2.2.1 Subshifts

Let $\mathcal {A}$ be a finite set that we call alphabet. Elements in $\mathcal {A}$ are called letters or symbols. The number of letters of $\mathcal {A}$ is denoted by $|\mathcal {A}|$ . The set of finite sequences or words of length $\ell \in \mathbb {N}$ with letters in $\mathcal {A}$ is denoted by $\mathcal {A}^\ell $ and the set of two-sided sequences $(x_n)_{n \in \mathbb {Z}}$ in $\mathcal {A}$ is denoted by $\mathcal {A}^{\mathbb {Z}}$ . A word $w = w_0 w_1 \ldots w_{\ell - 1} \in \mathcal {A}^\ell $ can be seen as an element of the free monoid $\mathcal {A}^\ast $ endowed with the operation of concatenation (whose neutral element is $\varepsilon $ , the empty word). The integer $\ell $ is the length of the word w and is denoted by $|w| = \ell $ ; the length of the empty word is zero. A word v is a power of a word u if $v = u^n$ for some $n \in \mathbb {N}$ .

For finite words p and s in $\mathcal {A}^\ast $ , we say that they are a prefix and a suffix, respectively, of the word $p s$ . For $x \in \mathcal {A}^{\mathbb {Z}}$ and integers $N> n$ we define the word ${x_{[n,N)} = x_n x_{n + 1} \ldots x_{N - 1}}$ . For a non-empty word $w \in \mathcal {A}^\ast $ and a point $x \in \mathcal {A}^{\mathbb {Z}}$ , we say that w occurs in x if there exists $n \in \mathbb {Z}$ such that $x_n x_{n+1} \ldots x_{n + |w| - 1} = w$ . In this case, we say that the index n is an occurrence of w in x. We use the same notion for finite non-empty words x. We say that a non-empty word $w = w_0 w_1 \ldots w_{\ell - 1} \in \mathcal {A}^\ast $ starts (respectively, ends) with a non-empty word $u \in \mathcal {A}^\ast $ if $u = w_0 \ldots w_{i - 1}$ for some $i \le \ell $ (respectively, $u = w_j \ldots w_{\ell - 1}$ for some $j \ge 0$ ).

The shift map $S : \mathcal {A}^{\mathbb {Z}} \to \mathcal {A}^{\mathbb {Z}}$ is defined by $S ((x_n)_{n \in \mathbb {Z}}) = (x_{n+1})_{n \in \mathbb {Z}}$ . A subshift is a topological dynamical system $(X, S)$ where X is a closed and S-invariant subset of $\mathcal {A}^{\mathbb {Z}}$ . Here, we consider the product topology on $\mathcal {A}^{\mathbb {Z}}$ . Classically, one identifies $(X, S)$ with X, so one says that X itself is a subshift. When we say that a sequence x in a subshift is aperiodic, we implicitly mean that x is aperiodic for the action of the shift.

Let $(X, S)$ be a subshift. The language of $(X, S)$ is the set $\mathcal {L}(X)$ containing all words $w \in \mathcal {A}^\ast $ such that $w = x_{[m, m + |w|)}$ for some $x = (x_n)_{n \in \mathbb {Z}} \in X$ and $m \in \mathbb {Z}$ . In this case, we also say that w is a factor (also called subword) of x. We denote by $\mathcal {L}_\ell (X)$ the set of words of length $\ell $ in $\mathcal {L}(X)$ . Given $x \in X$ , the language $\mathcal {L}(x)$ is the set of all words that occur in x. As before, we define $\mathcal {L}_\ell (x)$ . For two words $u,v \in \mathcal {L}(X)$ , the cylinder set $[u.v]$ is the set $\{x \in X : x_{[-|u|,|v|)} = u v\}$ . When u is the empty word we only write $[v]$ , erasing the dot. We remark that cylinder sets are clopen sets and they form a base for the topology of the subshift.

2.2.2 Morphisms

Let $\mathcal {A}$ and $\mathcal {B}$ be finite alphabets and $\tau : \mathcal {A}^\ast \to \mathcal {B}^\ast $ be a morphism. We say that $\tau $ is erasing whenever there exists a letter $a \in \mathcal {A}$ such that $\tau (a)$ is the empty word. Otherwise, we say it is non-erasing. When the morphism $\tau $ is non-erasing, it extends naturally to a map from $\mathcal {A}^{\mathbb {Z}}$ to $\mathcal {B}^{\mathbb {Z}}$ by concatenation (we apply $\tau $ to positive and negative coordinates separately and we concatenate the results at coordinate zero). We continue to call this map $\tau $ . We observe that any map $\tau : \mathcal {A} \to \mathcal {B}^\ast $ can be naturally extended to a morphism (that we also denote by $\tau $ ) from $\mathcal {A}^\ast $ to $\mathcal {B}^\ast $ by concatenation.

The composition matrix of a morphism $\tau : \mathcal {A}^\ast \to \mathcal {B}^\ast $ is given for each $a \in \mathcal {A}$ and $b \in \mathcal {B}$ by $M_\tau (b,a) = |\tau (a)|_b$ , where $|\tau (a)|_b$ counts the number of occurrences of the letter b in the word $\tau (a)$ . The morphism $\tau $ is said to be positive if $M_\tau $ has positive entries and proper if there exist $p, s \in \mathcal {B}$ such that for all $a \in \mathcal {A}$ the word $\tau (a)$ starts with p and ends with s.

The minimum and maximal lengths of $\tau $ are, respectively, the numbers

$$ \begin{align*} \langle \tau \rangle = \min_{a \in \mathcal{A}} |\tau(a)| \quad \text{and} \quad |\tau| = \max_{a \in \mathcal{A}} |\tau(a)|. \end{align*} $$

We say that a morphism $\tau $ is of constant length if $\langle \tau \rangle = |\tau |$ . Observe that if $\tau : \mathcal {A}^\ast \to \mathcal {B}^\ast $ and $\tau ' : \mathcal {B}^\ast \to \mathcal {C}^\ast $ are two constant length morphisms, then $\tau ' \circ \tau $ is also of constant length and

(2) $$ \begin{align} |\tau' \circ \tau| = |\tau'| |\tau|. \end{align} $$

Following [Reference Berthé, Steiner, Thuswaldner and YassawiBSTY19], a morphism $\tau : \mathcal {A}^\ast \to \mathcal {B}^\ast $ is left permutative (respectively, right permutative) if the first (respectively, last) letters of $\tau (a)$ and $\tau (b)$ are different, for all distinct letters $a,b \in \mathcal {A}$ . Two morphisms $\tau , \widetilde {\tau } : \mathcal {A}^\ast \to \mathcal {B}^\ast $ are said to be rotationally conjugate if there is a word $w \in \mathcal {B}^\ast $ such that $\tau (a) w = w \widetilde {\tau }(a)$ for all $a \in \mathcal {A}$ or ${\widetilde {\tau }(a) w = w \tau (a)}$ for all $a \in \mathcal {A}$ .

2.2.3 $\mathcal {S}$ -adic subshifts

We recall the definition of $\mathcal {S}$ -adic subshifts as stated in [Reference Berthé, Steiner, Thuswaldner and YassawiBSTY19]. A directive sequence is a sequence of non-erasing morphisms. A slightly more general definition is given in [Reference Durand and PerrinDP22] including the case of erasing morphisms. When all morphisms $\tau _n$ for $n \ge 0$ are proper, we say that is proper. For $0 \le n \le N$ , we denote by $\tau _{[n,N)}$ the morphism $\tau _n \circ \tau _{n+1} \circ \cdots \circ \tau _{N-1}$ , where $\tau _{[n,n)} : \mathcal {A}_n^\ast \to \mathcal {A}_n^\ast $ is the identity map for each $n \ge 0$ . We say is everywhere growing if $\langle \tau _{[0,n)} \rangle \to +\infty $ as $n \to +\infty $ and say that it is primitive if for any $n \in \mathbb {N}$ there exists $N> n$ such that $M_{\tau _{[n,N)}} $ has positive entries, that is, for every $a \in \mathcal {A}_N$ the word $\tau _{[n,N)}(a)$ contains all letters in $\mathcal {A}_n$ . Observe that primitivity implies everywhere growing. If is primitive, then the subshift is minimal (see, for instance, [Reference Durand and PerrinDP22, Proposition 6.4.5]). However, there are minimal subshifts that are generated by non-everywhere-growing directive sequences, as for the Chacon subshift generated by a constant directive sequence given by the morphism $0 \mapsto 0010$ , $1 \mapsto 1$ .

For $n \in \mathbb {N}$ , the language

of level n associated with

is defined by

and let

be the set of points $x \in \mathcal {A}_n^{\mathbb {Z}}$ such that

. This set clearly defines a subshift that we call the subshift generated by

. We set

and call

or

the $\mathcal {S}$ -adic subshift generated by the directive sequence

.

A contraction of

is a directive sequence of the form

where the sequence $(n_k)_{k \ge 0}$ is such that $n_0 = 0$ and $n_k < n_{k+1}$ for all $k \ge 0$ . Observe that any contraction of

generates the same $\mathcal {S}$ -adic subshift

.

We say that a directive sequence is invertible if the linear map $M_{\tau _n} : \mathbb {R}^{\mathcal {A}_n} \to \mathbb {R}^{\mathcal {A}_{n+1}}$ (acting on row vectors) is invertible for all $n \ge 0$ . Observe that this implies that the sequence $(|\mathcal {A}_n|)_{n \ge 0}$ is constant.

The following proposition generalizes [Reference Berthé, Cecchi Bernales, Durand, Leroy, Perrin and PetiteBCBD+21, Lemma 3.3]. The proof is similar and we include it here for the sake of completeness.

2.2.4 Recognizability

Let $\tau : \mathcal {A}^\ast \to \mathcal {B}^\ast $ be a non-erasing morphism and $X \subseteq \mathcal {A}^{\mathbb {Z}}$ be a subshift. For $x \in X$ and $k \in \mathbb {N}$ with $0 \le k < |\tau (x_0)|$ , the cutting points of the pair $(k, x)$ are defined as follows. If $\ell \ge 0$ , we define the $\ell $ th cutting point of $(k, x)$ as

$$ \begin{align*} C_\tau^\ell (k, x) = |\tau(x_{[0, \ell)})| - k. \end{align*} $$

Similarly, if $\ell < 0$ the $\ell $ th cutting point of $(k, x)$ is $C_\tau ^\ell (k, x) = -|\tau (x_{[\ell , 0)})| - k$ . Define $\mathcal {C}_\tau ^+(k, x) = \{C_\tau ^\ell (k, x) : \ell> 0\}$ .

If $y = S^k \tau (x)$ with $x \in X$ and $k \in \mathbb {N}$ , $0 \le k < |\tau (x_0)|$ , we say that $(k, x)$ is a centered $\tau $ -representation of y. The centered $\tau $ -representation $(k, x)$ is in X if x belongs to X. The morphism $\tau $ is recognizable in X (respectively, recognizable in X for aperiodic points) if any point $y \in \mathcal {B}^{\mathbb {Z}}$ (respectively, any aperiodic point $y \in \mathcal {B}^{\mathbb {Z}}$ ) has at most one centered $\tau $ -representation in X. If $\tau $ is recognizable in $\mathcal {A}^{\mathbb {Z}}$ (for aperiodic points), we say that $\tau $ is fully recognizable (for aperiodic points).

In what follows, we use the following results [Reference Berthé, Steiner, Thuswaldner and YassawiBSTY19, Theorem 3.1, Lemma 3.5].

We also need the following straightforward lemma [Reference Durand and PerrinDP22, Proposition 1.4.30].

2.2.5 Recognizability for sequences of morphisms

Following [Reference Berthé, Steiner, Thuswaldner and YassawiBSTY19], a directive sequence is said to be recognizable at level n if the morphism $\tau _n$ is recognizable in . We say that the directive sequence is recognizable if it is recognizable at level n for each $n \ge 0$ .

We have that is recognizable if and only if for all $0 \le n < N$ and any point there is a unique couple $(k, x)$ with and $0 \le k < |\tau _{[n,N)}(x_0)|$ such that $y = S^k \tau _{[n,N)}(x)$ . This is the content of [Reference Berthé, Steiner, Thuswaldner and YassawiBSTY19, Lemmas 3.5 and 4.2]. Indeed, is recognizable if and only if for all $n \ge 0$ and any point there is a unique couple $(k, x)$ with and $0 \le k < |\tau _{[0,n)}(x_0)|$ such that $y = S^k \tau _{[0,n)}(x)$ .

Lemma 2.4 implies the following.

2.3.1 CKR partitions of minimal Cantor systems

A clopen Kakutani–Rokhlin (CKR) partition $\operatorname {\mathcal {T}}$ of $(X, T)$ is a partition of X of the form

$$ \begin{align*} \operatorname{\mathcal{T}} = \{T^k B(a) : a \in \mathcal{A}(\operatorname{\mathcal{T}}\,), 0 \le k < h(a)\}, \end{align*} $$

where $\mathcal {A}(\operatorname {\mathcal {T}}\,)$ is a non-empty finite alphabet, the value $h(a)$ is a positive integer and $B(a)$ is a clopen set for all $a \in \mathcal {A}(\operatorname {\mathcal {T}}\,)$ . Observe that

$$ \begin{align*} \bigcup_{a \in \mathcal{A}(\operatorname{\mathcal{T}}\,)} T^{h(a)} B(a) = \bigcup_{a \in \mathcal{A}(\operatorname{\mathcal{T}}\,)} B(a). \end{align*} $$

The base of $\operatorname {\mathcal {T}}$ is the set $B(\operatorname {\mathcal {T}}\,) = \bigcup _{a \in \mathcal {A}(\operatorname {\mathcal {T}}\,)} B(a)$ . The set $\operatorname {\mathcal {T}}(a) = \bigcup _{0 \le k < h(a)} T^k B(a)$ is called the tower indexed by $a \in \mathcal {A}(\operatorname {\mathcal {T}}\,)$ of $\operatorname {\mathcal {T}}$ with base $B(a)$ and height $h(a)$ .

Let

$$ \begin{align*} \operatorname{\mathcal{T}}_n = \{T^k B_n(a) : a \in \mathcal{A}(\operatorname{\mathcal{T}}_n), 0 \le k < h_n(a)\}, \quad n \ge 0 \end{align*} $$

be a sequence of CKR partitions of $(X, T)$ . It is nested if for any $n \ge 0$ :

1. (KR1) $B(\operatorname {\mathcal {T}}_{n+1}) \subseteq B(\operatorname {\mathcal {T}}_n)$ ;

2. (KR2) $\operatorname {\mathcal {T}}_n \preceq \operatorname {\mathcal {T}}_{n+1}$ , that is, for every $A \in \operatorname {\mathcal {T}}_{n+1}$ there exists $B \in \operatorname {\mathcal {T}}_n$ such that $A \subseteq B$ ;

3. (KR3) $\bigcap _{n \ge 0} B(\operatorname {\mathcal {T}}_n) = \{x\}$ for some point $x\in X$ ; and

4. (KR4) the atoms of $\bigcup _{n \ge 0} \operatorname {\mathcal {T}}_n$ generate the topology of X.

We remark that nested sequences always exist [Reference Herman, Putnam and SkauHPS92, Theorem 4.2].

For each $n \ge 0$ , the incidence matrix $M_n$ between the partitions $\operatorname {\mathcal {T}}_{n+1}$ and $\operatorname {\mathcal {T}}_n$ is given for each $a \in \mathcal {A}(\operatorname {\mathcal {T}}_n)$ and $b \in \mathcal {A}(\operatorname {\mathcal {T}}_{n+1})$ by

(3) $$ \begin{align} M_n(a,b) = \# \{0 \le k < h_{n+1}(b) : T^k B_{n+1}(b) \subseteq B_n(a)\}. \end{align} $$

For $n \ge 0$ let $h_n$ be the row vector called height vector and defined by

$$ \begin{align*} h_n = (h_n(a))_{a \in \mathcal{A}(\operatorname{\mathcal{T}}_n)}. \end{align*} $$

We define $P_{m,n} = M_m M_{m+1} \ldots M_{n-1}$ for $0 \le m < n$ . Observe that $P_{n,n+1} = M_n$ . By means of a simple induction argument, we have $h_n = h_m P_{m,n}$ and

(4) $$ \begin{align} P_{m,n}(a,b) = \# \{0 \le k < h_n(b) : T^k B_n(b) \subseteq B_m(a)\}, \end{align} $$

$a \in \mathcal {A}(\operatorname {\mathcal {T}}_m)$ , $b \in \mathcal {A}(\operatorname {\mathcal {T}}_n)$ , $0 \le m < n$ .

The topological rank of $(X, T)$ is the value

(5) $$ \begin{align} \operatorname{rank} (X, T) = \inf_{\substack{\text{nested sequence }(\operatorname{\mathcal{T}}_n)_{n \ge 0} \\ \text{of CKR partitions of }(X, T)}} \liminf_{n \to +\infty} |\mathcal{A}(\operatorname{\mathcal{T}}_n)|. \end{align} $$

Roughly speaking, the topological rank of $(X, T)$ is the smallest number of CKR towers needed to describe $(X, T)$ . The topological rank is invariant under topological conjugacy. See [Reference Bressaud, Durand and MaassBDM10, Reference Downarowicz and MaassDM08] for more details.

2.3.2 Invariant measures through CKR partitions

Let $(\operatorname {\mathcal {T}}_n)_{n \ge 0}$ be a nested sequence of CKR partitions. Any T-invariant probability measure $\mu $ of $(X,T)$ is uniquely determined by the values it assigns to atoms of the partitions, hence to the bases $B_n(a)$ , $a \in \mathcal {A}(\operatorname {\mathcal {T}}_n)$ and $n \ge 0$ .

For $n \ge 0$ let $\mu _n$ be the column vector called measure vector and defined by

$$ \begin{align*} \mu_n = (\mu_n(a))_{a \in \mathcal{A}(\operatorname{\mathcal{T}}_n)}, \quad \text{where } \mu_n(a) = \mu(B_n(a)). \end{align*} $$

Therefore, the measure $\mu $ is completely determined by the sequence of measure vectors $(\mu _n)_{n \ge 0}$ . As $\mu $ is a probability measure, we have

(6) $$ \begin{align} \mu(\operatorname{\mathcal{T}}_n(a)) = h_n(a) \mu_n(a) \quad \text{and} \quad \sum_{a \in \mathcal{A}(\operatorname{\mathcal{T}}_n)} \mu(\operatorname{\mathcal{T}}_n(a)) = 1. \end{align} $$

In addition, by (4) we have

(7) $$ \begin{align} \mu_m = P_{m,n} \mu_n, \quad 0 \le m < n. \end{align} $$

2.3.3 CKR partitions of $\mathcal {S}$ -adic subshifts

Let be a primitive, proper and recognizable directive sequence which generates the $\mathcal {S}$ -adic subshift . Define the sequence $(\operatorname {\mathcal {T}}_n)_{n \ge 0}$ as follows:

(8) $$ \begin{align} \operatorname{\mathcal{T}}_n = \{S^k \tau_{[0,n)}([a]) : a \in \mathcal{A}_n, 0 \le k < |\tau_{[0,n)}(a)|\}, \quad n \ge 0. \end{align} $$

The following result proved in [Reference Durand and LeroyDL12, Proposition 2.2] shows that $(\operatorname {\mathcal {T}}_n)_{n \ge 0}$ defines a nested sequence of CKR partitions. We include a proof for the sake of completeness.

Proposition 2.6. The sequence $(\operatorname {\mathcal {T}}_n)_{n \ge 0}$ is a nested sequence of CKR partitions of . Moreover, for each $n \ge 0$ the incidence matrix $M_n$ between the partitions $\operatorname {\mathcal {T}}_{n+1}$ and $\operatorname {\mathcal {T}}_n$ coincides with the composition matrix $M_{\tau _n}$ of the morphism $\tau _n$ :

$$ \begin{align*} M_n = M_{\tau_n}. \end{align*} $$

Proof. As is recognizable, $\operatorname {\mathcal {T}}_n$ is a CKR partition of for each $n \ge 0$ . Observe that the tower $\operatorname {\mathcal {T}}_n(a)$ has base $B_n(a) = \tau _{[0,n)}([a])$ for $a \in \mathcal {A}_n$ . Clearly we have $B(\operatorname {\mathcal {T}}_{n+1}) \subseteq B(\operatorname {\mathcal {T}}_n)$ for $n \ge 0$ .

Claim 2.6.1. We claim that $\operatorname {\mathcal {T}}_n \preceq \operatorname {\mathcal {T}}_{n+1}$ .

Indeed, let $S^k \tau _{[0, n+1)}([a])$ be an atom of $\operatorname {\mathcal {T}}_{n+1}$ , $a \in \mathcal {A}_{n+1}$ , $0 \le k < |\tau _{[0, n+1)}(a)|$ . Let $\tau _n(a) = b_0 b_1 \ldots b_{i-1}$ with $b_j \in \mathcal {A}_n$ , $0 \le j < i$ . Then, there exists $j \in [0, i-1)$ satisfying

$$ \begin{align*} |\tau_{[0,n)}(b_0 b_1 \ldots b_j)| \le k < |\tau_{[0,n)}(b_0 b_1 \ldots b_{j+1})|. \end{align*} $$

We deduce that if $k' = |\tau _{[0,n)}(b_0 b_1 \ldots b_j)|$ , then $S^k \tau _{[0,n+1)}([a]) \subseteq S^{k - k'} \tau _{[0,n)}([b_{j+1}])$ with $0 \le k - k' < |\tau _{[0,n)}(b_{j+1})|$ . This proves the claim.

Claim 2.6.2. The atoms of $\bigcup _{n \ge 0} \operatorname {\mathcal {T}}_n$ generate the topology of .

Indeed, let $n \ge 1$ , $a \in \mathcal {A}_n$ , $0 \le k < |\tau _{[0,n)}(a)|$ and $\ell $ be a non-negative integer. As $\tau _n$ is proper, there exist two letters $p_n$ and $s_n$ in $\mathcal {A}_n$ such that $\tau _n(a)$ starts with $p_n$ and ends with $s_n$ for all $a \in \mathcal {A}_{n+1}$ and $n \ge 0$ . As is primitive, there exists $N \in \mathbb {N}$ such that if $n \ge N$ then $\langle \tau _{[0,n-1)} \rangle \ge \ell $ . Let $x', y' \in \tau _{[0,n)}([a])$ , $u_n = \tau _{[0,n-1)}(s_{n-1})$ and $v_n = \tau _{[0,n)}(a) \tau _{[0,n-1)}(p_{n-1})$ . We have

$$ \begin{align*} x^{\prime}_{[-|u_n|, |v_n|)} = y^{\prime}_{[-|u_n|, |v_n|)} = u_n v_n, \end{align*} $$

so that $x^{\prime }_{[-\ell , \ell + k]} = y^{\prime }_{[-\ell , \ell + k]}$ . If $x,y$ belong to $S^k \tau _{[0,n)}([a])$ , $n \ge N$ , then $x_{[-\ell - k, \ell ]} = y_{[-\ell - k, \ell ]}$ and, in particular, $x_{[-\ell , \ell ]} = y_{[-\ell , \ell ]}$ . Therefore, $\operatorname {diam}(S^k \tau _{[0,n)}([a])) \to 0$ as ${n \to +\infty }$ . This proves the claim. As the bases $(B(\operatorname {\mathcal {T}}_n))_{n \ge 0}$ are nested, they converge to some point. This finishes the proof of the first statement.

The second statement follows easily from the recognizability of $\boldsymbol{\tau }$ .

We remark that the height vectors $(h_n)_{n \ge 0}$ of $(\operatorname {\mathcal {T}}_n)_{n \ge 0}$ defined by (8) satisfy

(9) $$ \begin{align} h_n(a) = |\tau_{[0, n)}(a)|, \quad a \in \mathcal{A}_n, \quad n \ge 0. \end{align} $$

2.4.1 Direct limits

Let $(G_n)_{n \ge 0}$ be a sequence of abelian groups and let $i_{n+1, n} : G_n \to G_{n+1}$ for each $n \ge 0$ be a morphism. Define the subgroups $\Delta $ and $\Delta ^0$ of the direct product $\prod _{n \ge 0} G_n$ by

$$ \begin{align*} \Delta = \{(g_n)_{n \ge 0} \in \textstyle\prod_{n \ge 0} G_n : g_{n+1} = i_{n+1, n}(g_n)\ \text{for every large enough }n\} \end{align*} $$

and

$$ \begin{align*}\Delta^0 = \{(g_n)_{n \ge 0} \in \textstyle\prod_{n \ge 0} G_n : g_n = 0\ \text{for every large enough }n\}. \end{align*} $$

Let $G = \Delta / \Delta ^0$ be the quotient group and $\pi : \Delta \to G$ be the natural projection. The group G is called the direct limit of $(G_n)_{n \ge 0}$ and we write $G = \varinjlim G_n$ . If $g \in G_n$ , then all sequences $(g_k)_{k \ge 0}$ such that $g_n = g$ and $g_{k+1} = i_{k+1, k}(g_k)$ for all $k \ge n$ belong to $\Delta $ and have the same projection in G, denoted by $i_n(g)$ . This defines a group morphism ${i_n : G_n \to G}$ , which we call the natural morphism from $G_n$ to G. For $0 \le m < n$ define

$$ \begin{align*} i_{n, m} = i_{n-1, n} \circ i_{n, n+1} \circ \cdots \circ i_{m+1, m}. \end{align*} $$

We have $i_m = i_n \circ i_{n, m}$ and $G = \bigcup _{n \ge 0} \operatorname {Im} i_n$ .

We can also define direct limits of vector spaces. Let $\mathbb {K}$ be a field. For each $n \ge 0$ , let $V_n$ be a vector space over $\mathbb {K}$ and $i_{n+1, n} : V_n \to V_{n+1}$ be a linear map. The direct limit $V = \varinjlim V_n$ is the vector space over $\mathbb {K}$ , where the group structure on V is that given by the direct limit of the abelian groups $V_n$ and the scalar multiplication is given by pointwise scalar multiplication on each coordinate.

2.4.2 Orbit equivalence

Two minimal Cantor systems $(X, T)$ and $(X', T')$ are orbit equivalent if there exists a homeomorphism $\Phi : X \to X'$ which sends orbits onto orbits, that is,

$$ \begin{align*} \Phi(\{T^n x : n \in \mathbb{Z}\}) = \{(T')^n \circ \Phi (x) : n \in \mathbb{Z}\}, \quad x \in X. \end{align*} $$

This implies that there exist two maps $\alpha : X \to \mathbb {Z}$ and $\beta : X' \to \mathbb {Z}$ , uniquely defined by aperiodicity, such that

$$ \begin{align*} \Phi \circ T (x) = (T')^{\alpha(x)} \circ \Phi(x) \quad \text{and} \quad \Phi \circ T^{\beta(x)} (x) = T' \circ \Phi(x), \quad x \in X. \end{align*} $$

The minimal Cantor systems $(X, T)$ and $(X', T')$ are strongly orbit equivalent if $\alpha $ and $\beta $ both have at most one point of discontinuity.

2.4.3 Dimension groups of minimal Cantor systems

Denote by $C(X, \mathbb {Z})$ (respectively, $C(X, \mathbb {N})$ ) the group (respectively, monoid) of continuous functions from X to $\mathbb {Z}$ (respectively, $\mathbb {N}$ ) with the addition operation. Consider the map $\partial : C(X, \mathbb {Z}) \to C(X, \mathbb {Z})$ defined by $\partial f = f \circ T - f$ .

A map f is called a coboundary if there exists $g \in C(X, \mathbb {Z})$ such that $f = \partial g$ . Two maps $f, f' \in C(X, \mathbb {Z})$ are said to be cohomologous if $f - f'$ is a coboundary.

Define the quotient group $H(X, T) = C(X, \mathbb {Z}) / \partial C(X, \mathbb {Z})$ . Let $[f]$ be the class of $f \in C(X, \mathbb {Z})$ in $H(X,T)$ and $\pi : C(X, \mathbb {Z}) \to H(X,T)$ be the projection map. Define $H^+(X, T) = \pi (C(X, \mathbb {N}))$ and denote by ${\boldsymbol {1}_X}$ the constant one valued function.

Consider the triple

$$ \begin{align*} K^0(X, T) = (H(X, T), H^+(X, T), [\boldsymbol{1}_X]). \end{align*} $$

It is an ordered group with order unit $[\boldsymbol {1}_X]$ . As $(X, T)$ is minimal, it is a dimension group. See [Reference Durand and PerrinDP22, Reference Giordano, Putnam and SkauGPS95] for the definitions and more details. We call it the dimension group of $(X, T)$ .

It is classical to observe that if $(X, T)$ is topologically conjugate to $(X', T')$ , then the ordered groups with order units $K^0(X, T)$ and $K^0(X', T')$ are unital order isomorphic, i.e., there exists a group morphism $\delta : H(X, T) \to H(X', T')$ such that $\delta (H^+(X, T))= H^+(X', T')$ and $\delta ([\boldsymbol {1}_X]) = [\boldsymbol {1}_{X'}]$ .

Denote by $\mathcal {M}(X, T)$ the set of invariant probability measures of $(X, T)$ . We define the set of infinitesimals of $H(X, T)$ as

$$ \begin{align*} \operatorname{Inf} H(X, T) = \bigg\{[f] \in H(X, T) : \int f \,d\mu = 0\ \text{for all}\ \mu \in \mathcal{M}(X, T)\bigg\}. \end{align*} $$

We have that $H(X, T) / \operatorname {Inf} H(X, T)$ with the induced order is also a dimension group. We denote it by $K^0(X, T) / \operatorname {Inf} K^0(X, T)$ .

The dimension groups $K^0(X, T)$ and $K^0(X, T) / \operatorname {Inf} K^0(X, T)$ characterize strong orbit equivalence and orbit equivalence, respectively [Reference Giordano, Putnam and SkauGPS95].

Another description of the dimension group $K^0(X, T)$ is as follows. Let $(\operatorname {\mathcal {T}}_n)_{n \ge 0}$ be a nested sequence of CKR partitions of $(X, T)$ as defined in §2.3. Let $(\mathcal {A}(\operatorname {\mathcal {T}}_n))_{n \ge 0}$ , $(h_n)_{n \ge 0}$ and $(M_n)_{n \ge 0}$ be the associated sequences of alphabets, height vectors and incidence matrices, respectively.

For $n \ge 0$ we consider $\mathbb {Z}^{\mathcal {A}(\operatorname {\mathcal {T}}_n)}$ as an ordered group of row vectors with the usual order. Define the sequence of ordered groups with order units

$$ \begin{align*} \mathcal{G}_n = (\mathbb{Z}^{\mathcal{A}(\operatorname{\mathcal{T}}_n)}, \mathbb{Z}_+^{\mathcal{A}(\operatorname{\mathcal{T}}_n)}, h_n), \quad n \ge 0. \end{align*} $$

Let $\mathcal {G} = \varinjlim \mathcal {G}_n$ be the direct limit of the groups $\mathcal {G}_n$ with respect to the morphisms $M_n : \mathbb {Z}^{\mathcal {A}(\operatorname {\mathcal {T}}_n)} \to \mathbb {Z}^{\mathcal {A}(\operatorname {\mathcal {T}}_{n+1})}$ given by the incidence matrix $M_n$ (acting on row vectors). Let $\mathcal {G}^+$ be the projection in $\mathcal {G}$ of the set of points $(x_n)_{n \ge 0} \in \textstyle \prod _{n \ge 0} \mathbb {Z}^{\mathcal {A}(\operatorname {\mathcal {T}}_n)}$ for which there exists $N \in \mathbb {N}$ such that $x_N \in \mathbb {Z}_+^{\mathcal {A}(\operatorname {\mathcal {T}}_N)}$ and $x_{k+1} = x_k M_k$ , $k \ge N$ . Denote by u the projection in $\mathcal {G}$ of the sequence $(h_n)_{n \ge 0}$ .

The tuple $\mathcal {K} = (\mathcal {G}, \mathcal {G}^+, u)$ is a dimension group. The introduction of this dimension group is motivated by the following proposition [Reference Durand and PerrinDP22, Theorem 5.3.6].

2.4.4 Dimension groups of $\mathcal {S}$ -adic subshifts

Let $\boldsymbol {\tau } = (\tau _n : \mathcal {A}_{n+1}^\ast \to \mathcal {A}_n^\ast )_{n \ge 0}$ be a primitive, proper and recognizable directive sequence. Suppose that $(X_{\boldsymbol {\tau }}, S)$ is aperiodic. Let $(\operatorname {\mathcal {T}}_n)_{n \ge 0}$ be the sequence of CKR partitions given in (8) and $\mathcal {K}$ be the dimension group associated to it. Recall that, by Proposition 2.6, the incidence matrix $M_n$ between the partitions $\operatorname {\mathcal {T}}_{n+1}$ and $\operatorname {\mathcal {T}}_n$ coincide with the composition matrix $M_{\tau _n}$ of the morphism $\tau _n$ .

We call $\mathcal {K}$ the dimension group of $\boldsymbol {\tau }$ . By Proposition 2.7, the dimension group of $(X_{\boldsymbol {\tau }}, S)$ is unital order isomorphic to $\mathcal {K}$ .

In the case where all the linear maps $M_{\tau _n}$ , $n \ge 1$ are invertible, it is easy to check from the definition that the dimension group is unital order isomorphic to $(\mathcal {G}, \mathcal {G}^+, u)$ , where

$$ \begin{align*} \mathcal{G} &= \{x \in \mathbb{R}^{\mathcal{A}_1} : x M_{\tau_1} M_{\tau_2} \ldots M_{\tau_n} \in \mathbb{Z}^{\mathcal{A}_{n+1}}\ \text{for large enough }n\},\\ \mathcal{G}^+ &= \{x \in \mathbb{R}^{\mathcal{A}_1} : x M_{\tau_1} M_{\tau_2} \ldots M_{\tau_n} \in \mathbb{Z}_+^{\mathcal{A}_{n+1}}\ \text{for large enough }n\}, \end{align*} $$

and $u = (|\tau _0(a)|)_{a \in \mathcal {A}_1} \in \mathbb {R}^{\mathcal {A}_1}$ .

4.4.1 Basics on tensor products

We need very classical facts on tensor products between abelian groups and $\mathbb {Q}$ . We recall what is needed to follow our arguments and refer to [Reference BourbakiBou62] for more details.

Let G be an abelian group. Then, it has a $\mathbb {Z}$ -module structure and we can define the tensor product $G \otimes \mathbb {Q}$ . Moreover, this product has the structure of a vector space over $\mathbb {Q}$ . Elements in $G \otimes \mathbb {Q}$ are linear combinations of the form

$$ \begin{align*} \sum_{k=0}^n g_k \otimes q_k, \quad n \in \mathbb{N}, \quad g_k \in G, \quad q_k \in \mathbb{Q}, \quad 0 \le k \le n. \end{align*} $$

An element of the form $g \otimes q$ with $g \in G$ and $q \in \mathbb {Q}$ is said to be a pure tensor.

4.4.2 Back to the computation of the topological rank

The following lemma gives a lower bound for the topological rank of a minimal Cantor system.

Lemma 4.7. Let $(X, T)$ be a minimal Cantor system. Then

(34) $$ \begin{align} \operatorname{rank}_{\mathbb{Q}} H(X, T) \le \operatorname{rank} (X, T). \end{align} $$

Proof. We can assume $\operatorname {rank} (X, T) < +\infty $ . Let $(\operatorname {\mathcal {T}}_n)_{n \ge 0}$ be any nested sequence of CKR partitions of $(X, T)$ such that $\liminf _{n \to +\infty } |\mathcal {A}(\operatorname {\mathcal {T}}_n)| < +\infty $ . Denote by $(M_n)_{n \ge 0}$ the sequence of incidence matrices of $(\operatorname {\mathcal {T}}_n)_{n \ge 0}$ . We can assume that $|\mathcal {A}(\operatorname {\mathcal {T}}_n)| = p$ , $n \ge 0$ for some $p \in \mathbb {N}$ .

By Proposition 2.7, the dimension group $H(X, T)$ can be seen as the direct limit $\varinjlim \mathbb {Z}^{\mathcal {A}(\operatorname {\mathcal {T}}_n)}$ with linear maps $M_n : \mathbb {Z}^{\mathcal {A}(\operatorname {\mathcal {T}}_n)} \to \mathbb {Z}^{\mathcal {A}(\operatorname {\mathcal {T}}_{n+1})}$ , $n \ge 0$ . Define the linear maps $j_{n+1, n} : \mathbb {Z}^{\mathcal {A}(\operatorname {\mathcal {T}}_n)} \otimes \mathbb {Q} \to \mathbb {Z}^{\mathcal {A}(\operatorname {\mathcal {T}}_{n+1})} \otimes \mathbb {Q}$ on pure tensors by

$$ \begin{align*} j_{n+1, n}(v \otimes q) = v M_n \otimes q, \quad v \in \mathbb{Z}^{\mathcal{A}(\operatorname{\mathcal{T}}_n)}, \quad q \in \mathbb{Q}, \quad n \ge 0 \end{align*} $$

and extend them by linearity to $\mathbb {Z}^{\mathcal {A}(\operatorname {\mathcal {T}}_n)} \otimes \mathbb {Q}$ . We consider $\varinjlim (\mathbb {Z}^{\mathcal {A}(\operatorname {\mathcal {T}}_n)} \otimes \mathbb {Q})$ with linear maps $(j_{n+1,n})_{n \ge 0}$ .

Proposition 4.6 implies that $H(X, T) \otimes \mathbb {Q}$ and $\varinjlim (\mathbb {Z}^{\mathcal {A}(\operatorname {\mathcal {T}}_n)} \otimes \mathbb {Q})$ are isomorphic vector spaces over $\mathbb {Q}$ . Each morphism $j_n : \mathbb {Z}^{\mathcal {A}(\operatorname {\mathcal {T}}_n)} \otimes \mathbb {Q} \to \varinjlim (\mathbb {Z}^{\mathcal {A}(\operatorname {\mathcal {T}}_n)} \otimes \mathbb {Q})$ is linear and we have

$$ \begin{align*} \dim_{\mathbb{Q}} \operatorname{Im} j_n \le \dim_{\mathbb{Q}} \operatorname{Im} j_n + \dim_{\mathbb{Q}} \ker j_n = p. \end{align*} $$

As $\operatorname {Im} j_m \subseteq \operatorname {Im} j_n$ for $m < n$ , there exists $N \in \mathbb {N}$ such that $\operatorname {Im} j_m = \operatorname {Im} j_n$ for all $m, n \ge N$ . This, together with the fact that $\varinjlim (\mathbb {Z}^{\mathcal {A}(\operatorname {\mathcal {T}}_n)} \otimes \mathbb {Q}) = \bigcup _{n \ge 0} \operatorname {Im} j_n$ , implies that $ \operatorname {rank}_{\mathbb {Q}} H(X, T) \le p$ .

As the choice of the sequence $(\operatorname {\mathcal {T}}_n)_{n \ge 0}$ is arbitrary, we deduce (34).

The proof of the next lemma is essentially given in [Reference Berthé, Cecchi Bernales, Durand, Leroy, Perrin and PetiteBCBD+21, Theorem 4.1].

Proposition 4.8. Let

be a primitive, proper and invertible directive sequence. Let $d = |\mathcal {A}_0|$ . Then

Proof. By Proposition 2.1 we have that is an aperiodic subshift. Let $(\operatorname {\mathcal {T}}_n)_{n \ge 0}$ be the sequence given by (8). As is recognizable [Reference Berthé, Steiner, Thuswaldner and YassawiBSTY19, Theorem 3.1], by Proposition 2.6 we have that $(\operatorname {\mathcal {T}}_n)_{n \ge 0}$ is a nested sequence of CKR partitions. Hence, by (5) we have .

Now we show that

is finite-dimensional. Indeed, we prove that

$$ \begin{align*} B = \{[\chi_{[a]}] \otimes 1 : a \in \mathcal{A}_0\} \end{align*} $$

is a basis of

, where $[\chi _{[a]}]$ denotes the class of the characteristic function of the cylinder $[a]$ in

. This will finish the proof because (34) implies

Following the same steps as in the proof of [Reference Berthé, Cecchi Bernales, Durand, Leroy, Perrin and PetiteBCBD+21, Theorem 4.1] and because the matrix $M_{\tau _{[0,n)}}^{-1}$ has rational entries, we deduce that B spans . Suppose that $\alpha = (\alpha _a)_{a \in \mathcal {A}_0} \in \mathbb {Z}^{\mathcal {A}_0}$ is such that

$$ \begin{align*} \sum_{a \in \mathcal{A}_0} \alpha_a [\chi_{[a]}] \otimes 1 = 0. \end{align*} $$

The fact that is a torsion-free abelian group [Reference Durand and PerrinDP22, Proposition 2.1.13] and Proposition 4.6 imply that $\sum _{a \in \mathcal {A}_0} \alpha _a [\chi _{[a]}] = 0$ in . Then, as in the proof of [Reference Berthé, Cecchi Bernales, Durand, Leroy, Perrin and PetiteBCBD+21, Theorem 4.1], we obtain $\alpha = 0$ and B is a basis.

Proposition 4.8 and (30) directly imply the following.

Corollary 4.9. Let $(X_{\mathcal {W}}, S)$ be a minimal Ferenczi subshift. Then

$$ \begin{align*} \operatorname{rank} (X_{\mathcal{W}}, S) = d_{\mathcal{W}}. \end{align*} $$

4.6.1 Facts about numeration systems

Let $(p_k)_{k \ge 0}$ be a sequence of positive integers with $p_k \ge 2$ , $k \ge 0$ . Then, for every real number x with $0 \le x \le 1$ , there exists a sequence $(f_k)_{k \ge 1}$ such that $0 \le f_k \le p_{k-1}$ for $k \ge 1$ and

$$ \begin{align*} x = \sum_{k = 1}^\infty \frac{f_k}{p_0 p_1 \ldots p_{k-1}}. \end{align*} $$

We say that $(f_k)_{k \ge 1}$ is the expansion of x in the base $(p_k)_{k \ge 0}$ .

4.6.2 Ferenczi-type dimension groups

Let $\mathcal {B}$ be a non-empty alphabet. We define

$$ \begin{align*} U^{\mathcal{B}} = \{u \in \mathbb{Z}_{>0}^{\mathcal{B}} : u(b) \not= u(b')\ \text{for}\ b, b' \in \mathcal{B}\}. \end{align*} $$

Observe that the unit $u_{\mathcal {W}}$ in Proposition 4.10 belongs to $U^{\mathcal {B}_{\mathcal {W}}} \times \mathbb {Z}_{>0}$ because all elements in $\mathcal {A}_{\mathcal {W}}$ are distinct. Define

$$ \begin{align*} \Delta^{\mathcal{B}} = \{z \in \mathbb{R}_{>0}^{\mathcal{B}} : \textstyle\sum_{b \in \mathcal{B}} z(b) < 1 \} \end{align*} $$

and let $(r_n)_{n \ge 0}$ be a sequence of integers with $r_n \ge 2$ .

We say that a dimension group $(\mathcal {G}, \mathcal {G}^+, u)$ is of Ferenczi type if there exist a non-empty alphabet $\mathcal {B}$ , a sequence $(r_n)_{n \ge 0}$ as before and

such that

Proposition 4.10 shows that the dimension group of a minimal Ferenczi subshift is of Ferenczi type. Conversely, let $(\mathcal {G}, \mathcal {G}^+, u)$ be a dimension group of Ferenczi type given by $\mathcal {B}$ , $(r_n)_{n \ge 0}$ and . Write $u = (v, w)$ , where $v \in U^{\mathcal {B}}$ and $w \in \mathbb {Z}_{>0}$ .

Let $a' = w - 1$ and $s(b) = a' + v(b)$ for $b \in \mathcal {B}$ . Observe that $s(b)> a'$ and ${s(b) \not = s(b')}$ for $b, b' \in \mathcal {B}$ . Define any sequence of generating words $\mathcal {W}$ such that:

1. (1) $n_0 = 1$ and $\mathcal {A}_{\mathcal {W}} = \{s(b) : b \in \mathcal {B}\} \cup \{a'\}$ ;

2. (2) $q_n = r_n$ for $n \ge 0$ ; and

3. (3) for $b \in \mathcal {B}_{\mathcal {W}}$ , let $(f_k(s(b)))_{k \ge 1}$ be the expansion of in the base $(q_k + 1)_{k \ge 0}$ , that is,

   

From Equation (35), we have thus proved the following.