1 Introduction
Tameness is known for its many facets related to deep theorems in topology, Banach space theory and model theory, such as Rosenthal’s $\ell ^1$ embedding theorem or the BFTdichotomy. It was introduced to dynamics by Köhler, under the name regularity [Reference Köhler34], and the joint efforts of the community helped to shed light on general structural properties of tame dynamical systems [Reference Fuhrmann, Glasner, Jäger and Oertel19, Reference Glasner24–Reference Glasner and Megrelishvili26, Reference Huang30, Reference Kerr and Li33], see also [Reference Glasner and Megrelishvili27] for an exposition and numerous further references. The opposite of tame is wild, but with mathematical modesty and to not overuse the word wild, the community simply calls wild dynamical systems nontame. One fascinating phenomenon is that nontame systems are not just not tame, but qualitatively very far from tame systems. This is most visibly reflected in the following characterization: a $\mathbb {Z}$ action defined by a homeomorphism T on a compact metrizable space X is tame if the cardinality of its Ellis semigroup $E(X,T)$ is at most that of the continuum c, whereas if it is nontame, $E(X,T)$ must contain a copy of $\beta \mathbb {N}$ .
To put our results in context, it is important to say a few words about the relation between tameness and almost automorphy—a classical notion extensively studied by Veech [Reference Veech40]. Huang realized that tame minimal $\mathbb {Z}$ actions are necessarily uniquely ergodic and almost automorphic [Reference Huang30], that is, the maximal equicontinuous factor $\mathcal Z$ contains a point which has a unique preimage under the factor map $\pi :X\to \mathcal Z$ . Such a point is called regular and correspondingly, points $z\in \mathcal Z$ for which $\pi ^{1}(z)$ consists of more than one element are called singular. Recently, Glasner extended Huang’s result to minimal actions of general groups possessing an invariant measure [Reference Glasner25]. Based on these results, it was proved in [Reference Fuhrmann, Glasner, Jäger and Oertel19] that tame minimal systems are actually regularly almost automorphic, which means that the set of regular points has full (Haar) measure.
The vanishing, in measure, of the singular points is thus a necessary condition for tameness, but it is far from being sufficient. For shift dynamical systems, the set of singular points is the union of the $\mathbb {Z}$ orbits of its discontinuity points: these are the points in the maximal equicontinuous factor whose fibre contains two points which disagree on their $0$ coordinate. A binary almost automorphic shift whose maximal equicontinuous factor is an irrational rotation on the circle is nontame if its set of discontinuity points is a Cantor set [Reference Fuhrmann, Glasner, Jäger and Oertel19, Proposition 3.3]. Note that among such systems, one easily finds examples for which the set of singular points is a zero measure set. Furthermore, it is possible to construct nontame almost automorphic systems for which the set of singular points consists of a single orbit [Reference Fuhrmann and Kwietniak22]. However, in this case, the preimage of a singular point under the factor map has to be uncountable. These results suggest that nontameness is related to the smallness of the difference between X and its maximal equicontinuous factor, where smallness is computed either via a measure or via cardinality, either in X or its maximal equicontinuous factor. Our results largely affirm this suggestion but emphasize that this relation is generally speaking more subtle.
We investigate the notion of tameness for the class of Toeplitz shifts, which are almost automorphic extensions of odometers (procyclic group rotations). In this class, Oxtoby found a first example of a minimal system which is not uniquely ergodic [Reference Oxtoby37]. Jacobs and Keane defined and studied Toeplitz shifts systematically in [Reference Jacobs and Keane32], recognising their close relation to Toeplitz’s constructions in [Reference Toeplitz39]. Toeplitz shifts have since enjoyed ample attention due to their dynamical diversity and their relevance in measurable and topological dynamics, see e.g. [Reference Baake, Jäger and Lenz3, Reference Bułatek and Kwiatkowski4, Reference Downarowicz9, Reference Downarowicz and Kasjan12, Reference Downarowicz and Lacroix13, Reference Downarowicz and Serafin15, Reference Gjerde and Johansen23, Reference Iwanik31, Reference Markley and Paul36, Reference Williams42] as well as [Reference Downarowicz10] for a survey and further references. Gjerde and Johansen [Reference Gjerde and Johansen23] represent Toeplitz shifts as Bratteli–Vershik systems where the Bratteli diagrams are appropriately constrained, and it is this representation that is particularly useful to us. A Toeplitz shift has finite Toeplitz rank if it has such a representation for which the number of vertices at each level of the Bratteli diagram is uniformly bounded. Note that, on the face of it, the class of finite Toeplitz rank systems is possibly smaller than the class of Toeplitz shifts of finite topological rank, that is, the class of Toeplitz shifts which have some proper Bratteli–Vershik representation (which not necessarily satisfies the constraints of Gjerde and Johansen) with a bounded number of vertices at each level. It is known that the class of finite Topelitz rank systems is a subset of the class of shifts with nonsuperlinear complexity, [Reference Donoso, Durand, Maass and Petite8, Corollary 6.7]. We show the following theorem.
Theorem 1.1. Let $(X, T)$ be a Toeplitz shift of finite Toeplitz rank. Then $(X,T)$ is tame if and only if its maximal equicontinuous factor has only countably many singular points.
This result should be contrasted with the work of Aujogue [Reference Aujogue1], who shows that a family of tiling systems is tame despite having uncountably many singular points.
As a corollary to Theorem 1.1, we get the following necessary and sufficient criterion for tameness of substitution shifts of constant length. Note that, in this case, the property of having only finitely many orbits of singular points can be easily read off from an associated graph introduced in [Reference Coven, Quas and Yassawi5]. We elaborate on this in the main body of this work.
Theorem 1.2. Let $\theta $ be a primitive aperiodic substitution of constant length. The associated shift $(X_{\theta }, T)$ is tame if and only if $\theta $ has a coincidence and the maximal equicontinuous factor contains only finitely many orbits of singular points.
Against the background of Theorem 1.1, one might be tempted to guess that the presence of uncountably many singular fibres implies nontameness for general Toeplitz shifts, that is, also for those of infinite rank. However, it turns out that in spite of Theorem 1.1, neither the cardinality of the singular points nor that of the fibres decide whether $(X,T)$ is tame or otherwise. We show the following theorem.
Theorem 1.3. There is a tame binary Toeplitz shift whose set of discontinuity points is a Cantor set and so its maximal equicontinuous factor has uncountably many singular points. Moreover, there exist tame as well as nontame binary Toeplitz shifts with a unique singular orbit whose fibres are uncountable.
The first example shows that the aforementioned sufficient criterion for nontameness of almost automorphic shifts over irrational rotations of the circle, namely that the set of discontinuity points is a Cantor set, does not generalize when we replace the circle by a totally disconnected set. The proof of the second part of Theorem 1.3 is based on a slightly refined version of the constructions carried out in [Reference Fuhrmann and Kwietniak22], where the possible nontameness of systems with a unique singular orbit was already observed. Somewhat surprisingly, our constructions allow us to deduce the fact that it is possible for a minimal system $(X,T)$ to be nontame even if it is forward tame, that is, the corresponding forward motion is tame (see §1.1 for definitions).
In light of Theorem 1.3, it is not straightforward to identify a property that implies nontameness for Toeplitz shifts. Whilst stipulating that a Cantor set of singular fibres exists is not sufficient, additionally controlling the points in such a set of fibres does the trick. This control is granted by a property we refer to as thickness. Its definition necessitates extending the Bratteli–Vershik diagrams associated to Toeplitz shifts, see Definitions 2.7 and 2.10. Our main result, from which Theorems 1.1 and 1.2 follow, is the following theorem.
Theorem 1.4. Every thick Toeplitz shift is nontame.
Let us close the introduction by pointing out an interesting connection between tameness of substitution shifts and amorphic complexity, a topological invariant which was introduced to detect complex behaviour in the zero entropy regime [Reference Fuhrmann, Gröger and Jäger21]. In the case of symbolic systems, it coincides with the box dimension of the maximal equicontinuous factor, the box dimension being determined through an averaging metric which is defined by the dynamics. For constant length substitution shifts on a binary alphabet, Theorem 1.2 and [Reference Fuhrmann and Gröger20, Theorem 1.1] yield that $(X_\theta ,T)$ is tame if and only if its amorphic complexity is $1$ .
The work is organized as follows. In the remainder of this section, we collect some basic notions which are needed throughout the article. In §2, we give the Bratteli–Vershik and Toeplitz background needed and prove Theorem 1.4. We take some time to explain our results specified to the family of substitution shifts as this is an important family where the proofs are simpler and motivational for the following notions. We recall the general construction of semicocycle extensions in §3, and we obtain criteria for tameness of almost automorphic shifts by investigating their discontinuity points and their formulation as semicocycle extensions in §4. While our examples and main results solely deal with symbolic shifts on finite alphabets, it turns out that the extra effort due to treating general almost automorphic systems in that section is almost negligible. Finally, the first part of Theorem 1.3 is proven in §5 and the second part is dealt with in the last section.
1.1 Basic notions and notation
Most of this section is standard, see e.g. [Reference Auslander2, Reference de Vries6, Reference Gottschalk and Hedlund28]. We provide additional references for less standard material in the text.
We denote by $\mathbb {N}$ the positive integers and by $\mathbb {N}_0$ the nonnegative integers. A dynamical system is a continuous $\mathbb {Z}$ action (or $\mathbb {N}$ action) on a compact metric space X. Such a system is specified by a pair $(X,T)$ , where T is a continuous selfmap on X. Clearly, T is invertible when dealing with a $\mathbb {Z}$ action. Further, every $\mathbb {Z}$ action restricts to two $\mathbb {N}$ actions, its socalled forward motion given by positive powers of T, and its backward motion given by negative powers of T. Notions such as subsystem, minimality, (topological) factor, extension, conjugacy etc. have their standard meaning.
A $\mathbb {Z}$ action is called equicontinuous if the family $\{T^n : n\in \mathbb {Z}\}$ is equicontinuous. It is well known, see e.g. [Reference Downarowicz10], that a minimal equicontinuous system $(\mathcal Z,S)$ is a minimal rotation, that is, there is a continuous abelian group structure on $\mathcal Z$ and an element $g\in \mathcal Z$ such that the homeomorphism S is given by adding g, $S(z) = z+g$ . We hence refer to such a system by $(\mathcal Z,+g)$ . Minimality implies that g is a topological generator of $\mathcal Z$ , that is, $\{ng : n\in \mathbb {Z}\}$ is dense in $\mathcal Z$ .
An equicontinuous factor of $(X,T)$ is maximal if any other equicontinuous factor of $(X,T)$ factors through it. Here, $(X,T)$ is an almost onetoone extension of a system $(Y, S)$ if the associated factor map $\pi \colon X\to Y$ is almost onetoone, that is, if $\{x\in X: \pi ^{1}(\{\pi (x) \})=\{x\}\}$ is $G_\delta $ dense in X. A system $(X,T)$ is almost automorphic if it is an almost onetoone extension of a minimal equicontinuous factor. Almost automorphic systems are necessarily minimal and for minimal systems, $\{x\in X: \pi ^{1}(\{\pi (x) \})=\{x\}\}$ is a dense $G_\delta $ if it is nonempty. Given an almost automorphic system $\pi :(X,T)\to (\mathcal Z,+g)$ , we call the points $z\in \mathcal Z$ with a unique $\pi $ preimage regular, and those which have multiple preimages singular. Correspondingly, we call a $\pi $ fibre $\pi ^{1}(z)$ regular if it is a singleton and otherwise singular. A point $x\in X$ in a regular fibre is also called an injectivity point.
In §3, we discuss a natural representation of almost automorphic $\mathbb {Z}$ actions as (bilateral) shifts. Here, by shift, we mean a subsystem of the system $(K^{\mathbb {Z}},\sigma )$ given by the set of Kvalued bilateral sequences $(x_n)_{n\in \mathbb {Z}}\in K^{\mathbb {Z}}$ , equipped with the product topology, where K is a compact metric space and $\sigma $ is the left shift, $\sigma (x)_n = x_{n+1}$ .
The theory of topological independence allows for an alternative characterization of (non)tameness which turns out to be particularly convenient in explicit computations and does not explicitly involve the Ellis semigroup [Reference Kerr and Li33]. Given a system $(X,T)$ and subsets $A_0,A_1 \subseteq X$ , we say that $J\subseteq \mathbb {Z}$ is an independence set for $(A_0,A_1)$ if for each finite subset $I\subseteq J$ and every choice function $\varphi \in \{0,1\}^I$ , there exists $x\in X$ such that $T^i(x)\in A_{\varphi (i)}$ for each $i\in I$ . By combining the results from [Reference Kerr and Li33] and the aforementioned shift representation of almost automorphic systems, we obtain the following characterization of nontameness which we actually understand as its definition in the following proposition.
Proposition 1.5. (Cf. [Reference Kerr and Li33, Proposition 6.4] and [Reference Fuhrmann and Kwietniak22, Proposition 3.1])
A shift $(X,\sigma )\subseteq (K^{\mathbb {Z}},\sigma )$ is nontame (or wild) if and only if there are disjoint compact sets $V_0,\, V_1\subseteq K$ and a sequence of integers $(t_n)_{n\in \mathbb {N}}$ such that for each choice function $\varphi \in \{0,1\}^{\mathbb {N}}$ , there is $(x_n)_n\in X$ for which $x_{t_n}\in V_{\varphi (n)}$ for all $n\in \mathbb {N}$ .
We call the infinite sequence $(t_n)$ of the proposition an independence sequence for the pair $(V_0,V_1)$ and note that its elements must be pairwise distinct. In the terminology of [Reference Kerr and Li33], the elements of $(t_n)$ form an independence set for the pair of cylinder sets
In line with the above characterization of nontameness, we call a shift $(X,\sigma )$ forward nontame if it allows for an infinite independence sequence (for some disjoint cylinder sets) of positive integers. Similar to the characterization of nontameness of $\mathbb {Z}$ actions given in the introduction, this is the case if and only if the Ellis semigroup of the $\mathbb {N}$ action given by the forward motion of $(X,\sigma )$ contains a copy of $\beta \mathbb {N}$ .
Remark 1.6. In the case of symbolic shifts, that is, when $K=\{a_0,\ldots ,a_n\}$ is finite, it is important to note that $V_0$ and $V_1$ can be chosen to be singletons $\{a_k\}$ and $\{a_\ell \}$ , respectively, and we simply write $[a_k],[a_\ell ]$ for the corresponding cylinder sets. Indeed, with [Reference Kerr and Li33, Proposition 6.4], we have that if there exist $V_0$ and $V_1$ as in Proposition 1.5, then there are $x\in [V_0]$ and $y\in [V_1]$ such that for any pair of neighbourhoods $U_x$ and $U_y$ of x and y, respectively, there is an infinite independence set. One may hence choose disjoint cylinder sets $U_x=[\ast \cdots \ast a_k\ast \cdots \ast ]$ and $U_y=[\ast \cdots \ast a_\ell \ast \cdots \ast ]$ (where $\ast $ is a placeholder and $a_k\neq a_\ell $ ) to see that actually, we have an independence sequence for $(\{a_k\},\{a_\ell \})$ .
In particular, to prove tameness (or forward tameness) of binary shifts, that is, where $K = \{a,b\}$ , it suffices to show that $([a],[b])$ does not allow for an infinite independence set (of positive integers).
2 Toeplitz shifts
Our main goal in this work is to characterize those Toeplitz shifts which are tame. In the first two parts of this section, we define Toeplitz shifts, and in particular a simple and ubiquitous subclass, which is defined by (primitive, aperiodic) substitutions of constant length which have a coincidence. While our results are far more general, we will later use examples from this class to illustrate our constructions. These constructions are carried out in §2.4, where we turn to the Bratteli–Vershik representation of Toeplitz shifts to prove Theorems 1.1, 1.2 and 1.4.
2.1 Odometers and Toeplitz shifts
Given a sequence $(\ell _n)$ of natural numbers, we work with the group
where the group operation is given by coordinatewise 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 leftinfinite 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 biinfinite 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 $\sigma ^i(\theta ^n(X_\theta ))$ , $i=0,\ldots ,\ell ^n1$ , 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 onetoone 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
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 nonempty. 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.
Lemma 2.1. Let $\theta $ be a primitive aperiodic substitution of constant length with pure base $\theta '$ . Its maximal equicontinuous factor contains either finitely many, or uncountably many orbits of singular points. The latter is the case if and only if $\mathcal G_{\theta '}$ contains two distinct cycles which share a common vertex.
Proof. Observe that besides the paths corresponding to the orbit through $0$ , two infinite paths in $\mathcal G_{\theta '}$ belong to the same $\mathbb {Z}$ orbit of $\mathbb {Z}_\ell $ if they differ only on a finite initial segment, that is, if they are tail equivalent. Therefore, the statement comes down to showing that there are finitely many or uncountably many distinct infinite paths up to tail equivalence in $\mathcal G_{\theta '}$ .
Clearly, $\mathcal G_{\theta '}$ must contain cycles as it contains infinite paths. If we have a vertex in two different cycles then, starting from that vertex, we can follow through the two cycles in any order we wish and therefore the number of paths in $\mathcal G_{\theta '}$ is uncountable.
Now assume that there is no vertex in two different cycles. An infinite path must visit some vertex infinitely often. As this vertex is not part of more than one cycle, the path must eventually follow the same cycle. As there are only finitely many vertices and edges, there can only be finitely many cycles. Hence, up to tail equivalence, there are only finitely many infinite paths in $\mathcal G_{\theta '}$ .
Let us anticipate the following important consequence of Lemma 2.1 combined with Theorem 2.18 and the discussion in §2.3.4.
Theorem 2.2. Let $\theta $ be a primitive aperiodic substitution of constant length with pure base $\theta '$ . Then $(X_\theta ,\sigma )$ is tame if and only if it has a coincidence and $\mathcal G_{\theta '}$ does not contain two distinct cycles which share a common vertex.
Example A. Let $\theta $ be the substitution
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 lefthand 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 nontame.
Example B. We modify slightly the above example and define $\theta $ as
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.
2.3.1 Bratteli–Vershik systems
A Bratteli diagram is an infinite graph $B=(V,E)$ , where the vertex set $V=\bigsqcup _{n\geq 0}V_n$ and the edge set $E=\bigsqcup _{n\geq 0}E_n$ are equipped with a range map $r\colon E\to V$ and a source map $s\colon E\to V$ such that:

(1) $V_0=\{v_0\}$ is a singleton;

(2) $V_n$ and $E_n$ are finite sets;

(3) $r(E_n)= V_{n+1}$ , $s(E_n)= V_{n}$ ;

(4) $r^{1}(v)\neq \emptyset $ for all $v\neq v_0$ .
The pair $(V_n,E_n)$ or just $V_n$ is called the nth level of the diagram B. A finite or infinite sequence of edges $(\gamma _n : \gamma _n\in E_n)$ such that $r(\gamma _{n})=s(\gamma _{n+1})$ is called a finite or infinite path, respectively. The source map and the range map extend to paths in the obvious way. For a Bratteli diagram B, let $X_B$ be the set of infinite paths $(\gamma _n)_{n\geq 0}$ starting at the top vertex $v_0$ . Given a path $(\gamma _n)_{n\geq 0}$ and $m\geq 0$ , we call $(\gamma _n)_{n\geq m}$ a tail of $(\gamma _n)$ and $(\gamma _n)_{n\leq m}$ a head of $(\gamma _n)$ . Two paths $(\gamma _n)$ and $ (\gamma ^{\prime }_n)$ are called tail equivalent (or cofinal) if they share a common tail.
We shall constantly use the telescoping procedure. Let B be a Bratteli diagram and $n_0 = 0 <n_1<n_2 < \cdots $ be a strictly increasing sequence of integers. The telescoping of B to $(n_k)$ is the Bratteli diagram $B'$ , whose klevel vertex set is $V_k':= V_{n_k}$ , and where the set of edges between $v\in V_k'$ and $w\in V_{k+1}'$ are in onetoone correspondence with the set of paths in B between $v\in V_{n_k}$ and $w\in V_{n_{k+1}}$ . There is then an obvious bijection between $X_B$ and $X_{B'}$ .
A Bratteli diagram B has rank d if there is a telescoping $B'$ of B such that $B'$ has exactly d vertices at each level. We say that B is simple if for any level m, there is $n>m$ such that for any two vertices $v\in E_n$ and $w\in V_m$ , there is a path with source w and range v. This is equivalent to saying that by telescoping, we can arrive at a Bratteli diagram $B'$ such that any two vertices in consecutive levels are connected by an edge.
Recall also that the path space $X_B$ comes with a totally disconnected compact metrizable topology, and if B is simple and $E_n>1$ infinitely often, then $X_B$ is a Cantor set.
Definition 2.3. We say that $B=(V,E)$ has the equal path number property if there is a sequence $(\ell _n)_{n\geq 0}$ such that for each $v\in V_{n+1}$ , there are $\ell _n$ edges with range v. We call $(\ell _n)_{n\geq 0}$ the characteristic sequence of B.
An ordered Bratteli diagram is a Bratteli diagram together with a total order on each set of edges which end at the same vertex. In other words, for each $r^{1}(v)$ , $v\in V$ , the order naturally defines a bijection $\omega :r^{1}(v)\to \{0,\ldots ,r^{1}(v)1\}$ . We refer to $\omega (e)$ also as the label of e. Under certain circumstances, given in detail in [Reference Gjerde and Johansen23], this order induces a proper order $\omega $ on $X_B$ . In a nutshell, the successor of an infinite path $(\gamma _n)_{n\geq 0}\in X_B$ with respect to $\omega $ , when this exists, is a tailequivalent path whose order labelling corresponds to an addition of $1$ to that of $(\gamma _n)_{n\geq 0}$ . The notion of a predecessor is defined analogously. The order being proper means that there is a unique path (the maximal path) which has no successor in this order, and a unique path (the minimal path) which has no predecessor. In this case, one can define the Vershik map $\varphi _\omega : X_B\rightarrow X_B$ , which sends a nonmaximal path to its successor and which sends the unique maximal path to the unique minimal path. It is a homeomorphism and thus defines a dynamical system $(X_B,\varphi _\omega )$ referred to as a Bratteli–Vershik system. If $(B,\omega )$ is a properly ordered Bratteli diagram and $B'=(V',E')$ is the telescoping of B to levels $(n_k)$ , then the order $\omega $ defines a natural proper order $\omega '$ on $B'$ , and $(X_B,\varphi _\omega )$ is topologically conjugate to $(X_{B'},\varphi _{\omega '})$ . Properly ordered simple Bratteli diagrams define minimal Bratteli–Vershik systems. Conversely, any Cantor minimal dynamical system $(X,T)$ is conjugate to a Bratteli–Vershik system where B is simple [Reference Herman, Putnam and Skau29]; the latter is called a Bratteli–Vershik representation of $(X,T)$ . Not every such dynamical system has a Bratteli–Vershik representation with finite rank, but if this is the case, one says that $(X,T)$ has finite topological rank. More precisely, the topological rank of such a system is the smallest rank among its Bratteli–Vershik representations.
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 levelwise application of the edge order map
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
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
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
is given by the composition
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,
(with the understanding that $\prod _{m=n_k}^{n_k1} \ell _{m}=1$ ). For instance, if we telescope level n with level $n+1$ , we get ${\theta '}^{(n)} = \theta ^{(n)} \theta ^{(n+1)}$ and
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 biinfinite 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 lengthk 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
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 onetoone 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 builtin 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 $x_n x_{n+1}\cdots x_{m1}$ .
Lemma 2.6. Let $(X_B,\varphi _\omega )$ be a Toeplitz Bratteli–Vershik system with characteristic sequence $(\ell _n)$ where $\ell _0=1$ . Set $\ell ^{(n)} = \prod _{k=0}^{n1}\ell _k$ . For all $\gamma \in X_B$ and $n\in \mathbb {N}$ , we have
where $z^{(n)} = \sum _{k=0}^{n1} \omega (\gamma _k) \ell ^{(k)}$ . In particular,
2.3.4 Toeplitz Bratteli–Vershik diagrams for constant length substitutions
A stationary Toeplitz Bratteli–Vershik system is one where for all $n\geq 1$ , $V_n=V_1$ , $E_n=E_1$ , and the order structure on $E_n$ is the same as that on $E_1$ . In this case, the morphisms $\theta ^{(n)}$ of equation (2.2) all agree and hence define a single morphism $\theta :V_1\to V_1^{\ell _1}$ . If the range map is a bijection between $E_0$ and $V_1$ (there is a single edge between $v_0$ and each of the vertices of $V_1$ ), we can identify the space $X_1$ with the substitution shift space $X_\theta $ of $\theta $ . In other words, a stationary Toeplitz Bratteli–Vershik system for which $E_0\cong V_1$ defines a primitive substitution of constant length.
The converse, associating a Toeplitz Bratteli–Vershik system to a primitive substitution $\theta $ of constant length $\ell $ over an alphabet ${\mathcal A}$ , is subtle. The natural approach [Reference Vershik and Livshits41], which consists of defining a stationary Bratteli–Vershik system by setting $V_n={\mathcal A}$ and defining the edges and their order with $\theta ^{(n)}=\theta $ as in equation (2.2), works well if all substitution words start with the same letter and end with the same letter, that is, $\theta _0(a)$ and $\theta _{\ell 1}(a)$ are independent of a. Indeed, if that is the case, then the order on the Bratteli diagram is proper. However, for general $\theta $ , the respectively defined order may fail to be proper and the arguments in [Reference Vershik and Livshits41] only give a measurable Bratteli–Vershik representation. While there are classical methods to rewrite the substitution to obtain a stationary Bratteli–Vershik representation for $(X_\theta ,\sigma )$ [Reference Durand, Host and Skau16, Reference Forrest17], those procedures do not necessarily give a Toeplitz Bratteli–Vershik representation. Instead, one needs to follow the approach of [Reference Gjerde and Johansen23] to obtain a properly ordered Toeplitz Bratteli–Vershik system such that $(X_1,\sigma )$ equals $(X_\theta , \sigma )$ .
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.
Definition 2.7. Let $(B, \omega )$ be an ordered Bratteli diagram with the equal path number property and characteristic sequence $(\ell _n)_{n \geq 0}$ . The extended Bratteli diagram is an infinite graph which satisfies the properties (1)–(3) of a Bratteli diagram, but not necessarily property (4). The extended Bratteli diagram $\tilde B=(\tilde V_n,\tilde E_n)$ associated to B has the following vertices and edges.

(1) The level n vertex set $ \tilde V_n$ is the set of all nonempty subsets of $V_n$ .

(2) For $0\leq i < \ell _n$ , $\tilde E_n$ contains an edge labelled i with source $A\in \tilde V_{n}$ and range $B\in \tilde V_{n+1}$ if and only if $\theta ^{(n)}_{i}(B) = A$ .
Identifying singleton sets with the element they contain, we see that $\tilde B$ contains B as a subdiagram. 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
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 oneelement 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 nondecreasing cardinality. This motivates the following definition.
Definition 2.8. If a path of $X_{\tilde B}$ eventually goes through vertices $A_n\in \tilde V_n$ with $A_n= k<\infty $ for all n large, we will say that the path has thickness k. Otherwise, we say that the path has infinite thickness.
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 subdiagram $X_{\tilde B}^{1}$ 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,
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.
Proof. Recall the definition of the maps ${\theta ^{({n})}_{\tilde \gamma }}$ and subsets $A_n:=s(\gamma ^{(n)}) \subseteq V_n$ associated to a path $\tilde \gamma \in X_{\tilde B}$ in equation (2.5). If $\mathrm {thk}(\tilde \gamma )\geq k$ , then there exists $n_0$ such that $A_{n}\geq k$ for $n\geq n_0$ . Since $ {\theta ^{({n})}_{\tilde \gamma }} (A_{n+1})=A_{n}$ , there are at least k paths $\gamma $ in the original path space $X_B$ such that $\omega (\tilde \gamma ) = \omega (\gamma )$ . This shows the inequality ‘ $\geq $ ’.
Now suppose that $\{\gamma \in {X_{B}}: \omega (\gamma )=z\} \geq k$ , so that there are at least k distinct paths $\gamma {\in X_{ B}}$ with the same edge labels. We need to make sure that there is at least one n such that they go through k different vertices at level n. Note that if two paths with equal edge labels agree on a vertex at level n, then their head agrees up to level n. Thus, k distinct paths must at some level go through k distinct vertices. This implies that there is an $A_{n}$ with $A_{n}\geq k$ which is a vertex of a path in $X_{\tilde B}$ which has edge labels z. Thus, $\sup \{\mathrm {thk}(\tilde \gamma ):\tilde \gamma \in X_{\tilde B}, \omega (\tilde \gamma ) = z\}\geq k$ .
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.
Definition 2.10. The largest k such that $X_{\tilde B}^{k}$ is uncountable is called the essential thickness of $(X_B,\varphi _\omega )$ . We say that $(X_B,\varphi _\omega )$ is thick if its essential thickness k is strictly larger than $1$ and finite, and if there is a double path of thickness k. A Toeplitz shift is thick if it has a thick Toeplitz Bratteli–Vershik representation.
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
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 $2$ .
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 B. (Continued)
It is not difficult to derive an extended Bratteli diagram for the substitution of Example B as well. What one will find is that there is one edge between two consecutive vertices $\{a,b,c\}$ and one edge between two consecutive vertices $\{b,c\}$ . It follows that the diagram has thickness $3$ . However, there is only one path which goes infinitely often through $\{a,b,c\}$ and only countably many which go infinitely often through $\{b,c\}$ . It follows that the essential thickness of the diagram is $1$ . The system is therefore not thick.
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 onesided versions of this family to maximize the similarity to his original description. Given a sequence $(\ell _n)$ of natural numbers, define the substitutions
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 $\ell _1=3$ , $\ell _2=5$ .
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 onesided Bratteli–Vershik system is conjugate to the onesided shift defined by Oxtoby. Oxtoby showed that if $(\ell _n)$ grows fast enough, in particular if $\sum ({\ell _{k1}}/{\ell _k})<1$ , then the resulting system is not uniquely ergodic and thus it cannot be tame. On the righthand 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 onesided representation of the perioddoubling 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.
Lemma 2.11. Let $(X_B,\varphi _\omega )$ be a Toeplitz Bratteli–Vershik system.

(1) If all paths in $X_{\tilde B}$ have finite thickness, the maximal equicontinuous factor contains uncountably many singular points if and only if there is $j>1$ such that $X_{\tilde B}^{j}$ is uncountable.

(2) If $X_{\tilde B}^{j}$ contains a double path, then it is uncountable.

(3) If $X_{\tilde B}$ has finite rank and $X_{\tilde B}^{j}$ is uncountable, then $X_{\tilde B}^j$ contains a double path.
Proof. For $z\in \mathbb {Z}_{(\ell _n)}$ and $n\in \mathbb {N}$ , let $B^z_n\subseteq V_n$ be of maximal cardinality among those elements of $\tilde V_n$ which are traversed by some path $\tilde \gamma \in X_{\tilde B}$ with $\omega (\tilde \gamma )=z$ . Observe that $B^z_n$ is uniquely determined because if $B,B'\in V_n$ are traversed by such a $\tilde \gamma $ , then $B\cup B'$ is extendable and there is such a $\tilde \gamma $ going through $B\cup B'$ . This shows that the supremum in the formula of Lemma 2.9 is attained at some path $\tilde \gamma $ . Since all paths in $X_{\tilde B}$ have finite thickness, Lemma 2.9 implies that the restriction of $\omega $ to $X_B$ is finitetoone, and this implies that the restriction of $\omega $ to $X^j_{\tilde B}$ is finitetoone. Hence, if $X_{\tilde B}^{j}$ is uncountable, then its image under $\omega $ must be uncountable. The converse follows from Lemma 2.9, which tells us that the singular points of $\mathbb {Z}_{(\ell _n)}$ are given by the image of $X_{\tilde B}\setminus X_{\tilde B}^1=\bigcup _{j\geq 2} X_{\tilde B}^j$ .
Suppose that $X_{\tilde B}^{j}$ contains a double path. Then there is $n_0$ such that for all $n\geq n_0$ , there is $A_n\subseteq V_n$ containing j elements such that between $A_n$ and $A_{n1}$ , there are at least $2$ edges in the extended Bratteli diagram. The set of paths in $X_{\tilde B}^{j}$ obtained by choosing one of the two edges at each level is uncountable.
Suppose now that $X_{\tilde B}^{j}$ is uncountable. Then the set of condensation points $X_{\tilde B}^{j}\cap \mathcal C $ of $X_{\tilde B}^{j}$ is uncountable. Since $X_{\tilde B}^{j}\cap \mathcal C $ has no isolated points, for any given path $\gamma \in X_{\tilde B}^{j}\cap \mathcal C $ and $n\geq 0$ , there are infinitely many distinct paths in $X_{\tilde B}^{j}\cap \mathcal C $ which agree with $\gamma $ on its head of length n. Let K be the rank of the Toeplitz Bratteli–Vershik system and pick some $m_1\geq 1$ . There is $m> m_1$ and $K+1$ paths of $X_{\tilde B}^{j}\cap \mathcal C $ which agree with $\gamma $ ’s head of length $m_1$ but pairwise disagree on the head of length m. Since infinitely often $V_n=K$ , the pigeon hole principle requires that two of the distinct paths must meet a common vertex of level $m_2>m$ . Telescoping the levels $m_1$ through $m_2$ , the part of these two paths between level $m_1$ and level $m_2$ defines a parallel edge. Iterating this procedure ( $m_2$ playing the role of $m_1$ etc.) proves the statement.
The following corollary can be seen as a generalization of Lemma 2.1 to all Toeplitz shifts with finite Toeplitz rank.
Corollary 2.12. Let $(X_B,\varphi _\omega )$ be a Toeplitz Bratteli–Vershik system with finite Toeplitz rank. The maximal equicontinuous factor contains uncountably many singular points if and only if $(X_B,\varphi _\omega )$ is thick.
Proof. Finite rank implies that all paths in $X_{\tilde B}$ are of finite thickness. Now, part (1) of Lemma 2.11 gives that we have uncountably many singular points if and only if the essential thickness is strictly greater than $1$ . Part (3) of the same lemma gives that if the essential thickness is k, then there is a double path of thickness k.
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
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
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 $\ell =0,\ldots ,n1$ 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
Indeed, the contrary, that is, the assumption that for all but at most countably many $\varphi $ , we have
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.
Corollary 2.14. Let $(X_B,\varphi _\omega )$ be a thick Toeplitz Bratteli–Vershik system with essential thickness k. Possibly, after telescoping, there exists a choice function $\varphi $ such that $ {\theta ^{({m})}_{\varphi (\bar {\gamma })}} (V_{m+1})=k$ for all large enough m.
2.6 Toeplitz systems and nontameness
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
Proof. Observe that the Haar probability measure $\mu $ of the set $D\subseteq \mathbb {Z}_{\ell _n}$ of singular points is
Let $n\geq m\in \mathbb {N}_0$ and $w= w_n\cdots w_m$ , with $0\leq w_i \leq \ell _i1$ . 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
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.
Proposition 2.16. Let $(X_B,\varphi _\omega )$ be a thick Toeplitz Bratteli–Vershik system. Suppose that the set of singular points has Haar measure $0$ . Possibly, after telescoping, there exist:

(1) for any $n\geq 1$ , an arithmetic progression $j_0^{(n)}, j_1^{(n)}, j_2^{(n)}\in \{0,\ldots ,\ell _n1\}$ (that is, $j_2j_1=j_1j_0$ ), and sets $A_n\subseteq V_n$ , such that $\theta ^{(n)}_{j_1^{(n)}}$ and $\theta ^{(n)}_{j_2^{(n)}}$ restrict to the same bijection from $A_{n+1}$ to $A_{n}$ , while $B_{n} :={ \theta ^{(n)}_{j_0^{(n)}}(V_{n+1})}$ is a proper subset of $ A_{n}$ ;

(2) for any $n>1$ , an $i_n\in \{0,\ldots ,\ell _n1\}$ such that ${\theta ^{({n})}_{i_n}}(V_{n+1})$ is contained in $\{a\in A_{n} : {{\theta ^{({n1})}_{j_1^{(n)}}}}(a) \notin B_{n1}\}$ .
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,
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
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 $(\omega (\gamma _{n,1})+\Delta _{n}\mathbb {Z}) \cap \{0,\ldots ,\ell _n1\}$ 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, $\kappa _n\in \{0,\ldots ,\ell _n1\}$ 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 $\psi ^{(n)}_j:\tilde V_{n+2}\to \tilde V_{n1}$ through
and note that $\psi ^{(n)}_j{\kern1pt}(A_{n+2}) {\kern1.2pt}\subseteq{\kern1.2pt} A_{n1}$ . Moreover, the inclusion is proper if ${\theta ^{({n})}_{j}}{\kern1pt}(A_{n+1})<k$ , while $\psi ^{(n)}_j$ is a bijection if $j=\omega (\gamma _{n,1})$ or $j=\omega (\gamma _{n,2})$ . Thus, we can find $j_0^{(n)},j_1^{(n)},j_2^{(n)} \in (\omega (\gamma _{n,1})+\Delta _{n}\mathbb {Z}) \cap \{0,\ldots ,\ell _n1\}$ such that $j_2^{(n)}j_1^{(n)} = j_1^{(n)}j_0^{(n)}$ and $ B_{n1} = \psi ^{(n)}_{j_0^{(n)}}(A_{n+2}) $ is a proper subset of $ A_{n1}$ , while $\psi ^{(n)}_{j_1^{(n)}}(A_{n+2}) = A_{n1}$ and $\psi ^{(n)}_{j_2^{(n)}}(A_{n+2}) = A_{n1}$ . 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
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 nonsurjective 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
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
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 ${{\theta ^{({n1})}_{j_1^{(n)}}}}(a) \notin B_{n1}$ . By minimality (and perhaps further telescoping), we find $i_n$ such that ${\theta ^{({n})}_{{i_n}}}(V_{n+1})=\{a\}$ .
Theorem 2.17. Every thick Toeplitz shift is nontame.
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 nontame [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 ${f_1^{(n1)}} h_{n}(A_{n+1})\subseteq A_{n1}\backslash B_{n1}$ . Set
$t_0=0$ , $t_1 = (j^{(2)}_1 i_{2} ) \ell _1 + \Delta _1$ , and for $n\geq 2$ ,
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
whereas if $\varphi _0 = 1$ , then
Furthermore,
Hence, taking into account that $f^{(n)}_1=f^{(n)}_2$ , we have
Since
it follows that
Similarly, we find for all $n\geq 2$ ,
By finiteness of $A_{1}$ , there is a subsequence of $B_{2n+1}$ such that