2.1 Odometers and Toeplitz shifts

Given a sequence $(\ell _n)$ of natural numbers, we work with the group

$$ \begin{align*}\mathbb{Z}_{(\ell_n)} :=\prod_{n} \mathbb{Z}/\ell_n\mathbb{Z},\end{align*} $$

where the group operation is given by coordinate-wise addition with carry. For a detailed exposition of equivalent definitions of $\mathbb {Z}_{(\ell _n)}$ , we refer the reader to [Reference Downarowicz10]. Endowed with the product topology over the discrete topology on each $\mathbb {Z}/\ell _n\mathbb {Z}$ , the group $\mathbb {Z}_{(\ell _n)}$ is a compact metrizable topological group, where the unit $z=\ldots 001$ , which we simply write as $z=1$ , is a topological generator. We write elements $(z_n)$ of $\mathbb {Z}_{(\ell _n)}$ as left-infinite sequences $\ldots z_2z_1$ , where $z_n\in \mathbb {Z}/\ell _n\mathbb {Z}$ , so that addition in $\mathbb {Z}_{(\ell _n)}$ has the carries propagating to the left as usual in $\mathbb {Z}$ . If $\ell _n=\ell $ is constant, then $\mathbb {Z}_{(\ell _n)}= \mathbb {Z}_\ell $ is the classical ring of $\ell $ -adic integers.

With the above notation, an odometer is a dynamical system $(\mathcal Z, +1)$ , where $\mathcal Z=\mathbb {Z}_{(\ell _n)}$ for some sequence $(\ell _n)$ . A Toeplitz shift is a symbolic shift $(X,\sigma )$ , $X\subseteq \mathcal A^{\mathbb {Z}}$ with $\mathcal A$ finite, which is an almost automorphic extension of an odometer and hence minimal.

2.2 Constant length substitutions

A special class of almost automorphic extensions of odometers is the class of primitive aperiodic constant length substitutions which possess a coincidence. Since we illustrate our theory mainly with examples from this class, and because the proof of our main result simplifies for this class, we give the reader both a brief exposition of substitutions and also a flavour of our main result for this important class of almost automorphic extensions.

Let ${\mathcal A}$ be a finite set, referred to as an alphabet. A substitution of (constant) length $\ell $ over ${\mathcal A}$ is a map $\theta :\mathcal A\rightarrow \mathcal A^{\ell }.$ We can write such a substitution as follows: there are $\ell $ maps $\theta _i:\mathcal A \rightarrow \mathcal A$ , $0\leq i \leq \ell -1$ such that $\theta (a) = \theta _0(a)\mid \cdots \mid \theta _{\ell -1}(a)$ for all $a\in \mathcal A$ , where $\mid $ is to separate the concatenated letters.

We use concatenation to extend $\theta $ to a map on finite and infinite words in $\mathcal A$ . We say that $\theta $ is primitive if there is some $k\in \mathbb {N}$ such that for any $a,a'\in \mathcal A$ , the word $\theta ^k(a)$ contains at least one occurrence of $a'$ . We say that a finite word is allowed for $\theta $ if it appears as a subword in some $\theta ^k(a)$ , $a\in \mathcal A, k\in \mathbb {N}$ .

Let $X_\theta \subseteq {\mathcal A}^{\mathbb {Z}}$ be the set of bi-infinite sequences all of whose finite subwords are allowed for $\theta $ . Then $( X_\theta , \sigma )$ is the substitution shift defined by $\theta $ . Primitivity of $\theta $ implies that $(X_\theta ,\sigma )$ is minimal. We say that a primitive substitution is aperiodic if $X_\theta $ does not comprise $\sigma $ -periodic sequences. This is the case if and only if $X_\theta $ is an infinite space.

The shift $(X_\theta ,\sigma )$ of a primitive aperiodic substitution of constant length $\ell $ factors onto the odometer $(\mathbb {Z}_\ell ,+1)$ . Indeed, for any $n\geq 1$ , the space $X_\theta $ can be partitioned into $\ell ^n$ clopen subsets $\sigma ^i(\theta ^n(X_\theta ))$ , $i=0,\ldots ,\ell ^n-1$ , and the factor map is given by $X_\theta \ni x\mapsto \ldots z_2 z_1\in \mathbb {Z}_\ell $ , where $z_n$ is the unique i such that $x\in \sigma ^i(\theta ^n(X_\theta ))$ [Reference Dekking7]. The maximal equicontinuous factor of $(X_\theta ,\sigma )$ is therefore a covering of $(\mathbb {Z}_\ell ,+1)$ and the degree of this covering is called the height of the substitution. The height h is always finite. Given $\theta $ , there is a primitive aperiodic substitution $\theta '$ , referred to as the pure base of $\theta $ , which is of the same length $\ell $ , has height $1$ and is such that $(X_\theta ,\sigma )$ is a $\mathbb {Z}/h\mathbb {Z}$ -suspension over $(X_{\theta '},\sigma )$ . That is, $X_\theta \cong X_{\theta '}\times \mathbb {Z}/{\sim} $ with $(x,n+h)\sim (\sigma (x),n)$ and the action is induced by $\mathrm {id}\times (+1)$ . There is an explicit construction of $\theta '$ which, in fact, equals $\theta $ if $h=1$ . Clearly, the maximal equicontinuous factor of the pure base system is $(\mathbb {Z}_\ell ,+1)$ . For all details, see [Reference Dekking7].

Let $\theta $ have as pure base the substitution $\theta '$ defined on the alphabet $\mathcal A'$ . We say that $\theta $ has a coincidence if for some $k\in \mathbb {N}$ and some $i_1, \ldots ,i_k\in \{0,\ldots ,\ell -1\}$ , we have $|\theta ^{\prime }_{i_1} \ldots \theta ^{\prime }_{i_k}(\mathcal A')|=1$ . The importance of this notion lies in the theorem of Dekking stating that the substitution shift $( X_\theta , \sigma )$ is almost automorphic if and only if $\theta $ has a coincidence [Reference Dekking7].

We shall see below that the question of whether $(X_\theta ,\sigma )$ is tame or not is governed by the cardinality of the set of orbits of singular points in the maximal equicontinuous factor of $(X_\theta ,\sigma )$ . Since $X_\theta \cong X_{\theta '}\times \mathbb {Z}/{\sim} $ , the orbits of singular points in the maximal equicontinuous factor of $(X_\theta ,\sigma )$ are in one-to-one correspondence with the orbits of singular points in the maximal equicontinuous factor of the pure base system $(X_{\theta '},\sigma )$ . We may therefore just determine the cardinality of the latter. There is an effective procedure which achieves this [Reference Coven, Quas and Yassawi5]. We recapitulate a slightly modified version here in the only case which concerns us, which is when $\theta $ has a coincidence and, by going over to the pure base of the substitution if needed, its height is $1$ .

Consider the graph $\mathcal G_\theta $ whose vertices are the sets

$$ \begin{align*}\{\mathcal A\}\cup \{A:=\theta_{w_1}\cdots \theta_{w_k}(\mathcal A)\colon k\ge 1,\; w_1,\ldots,w_k\in \{0,1,\ldots,\ell-1\}, \mbox{ and } |A|>1\}\end{align*} $$

and whose edges are defined as follows: if A and B are vertices in $\mathcal G_\theta $ , then there is an oriented edge from B to A, labelled i, if and only if $\theta _i(A)=B$ . An infinite path in $\mathcal G_\theta $ defines a point $\ldots z_2z_1\in \mathbb {Z}_\ell $ , where $z_i$ is the label of the ith edge in the path.

We define $\hat \Sigma _\theta $ to be the set of all sequences $(z_i)\in \mathbb {Z}_\ell $ obtained as above from infinite paths in $\mathcal G_\theta $ . If $\theta $ is minimal, aperiodic and has a coincidence, then $\hat \Sigma _\theta $ is non-empty. The $\mathbb {Z}$ -orbit of $\hat \Sigma _\theta $ under $+$ 1 equals $ \{ z\in \mathbb {Z}_\ell : |\pi ^{-1}(z)|>1\}$ , which is the set of singular points in $\mathbb {Z}_\ell $ . This set is a proper subset of $\mathbb {Z}_\ell $ , as $(X_\theta ,\sigma )$ is almost automorphic.

Recall that a cycle on an oriented graph is a finite path which is closed and minimal in the sense that it is not the concatenation of smaller closed paths.

Let us anticipate the following important consequence of Lemma 2.1 combined with Theorem 2.18 and the discussion in §2.3.4.

Example A. Let $\theta $ be the substitution

$$ \begin{align*}\begin{array}{c c l} a & \mapsto & aaca\\ b & \mapsto & abba \\ c& \mapsto & aaba, \end{array} \end{align*} $$

on the alphabet ${\mathcal A}=\{a,b,c\}$ . It is primitive, aperiodic and has trivial height. Its graph $\mathcal G_\theta $ is depicted on the left-hand side in Figure 1. Note that $\{ a,c\}$ is not a vertex in $\mathcal G_\theta $ because it cannot be expressed as $\{ a,c\}= \theta _{w_1}\cdots \theta _{w_k}(\mathcal A)$ for any word $w_1 \cdots w_k$ . Here, $\mathcal G_\theta $ has two different cycles at $\{a,b\}$ , so by Theorem 2.2, $(X_\theta ,\sigma )$ is non-tame.

Figure 1 The graph $\mathcal G_\theta $ for the substitutions from Example A (left) and Example B (right). We used blue dotted and violet dashed lines for better comparison with Figure 2.

Example B. We modify slightly the above example and define $\theta $ as

$$ \begin{align*}\begin{array}{c c l} a &\mapsto & aaca\\ b & \mapsto & abba \\ c& \mapsto & acba, \end{array} \end{align*} $$

on the alphabet ${\mathcal A}=\{a,b,c\}$ . It still is primitive, aperiodic and has trivial height. The graph $\mathcal G_{\theta }$ is shown on the right in Figure 1, and as it has only one cycle about any vertex, by Theorem 2.2, $(X_\theta ,\sigma )$ is tame.

2.3 The Bratteli–Vershik representation of a Toeplitz shift

In this section, we briefly discuss the Bratteli–Vershik representation of Toeplitz shifts. As Bratteli–Vershik systems are well documented in the literature, we keep this to a minimum. Interested readers may consult classical references on Bratteli–Vershik systems, such as [Reference Herman, Putnam and Skau29]. In particular, since we will only be concerned with Bratteli–Vershik systems that are conjugate to Toeplitz shifts, we refer the reader to the work by Gjerde and Johansen [Reference Gjerde and Johansen23]; unless stated otherwise, we adopt the latter’s notational conventions.

2.3.2 Toeplitz Bratteli–Vershik systems

We now focus on Bratteli diagrams which have the equal path number property. Here, if $v\in V_n$ , then $|r^{-1}(v)|=\ell _n$ . Recall that a Bratteli–Vershik system is expansive if and only if there is $k\in \mathbb {N}$ such that for distinct $x,y\in X_B$ , the head of length k of $\phi _\omega ^n(x)$ differs from that of $\phi _\omega ^n(y)$ for some $n\in \mathbb {Z}$ . Downarowicz and Maass [Reference Downarowicz and Maass14] show that a simple properly ordered Bratteli–Vershik system with finite topological rank is expansive if and only if its topological rank is strictly larger than $1$ . A simple properly ordered Bratteli–Vershik system with topological rank $1$ is an odometer.

Theorem 2.4. [Reference Gjerde and Johansen23]

The family of expansive, simple, properly ordered Bratteli–Vershik systems with the equal path number property coincides with the family of Toeplitz shifts up to conjugacy.

In view of this result, we call an expansive simple properly ordered Bratteli–Vershik system with the equal path number property a Toeplitz Bratteli–Vershik system. Moreover, we say that a Toeplitz shift has finite Toeplitz rank if it is conjugate to a Toeplitz Bratteli–Vershik system which has finite rank. Note that having finite Toeplitz rank is stronger than having a Toeplitz shift with finite topological rank as we cannot rule out that a Toeplitz system with infinite Toeplitz rank has a Bratteli–Vershik representation of finite rank but without the equal path number property.

Lemma 2.5. [Reference Gjerde and Johansen23]

Let $(X_B,\varphi _\omega )$ be a Toeplitz Bratteli–Vershik system with characteristic sequence $(\ell _n)$ . The level-wise application of the edge order map

$$ \begin{align*} \omega :(X_B,\varphi_\omega) \to (\mathbb{Z}_{(\ell_n)},+1), \omega((\gamma_n)) = (\omega(\gamma_n))\end{align*} $$

is a factor map to the maximal equicontinuous factor.

For Bratteli diagrams that have the equal path number property, it is standard to describe the ordering $\omega $ using a sequence of constant length morphisms. To describe the ordering of the edge set $E_n$ between $V_n$ and $V_{n+1}$ , we use the morphism

(2.1) $$ \begin{align} \theta^{(n)}:V_{n+1}\rightarrow V_n^{\ell_n}\end{align} $$

which, when written as a concatenation of maps $\theta ^{(n)}=\theta ^{(n)}_0|\cdots |\theta ^{(n)}_{\ell _n -1}$ , where $\theta ^{(n)}_i:V_{n+1}\rightarrow V_n$ (similarly as in §2.2), is given by

$$ \begin{align*}\theta^{(n)}_i(v) = s(e), \ \mbox{ with }\; e\in \omega^{-1}(i)\cap r^{-1}(v) ,\end{align*} $$

that is, the ith morphism reads the source of the unique edge with range v and label i.

If $(B,\omega )$ has the equal path number property and $B'=(V',E')$ is the telescoping of B to levels $(n_k)$ , then $B'$ also has the equal path number property, and the corresponding morphism

(2.2) $$ \begin{align} \theta^{'(k)}:V_{k+1}' \rightarrow {V_k'}^{ \ell^{\prime}_k} \end{align} $$

is given by the composition

$$ \begin{align*} \theta^{'(k)}= \theta^{(n_k)} \cdots \theta^{(n_{k+1}-1)}. \end{align*} $$

We can again write $\theta ^{'(k)}$ as a succession of maps $\theta ^{'(k)}_i:V_{k+1}'\rightarrow V_k'$ which are each a composition of maps $\theta ^{(n)}_{i_n}:V_{n+1}\to V_n$ , $n_k\leq n<n_{k+1}$ , where the labels $i_n$ are those of the edges in $(B,\omega )$ which constitute the edge $e\in E^{\prime }_k$ with label i. More precisely,

$$ \begin{align*} \theta^{'(k)}_{i} = \theta^{(n_k)}_{i_{n_k}} \cdots \theta^{(n_{k+1}-1)}_{i_{n_{k+1}-1}} \quad \text{where } i = \sum_{n=n_k}^{n_{k+1}-1} i_n \prod_{m=n_k}^{n-1} \ell_{m} \end{align*} $$

(with the understanding that $\prod _{m=n_k}^{n_k-1} \ell _{m}=1$ ). For instance, if we telescope level n with level $n+1$ , we get ${\theta '}^{(n)} = \theta ^{(n)} \theta ^{(n+1)}$ and

$$ \begin{align*} {\theta'}_{i+j\ell_n}^{(n)} = \theta^{(n)}_i\theta^{(n+1)}_j \quad \text{for } 0\leq i < \ell_{n},\; 0\leq j < \ell_{n+1}. \end{align*} $$

By telescoping if necessary, we can assume that $\ell _n>1$ for each n. For otherwise, $\ell _n=1$ for almost all n and this implies that $X_B$ is not a Cantor space.

2.3.3 Shift interpretation

There is a strong connection between expansive Bratteli– Vershik systems and shifts [Reference Downarowicz and Maass14]. Suppose $(X_B,\varphi _\omega )$ is expansive and $k\in \mathbb {N}$ is such that the heads of length k suffice to separate orbits of distinct elements in $X_B$ . Given a path $x\in X_B$ , associate the bi-infinite sequence whose nth entry consists of the head of length k of $\varphi _{\omega }^n(x)$ . Let $(X_k,\sigma )$ be the shift whose space consists of all such sequences of length-k heads. Then $(X_k,\sigma )$ and $(X_B,\phi _\omega )$ are conjugate. By telescoping the diagram to the kth level, we may assume that $k = 1$ . Given a Toeplitz Bratteli–Vershik system, we may (and will throughout this article) therefore assume that

(2.3) $$ \begin{align} p_1:(X_B,\varphi_\omega) \rightarrow (X_1,\sigma),\quad (\gamma_\ell)_{\ell\geq 0} \mapsto {((\varphi^n_\omega(\gamma))_0)}_{n\in \mathbb{Z}} \end{align} $$

is a conjugacy.

If there is only one edge between the top vertex $v_0$ and each vertex in $V_1$ , then the range map r is a bijection between $E_0$ and $V_1$ so that $(X_B,\phi _\omega )$ is conjugate to a shift over the alphabet $V_1$ . We denote the respective conjugacy also by r. Observe that this situation can always be enforced by insertion of an extra level. Namely, we introduce an intermediate vertex set $\mathcal V$ between $V_0$ and $V_1$ , which is in one-to-one correspondence with $E_0$ . We introduce one edge from $v_0$ to each vertex of $\mathcal V$ and then for each $e\in E_0=\mathcal V$ , an edge with source e and range $r(e)\in V_1$ with the same order label as e. Clearly, the resulting Bratteli–Vershik system is topologically conjugate to the old one. For Toeplitz Bratteli–Vershik systems, this means that $\ell _0=1$ .

The following lemma is elementary to verify; its proof follows from the built-in recognizability of Bratteli–Vershik systems, by which we mean that for each n, the towers defined by the first n levels of B form a partition of $p_1(X_B)$ , where $p_1$ is defined in equation (2.3) and which we assume, without loss of generality, to be a conjugacy. Given a sequence $(x_n)$ and $n<m$ , we denote by $x_{[n,m[}$ the finite word $x_n x_{n+1}\cdots x_{m-1}$ .

Lemma 2.6. Let $(X_B,\varphi _\omega )$ be a Toeplitz Bratteli–Vershik system with characteristic sequence $(\ell _n)$ where $\ell _0=1$ . Set $\ell ^{(n)} = \prod _{k=0}^{n-1}\ell _k$ . For all $\gamma \in X_B$ and $n\in \mathbb {N}$ , we have

$$ \begin{align*} r\circ p_1(\gamma)_{[-z^{(n)},\ell^{(n)}-z^{(n)}[} = \theta^{(1)}\cdots \theta^{(n)}\circ r(\gamma_{n}),\end{align*} $$

where $z^{(n)} = \sum _{k=0}^{n-1} \omega (\gamma _k) \ell ^{(k)}$ . In particular,

(2.4) $$ \begin{align} \{x_0 : x\in r\circ p_1 (\omega^{-1}(z))\} \subseteq \bigcap_n \theta^{(1)}_{z_1} \cdots \theta^{(n)}_{z_n} (V_{n+1}). \end{align} $$

2.4 The extended Bratteli diagram

In this section, we introduce the notion of the extended Bratteli diagram and its essential thickness, concepts which are fundamental for the proofs of Theorems 1.4 and 1.1. The extended Bratteli diagram can be seen as a generalization of the graph $\mathcal G_\theta $ , introduced in §2 for constant length substitution shifts.

Identifying singleton sets with the element they contain, we see that $\tilde B$ contains B as a sub-diagram. The labelling of the edges defines an order on $\tilde B$ which extends the order on B. We also consider the space $X_{\tilde B}$ of infinite paths over $\tilde B$ starting at the top vertex $v_0$ . Clearly, $X_{\tilde B}$ contains $X_B$ . The edge order map $\omega $ from Lemma 2.5 extends to a map from $X_{\tilde B}$ to the maximal equicontinuous factor $\mathbb {Z}_{\ell _n}$ which we denote by the same letter $\omega $ . Due to the lack of property (4), the vertices in the extended diagram need not to have any outgoing edges. Infinite paths ignore such vertices and we call vertices extendable if they are traversed by a path in $X_{\tilde B}$ .

To a path $\gamma $ in $X_{\tilde B}$ , we associate the sequence of maps

(2.5) $$ \begin{align} {\theta^{({n})}_{\gamma}}:={\theta^{({n})}_{\omega(\gamma)_n }}: \tilde V_{n+1}\to \tilde V_n \end{align} $$

and the sequence of subsets $A_n:=s(\gamma ^{(n)}) \subseteq V_n$ . Then ${\theta ^{({n})}_{\omega (\gamma )_n }}(A_{n+1})=A_{n}$ . Recall that we can arrange for $\ell _0=1$ in the characteristic sequence of the original Toeplitz Bratteli system. This implies that in the extended Bratteli diagram, the top vertex is linked to any vertex of $\tilde V_1$ by exactly one edge.

We remark that in the case where the ordered diagram B is stationary, that is, $E_n=E_1$ and $\theta ^{(n)}=\theta ^{(1)}$ for $n\geq 1$ , then the graph $\mathcal G_{\theta ^{(1)}}$ defined in §2 is an abbreviated form of the extended Bratteli diagram $(\tilde B ,\tilde \omega )$ . The extended Bratteli diagram will also be stationary and so can be described by the edge and order structure of its first level. The only other difference is that in $\mathcal G_{\theta ^{(1)}}$ , we chose to exclude vertices indexing one-element sets, as $\mathcal G_{\theta ^{(1)}}$ is only to identify the singular fibres.

Note that since $|{\theta ^{({n})}_{i}}(A)|\leq |A|$ for each n and i, a path in $X_{\tilde B}$ must pass through vertices of non-decreasing cardinality. This motivates the following definition.

If the diagram has finite rank, then there are no paths with infinite thickness. We denote by $X_{\tilde B}^{k}$ the infinite paths of thickness $k\in \mathbb {N}\cup \{\infty \}$ . Note that the sub-diagram $X_{\tilde B}^{1}$ corresponds to the original path space $X_B$ .

The following lemma tells us that $z\in \mathbb {Z}_{(\ell _n)}$ is singular if and only if it is the image of a path of thickness $k>1$ . Let $\mathrm {thk}(\gamma )$ denote the thickness of the path $\gamma $ .

Lemma 2.9. Let $(X_B,\varphi _\omega )$ be a Toeplitz Bratteli–Vershik system with extended path space $X_{\tilde B}$ . Let $z\in \mathbb {Z}_{(\ell _n)}$ . Then,

$$ \begin{align*}|\{\gamma\in X_{B} : \omega(\gamma)=z\}| = \sup\{\mathrm{thk}(\tilde\gamma) : \tilde\gamma\in X_{\tilde B}, \omega(\tilde\gamma) = z\}.\end{align*} $$

In particular, the set of singular points in $\mathbb {Z}_{(\ell _n)}$ coincides with the union $\bigcup _{j\geq 2}\omega (X^j_{\tilde B})$ , and the rank of the Toeplitz Bratteli–Vershik system is an upper bound for the maximal number of elements in a fibre of the factor map to the maximal equicontinuous factor.

2.5 Thick Toeplitz shifts

A pair of parallel edges in $\tilde E_n$ is a pair of edges $(e_{1}, e_{2})\in \tilde E_n\times \tilde E_n $ with the same source and range but distinct labels according to the order. A double path in $X_{\tilde B}$ is a pair of paths consisting of parallel edges at each level $n> 0$ . We write $\bar {\gamma }=( \bar {\gamma }_n)=(\gamma _{n,1}, \gamma _{n,2})$ to denote a double path.

Example A. (Continued)

To illustrate the above notions, we apply them to the first substitution in Example A. While this example is not sensitive to some of the subtleties that we will meet later (because of its stationarity), it can at least be described explicitly.

Recall that $\theta : \{a,b,c \}\rightarrow \{a,b,c \}^{4}$ is the substitution

$$ \begin{align*}\begin{array}{c c l} a & \mapsto & aaca\\ b & \mapsto & abba \\ c & \mapsto & aaba. \\ \end{array} \end{align*} $$

Since all substitution words begin and end on a, the approach of [Reference Vershik and Livshits41] to define the Toeplitz Bratteli–Vershik system works here and it is not difficult to derive the extended system as well. The extended Bratteli diagram is stationary and we have drawn one level in Figure 2. We follow the convention of reading levels from top to bottom. Note that the vertices $\{a,b,c\}$ and $\{ a,c\}$ have no outgoing edges, so no infinite path will go through them and, in particular, there are no paths of thickness $3$ . There are uncountably many paths of thickness $2$ , namely those which keep going through vertices $\{a,b\}$ or $\{b,c\}$ . Hence, all singular fibres consist of two elements and the essential thickness is $2$ .

Figure 2 One level of the stationary extended Bratteli diagram of Example A. The order is indicated through colour: black, blue dotted, violet dashed and red edges correspond to order label 0, 1, 2 and 3, respectively. The grey vertices are not extendable. Red edges are finer than black edges if viewed without colour.

If we telescope the extended Bratteli diagram to even levels, we will find that there are two edges between two consecutive vertices $\{a,b\}$ . These two edges form a pair of parallel edges and consequently the telescoped diagram admits a double path of thickness $2$ . In particular, the Toeplitz Bratteli–Vershik system is thick.

Notice the connections between Figure 2 and the graph $\mathcal G_\theta $ in Figure 1. The infinite paths on $\mathcal G_\theta $ correspond to paths in the stationary extended Bratteli diagram which start at the top vertex $v_0$ (the first level consists of one edge between $v_0$ and each of the seven vertices of $\tilde V_1$ ) and go downwards without ever passing through a vertex which is a singleton nor through a vertex which does not have an outgoing edge. The fact that the telescoped extended Bratteli diagram admits a double path of thickness $2$ going through the vertices $\{a,b\}$ is equivalent to the fact that the vertex $\{a,b\}$ of $\mathcal G_\theta $ belongs to two distinct cycles.

Example C. Oxtoby [Reference Oxtoby37] described a family of minimal binary Toeplitz shifts that are not uniquely ergodic and hence cannot be tame. We describe the Bratteli–Vershik representations for the one-sided versions of this family to maximize the similarity to his original description. Given a sequence $(\ell _n)$ of natural numbers, define the substitutions

$$ \begin{align*} a&\stackrel{\theta^{(n)}}{\mapsto} ab^{\ell_n -1}\\ b&\stackrel{\theta^{(n)}}{\mapsto} aa^{\ell_n-1}, \end{align*} $$

and define an ordered Bratteli diagram with the sequence $\{\theta ^{(n)} \}$ as in equation (2.1); see Figure 3 for two examples, the one on the left with $\ell _1=\ell _2=2$ , the second with $\ell _1=3$ , $\ell _2=5$ .

Figure 3 On the right, we see the first levels of the extended Bratteli diagram of the one-sided shifts for Example C with $\ell _1=3$ and $\ell _2=5$ . The order is indicated through colour: black, blue dotted, violet dashed, green densely dotted and red edges correspond to order label 0, 1, 2, 3 and $4$ , respectively. Red edges are finer than black edges if viewed without colour. As more levels are added, there are increasingly many edges between vertices labelled $\{a,b\}$ in consecutive levels. This is to be contrasted with the one-sided period-doubling substitution shift (on the left), where $\ell _n=2$ for all n, and which is tame (again, black and blue dotted edges correspond to order label $0$ and $1$ , respectively). It has thickness one.

Note that these ordered Bratteli diagrams each have two maximal paths and one minimal path; this means that we cannot define a Vershik map which is a homeomorphism. Nevertheless, we can still define a continuous Vershik map by sending the two maximal paths to the unique minimal path. This one-sided Bratteli–Vershik system is conjugate to the one-sided shift defined by Oxtoby. Oxtoby showed that if $(\ell _n)$ grows fast enough, in particular if $\sum ({\ell _{k-1}}/{\ell _k})<1$ , then the resulting system is not uniquely ergodic and thus it cannot be tame. On the right-hand side of Figure 3, one clearly sees the beginning of a double path of thickness two. This should be contrasted with the stationary figure on the left, which is a one-sided representation of the period-doubling substitution shift and which has thickness one, and so is tame, as we will see below.

Recall that $x\in X$ is a condensation point if every neighbourhood of x is uncountable. Let $\mathcal C\subseteq X$ be the set of its condensation points. The Cantor–Bendixson theorem gives that for second countable spaces, $X\backslash \mathcal C$ is countable.

The following corollary can be seen as a generalization of Lemma 2.1 to all Toeplitz shifts with finite Toeplitz rank.

Given a choice function $\varphi \in \{0,1\}^{\mathbb {N}_0}$ and a double path $\bar {\gamma }$ , let $\varphi (\bar {\gamma })$ be the (single edge) path $ ({\gamma }_{n,\varphi (n)})$ . Such a single edge path defines a sequence of maps ${\theta ^{({n})}_{\varphi (\bar {\gamma })}}$ , see equation (2.5). The second part of Lemma 2.11 implies that the essential thickness is an upper bound for the maximal thickness a double path can have. From the next results, we obtain more delicate information: the size of the image $ | {\theta ^{({m})}_{\varphi (\bar {\gamma })}}\cdots {\theta ^{({n})}_{\varphi (\bar {\gamma })}} (V_{n+1})|$ of sufficiently long finite paths is also bounded above by the essential thickness.

In the proof of the next statement, we denote the set of all finite and infinite paths in $\tilde B$ which start at the top vertex $v_0$ by $Y_{\tilde B}$ . Observe that $Y_{\tilde B}$ can naturally be seen as a compact space, where finite paths are seen as infinite paths which eventually pass through a placeholder vertex.

Lemma 2.13. Let $(X_B,\varphi _\omega )$ be a Toeplitz Bratteli–Vershik system with essential thickness k. Consider $z^0,z^1\in \mathbb {Z}_{(\ell _n)} $ with $z^0_n\neq z^1_n$ for each n. Then for all but at most countably many $\varphi \in \{0,1\}^{\mathbb {N}_0}$ , we have

$$ \begin{align*} \text{ for all } m\in \mathbb{N} \,\, \text{ there exists } n_0>m \,\, \text{ for all } n\geq n_0 \colon | {\theta^{({m})}_{z^{\varphi(m)}_m}}\cdots {\theta^{({n})}_{z^{\varphi(n)}_n}} (V_{n+1})| \leq k. \end{align*} $$

Proof. We only have to consider the case of finite k as the statement is trivially true otherwise. Suppose there are uncountably many $\varphi $ such that

$$ \begin{align*} \text{ there exists } m\in \mathbb{N}\,\, \text{ for all } n_0>m\,\, \text{ there exists } n\geq n_0\colon |{\theta^{({m})}_{z^{\varphi(m)}_m}}\cdots {\theta^{({n})}_{z^{\varphi(n)}_n}} (V_{n+1})|> k.\end{align*} $$

Then for each such $\varphi $ , there are arbitrarily large n and paths $\gamma ^{\varphi ;n}\in Y_{\tilde B}$ of length n with $\omega (\gamma ^{\varphi ;n}_\ell )=z^{\varphi (\ell )}_\ell $ for $\ell =0,\ldots ,n-1$ which traverse a subset of $V_m$ of size bigger than k. Due to the compactness of $Y_{\tilde B}$ , there must hence be an infinite path $\gamma ^\varphi $ (that is, an element of $X_{\tilde B}$ ) with $\omega (\gamma ^{\varphi }_\ell )=z^{\varphi (\ell )}_\ell $ ( $\ell \in \mathbb {N}_0$ ) which traverses a subset of $V_m$ of size bigger than k. It follows that $X^{>k}_{\tilde B} $ is uncountable, contradicting our assumption that k is the essential thickness.

Note that if a double path $\bar \gamma $ has thickness $k<\infty $ , then there exists $m_0$ such that also the opposite inequality is true. More precisely, for uncountably many $\varphi \in \{0,1\}^{\mathbb {N}_0}$ , we have

$$ \begin{align*}\text{ there exists } m\, \text{ for all } n\geq m\colon | {\theta^{({m})}_{\varphi(\bar{\gamma})}}\cdots {\theta^{({n})}_{\varphi(\bar{\gamma})}}(V_{n+1})| \geq k .\end{align*} $$

Indeed, the contrary, that is, the assumption that for all but at most countably many $\varphi $ , we have

$$ \begin{align*} \text{ for all } m\,\text{ there exists } n\geq m\colon | {\theta^{({m})}_{\varphi(\bar{\gamma})}}\cdots {\theta^{({n})}_{\varphi(\bar{\gamma})}}(V_{n+1})| < k, \end{align*} $$

implies that all but at most countably $\varphi (\bar {\gamma })$ belong to $X_{\tilde B}^{<k}$ , which is a contradiction.

We immediately obtain the following corollary.

2.6 Toeplitz systems and non-tameness

In this section, we prove one of our main results, Theorem 2.17. It applies to all finite rank Toeplitz systems with finite Toeplitz rank, as stated in Theorem 2.18.

As before, we identify subsets $A_n\subseteq V_n$ with vertices $A_{n}\in \tilde V_n$ . For the convenience of the reader, we provide a proof of the next statement which is reminiscent of [Reference Downarowicz10, Theorem 13.1].

Lemma 2.15. Consider a Toeplitz Bratteli–Vershik system with characteristic sequence $(\ell _n)$ and let $(A_n)$ be some sequence of extendable vertices $A_{n}\subseteq V_{n}$ . Suppose that the set of singular points of $\mathbb {Z}_{(\ell _n)}$ has Haar measure $0$ .

By telescoping, we can ensure that

$$ \begin{align*} \lim_{n\to\infty} \frac{ | \{i\in [0,\ell_n-1]: |\theta_i^{(n)}(A_{n+1})|\geq 2 \}|}{ \ell_n} = 0.\end{align*} $$

Proof. Observe that the Haar probability measure $\mu $ of the set $D\subseteq \mathbb {Z}_{\ell _n}$ of singular points is

$$ \begin{align*}\mu(D) = \lim_{n\to +\infty} \frac1{\prod_{k=0}^{n}\ell_k} |\{z_n\cdots z_0:z\in D\}|=0.\end{align*} $$

Let $n\geq m\in \mathbb {N}_0$ and $w= w_n\cdots w_m$ , with $0\leq w_i \leq \ell _i-1$ . If there is an extendable $A_{n+1}\subseteq V_{n+1}$ such that $|{\theta ^{({m})}_{{w_m}}}\cdots {\theta ^{({n})}_{{w_n}}}(A_{n+1})|\geq 2$ , then w is a subword of some singular point $z\in D$ , see Lemma 2.9. Hence, given any sequence of extendable vertices $A_{n}\in \tilde V_{n}$ and any $m\geq 0$ , we have

$$ \begin{align*} \hspace{-3pt}&\limsup_{n\to+\infty} \frac1{\prod_{k=m}^{n}\ell_k} |\{w_n\cdots w_m\colon |{\theta^{({m})}_{w_m}}\cdots {\theta^{({n})}_{w_n}}(A_{n+1})|\geq 2\}|\\ \hspace{-3pt}&\leq\! \limsup_{n\to+\infty}\! \frac1{\prod_{k=m}^{n}\ell_k} |\{z_n\cdots z_m\colon z\in D\}| \!\leq\! \prod_{k=0}^{m-1}\!\ell_k \!\cdot\! \limsup_{n\to+\infty}\! \frac1{\prod_{k=0}^{n}\ell_k} |\{z_n\cdots z_0\colon z\in D\}| = 0. \end{align*} $$

Telescoping from level m to n, the statement follows.

Recall that for systems with finite topological rank, Lemma 2.11 tells us that essential thickness $k>1$ is equivalent to the existence of a double path of thickness k. Therefore, the following technical proposition applies to all finite rank Toeplitz systems with uncountably many singular points.

Proof. Let $k>1$ be the essential thickness of the Toeplitz Bratteli–Vershik system and $\bar {\gamma }$ be a double path of thickness k. It is a sequence of parallel edges $\bar {\gamma }_{n} = (\gamma _{n,1},\gamma _{n,2})$ , that is,

$$ \begin{align*}s(\gamma_{n,1}) = s(\gamma_{n,2}) = A_{n}\subseteq V_{n},\quad r(\gamma_{n,1}) = r(\gamma_{n,2}) = A_{n+1}\subseteq V_{n+1}\end{align*} $$

such that, for large enough n, $|A_n|=k$ , and furthermore, the maps ${\theta ^{({n})}_{\omega (\gamma _{n,1})}}$ and ${\theta ^{({n})}_{\omega (\gamma _{n,2})}}$ each map $A_{n+1} $ bijectively to $A_{n}$ . Note that, by definition, all $A_{n+1}$ are extendable.

Take the nth parallel edge $(\gamma _{n,1},\gamma _{n,2})$ of our double path and set

$$ \begin{align*}\Delta_{n}(\gamma_{n,1},\gamma_{n,2}) := \omega(\gamma_{n,2})-\omega(\gamma_{n,1}).\end{align*} $$

Telescoping with the next level $n+1$ will produce four parallel edges. It is crucial to observe that at least two of these four edges have the same value of $\Delta _{n}$ as the two above. Hence, we can apply Lemma 2.15 to conclude that, possibly after telescoping, there is j in $(\omega (\gamma _{n,1})+\Delta _{n}\mathbb {Z}) \cap \{0,\ldots ,\ell _n-1\}$ such that $|{\theta ^{({n})}_{j}}(A_{n+1})|<k$ . We next need to find such a j where moreover ${\theta ^{({n})}_{j}}(A_{n+1})\subseteq A_n$ .

By Corollary 2.14, we can find for each n large enough, $\kappa _n\in \{0,\ldots ,\ell _n-1\}$ such that ${\theta ^{({n})}_{\kappa _n}}(A_{n+1}) = A_{n}$ and $| {\theta ^{({n})}_{\kappa _n}} (V_{n+1})|= k$ , and hence ${\theta ^{({n})}_{\kappa _n}} (V_{n+1})= A_{n}$ . Define $\psi ^{(n)}_j:\tilde V_{n+2}\to \tilde V_{n-1}$ through

$$ \begin{align*}\psi^{(n)}_j := {\theta^{({n-1})}_{{\kappa_{n-1}}}} {\theta^{({n})}_{j}} {\theta^{({n+1})}_{\kappa_{n+1}}}\end{align*} $$

and note that $\psi ^{(n)}_j{\kern-1pt}(A_{n+2}) {\kern-1.2pt}\subseteq{\kern-1.2pt} A_{n-1}$ . Moreover, the inclusion is proper if $|{\theta ^{({n})}_{j}}{\kern-1pt}(A_{n+1})|<k$ , while $\psi ^{(n)}_j$ is a bijection if $j=\omega (\gamma _{n,1})$ or $j=\omega (\gamma _{n,2})$ . Thus, we can find $j_0^{(n)},j_1^{(n)},j_2^{(n)} \in (\omega (\gamma _{n,1})+\Delta _{n}\mathbb {Z}) \cap \{0,\ldots ,\ell _n-1\}$ such that $j_2^{(n)}-j_1^{(n)} = j_1^{(n)}-j_0^{(n)}$ and $ B_{n-1} = \psi ^{(n)}_{j_0^{(n)}}(A_{n+2}) $ is a proper subset of $ A_{n-1}$ , while $\psi ^{(n)}_{j_1^{(n)}}(A_{n+2}) = A_{n-1}$ and $\psi ^{(n)}_{j_2^{(n)}}(A_{n+2}) = A_{n-1}$ . We now telescope the three floors together and thus the $\psi ^{(n)}_{j_0^{(n)}}$ , $\psi ^{(n)}_{j_1^{(n)}}$ , $\psi ^{(n)}_{j_2^{(n)}}$ can be realized as $\theta ^{(n)}_{j_0^{(n)}}$ , $\theta ^{(n)}_{j_1^{(n)}}$ and $\theta ^{(n)}_{j_2^{(n)}}$ .

This shows the first statement except for the fact that we only know that $\theta ^{(n)}_{j_1^{(n)}}$ and $\theta ^{(n)}_{j_2^{(n)}}$ restrict to bijections $f^{(n)}_1$ and $f^{(n)}_2$ from $A_{n+1}$ to $A_{n}$ , and it remains to show that, possibly after telescoping, $f^{(n)}_1=f^{(n)}_2$ . It is convenient to identify the $A_{n+1}$ with $A_{1}$ . We do this using the bijection $ \tau ^{(n)}:= f^{(1)}_1\cdots f^{(n)}_1 $ . With

(2.6) $$ \begin{align}I_n := \tau^{(n-1)} f^{(n)}_2{\tau^{(n)}}^{-1},\end{align} $$

our aim is thus to show that, possibly after telescoping, all $I_n$ are the identity.

Let further $f^{(n)}_0:A_{n+1}\to A_{n}$ be the restriction of $\theta ^{(n)}_{j_0^{(n)}}$ to $A_{n+1}$ ; it is non-surjective with image $B_{n}$ . If we telescope level n with level $n+1$ , we get nine compositions $f^{(n)}_i f^{(n+1)}_j$ for the three possible values of i and j. Let us take a closer look at two sets of choices for them.

Consider first the maps $f^{(n)}_0 f^{(n+1)}_1$ , $f^{(n)}_{1} f^{(n+1)}_1$ , $f^{(n)}_2 f^{(n+1)}_1$ . These correspond, after telescoping of the two levels, to the restriction to $A^{(n+1)}$ of maps $\theta ^{(n)}_{j_0^{(n)}}$ , $\theta ^{(n)}_{j_1^{(n)}}$ and $\theta ^{(n)}_{j_2^{(n)}}$ , with $j_2^{(n)}-j_1^{(n)} = j_1^{(n)}-j_0^{(n)}$ , and $f^{(n)}_1 f^{(n+1)}_1$ , $f^{(n)}_2 f^{(n+1)}_1$ are bijections while $f^{(n)}_0 f^{(n+1)}_1$ is not. It is quickly seen that under this choice, the map $I_n$ after telescoping coincides with the map before telescoping. In other words, if we replace $f^{(n)}_2$ with $f^{(n)}_2 f^{(n+1)}_1$ in equation (2.6), then

$$ \begin{align*} \tau^{(n-1)} f^{(n)}_2 f^{(n+1)}_1{\tau^{(n+1)}}^{-1}= I_n.\end{align*} $$

Now consider the maps $f^{(n)}_0 f^{(n+1)}_0$ , $f^{(n)}_1 f^{(n+1)}_1$ , $f^{(n)}_2 f^{(n+1)}_2$ . Again these correspond, after telescoping of the two levels, to the restriction to $A_{n+1}$ of maps $\theta ^{(n)}_{j_0^{(n)}}$ , $\theta ^{(n)}_{j_1^{(n)}}$ and $\theta ^{(n)}_{j_2^{(n)}}$ , with $j_2^{(n)}-j_1^{(n)} =j_1^{(n)}-j_0^{(n)}$ , and $f^{(n)}_1 f^{(n+1)}_1$ , $f^{(n)}_2 f^{(n+1)}_2$ are bijections while $f^{(n)}_0 f^{(n+1)}_0$ is not. The telescoping, however, affects the map $I_n$ . The new map $\tilde I_n$ becomes

$$ \begin{align*} \tilde I_n = \tau^{(n-1)} f^{(n)}_2 f^{(n+1)}_2 {\tau^{(n+1)}}^{-1}=I_nI_{n+1}.\end{align*} $$

As $A_{1}$ is finite, the sequence $I_n$ admits a constant subsequence $g^{(n_k)} = g$ . Telescoping the levels from $n_k$ to $n_{k+1}-1$ in the first way described above, we arrive at a situation where all $I_n$ coincide with g. Let N be the order of g. Now, telescoping N consecutive levels together in the second way above, we arrive at a situation where all $I_n$ are equal to the identity. While all this telescoping has an effect on $\theta ^{(n)}_{j_0^{(n)}}$ , it does not change its crucial property, namely that it maps $A_{n+1}$ to a proper subset of $A_{n}$ , and that $j_0^{(n)},j_1^{(n)},j_2^{(n)}$ form an arithmetic progression.

It remains to show the second property. Since the order $\omega $ is proper, we can assume, by telescoping if necessary, that for each n, $|\theta ^{(n)}_0(V_{n+1})|=|\theta ^{(n)}_{\ell _n -1}(V_{n+1})|= 1$ . Take $a\in A_{n}$ such that ${{\theta ^{({n-1})}_{j_1^{(n)}}}}(a) \notin B_{n-1}$ . By minimality (and perhaps further telescoping), we find $i_n$ such that ${\theta ^{({n})}_{{i_n}}}(V_{n+1})=\{a\}$ .

Proof. Let $(X_B,\varphi _\omega )$ be a thick Bratteli–Vershik representation of the given Toeplitz shift which we assume to be conjugate to $(X_1,\sigma )$ , see §2.3.3 . We will construct an infinite independence set for two cylinder sets in $X_1$ which we define in the proof.

Observe that if the set of singular points in $\mathcal Z$ has positive Haar measure, then $(X_B, \varphi _\omega )$ is non-tame [Reference Fuhrmann, Glasner, Jäger and Oertel19, Theorem 1.2]. We hence assume the set of singular points in $\mathcal Z$ to have zero Haar measure so that we can apply Proposition 2.16 in the following.

Let $\varphi \in \{0,1\}^{\mathbb {N}_0}$ be a choice function. We use the notation of the proof of Proposition 2.16 and let $h_{n}$ be the restriction of ${\theta ^{({n})}_{i_n}}$ to $A_{n+1}$ . Recall that ${f_1^{(n-1)}} h_{n}(A_{n+1})\subseteq A_{n-1}\backslash B_{n-1}$ . Set

$$ \begin{align*} z = \cdots i_{2n+2} j^{(2n+1)}_{\varphi_n}\cdots \cdots i_{2} j^{(1)}_{\varphi_0}, \end{align*} $$

$t_0=0$ , $t_1 = (j^{(2)}_1- i_{2} ) \ell _1 + \Delta _1$ , and for $n\geq 2$ ,

$$ \begin{align*}t_{n} = t_{n-1} + (j^{(2n)}_1- i_{2n})\prod_{j=1}^{2n-1}\ell_{j}+ \Delta_{2n-1} \prod_{j=1}^{2n-2}\ell_{j}.\end{align*} $$

Choose $x\in \omega ^{-1}(z)$ . By equation (2.4), we have $x_0 \in \theta ^{(1)}_{j^{(1)}_{\varphi _0}}\theta ^{(2)}_{i_{2}}(A_{3})=f^{(1)}_{\varphi _0} h_{2}(A_{3})$ . If $\varphi _0 = 0$ , then

$$ \begin{align*}f^{(1)}_{0} h_{2}(A_{3})\subseteq \mathrm{im} f^{(1)}_{0} = B_{1},\end{align*} $$

whereas if $\varphi _0 = 1$ , then

$$ \begin{align*}f^{(1)}_{1} h_{2}(A_{3}) \subseteq A_{1}\backslash B_{1}.\end{align*} $$

Furthermore,

$$ \begin{align*} z + t_1 = \cdots i_{4} j^{(3)}_{\varphi_1}j_1^{(2)} j^{(1)}_{\varphi_0+1}.\end{align*} $$

Hence, taking into account that $f^{(n)}_1=f^{(n)}_2$ , we have

$$ \begin{align*}x_{t_1} \in \theta^{(1)}_{j^{(1)}_{\varphi_0+1}}\theta^{(2)}_{j_1^{(2)}} \theta^{(3)}_{j^{(3)}_{\varphi_1}}\theta^{(4)}_{i_{4}}(A_{5}) =f^{(1)}_{1} f^{(2)}_1 f^{(3)}_{\varphi_1} h_{4}(A_{5}).\end{align*} $$

Since

$$ \begin{align*} f^{(3)}_{\varphi_1} h_{4}(A_{5}) = \left\{\begin{array}{ll} f^{(3)}_{0} h_{4}(A_{5}) \subseteq B_{3} & \mbox{if } \varphi_1=0,\\ f^{(3)}_{1} h_{4}(A_{5}) \subseteq A_{3}\backslash B_{3} & \mbox{if } \varphi_1=1, \end{array} \right. \end{align*} $$

it follows that

$$ \begin{align*} \begin{array}{ll} x_{t_1} \in \tau^{(2)}(B_{3}) & \mbox{ if } \varphi_1=0,\\ x_{t_1} \in A_{1}\backslash \tau^{(2)}(B_{3}) & \mbox{ if } \varphi_1=1. \end{array} \end{align*} $$

Similarly, we find for all $n\geq 2$ ,

$$ \begin{align*} \begin{array}{ll} x_{t_n} \in \tau^{(2n)}(B_{2n+1}) & \mbox{ if } \varphi_n=0,\\ x_{t_n} \in A_{1}\backslash \tau^{(2n)}(B_{2n+1}) & \mbox{ if } \varphi_n=1. \end{array} \end{align*} $$

By finiteness of $A_{1}$ , there is a subsequence of $B_{2n+1}$ such that $\tau ^{2n}(B^{2n+1})$ is constant, say equal to B. Restricting to choice functions with support in this subsequence, we obtain an independence sequence for the cylinder set $[B]$ and its complement in $[A_{1}]$ .

For non-tame constant length substitution shifts with a coincidence, an independence set can be explicitly computed, as the following example illustrates.

Example A. (Continued)

We illustrate Proposition 2.16 and Theorem 2.17 with our the substitution in Example A. The graph of Figure 2 has no parallel edges of thickness $2$ , but if we telescope two levels together, we get a pair of parallel edges between the vertex $\{a,b\}$ above and the vertex $\{a,b\}$ below. For substitutions, telescoping amounts to taking powers of the substitution. We therefore need to work (at least) with the second power $\theta ^2$ of the substitution:

We see that the restrictions of ${\theta ^2}_5$ and ${\theta ^2}_9$ to $\{a,b\}$ are both equal to the identity. Furthermore, ${\theta ^2}_1$ is the projection onto letter a. (In the coloured version, the images of ${\theta ^2}_1$ , ${\theta ^2}_5$ and ${\theta ^2}_9$ are marked blue, and that of ${\theta ^2}_{10}$ is in red.) We may therefore chose $\theta ^{(1)}=\theta ^2$ , $A_{2} = \{a,b\} = A_{1}$ and the arithmetic progression $j_0=1$ , $j_1=5$ , $j_2=9$ . Since here the extended Bratteli diagram is stationary, all $\theta ^{(n)}$ and $A_{n}$ can be taken to be equal. As ${\theta ^2}_{10}$ is the projection onto the letter b, we may take $i_1=10$ . Therefore, the independence sequence from the proof of Theorem 2.17 is given by $t_0=0$ , and $t_{n} = t_{n-1} -5 \cdot 16^{2n-1}+ 4\cdot 16^{2n-2}$ for $n\geq 1$ .

Example D. As is the case with many properties, tameness is not preserved under strong orbit equivalence (see [Reference Gjerde and Johansen23] for a definition). Take the two substitutions

$$ \begin{align*}\begin{array}{c c l} a &\mapsto & aabaa\\ b & \mapsto & abbaa \\ \end{array} \end{align*} $$

and

$$ \begin{align*}\begin{array}{c c l} a &\mapsto & aaaba\\ b & \mapsto & abbaa. \\ \end{array} \end{align*} $$

Then both substitution shifts are conjugate to the stationary Bratteli–Vershik system they define, see §2.3.4. Furthermore, since the underlying unordered Bratteli diagrams are identical, these two shifts are strong orbit equivalent. However, by computing the graphs $\mathcal G_\theta $ , we see that the first has one orbit of singular fibres, whereas the second has uncountably many. In the language of this section, the second has essential thickness two. Hence, by Theorem 2.18, the second shift is non-tame. As the first Toeplitz system has one singular fibre, it is tame by Theorem 2.18.

5.1 Set of discontinuities D and its properties

Given $z\in \mathcal Z=\mathbb {Z}_{(4^n)}$ , we denote by ${\mathrm {head}}_i(z) = z_i \cdots z_1$ the head of length i. We use the common head length function

$$ \begin{align*}L(z,z') := \sup\{ i\in \mathbb{N}\colon {\mathrm{head}}_i(z)={\mathrm{head}}_i(z')\}\end{align*} $$

and further set $L(z,A) = \sup \{L(z,z')\colon z'\in A\}$ for $A\subseteq \mathcal Z$ . Recall that the topology of $\mathbb {Z}_{(4^n)}$ has a base of clopen sets $U_w$ labelled by finite words $w=w_k\cdots w_1$ , $w_i\in \{0,\ldots , 4^i-1\}$ , where

$$ \begin{align*}U_w = \{z\in \mathbb{Z}_{(4^n)} : {\mathrm{head}}_k(z)=w\}.\end{align*} $$

In the following, integers $t\in \mathbb {Z}$ are identified with their natural representation in $\mathcal Z$ and we write $z_n(t)$ for the nth entry in that representation.

We recursively define a chain of subsets $D^i$ of $\mathcal Z$ . The subset $D^i$ will contain $m_i:=2^{i}$ elements which are all of the form $z=\cdots z_2 z_1$ with $z_i = 3^{k_i}$ a power of $3$ . We write $\bar {\cdot }$ to indicate infinite repetition to the left.

1. (0) $D^0=\{\overline {3^0}\}$ .

2. ( $\tilde 0$ ) $\tilde D^0 = \{\overline {3^1} 3^0\}$ .

3. (1) $D^1 = D^0\cup \tilde D^0 = \{\overline {3^0},\overline {3^1} 3^0\}$ .

4. (i+1) Suppose that we have constructed $D^{i}$ and ordered its elements (in some fashion)

   $$ \begin{align*}D^{i} = \{d^{(i,0)},\ldots,d^{(i,m_i-1)}\}.\end{align*} $$

5. For each $l\in \{0,\ldots ,m_i-1\}$ , we define ${\tilde d^{(i,l)}\in }\tilde D^{i}$ to be

   $$ \begin{align*} \tilde d^{(i,l)} = \overline{3^{m_i+l}}\; {\mathrm{head}}_{m_i+l} d^{(i,l)} \end{align*} $$

6. and $D^{i+1} := D^{i}\cup \tilde D^{i}$ .

We may order $D^{i+1}$ in a way where we first take the elements of $D^{i}$ and then those of $\tilde D^{i}$ . For example, this gives

$$ \begin{align*} {D^2} =&\{\overline{3^0},\overline{3^1}3^0,\overline{3^2}3^03^0,\overline{3^3}3^13^13^0\} \end{align*} $$

and

$$ \begin{align*} {D^3} =&\{\overline{3^0},\,\,\overline{3^1}3^0,\,\,\overline{3^2}3^03^0,\,\,\overline{3^3}3^13^13^0\} \\ &\cup \{\overline{3^4}3^03^03^03^0,\,\, \overline{3^5}3^13^13^13^13^0,\,\,\overline{3^6}3^23^23^23^23^03^0,\,\,\overline{3^7}3^33^33^33^33^13^13^0 \}. \end{align*} $$

Define

$$ \begin{align*} D=\overline{\bigcup_{i\in \mathbb{N}} D^i}. \end{align*} $$

We denote by $\mathrm {Head}_m:=\{{\mathrm {head}}_m(z) \colon z\in D\}$ , the heads of length m of elements of D and by $q:\mathrm {Head}_{m+1}\to \mathrm {Head}_m$ the obvious surjective map given by shortening the head by one element. A word $w\in \mathrm {Head}_m$ is (left) special if it does not have a unique preimage under q.