is any finite directed graph, we may consider the free abelian group on the vertex set $G^{0}$ , denoted $\mathbb {Z} G^{0}$ . This comes with two canonical endomorphisms:

$$ \begin{align*} \gamma_{G}^{s}(v) = \sum_{i(e) = v} t(e), \quad \gamma_{G}^{u}(v) = \sum_{t(e) = v} i(e), \end{align*} $$

for any v in $G^{0}$ . This is needed at various places in the theory and in our proofs below. However, for the statements of the results, we can use the more familiar group $\mathbb {Z}^{G^{0}}$ , which is isomorphic to $\mathbb {Z} G^{0}$ in an obvious way. Under this isomorphism, the map $\gamma _{G}^{s}$ becomes multiplication by the matrix $A_{G}$ , while $\gamma ^{u}_{G}$ is multiplication by its transpose. Then we define $ D^{s}(G)$ as the inductive limit of the sequence:

$$ \begin{align*} \mathbb{Z}^{G^{0}} \stackrel{A_{G}}{\rightarrow} \mathbb{Z}^{G^{0}} \stackrel{A_{G}}{\rightarrow} \mathbb{Z}^{G^{0}} \stackrel{A_{G}}{\rightarrow} \cdots\!. \end{align*} $$

Similarly, the invariant $ D^{u}(G)$ is computed in a similar way, but with the transpose, $A_{G}^{T}$ , replacing $A_{G}$ .

With this notation, we can summarize our results with the following theorem.

Theorem 5.1. Let $G, H, \xi $ satisfy the standing hypotheses. We have

$$ \begin{align*} &H^{s}_{0}(X_{\xi}, \sigma_{\xi}) \cong D^{s}(G),\quad (\sigma_{\xi})_{*}^{-1} \cong A_{G}, \\ &H^{s}_{1}(X_{\xi}, \sigma_{\xi}) \cong D^{s}(H),\quad (\sigma_{\xi})_{*}^{-1} \cong A_{H},\\ &H^{u}_{0}(X_{\xi}, \sigma_{\xi}) \cong D^{u}(G),\quad (\sigma_{\xi})_{*} \cong A_{G}^{T}, \\ &H^{u}_{1}(X_{\xi}, \sigma_{\xi}) \cong D^{u}(H),\quad (\sigma_{\xi})_{*} \cong A_{H}^{T},\\ &H^{s}_{k}(X_{\xi}, \sigma_{\xi}) \cong 0,\quad k \neq 0,1, \\ &H^{u}_{k}(X_{\xi}, \sigma_{\xi}) \cong 0,\quad k \neq 0,1. \end{align*} $$

In the first two lines, we regard $\sigma _{\xi }$ as an s-bijective factor map from $(X_{\xi }, \sigma _{\xi })$ to itself and our description of the induced map on homology is interpreted via the description which precedes it. In the next two lines, we regard $\sigma _{\xi }$ as a u-bijective factor map.

In fact, we will prove only the first, second and fifth parts as the other three are quite similar.

The starting point for computation of the homology theory for a general Smale space, $(X, \varphi )$ , is to find an $s/u$ -bijective pair: Smale spaces $(Y, \psi )$ and $(Z, \zeta )$ such that $Y^{u}(y)$ and $Z^{s}(z)$ are totally disconnected, for all y in Y and z in Z, along with an s-bijective factor map $\pi _{s}: (Y, \psi ) \rightarrow (X, \varphi )$ and a u-bijective factor map $\pi _{u}: (Z, \zeta ) \rightarrow (X, \varphi )$ . Usually, the collection $(Y, \psi , \pi _{s}, Z, \zeta , \pi _{u})$ is written as $\pi $ .

Given such an $s/u$ -bijective pair, one proceeds to define dynamical systems as fibred products: for $L, M \geq 0$ ,

$$ \begin{align*} \Sigma_{L,M}(\pi) = \{ (y^{0}, \ldots, y^{L}, z^{0}, \ldots, z^{M}) \in Y^{L+1} \times Z^{M+1} \mid \pi_{s}(y^{l}) = \pi_{u}(z^{m}), \text{ all } l, m \}. \end{align*} $$

Each of these systems is a shift of finite type and $\Sigma _{L,M}(\pi )$ carries an obvious action of the group $S_{L+1} \times S_{M+1}$ .

The homology $H^{s}_{*}$ is computed from a double complex whose groups are denoted $C^{s}_{\mathcal {Q}, \mathcal {A}}(\pi _{\xi })_{L,M}$ $L, M \geq 0$ . These are obtained by considering the dimension groups $D^{s}(\Sigma _{L,M}(\pi ))$ and taking first subgroups and then quotient groups obtained from the actions of $S_{M+1}$ and $S_{L+1}$ , respectively. This is an analogy with full and alternating complexes in algebraic topology.

The next step is to find a graph which gives a symbolic presentation for $\Sigma _{0,0}(\pi _{\xi }) $ . This requires that, for any $x,y$ in $X_{G}$ satisfying $t(x_{0}) = t(y_{0})$ , then $\pi _{\xi }[x,y] = [ \pi _{\xi }(x), \pi _{\xi }(y)]$ , meaning that both sides are defined. Unfortunately, G will not suffice but this is not a major obstacle.

We review the standard notion of the higher block coding of G (see [Reference Lind and Marcus18, §2.3]). We let $G^{K}$ be the set of all paths of length K in G, for any $K \geq 2$ , and define $i,t :G^{K} \rightarrow G^{K-1}$ by

$$ \begin{align*} i(x_{1}, \ldots, x_{K}) = (x_{1}, \ldots, x_{K-1}), \quad t(x_{1}, \ldots, x_{K}) = (x_{2}, \ldots, x_{K}), \end{align*} $$

for $(x_{1}, \ldots , x_{K})$ in $G^{K}$ . In this way, we interpret $G^{K},G^{K-1},i,t$ as a graph. For any $1 \leq k \leq K$ , the map sending $(x_{1}, \ldots , x_{K})$ in $G^{K}$ to $x_{k}$ induces a homeomorphism between $X_{G^{K}}$ and $X_{G}$ which intertwines the shift maps. It will not be necessary for us to write this map explicitly: we simply regard $X_{G^{K}}$ and $X_{G}$ as equal, in an obvious way.

We use $K=7$ and $k=4$ . The point is that, if $x,y$ are in $X_{G}$ and $t(x_{0}) = t(y_{0})$ , then regarding them as elements in $X_{G^{7}}$ means $x_{[-2,3]} = y_{[-2,3]}$ , which implies $d(x,y) \leq 2^{-3}$ and so part (2) of Theorem 4.3 applies and $G^{7}$ satisfies the conditions of [Reference Putnam24, Definition 2.6.8].

We remark that the map $t: G^{6} \rightarrow G^{0}$ induces an isomorphism between $D^{s}(G^{7})$ and $D^{s}(G)$ .

Our two maps are $\pi _{s} = \pi _{\xi }$ and $\pi _{u}$ being the identity on $X_{\xi }$ . Due to part (3) of Theorem 4.3, we see that $\# \pi _{s}\{ x \} \leq 2 =L_{0}$ and obviously $\# \mathrm {id}^{-1}\{ x \} \leq 1 = M_{0}$ , for all x in $X_{\xi }$ , so by [Reference Putnam24, Theorem 5.1.10], our complex consists of only two non-zero groups, $C^{s}_{\mathcal {Q}, \mathcal {A}}(\pi _{\xi })_{0,0}$ and $C^{s}_{\mathcal {Q}, \mathcal {A}}(\pi _{\xi })_{1,0}$ , and a single homomorphism from the latter to the former. So what are these groups, how do we compute them and what is the homomorphism between them? The first is given already in Theorem 5.2. In the case $L=M=0$ , the permutation groups are both trivial and $C^{s}_{\mathcal {Q}, \mathcal {A}}(\pi _{\xi })_{0,0} = D^{s}(G)$ .

We now come to the second group, $C^{s}_{\mathcal {Q}, \mathcal {A}}(\pi _{\xi })_{1,0}$ , where it will be necessary to use our higher block presentation. The system $\Sigma _{(1,0)}(\pi _{\xi })$ of [Reference Putnam24] consists of pairs

$$ \begin{align*} \Sigma_{(1,0)}(\pi_{\xi}) = \{ (x^{0}, x^{1}) \mid x^{0}, x^{1} \in X_{G}, \pi_{\xi}(x^{0}) = \pi_{\xi}(x^{1}) \}. \end{align*} $$

Fortunately, Theorem 4.3 gives a nice description of these. The fact that our map is regular means that this is the shift of finite type associated with an obvious subgraph of $G^{7}_{1} \subseteq G^{7} \times G^{7}$ , whose vertex set is $G^{6}_{1} \subseteq G^{6} \times G^{6}$ , with obvious maps $i,t$ . (This suppresses the fact that an infinite sequence of pairs can also be seen as a pair of infinite sequences.) This graph is obtained by simply taking pairs in $\Sigma _{(1,0)}(\pi _{\xi })$ and finding all words of length $6$ and $7$ .

We may partition our vertex set $G^{6}_{1} = V_{0} \cup V_{1} \cup \cdots \cup V_{6}$ as follows. First, $V_{0}$ consists of all pairs $(x,x)$ , where x is in $G^{6}$ . Also, $V_{6}$ consists of all pairs $(\xi ^{0}(y), \xi ^{1}(y)), (\xi ^{1}(y), \xi ^{0}(y))$ , where y is in $H^{6}$ . For $ 1 \leq k < 6$ , we let $V_{k}$ be all pairs of the following types:

$$ \begin{align*} (x\xi^{0}(y), x\xi^{1}(y) ), ( x\xi^{1}(y), x\xi^{0}(y) ), \end{align*} $$

where $x $ is in $G^{6-k}$ , y is in $H^{k}$ and $t(x) = \xi ^{0}(i(y))$ , along with

$$ \begin{align*} ( x \xi^{1}(y_{1})\xi^{0}(y_{2})\ldots \xi^{0}(y_{k+1}), & \ x\xi^{0}(y_{1})\xi^{1}(y_{2}) \ldots \xi^{1}(y_{k+1}) ),\\ ( x\xi^{0}(y_{1})\xi^{1}(y_{2}) \ldots \xi^{1}(y_{k+1}), &\ x\xi^{1}(y_{1})\xi^{0}(y_{2}) \ldots \xi^{0}(y_{k+1}) ), \end{align*} $$

where $ x $ is in $G^{n-k-1}$ , y is in $ H^{k+1}$ and $ t(x) = \xi ^{0}(i(y))$ .

There is an analogous partition $G^{7}_{1} = E_{0} \cup \cdots \cup E_{7}$ . Notice that $i(E_{0}) \subseteq V_{0}$ , $i(E_{i}) \subseteq V_{i-1}$ , if $1 \leq i \leq 7$ , $ t(E_{i}) \subseteq V_{i}$ , if $0 \leq i \leq 6$ , and $t(E_{7}) \subseteq V_{6}$ . That is, $V_{0}$ and $V_{6}$ are two components of $G^{7}_{1}$ and the remaining edges move from $V_{0}$ to $V_{6}$ . We consider the free abelian group on $G_{1}^{6}$ with an inductive limit given by $G^{7}_{1}$ .

Observe that the permutation group on $2$ -symbols acts in an obvious manner on all these objects.

As mentioned above, we obtain $C^{s}_{\mathcal {Q}, \mathcal {A}}(\pi _{\xi })_{1,0}$ from $D^{s}(G^{6})$ by taking a subgroup and then a quotient. As $M=0$ , the subgroup is the entire group. The quotient is obtained using the action of $S_{L+1}=S_{2}$ , which we regard as permutations of $0, 1$ . Let $\alpha $ denote the non-trivial element of this group. We mod out by the subgroup generated by vertices v satisfying $v = v \cdot \alpha $ and all elements of the form $v - v\cdot \alpha $ .

The first means that we are removing $V_{0}$ from consideration. Among the remaining vertices, only the ones in $V_{6}$ are the terminus of a path of any length greater than $6$ and it follows that the inductive limit only uses that part. However, when we quotient by $v - v \cdot \alpha $ , the result is clearly isomorphic to the free abelian group on $H^{6}$ , with generating set $ (\xi ^{0}(y), \xi ^{1}(y))$ , $y $ in $H^{6}$ . We conclude that $C^{s}_{\mathcal {Q}, \mathcal {A}}(\pi _{\xi })_{1,0}$ is isomorphic to to the inductive limit of this group under the map induced by $H^{7}$ . We have proved the following lemma.

We are now left to consider the boundary map between the two groups and a key technical point in its computation is the following lemma. Our notation is slightly different from [Reference Putnam24], so we give a self-contained statement here.

Lemma 5.5. The boundary map

$$ \begin{align*} d^{s}_{\mathcal{Q}} : C^{s}_{\mathcal{Q}, \mathcal{A}}(\pi_{\xi})_{1,0} \rightarrow C^{s}_{\mathcal{Q}, \mathcal{A}}(\pi_{\xi})_{0,0} \end{align*} $$

is the zero map.

Proof. Via the isomorphism of Lemma 5.3, this group is generated by classes of the form $ (\xi ^{0}(y), \xi ^{1}(y))$ , $y $ in $H^{6}$ . Using the formula given in [Reference Putnam24, Definitions 4.2.1 and 4.2.10], we have

$$ \begin{align*} d^{s}_{\mathcal{Q}}(\xi^{0}(y), \xi^{1}(y)) = \xi^{0}(y) - \xi^{1}(y), \end{align*} $$

for each y in $H^{6}$ , where the term on the right is interpreted as an element of $\mathbb {Z} G^{6}$ . We know that the map t induces an isomorphism between $D^{s}(G^{7})$ and $D^{s}(G)$ and we have

$$ \begin{align*} t( \xi^{0}(y) - \xi^{1}(y) ) = t(\xi^{0}(y)) - t(\xi^{1}(y)) = 0, \end{align*} $$

by hypothesis (H0).

The computations of the homology groups, $H^{s}_{*}(X_{\xi }, \sigma _{\xi })$ , as summarized in Theorem 5.1, follow from Theorem 5.2 and Lemmas 5.3 and 5.5.

Let us mention an alternate proof of these results. This is based on two sets of results. The first are those of Proietti and Yamashita [Reference Proietti and Yamashita21, Reference Proietti and Yamashita22], who show that the Smale space homology here agrees with the groupoid homology as studied by Matui. Further, Matui has recently adapted the results which are used in the next section for the K-theory of the $C^{*}$ -algebras to the case of groupoid homology: see [Reference Matui19]. These require the unit space to be totally disconnected and so only apply to parts (1), (2) and (5) of Theorem 5.1.

6 Groupoids, $C^*$ -algebras and K-theory

The goal of this section is to describe the stable and unstable equivalence relations (or groupoids) of our Smale space, $(X_{\xi }, d_{\xi }, \sigma _{\xi })$ , their associated $C^{*}$ -algebras and their K-theory.

We begin with a short review of what is involved for general Smale space. For any Smale space, $(X, d, \varphi )$ , two points $x,y$ in X are stably equivalent if

$$ \begin{align*} \lim_{n \rightarrow + \infty} d(\varphi^{n}(x), \varphi^{n}(y)) = 0. \end{align*} $$

Unstable equivalence is defined by simply replacing both occurrences of $\varphi $ by $\varphi ^{-1}$ . We define

$$ \begin{align*} G^{s}(X, \varphi) & = \{ (x,y) \in X \times X \mid \lim_{n \rightarrow +\infty} d(\varphi^{n}(x), \varphi^{n}(y)) = 0 \}, \\ G^{u}(X, \varphi) & = \{ (x,y) \in X \times X \mid \lim_{n \rightarrow +\infty} d(\varphi^{-n}(x), \varphi^{-n}(y)) = 0 \}. \end{align*} $$

Each is an equivalence relation and has a natural topology. It will not be necessary to describe this here in detail, but we refer the reader to [Reference Putnam23]. In addition, if $(X, \varphi )$ is non-wandering, then, as groupoids, each has a Haar system. Again, we do not need a detailed description. For any x in X, we let $X^{s}(x)$ and $X^{u}(x)$ be the equivalence classes of x in the two equivalence relations. Each carries a natural topology which is finer than the relative topology of X but makes each locally compact Hausdorff.

Generally speaking, étale groupoids [Reference Renault27] are much easier to deal with and, thanks to the work of Muhly, Renault and Williams [Reference Muhly, Renault and Williams20], our groupoids are equivalent to étale groupoids by reducing to an abstract transversal. With the slightly stronger hypothesis that $(X, \varphi )$ is irreducible, we select a finite set P with the property that $\varphi (P)=P$ (that is, consisting of periodic points), we let

$$ \begin{align*} X^{u}(P) = \bigcup_{p \in P} X^{u}(p), \quad X^{s}(P) = \bigcup_{p \in P} X^{s}(p) \end{align*} $$

and these function as natural transversals to $G^{s}(X, \varphi ), G^{u}(X, \varphi )$ , respectively. Note that these spaces are given locally compact topologies which are finer than the relative topology of X. Let us give a brief description of the former. First, the collection of local unstable sets provides a topology for the unit space of $G^{s}(X, \varphi )$ , which is $X^{u}(P)$ . Furthermore, if x and y are in the space and are stably equivalent, there is a positive integer n such that $\varphi ^{n}(x)$ and $\varphi ^{n}(y)$ lie in the same local stable set. It follows that the function $z \rightarrow \varphi ^{-n}[\varphi ^{n}(z), \varphi ^{n}(y)]$ is defined on some local unstable set of x and maps each z to a point stably equivalent to z. This is a typical open set in the base for the topology. We refer the reader to [Reference Putnam and Spielberg26].

We denote the reduction to these transversals by

$$ \begin{align*} G^{s}(X, \varphi, P) & = G^{s}(X, \varphi) \cap ( X^{u}(P) \times X^{u}(P) ), \\ G^{u}(X, \varphi, P) & = G^{u}(X, \varphi) \cap ( X^{s}(P) \times X^{s}(P) ). \end{align*} $$

Each is an étale equivalence relation. Different choices of P yield equivalent groupoids. In [Reference Putnam and Spielberg26], it is shown that these groupoids are amenable, so we do not need to make a distinction between the full and reduced $C^{*}$ -algebras. We define

$$ \begin{align*} S(X, \varphi, P) & = C^{*}(G^{s}(X, \varphi, P) ), \\ U(X, \varphi, P) & = C^{*}(G^{u}(X, \varphi, P) ). \end{align*} $$

The map which sends a continuous function of compact support, f, on either of the two groupoids, to $f \circ (\varphi \times \varphi )^{-1}$ extends to an automorphism of the respective $C^{*}$ -algebra, which is denoted by $\varphi $ . The Ruelle $C^{*}$ -algebras are defined as the associated crossed-products:

$$ \begin{align*} R^{s}(X, \varphi, P) & = S(X, \varphi, P) \rtimes_{\varphi} \mathbb{Z}, \\ R^{u}(X, \varphi, P) & = U(X, \varphi, P) \rtimes_{\varphi} \mathbb{Z}. \end{align*} $$

Let us review some basic properties of these $C^{*}$ -algebras. If $(X, \varphi )$ is mixing, then $S(X, \varphi ,P)$ and $U(X, \varphi , P)$ are separable, simple, finite, stable, satisfy the universal coef ficient theorem (UCT) [Reference Putnam and Spielberg26] and have finite nuclear dimension [Reference Deeley and Strung9]. If $(X, \varphi )$ is irreducible, then $R^{s}(X, \varphi , P) $ and $R^{u}(X, \varphi , P)$ are separable, simple, purely infinite, stable, satisfy the UCT [Reference Putnam and Spielberg26] and have finite nuclear dimension [Reference Deeley and Strung9]. All of these $C^{*}$ -algebras come under the Elliott classification scheme. (We include a small remark: the results of [Reference Deeley and Strung9] need the hypothesis that the $C^{*}$ -algebras contain a projection. This was later shown to hold in general [Reference Deeley, Goffeng and Yashinski10], but, in our case, we will explicitly provide clopen subsets of the unit spaces of both $G^{s}(X_{\xi }, \sigma _{\xi }, P_{\xi })$ and $G^{s}(X_{\xi }, \sigma _{\xi }, P_{\xi })$ which show this holds.)

In our particular case for $(X_{\xi }, d_{\xi }, \sigma _{\xi })$ , we can actually go a little further and the first step is to make a particular choice for P, as we describe.

Under the standing hypotheses, we select a periodic point as follows. We have already observed that $G^{1}-\xi (H^{1})$ is also primitive. We select a cycle $C^{1} \subseteq G^{1}-\xi (H^{1})$ of minimal length. This implies that $t|_{ C^{1}}$ is injective. We let $C^{0} = t(C^{1})$ . We let $P \subseteq X_{G}$ be the finite set of infinite paths which simply repeat the cycle $C^{1}$ . We note that an element p of P is uniquely determined in P by $i(p_{1})$ in $C^{0}$ . Also notice that $\pi _{\xi }$ is a bijection from P to $P_{\xi } = \pi _{\xi }(P)$ from part (3) of Theorem 4.3.

The s-bijective map $\pi _{\xi }:X_{G} \rightarrow X_{\xi }$ induces maps at the level of groupoids and also $C^{*}$ -algebras:

$$ \begin{align*} &(\pi_{\xi})^{*} : S(X_{\xi}, \sigma_{\xi}, P_{\xi}) \rightarrow S(X_{G}, \sigma, P), \\ & (\pi_{\xi})_{*} : U(X_{G}, \sigma, P) \rightarrow U(X_{\xi}, \sigma_{\xi}, P_{\xi}). \end{align*} $$

The $C^{*}$ -algebras $S(X_{G}, \sigma , P)$ and $U(X_{G}, \sigma , P)$ are both AF-algebras, their K-zero groups are $D^{u}(G)$ and $D^{s}(G)$ , respectively, while their $K_{1}$ -groups are trivial.

We now state the two main results of this section. Of course, they could easily be assembled into a single result, but we separate them as the proofs are rather long and also rather different, so they will be divided into two subsections.

Let us mention one other groupoid, without going into great detail. It follows from Theorems 3.17 and 3.19 that the map $\sigma _{\xi }$ of $X_{\xi }^{+}$ is locally expanding and also a local homeomorphism. We can associate to it its Deaconu–Renault groupoid, as follows. For each $x,y$ in $X_{\xi }^{+}$ and positive integers $m,n$ satisfying $\sigma _{\xi }^{m}(x) = \sigma _{\xi }^{n}(y)$ , we consider the triple $(x, m-n, y)$ . The set of all such triples is a groupoid with the product $(x, k, y)\cdot (x', k', y')$ defined when $y=x'$ and result $(x, k+k', y')$ . It can be given a topology which is étale [Reference Deaconu8]. The associated groupoid $C^{*}$ -algebra has the nice feature of being unital, and it is also equivalent to $R^{u}(X_{\xi }, \sigma _{\xi }, P_{\xi })$ .

The result of our K-theory computations has an interesting consequence that our construction exhausts all possible Ruelle algebras from mixing Smale spaces, as we describe below.

We begin with some simple observations. Let $(X, \varphi , d)$ be a mixing Smale space and Q be a finite $\varphi $ -invariant subset of X. The $C^{*}$ -algebras $R^{s}(X, \varphi , Q)$ and $R^{u}(X, \varphi , Q)$ are Spanier–Whitehead duals of each other (see [Reference Kaminker, Putnam and Whittaker15, Definition 4.1 and Theorem 1.1]). As noted in [Reference Kaminker, Putnam and Whittaker15, §4], this implies that their K-groups are finitely generated. Proietti and Yamashita have recently shown that $K_{*}(S(X, \varphi ))$ and $K_{*}(U(X, \varphi ))$ are finite rank ([Reference Proietti and Yamashita22, Theorem 5.1]) and, as explained in [Reference Kaminker, Putnam and Whittaker15, §4], this implies that $K_{0}(R^{s}(X, \varphi , Q))$ and $K_{1}(R^{s}(X, \varphi , Q))$ have the same rank. Putting all of this together, we conclude that there is an integer $k \geq 0$ and finite abelian groups $G_{0}$ and $G_{1}$ such that