2.1. Descriptive set theory

In the rest of the paper, we will be using some concepts, terminology, and notation from descriptive set theory. In this subsection, we review these concepts, terminology, and notation, which can be found in [Reference Kechris35].

A Polish space is a topological space that is separable and completely metrizable.

Let X be a Polish space and $d_X$ be a compatible complete metric on X. Let $K(X)$ be the space of all compact subsets of X, and let $d_H$ be the Hausdorff metric defined on $K(X)$ as follows. For $A\in K(X)$ and $x\in X$ , let $d(x,A)=\inf \{d(x,y): y\in A\}$ . Now for $A, B\in K(X)$ , let

$$ \begin{align*} d_H(A,B)=\max\{ \sup\{d(x, B): x\in A\},\sup\{d(y,A): y\in B\}\}. \end{align*} $$

Then $d_H$ is a metric on $K(X)$ that makes $K(X)$ a Polish space. Moreover, if X is compact, then $K(X)$ is compact.

Let X be a Polish space. A subset A of X is $G_{\delta }$ if A is the intersection of countably many open subsets of X. A subspace Y of X is Polish if and only if Y is a $G_{\delta }$ subset of X. We say that a subset A of X is generic, or the elements of A are generic in X, if A contains a dense $G_{\delta }$ subset of X.

More generally, by a transfinite induction on $1\leq \alpha <\omega _1$ , we can define the Borel hierarchy on X as follows:

$$ \begin{align*} \boldsymbol{\Sigma}^0_1&= \text{ the collection of all open subsets of } X, \\ \boldsymbol{\Pi}^0_1&= \text{the collection of closed subsets of } X, \\ \boldsymbol{\Sigma}^0_{\alpha}&= \bigg\{ \bigcup_{n\in\mathbb{N}} A_n: A_n\in \boldsymbol{\Pi}^0_{\beta_n} \text{ for some } \beta_n<\alpha\bigg\}, \\ \boldsymbol{\Pi}^0_{\alpha} &= \{ X\setminus A: A\in \boldsymbol{\Sigma}^0_{\alpha}\}. \end{align*} $$

We also define $\boldsymbol {\Delta }^0_{\alpha }=\boldsymbol {\Sigma }^0_{\alpha }\cap \boldsymbol {\Pi }^0_{\alpha }$ . Thus, $\boldsymbol {\Delta }^0_1$ is the collection of all clopen subsets of X. With this notation, $\bigcup _{\alpha <\omega _1}\boldsymbol {\Sigma }^0_{\alpha }=\bigcup _{\alpha <\omega _1}\boldsymbol {\Pi }^0_{\alpha }=\bigcup _{\alpha <\omega _1}\boldsymbol {\Delta }^0_{\alpha }$ is the collection of all Borel subsets of X. The collection of all $G_{\delta }$ subsets of X is exactly $\boldsymbol {\Pi }^0_2$ .

Let X be a topological space. Recall that a subset A of X is nowhere dense in X if the interior of the closure of A is empty. Here, A is meager in X if $A\subseteq \bigcup _{n\in \mathbb {N}} B_n$ , where each $B_n$ is nowhere dense in X; A is non-meager in X if it is not meager in X; and A is comeager in X if $X\setminus A$ is meager in X.

2.2. Topological dynamical systems

The concepts we review in this subsection are standard and can be found in any standard text on topological dynamics, e.g., [Reference Auslander4, Reference Kůrka37]. By a topological dynamical system, we mean a pair $(X, T)$ , where X is a compact metrizable space and $T: X\to X$ is a homeomorphism. If $(X, T)$ is a topological dynamical system and $Y\subseteq X$ satisfies $TY=Y$ , then Y is called a T-invariant subset.

If $(X, T)$ and $(Y, S)$ are topological dynamical systems and $\varphi : X\to Y$ is a continuous surjection satisfying $\varphi \circ T=S\circ \varphi $ , then $\varphi $ is called a factor map and $(Y, S)$ is called a (topological) factor of $(X, T)$ . If in addition $\varphi $ is a homeomorphism, then it is called a (topological) conjugacy (map), and we say that $(X, T)$ and $(Y,S)$ are (topologically) conjugate.

If $(X, T)$ is a topological dynamical system and $\rho $ is a compatible metric on X, then $\rho $ is necessarily complete since X is compact. Let $(X, T)$ be a topological dynamical system and fix $\rho $ a compatible metric on X. We say that $(X, T)$ is equicontinuous if for all $\epsilon>0$ , there is $\delta>0$ such that for all $n\in \mathbb {Z}$ , if $\rho (x,y)<\delta $ , then $\rho (T^nx, T^ny)<\epsilon $ . Since X is compact, the equicontinuity is a topological notion and does not depend on the compatible metric $\rho $ .

Every topological dynamical system $(X, T)$ has a maximal equicontinuous factor (or MEF), that is, an equicontinuous factor $(Y, S)$ with the factor map $\varphi $ such that if $(Z, G)$ is another equicontinuous factor of $(X, T)$ with factor map $\psi $ , then there is a factor map $\theta : (Y,S)\to (Z, G)$ such that $\psi =\theta \circ \varphi $ .

If $(X, T)$ is a topological dynamical system and $x\in X$ , the orbit of x is defined as $\{T^kx: k\in \mathbb {Z}\}$ . If A is a clopen subset of X, the return times of x to A is defined as $\mathrm {Ret}_A(x) =\{n\in \mathbb {Z}: T^nx\in A\}$ . We regard $\mathrm {Ret}_A(x)$ as an element of $2^{\mathbb {Z}}=\{0,1\}^{\mathbb {Z}}$ .

2.3. Minimal Cantor systems

Recall that a Cantor space is a zero-dimensional, perfect, compact metrizable space. Let X be a Cantor space and $T: X\to X$ be a homeomorphism. Then $(X, T)$ is called a Cantor system. Here, T is minimal if every orbit is dense, that is, for all $x\in X$ , $\{T^kx: k\in \mathbb {Z}\}$ is dense in X. A minimal Cantor system is a pair $(X, T)$ , where X is a Cantor space and $T: X\to X$ is a minimal homeomorphism.

Let $\mathcal {C}=2^{\mathbb {N}}=\{0,1\}^{\mathbb {N}}$ be the infinite product of the discrete space $\{0,1\}$ with the product topology. Then every Cantor space is homeomorphic to $\mathcal {C}$ . Let $d_{\mathcal {C}}$ be the canonical compatible complete metric on $\mathcal {C}$ , that is, for $x, y\in \mathcal {C}$ , if $x\neq y$ , then

$$ \begin{align*} d_{\mathcal{C}}(x,y)=2^{-n}\quad \text{where } n\in\mathbb{N} \text{ is the least such that } x(n)\neq y(n). \end{align*} $$

Let

$$ \begin{align*} \mathrm{Aut}(\mathcal{C})=\{T: T \text{ is a homeomorphism from } \mathcal{C} \text{ to } \mathcal{C}\} \end{align*} $$

be equipped with the compact-open topology, or equivalently the supnorm metric, that is, for $T, S\in \mathrm { Aut}(\mathcal {C})$ ,

$$ \begin{align*} d(T, S)=\sup\{d_{\mathcal{C}}(Tx, Sx): x\in \mathcal{C}\}. \end{align*} $$

Then $\mathrm {Aut}(\mathcal {C})$ is a Polish space (cf, e.g., [Reference Kechris and Rosendal36]). Let $M(\mathcal {C})$ be the set of all minimal homeomorphisms of $\mathcal {C}$ . Then for a $T\in \mathrm { Aut}(\mathcal {C})$ , $T\in M(\mathcal {C})$ if and only if for all non-empty clopen $U\subseteq \mathcal {C}$ , there is $N\in \mathbb {N}$ such that $\mathcal {C}=\bigcup _{-N\leq n\leq N} T^nU$ . This characterization implies that $M(\mathcal {C})$ is a $G_{\delta }$ subset of $\mathrm {Aut}(\mathcal {C})$ , and hence $M(\mathcal {C})$ is also a Polish space. Here, $M(\mathcal {C})$ is our coding space for all minimal Cantor systems.

We will also consider essentially minimal Cantor systems. A Cantor system $(X, T)$ is essentially minimal if it contains a unique minimal set, that is, a non-empty closed T-invariant set which is minimal among all such sets.

2.4. Ordered Bratteli diagrams

The concepts and terminology reviewed in this subsection are from [Reference Downarowicz and Maass14, Reference Giordano, Putnam and Skau30, Reference Herman, Putnam and Skau33]. Some notation are from [Reference Donoso, Durand, Maass and Petite13, Reference Durand16]. Recall that a Bratteli diagram is an infinite graph $(V,E)$ with the following properties:

• • the vertex set V is decomposed into pairwise disjoint non-empty finite sets $V=V_0\cup V_1\cup V_2\cup \cdots $ , where $V_0$ is a singleton $\{v_0\}$ ;

• • the edge set E is decomposed into pairwise disjoint non-empty finite sets $E=E_1 \cup E_2\cup \cdots $ ;

• • for any $n\geq 1$ , each $e\in E_n$ connects a vertex $u\in V_{n-1}$ with a vertex $v\in V_n$ . In this case, we write $\mathsf {s}(e)=u$ and $\mathsf {r}(e)=v$ . Thus, $\mathsf {s}, \mathsf {r}: E\to V$ are maps such that ${\mathsf {s}(E_n)=V_{n-1}}$ and $\mathsf {r}(E_n)=V_n$ for all $n\geq 1$ ;

• • $\mathsf {s}^{-1}(v)\neq \varnothing $ for all $v\in V$ and $\mathsf {r}^{-1}(v)\neq \varnothing $ for all $v\in V\setminus V_0$ .

An ordered Bratteli diagram is a Bratteli diagram $(V,E)$ together with a partial ordering $\preceq $ on E so that edges e and $e'$ are $\preceq $ -comparable if and only if $\mathsf {r}(e)=\mathsf {r}(e')$ .

A finite or infinite path in a Bratteli diagram $(V, E)$ is a sequence $(e_1, e_2, \ldots )$ , where $\mathsf {r}(e_i)=\mathsf {s}(e_{i+1})$ for all $i\geq 1$ . Given a Bratteli diagram $(V, E)$ and $0\leq n<m$ , let $E_{n,m}$ be the set of all finite paths connecting vertices in $V_n$ and those in $V_m$ . If $p=(e_{n+1},\ldots , e_m)\in E_{n,m}$ , define $\mathsf {r}(p)=\mathsf {r}(e_m)$ and $\mathsf {s}(p)=\mathsf {s}(e_{n+1})$ . If in addition the Bratteli diagram is partially ordered by $\preceq $ , then we also define a partial ordering ${p\preceq ' q}$ for $p=(e_{n+1},\ldots , e_m), q=(f_{n+1},\ldots , f_m)\in E_{n,m}$ as either $p=q$ or there exists ${n+1\leq i\leq m}$ such that $e_i\neq f_i$ , $e_i\preceq f_i$ , and $e_j=f_j$ for all $i<j\leq m$ . For an arbitrary strictly increasing sequence $(n_k)_{k\geq 0}$ of natural numbers with $n_0=0$ , we define the contraction or telescoping of a Bratteli diagram $(V,E)$ with respect to $(n_k)_{k\geq 0}$ as $(V',E')$ , where $V^{\prime }_k=V_{n_k}$ for $k\geq 0$ and $E^{\prime }_k=E_{n_{k-1}, n_k}$ for $k\geq 1$ . If in addition the given Bratteli diagram is ordered, then by contraction or telescoping, we also obtain an ordered Bratteli diagram $(V',E',\preceq ')$ with the order $\preceq '$ defined above. The inverse of the telescoping process is called microscoping. Two ordered Bratteli diagrams are equivalent if one can be obtained from the other by a sequence of telescoping and microscoping processes.

A Bratteli diagram $(V, E)$ is simple if there is a strictly increasing sequence $(n_k)_{k\geq 0}$ of natural numbers with $n_0=0$ such that the telescoping $(V', E')$ of $(V, E)$ with respect to $(n_k)_{k\geq 0}$ satisfies that for all $n\geq 1$ , $u\in V^{\prime }_{n-1}$ , and $v\in V^{\prime }_n$ , there is $e\in E^{\prime }_n$ with $\mathsf {s}(e)=u$ and $\mathsf {r}(e)=v$ . This is equivalent to the property that for any $n\geq 1$ , there is $m>n$ such that every pair of vertices $u\in V_n$ and $v\in V_m$ are connected by a finite path. It is obvious that if a Bratteli diagram B is simple, then any Bratteli diagram equivalent to it is also simple.

Given a Bratteli diagram $B=(V,E)$ , define

$$ \begin{align*} X_B=\{(e_n)_{n\geq 1}: e_n\in E_n, \mathsf{r}(e_n)=\mathsf{s}(e_{n+1}) \text{ for all } n\geq 1\}. \end{align*} $$

Since $X_B$ is a subspace of the product space $\prod _{n\geq 1}E_n$ , we equip $X_B$ with the subspace topology of the product topology on $\prod _{n\geq 1}E_n$ . An ordered Bratteli diagram ${B=(V, E,\preceq)}$ is essentially simple if there are unique elements such that for every $n\geq 1$ , $e_n$ is a $\preceq $ -maximal element and $f_n$ is a $\preceq $ -minimal element. Here, $B=(V, E, \preceq )$ is simple if $(V, E)$ is simple and B is essentially simple. If an ordered Bratteli diagram B is (essentially) simple, then any ordered Bratteli diagram equivalent to it is also (essentially) simple.

Given an essentially simple ordered Bratteli diagram $B=(V,E,\preceq )$ , we define the Vershik map $\unicode{x3bb} _B: X_B\to X_B$ as follows: ; if $(e_n)_{n\geq 1}\in X_B$ and , then let

where k is the least such that $e_k$ is not $\preceq $ -maximal, $f_k$ is the $\preceq $ -successor of $e_k$ , and $(f_1,\ldots , f_{k-1})$ is the unique path from $v_0$ to $\mathsf {s}(f_k)=\mathsf {r}(f_{k-1})$ such that $f_i$ is $\preceq $ -minimal for each $1\leq i\leq k-1$ . Then $(X_B, \unicode{x3bb} _B)$ is an essentially minimal Cantor system [Reference Herman, Putnam and Skau33], which we call the Bratteli–Vershik system generated by B. If $B=(V, E,\preceq )$ is a simple ordered Bratteli diagram and $X_B$ is infinite, then $(X_B,\unicode{x3bb} _B)$ is a minimal Cantor system [Reference Giordano, Putnam and Skau30]. If two simple ordered Bratteli diagrams are equivalent, then the Bratteli–Vershik systems generated by them are conjugate, with the conjugacy map sending to .

An essentially minimal Cantor system $(X, T)$ is of finite topological rank if it is conjugate to a Bratteli–Vershik system given by an essentially simple ordered Bratteli diagram $(V, E, \preceq )$ , where $(|V_n|)_{n\geq 1}$ is bounded by a natural number d. The minimum possible value of d is called the topological rank of the system, and is denoted by ${\mathrm {rank}_{\scriptsize \mathrm {top}}}(X,T)$ . Such Bratteli diagrams have been called finite rank Bratteli diagrams in the literature, but [Reference Durand16] appears to be the first place where the terminology of finite topological rank was introduced.

An essentially minimal Cantor system $(X, T)$ with topological rank $1$ is called an (infinite) odometer. It is easy to see that any ordered Bratteli diagram for such an odometer is necessarily simple, and therefore an odometer is in fact minimal. The infinite odometers coincide with all equicontinuous minimal Cantor systems.

2.5. Kakutani–Rohlin partitions

The concepts and terminology reviewed in this subsection are again from [Reference Giordano, Putnam and Skau30, Reference Herman, Putnam and Skau33], with some notation from [Reference Donoso, Durand, Maass and Petite13].

For an essentially minimal Cantor system $(X, T)$ , a Kakutani–Rohlin partition is a partition

$$ \begin{align*} \mathcal{P}=\{ T^jB(k): 1\leq k\leq d, 0\leq j<h(k)\} \end{align*} $$

of clopen sets, where $d, h(1),\ldots , h(d)$ are positive integers and $B(1),\ldots , B(d)$ are clopen subsets of X such that

$$ \begin{align*} \bigcup_{k=1}^d T^{h(k)}B(k)=\bigcup_{k=1}^d B(k). \end{align*} $$

The set $B(\mathcal {P})=\bigcup _{k=1}^d B(k)$ is called the base of $\mathcal {P}$ . For $1\leq k\leq d$ , the subpartition $\mathcal {P}(k)=\{T^jB(k): 0\leq j<h(k)\}$ is the kth tower of $\mathcal {P}$ , which has base $B(k)$ and height  $h(k)$ .

The following is a basic fact regarding the construction of Kakutani–Rohlin partitions.

The proof of the lemma gives a canonical construction of Kakutani–Rohlin partitions. Specifically, given $y\in Y$ and clopen set Z containing y, the function $Z\to \mathbb {N}$ , $x\mapsto n_x$ , where $n_x$ is the least positive integer n such that $T^nx\in Z$ , is continuous. Thus, by the compactness of X, $x\mapsto n_x$ is bounded. For any $h>0$ , let $A_h=\{x\in Z: n_x=h\}$ . Let $h(1),\ldots , h(d)$ enumerate all $h>0$ , where $A_h\neq \varnothing $ . Then $\{T_jA_{h(k)}: 1\leq k\leq d, 0\leq j<h(k)\}$ is a Kakutani–Rohlin partition with base Z.

Applying Lemma 2.1 repeatedly, one quickly obtains the following theorem.

We call the system of Kakutani–Rohlin partitions in Theorem 2.2 a nested system. From such a system, we define an ordered Bratteli diagram following [Reference Herman, Putnam and Skau33]. For each $n\geq 0$ , let

$$ \begin{align*} V_n=\{\mathcal{P}_n(k): 1\leq k\leq d_n\}. \end{align*} $$

For $n\geq 1$ , $1\leq k\leq d_n$ , $1\leq \ell \leq d_{n-1}$ , and $0\leq j<h_n(k)$ , there is an edge $e_j\in E_n$ connecting $\mathcal {P}_n(k)$ to $\mathcal {P}_{n-1}(\ell )$ if $T^jB_n(k)\subseteq \bigcup _{0\le i< h_n(\ell )}T^iB_{n-1}(\ell )$ . Then, if $e_{j_1},\ldots , e_{j_m}$ are all edges in $E_n$ connecting $\mathcal {P}_n(k)$ to some element of $V_{n-1}$ , we set the partial ordering $\preceq $ by letting $e_j\preceq e_{j'}$ if and only if $j\leq j'$ . It was proved in [Reference Herman, Putnam and Skau33] that this ordered Bratteli diagram is essentially simple and that the Bratteli–Vershik system generated by this ordered Bratteli diagram is conjugate to $(X, T)$ , with the conjugacy map sending to x. If in addition $(X, T)$ is a minimal Cantor system, then the resulting ordered Bratteli diagram is necessarily simple.

Thus, we have described a procedure to obtain an ordered Bratteli diagram given an essentially minimal Cantor system $(X, T)$ and a point x in the unique minimal set. It was proved in [Reference Herman, Putnam and Skau33] that the equivalence class of the ordered Bratteli diagram does not depend on the choice of the Kakutani–Rohlin partitions in the procedure, that is, all ordered Bratteli diagrams obtained through this procedure are equivalent.

Conversely, if $B=(V,E,\preceq )$ is an essentially simple ordered Bratteli diagram and $(X_B, \unicode{x3bb} _B)$ is the Bratteli–Vershik system generated by B, then there is a nested system of Kakutani–Rohlin partitions for $(X_B,\unicode{x3bb} _B)$ and such that the ordered Bratteli diagram $B'$ defined above is equivalent to B. Thus, if an essentially minimal Cantor system $(X, T)$ has finite topological rank d, then there is a nested system of Kakutani–Rohlin partitions $\{\mathcal {P}_n\}_{n\geq 1}$ , where $d_n=d$ for all $n\geq 1$ .

2.6. Subshifts

The concepts and notation reviewed in this subsection are from [Reference Gao and Hill25, Reference Gao, Jacoby, Johnson, Leng, Li, Silva and Wu27].

By a finite word, we mean an element of $2^{<\mathbb {N}}=\{0,1\}^{<\mathbb {N}}=\bigcup _{N\in \mathbb {N}}\{0,1\}^N$ . If v is a finite word, we regard it as a function with domain $\{0,1,\ldots , N-1\}$ for some $N\in \mathbb {N}$ , and call N its length, denoted as $|v|=N$ . The empty word is the unique finite word with length $0$ (or the empty domain), and we denote it as $\varnothing $ . If v is a finite word and $s,t$ are integers such that $0\leq s\leq t\leq |v|-1$ , then $v\!\upharpoonright \![s,t]$ denotes the finite word u of length ${t-s+1}$ , where for $0\leq i<t-s+1$ , $u(i)=v(s+i)$ ; $v\!\upharpoonright \! s$ denotes $v\!\upharpoonright \![0,s]$ , and is called a prefix or an initial segment of v. An end segment or a suffix of v is $v\!\upharpoonright \! [s, |v|-1]$ for some ${0\leq s\leq |v|-1]}$ . The empty word is both a prefix and a suffix of any word. Any word is also both a prefix and a suffix of itself. If $u, v$ are finite words, then $uv$ denotes the finite word w of length $|u|+|v|$ , where $w\!\upharpoonright \! [0,|u|-1]=u$ and $w\!\upharpoonright \![|u|,|u|+|v|-1]=v$ . For finite words $u, v$ with $|u|\leq |v|$ , we say that u is a subword of v if there is $0\leq s\leq |v|-|u|$ such that ${u=v\!\upharpoonright \![s,s+|u|-1]}$ ; when this happens, we also say that u occurs in v at position s.

An infinite word is an element of $2^{\mathbb {N}}$ , and a bi-infinite word is an element of $2^{\mathbb {Z}}$ . For any infinite word $V\in 2^{\mathbb {N}}$ and integers $s, t$ with $0\leq s\leq t$ , the notions $V\!\upharpoonright \![s,t]$ , $V\!\upharpoonright \! s$ , finite subwords, and their occurrences are similarly defined. For any bi-infinite word $x\in 2^{\mathbb {Z}}$ and integers $s, t$ with $s\leq t$ , the notions $x\!\upharpoonright \![s,t]$ , finite subwords, and their occurrences are also similarly defined.

We consider the Bernoulli shift on $2^{\mathbb {Z}}=\{0,1\}^{\mathbb {Z}}$ , which is the homeomorphism ${\sigma : 2^{\mathbb {Z}}\to 2^{\mathbb {Z}}}$ defined by

$$ \begin{align*} \sigma(x)(n)=x(n+1). \end{align*} $$

Since $2^{\mathbb {Z}}$ is homeomorphic to $\mathcal {C}=2^{\mathbb {N}}$ , $(2^{\mathbb {Z}}, \sigma )$ is a Cantor system. A subshift X is a closed $\sigma $ -invariant subset of $2^{\mathbb {Z}}$ . By a subshift, we also refer to the Cantor system $(X, \sigma \!\upharpoonright \! X)$ or simply $(X,\sigma )$ when there is no danger of confusion.

The following simple fact is a folklore.

If $V\in 2^{\mathbb {N}}$ is an infinite word, let

$$ \begin{align*} X_V=\{x\in 2^{\mathbb{Z}}: \text{ every finite subword of } x \text{ is a subword of } V\}. \end{align*} $$

Then $(X_V, \sigma )$ is a subshift and we call it the subshift generated by V. For any $V\in 2^{\mathbb {N}}$ , $X_V$ is always non-empty. Note that for any $x\in X_V$ and finite subword u of x, u must occur in V infinitely many times. We say that V is recurrent if every finite subword of V occurs in V infinitely many times. When V is recurrent, $X_V$ is either finite or a Cantor set, and $X_V$ is finite if and only if V is periodic, that is, there is a finite word v such that $V=vvv\cdots $ . Thus, an infinite subshift generated by a recurrent V is a Cantor system.

It is well known that all infinite odometers form a dense $G_{\delta }$ in the space $M(\mathcal {C})$ of all minimal Cantor systems. We give a proof of this fact in Corollary 3.4 and Proposition 3.6.

2.7. Subshifts of finite symbolic rank

Some of the concepts and notation reviewed in this subsection are from [Reference Gao and Hill25, Reference Gao, Jacoby, Johnson, Leng, Li, Silva and Wu27], and some are new.

Subshifts of finite symbolic rank are defined from infinite words of finite symbolic ranked constructions, whose definitions are inspired by the cutting-and-stacking processes that were used to construct measure-preserving transformations of finite rank [Reference Ferenczi23]. We first define (symbolic) rank-1 subshifts, which are also called Ferenczi subshifts in [Reference Arbulú and Durand2] to honor the fact that Ferenczi popularized the concept in [Reference Ferenczi23].

An infinite (symbolic) rank-1 word V is defined as follows. Given a sequence of positive integers $\{r_n\}_{n\geq 0}$ with $r_n>1$ for all $n\geq 0$ (called the cutting parameter) and a doubly indexed sequence of non-negative integers $\{s_{n,i}\}_{n\geq 0, 1\leq i<r_n}$ (called the spacer parameter), a (symbolic) rank-1 generating sequence given by the parameters is the recursively defined sequence of finite words:

$$ \begin{align*} v_0&= 0,\\ v_{n+1}&=v_n1^{s_{n,1}}v_n\cdots v_n1^{s_{n,r_n-1}}v_n. \end{align*} $$

Since $v_n$ is a prefix of $v_{n+1}$ , it makes sense to define $V=\lim _n v_n$ . This V is called a (symbolic) rank-1 word and $X_V$ is called a (symbolic) rank-1 subshift.

To generalize and define (symbolic) rank-n subshifts, we use the following concepts and notation. Let $\mathcal {F}$ be the set of all finite words in $2^{<\mathbb {N}}$ that begin and end with $0$ . For a finite set $S\subseteq \mathcal {F}$ and finite word $w\in \mathcal {F}$ , a building of w from S consists of a sequence $(v_1,\ldots , v_{k+1})$ of elements of S and a sequence $(s_1,\ldots , s_k)$ of non-negative integers for $k\geq 1$ such that

$$ \begin{align*} w=v_11^{s_1}v_2\cdots v_k1^{s_k}v_{k+1}. \end{align*} $$

The sequence $(s_1,\ldots , s_k)$ is called the spacer parameter of the building; it is bounded by M if $s_1,\ldots , s_k\leq M$ . We say that every word of S is used in this building if $\{v_1,\ldots , v_{k+1}\}=S$ . When there is a building of w from S, we also say that w is built from S; when the building consists of $(v_1,\ldots , v_{k+1})$ and $(s_1,\ldots , s_k)$ , we also say that w is built from S starting with $v_1$ . These notions can be similarly defined when the finite word w is replaced by an infinite word W.

For $n\geq 1$ , a (symbolic) rank-n generating sequence is a doubly indexed sequence $\{v_{i,j}\}_{i\geq 0, 1\leq j\leq n_i}$ of finite words satisfying, for all $i\geq 0$ :

• • $n_i\leq n$ ;

• • $v_{0,j}=0$ for all $1\leq j\leq n_0$ ;

• • $v_{i+1,1}$ is built from $S_i\triangleq \{v_{i,1},\ldots , v_{i, n_i}\}$ starting with $v_{i,1}$ ;

• • $v_{i+1,j}$ is built from $S_i$ for all $2\leq j\leq n_{i+1}$ .

A (symbolic) rank-n construction is the (symbolic) rank-n generating sequence $\{v_{i,j}\}_{i\geq 0, 1\leq j\leq n_i}$ together with exactly one building of $v_{i+1,j}$ from $S_i$ (for $v_{i+1,1}$ , the building should start with $v_{i,1}$ ) for all $i\geq 0, 1\leq j\leq n_i$ . We call $S_i$ the ith level of the construction. The spacer parameter of the rank-n construction is the collection of all spacer parameters of all the buildings in the construction; it is bounded if there is an $M>0$ such that all the spacer parameters of all the buildings in the construction are bounded by M. The (symbolic) rank-n construction is proper if for all $i\geq 0$ , $n_i=n$ , and for all $1\leq j\leq n$ , every word of $S_i$ is used in the building of each $v_{i+1,j}$ . Since each $v_{i,1}$ is a prefix of $v_{i+1,1}$ , it makes sense to define $V=\lim _{i}v_{i,1}$ .

Given a rank-n construction with associated rank-n generating sequence $\{v_{i,j}\}_{i\geq 0, 1\leq j\leq n_i}$ , we define the set of all expected subwords of $v_{i,j}$ , for $i\geq 0$ and $1\leq j\leq n_i$ , inductively as follows: for each $v_{0,j}$ , the set of all of its expected subwords is $\{v_{0,j}\}=\{0\}$ ; for $i\geq 0$ , the set of all expected subwords of $v_{i+1,j}$ consists of:

• • $v_{i+1,j}$ ;

• • $u_1,\ldots , u_{k+1}\in S_i$ , where $(u_1,\ldots , u_{k+1})$ and $(a_1,\ldots , a_k)$ give the building of $v_{i+1,j}$ from $S_i$ ;

• • all expected subwords of $u_1,\ldots , u_{k+1}\in S_i$ .

Finally, define the set of all expected subwords of $V=\lim _i v_{i,1}$ to be the union of the sets of all expected subwords of $v_{i,1}$ for all $i\geq 0$ . Without loss of generality, we may assume that for all $i\geq 0$ , all finite words in $S_i$ are expected subwords of V. It follows immediately from the construction that for all $i\geq 0$ , the infinite word V is built from $S_i$ starting with  $v_{i,1}$ .

Let $w\in \mathcal {F}$ and $S, T\subseteq \mathcal {F}$ are finite. Suppose w is built from S and that every word in S is built from T. Then by composing the building of w from S with the buildings of each element of S from T, we obtain a building of w from T, and thus w is also built from T. Given a rank-n construction with associated rank-n generating sequence $\{v_{i,j}\}_{i\geq 0, 1\leq j\leq n_i}$ , and given $i<i'$ , for all $1\leq j'\leq n_{i'}$ , we obtain a building of $v_{i',j'}$ from $S_i$ by composing the buildings of elements of $S_{\iota }$ from $S_{\iota -1}$ for all $i+1\leq \iota \leq i'$ . With this repeated composition process, we may obtain, for any increasing sequence $\{i_k\}_{k\geq 0}$ with $i_0=0$ , a rank-n construction with associated rank-n generating sequence $\{v_{i_k,j}\}_{k\geq 0, 1\leq j\leq n_{i_k}}$ . Since $\lim _i v_{i,1}=\lim _{k}v_{i_k,1}$ , the resulting infinite words are the same. We call this process telescoping.

An infinite word V is called a (symbolic) rank-n word if it has a rank-n construction but not a rank- $(n-1)$ construction. A subshift X has finite symbolic rank if for some $n\geq 1$ , $X=X_V$ , where V has a rank-n construction; the smallest such n is called the symbolic rank of X, and is denoted ${\mathrm {rank}_{\scriptsize \mathrm {symb}}}(X)={\mathrm {rank}_{\scriptsize \mathrm {symb}}}(X, \sigma )$ .

By definition, if ${\mathrm {rank}_{\scriptsize \mathrm {symb}}}(X)=n$ , then there is a rank-n word V such that $X=X_V$ .

Before closing this subsection, we give some examples. The best known example of a rank-1 generating sequence is the one coding the Chacón transformation:

$$ \begin{align*} v_0=0;\quad v_{n+1}=v_nv_n1v_n. \end{align*} $$

The Morse sequence is the infinite 0,1-word generated by the Thue–Morse substitution $0\mapsto 01$ and $1\mapsto 10$ . The subshift generated by the Morse sequence is an example of a minimal subshift of symbolic rank $2$ . More generally, any $\mathcal {S}$ -adic subshift of finite alphabet rank, for which the initial alphabet $A_0=\{0,1\}$ (defined in §6.4), is a natural example of subshifts of finite symbolic rank, where the symbolic rank is no more than the alphabet rank. Our main results in this paper will show that any minimal $\mathcal {S}$ -adic subshift of finite alphabet rank is conjugate to a minimal subshift of finite symbolic rank.

Finally, we turn to symbolic codings of interval exchange transformations, known as IET subshifts. When there are two intervals of irrational lengths, the interval exchange transformations are exactly irrational rotations and the corresponding IET subshifts are generated by Sturmian words. These subshifts have symbolic rank 2 [Reference Gao, Jacoby, Johnson, Leng, Li, Silva and Wu27]. In general, when there are more than two intervals and if the corresponding IET subshifts are minimal, they are known to have finite topological rank, and by our main results in this paper, they are conjugate to a minimal subshift of finite symbolic rank. If one considers the quotient systems of these subshifts where one of the intervals is coded by $0$ and the rest are coded by $1$ , then the subshifts are natural examples of subshifts of finite symbolic rank.

6.1. From finite symbolic rank to finite topological rank

We first consider minimal subshifts of finite symbolic rank.

The following concept of unique readability will be useful in our proofs to follow. Let $n\geq 1$ . Fix a symbolic rank-n construction with associated rank-n generating sequence $\{v_{i,j}\}_{i\geq 0, 1\leq j\leq n}$ . Let $V=\lim _i v_{i,1}$ . Without loss of generality, assume every $v_{i,j}$ is an expected subword of V, and that for each $i\geq 1$ , the words $v_{i,1},\ldots , v_{i,n}$ are distinct. For $x\in X_V$ , a reading of x is a sequence $\{E_i\}_{i\geq 0}$ satisfying, for each $i\geq 0$ :

1. (i) each element of $E_i$ is a pair $(k,j)$ , where $1\leq j\leq n$ and k is the starting position of an occurrence of $v_{i,j}$ in x;

2. (ii) if $(k_1,j_1), (k_2, j_2)\in E_i$ and $k_1<k_2$ , then $k_1+|v_{i,j_1}|\leq k_2$ ;

3. (iii) $E_0=\{(k,j): x(k)=0 \text { and } j=1\}$ ; and

4. (iv) for each $(k,j)\in E_{i}$ , there is exactly one $(k',j')\in E_{i+1}$ such that $k'\le k$ and $k'+|v_{i+1,j'}|\ge k+|v_{i,j}|$ .

If every $x\in X_V$ has a unique reading, we say that $\{v_{i,j}\}_{i\geq 0,1\le j\le m}$ has unique readability, and we call an occurrence (starting at position) k of $v_{i,j}$ in x expected if $(k,j)\in E_i$ for the unique reading of x. Every rank- $1$ generating sequence whose induced infinite rank- $1$ word is not periodic has unique readability [Reference Gao and Hill25, Proposition 2.29].

Next we define a concept that guarantees unique readability. Let $n\geq 1$ . We say a rank-n construction with associated rank-n generating sequence $\{v_{i,j}\}_{i\geq 0, 1\leq j\leq n}$ is good if it is proper and for any $i\geq 0$ and $1\le j\le n$ , $v_{i,j}$ is not of the form

$$ \begin{align*} \alpha 1^{s_1}v_{i,j_1}1^{s_2}v_{i,j_2}\cdots v_{i,j_{k-1}}1^{s_k}\beta, \end{align*} $$

where $k\geq 1$ , $\alpha $ is a non-empty suffix of some $v_{i,j_{k}}$ , and $\beta $ is a non-empty prefix of some $v_{i,j_{k+1}}$ . If a rank-n construction is good, we say that the infinite word $V=\lim _i v_{i,1}$ is good.

Note that the definition of goodness does not rule out the possibility that some $v_{i,j}$ is a subword of $v_{i,j'}$ for $j'\neq j$ .

Lemma 6.3. Suppose V has a good rank-n construction with associated rank-n generating sequence $\{v_{i,j}\}_{i\geq 0, 1\leq j\leq n}$ . Then for any $i\geq 0$ , $1\leq j\leq n$ , and $k\in \mathbb {Z}$ , the set

$$ \begin{align*} \{x\in X_V: \text{ there is an expected occurrence of } v_{i,j} \text{ at position } k\} \end{align*} $$

is clopen in $X_V$ .

Proof. Fix a proper rank-n construction of V with associated rank-n generating sequence $\{v_{i,j}\}_{i\geq 0,1\le j\le n}$ . Without loss of generality, assume $v_{i,1},\ldots , v_{i,n}$ are distinct for all $i\geq 1$ . By Corollary 5.3, there is a $k_0\in \mathbb {N}$ such that $0^{k_0}$ is not a subword of V; we fix such a $k_0$ . For all $i\geq 0$ , define $a_i=\max _{1\leq j\leq n}|v_{i,j}|$ and $b_i=\inf _{1\leq j\leq n}|v_{i,j}|$ . Then $b_{i+1}\geq nb_i$ for all $i\geq 0$ , and hence, in particular, $b_i\geq n^i$ for all $i\geq 0$ .

We define a rank- $2n$ generating sequence $\{w_{p,q}\}_{p\geq 0, 1\leq q\leq 2n}$ . Let $w_{0,q}=0$ for ${1\leq q\leq 2n}$ . To define $w_{1,q}$ , let $i_1\geq 0$ be such that $b_{i_1}>4k_0+4$ . Then define

$$ \begin{align*} w_{1,q}=\left\{\!\!\begin{array}{ll} v_{i_1,q} & \text{ if } 1\leq q\leq n, \\ 010^{|v_{i_1,q-n}|-4}10 &\text{ if } n+1\leq q\leq 2n.\end{array} \right. \end{align*} $$

Note that for all $1\leq j\leq n$ , $|w_{1,j}|=|w_{1,j+n}|=|v_{i_1,j}|$ .

For $p\geq 1$ , suppose $i_p$ has been defined and $w_{p,q}$ have been defined for all $1\leq q\leq 2n$ . We define $i_{p+1}$ and $w_{p+1,q}$ as follows. First set

$$ \begin{align*} m_p=\bigg\lceil\frac{a_{i_p+1}}{b_{i_p}}\bigg\rceil. \end{align*} $$

Then let $i_{p+1}>i_p$ be large enough such that by telescoping using the buildings in the proper rank-n construction $\{v_{i,j}\}_{i\geq 0, 1\leq j\leq n}$ , we can write, for all $1\le j\le n$ , $v_{i_{p+1},j}$ in the form

(1) $$ \begin{align} v_{i_p,j_1}1^{s_1}\cdots 1^{s_{\ell}}v_{i_p,j_{\ell+1}} \end{align} $$

with $\ell>12m_p+4n$ . This is doable since $\ell \geq n^{i_{p+1}-i_p}$ . Note that for $j=1$ , we have ${j_1=1}$ . We also note the following property (*) of the word in equation (1): for any $1\le t\le \ell +2-2m_p$ , $\{j_t,\ldots ,j_{t+2m_p-1}\}=\{1,\ldots ,n\}$ . This is because

(2) $$ \begin{align} u\triangleq v_{i_p, j_t}1^{s_t}\cdots v_{i_p,j_{t+2m_p-1}} \end{align} $$

consists of $2m_p$ many consecutive expected occurrences of subwords of $v_{i_{p+1},j}$ of the form $v_{i_p, j'}$ ; since $v_{i_{p+1}, j}$ is built from $\{v_{i_p+1,1},\ldots , v_{i_p+1,n}\}$ , by our definition of $m_p$ , u must contain some expected occurrence of $v_{i_p+1,j'}$ , where $1\leq j'\leq n$ , as a subword. Hence, property (*) holds by the properness of the construction.

We now fix $1\leq j\leq n$ and assume that $v_{i_{p+1},j}$ is in the form of equation (1). For ${1\leq t\leq \ell +1}$ , define

$$ \begin{align*} \phi(t)=\left\{\!\!\begin{array}{ll} j_t+n & \text{ if } 2\leq t\leq 2m_p+j+1 \text{ or } \ell-2m_p-j+1\leq t\leq \ell, \\ j_t & \text{ otherwise} \end{array} \right. \end{align*} $$

and

$$ \begin{align*} \psi(t)=\left\{\!\!\begin{array}{ll}j_t+n & \text{ if } 2\leq t\leq 2m_p+j+n+1 \text{ or } \ell-2m_p-j-n+1\leq t\leq \ell, \\ j_t & \text{ otherwise.} \end{array} \right. \end{align*} $$

Then define

$$ \begin{align*} w_{p+1,j}= w_{p,\phi(1)}1^{s_1}\cdots 1^{s_{\ell}}w_{p,\phi(\ell+1)} \end{align*} $$

and

$$ \begin{align*} w_{p+1,j+n}=w_{p,\psi(1)}1^{s_1}\cdots 1^{s_{\ell}}w_{p, \psi(\ell+1)}. \end{align*} $$

This finishes the definition of $\{w_{p,q}\}_{p\geq 0, 1\leq q\leq 2n}$ .

We verify that the construction defined is proper, that is, for all $p\geq 1$ and ${1\leq q\leq 2n}$ , all words in $\{w_{p,1},\ldots , w_{p,2n}\}$ are used in the building of $w_{p+1,q}$ . We first assume $1\leq q\leq n$ . Since $\ell>12m_p+4n$ , there exists $1\leq t_0\leq \ell +1$ such that for all $t_0\leq t\leq t_0+2m_p-1$ , $\phi (t)=j_t$ . By property (*), $\{\phi (t_0),\ldots , \phi (t_0+2m_p-1)\} =\{j_{t_0},\ldots , j_{t+2m_p-1}\}=\{1,\ldots , n\}$ . Thus, all words in $\{w_{p,1},\ldots , w_{p,n}\}$ are used in the building of $w_{p+1,q}$ . However, for $2\leq t\leq 2m_p+1$ , $\phi (t)=j_t+n$ . By property (*) again, $\{\phi (2),\ldots , \phi (2m_p+1)\}=\{j_2+n,\ldots , j_{2m_p+1}+n\}=\{n+1,\ldots , 2n\}$ . Hence, all words in $\{w_{p,n+1},\ldots , w_{p,2n}\}$ are also used in the building of $w_{p+1,q}$ . The case $n+1\leq q\leq 2n$ is similar.

Next we claim that:

• • for all $p\geq 2$ and $1\leq q\leq 2n$ , $w_{p,q}$ is not of the form

  (3) $$ \begin{align} \alpha 1^{r_1}w_{p,q_1}1^{r_2}w_{p,q_2}\cdots w_{p,q_{d-1}}1^{r_d}\beta, \end{align} $$
  where $d\geq 1$ , $\alpha $ is a non-empty suffix of some $w_{p,q_{d}}$ , and $\beta $ is a non-empty prefix of some $w_{p,q_{d+1}}$ ; and

• • for all $p\geq 2$ and $1\leq q, q'\leq 2n$ , if $q\neq q'$ , then $w_{p,q}$ is not a subword of $w_{p,q'}$ .

We prove this claim by induction on $p\geq 2$ .

First suppose $p=2$ . We observe that $w_{2,q}$ can be written as $u_1yu_2$ , where $0^{k_0}$ is not a subword of y, y begins and ends with 0, every word of $\{v_{i_1,1},\ldots , v_{i_1,n}\}$ occurs at least three different times in y, and both $u_1$ and $u_2$ are of the form

(4) $$ \begin{align} \alpha 01^{s_1}010^{h_1}101^{s_2}010^{h_2}10\cdots 010^{h_{2m_p+q}}101^{s_{2m_p+q+1}}0\beta, \end{align} $$

where $\alpha ,\beta $ have lengths at least $3k_0$ , $0^{k_0}$ is not a subword of either $\alpha $ or $\beta $ , and $h_t>4k_0$ for all $1\leq t\leq 2m_p+q$ . The statement about y is based on the observation that, by equation (1), y can be taken to contain subwords of the form in equation (2) for three different values $2m_p+2n+1< t_1<t_2<t_3< \ell -4m_p-2n$ , where $t_3-t_2, t_2-t_1> 2m_p$ ; by property (*), for each value $t_1, t_2, t_3$ , the subword of the form in equation (2) contains a distinct occurrence of each word in $\{v_{i_1,1},\ldots , v_{i_1,n}\}$ . Note that for $q'\neq q$ , $w_{2,q'}$ does not have a subword of the form in equation (4), and hence $w_{2,q}$ is not a subword of $w_{2,q'}$ . Now suppose $w_{2,q}$ can be written in the form of equation (3), then by the above observation, there are $1\leq q_1, q_2\leq 2n$ , a non-empty suffix $y_1$ of $w_{2,q_1}$ , and a non-empty prefix $y_2$ of $w_{2,q_2}$ , such that $w_{2,q}=y_11^{s}y_2$ for some non-negative integer s. First suppose $q_1=q_2=q$ . Then $w_{2,q}$ must have a subword of the form $z\triangleq 0^{2k_0}101^{s_1}v_{i_1,j_1}1^{s}v_{i_1,j_2}1^{s_2}010^{2k_0}$ and, in fact, y must be a subword of z. However, note that y has at least three different occurrences of each word in $\{v_{i_1,1},\ldots , v_{i_1,n}\}$ , while z does not have this property, which is a contradiction. Next suppose $q_1\ne q$ . Then $u_1$ is not a subword of $w_{2,q_1}$ , so $y_1$ is a prefix of $u_1$ . It follows that $yu_2$ is a suffix of $y_2$ , $q_2=q$ , and $y_2$ must be $w_{2,q}$ itself, which contradicts the assumption that $y_1$ is non-empty. The case $q_2\ne q$ is similar. This completes the proof of the claim for $p=2$ .

Suppose the claim holds for $p\geq 2$ . We verify it for $p+1$ . First we observe that for any $1\leq q\leq 2n$ , $w_{p+1,q}$ can be written as $u_1yu_2$ , where $u_1$ and $u_2$ are of the form

(5) $$ \begin{align} w_{p,q_1}1^{s_1}w_{p,q_2}1^{s_2}\cdots w_{p, q_{2m_p+q+1}}1^{s_{2m_p+q+1}}w_{p,q_{2m_p+q+2}}, \end{align} $$

where $1\leq q_1, q_{2m_p+q+2}\leq n$ , $n+1\leq q_t\leq 2n$ for all $2\leq t\leq 2m_p+q+1$ , and by inductive hypothesis, if $w_{p,\kappa }$ is a subword of y, then $1\leq \kappa \leq n$ . By the inductive hypothesis, if $q\neq q'$ , then $w_{p+1,q'}$ does not contain a subword of the form in equation (5), and hence $w_{p+1,q}$ is not a subword of $w_{p+1,q'}$ . Next assume $w_{p+1,q}$ can be written in the form in equation (3) with $p+1$ replacing p. Then by the above observation, there are $1\leq q_1, q_2\leq 2n$ , a non-empty suffix $y_1$ of $w_{p+1,q_1}$ , and a non-empty prefix $y_2$ of $w_{p+1, q_2}$ such that $w_{p+1,q}=y_11^sy_2$ for some non-negative integer s. First suppose $q_1=q_2=q$ . Then $w_{p+1,q}$ has a subword of the form $z\triangleq w_{p, j_1}1^{s_1}w_{p,j_2}1^{s}w_{p, j_3}1^{s_2}w_{p, j_4}$ , where $n+1\leq j_1, j_4\leq 2n$ and $1\leq j_2, j_3\leq n$ . In fact, y must be a subword of z. However, y contains at least three different occurrences of words in $\{w_{p, 1},\ldots , w_{p,n}\}$ , which is a contradiction. Next suppose $q_1\ne q$ , then $u_1$ is not a subword of $w_{p+1,q_1}$ , so $y_1$ is a prefix of $u_1$ . It follows that $yu_2$ is a suffix of $y_2$ , $q_2=q$ , and $y_2$ must be $w_{p+1,q}$ itself, which contradicts the assumption that $y_1$ is non-empty. The case $q_2\ne q$ is similar. This completes the proof of the claim.

In view of the claim, if we define $\{w^{\prime }_{p,q}\}_{p\geq 0, 1\leq q\leq 2n}$ by letting $w^{\prime }_{0,q} =0$ and ${w^{\prime }_{p,q}=w_{p+1,q}}$ for $p\geq 1$ and $1\leq q\leq 2n$ , then we obtain a good proper rank- $2n$ construction. Let $W=\lim _p w^{\prime }_{p,1}$ . Note that $W=\lim _p w_{p, 1}$ .

We define a factor map $\varphi : X_W\to X_V$ . For $x\in X_W$ and $k\in \mathbb {Z}$ , if there is $1\leq j\leq n$ such that the position k is part of an expected occurrence of $w^{\prime }_{1,j}$ or $w^{\prime }_{1,j+n}$ which starts at position $k'\le k$ , then let $\varphi (x)(k)=v_{i_2,j}(k-k')$ ; otherwise, let $\varphi (x)(k)=1$ . By the unique readability, and since for all $1\leq j\leq n$ , $|w^{\prime }_{1,j}|=|w^{\prime }_{1,j+n}|=|w_{2,j}|=|w_{2,j+n}|=|v_{i_2,j}|$ , $\varphi $ is well defined. By Lemma 6.3, $\varphi $ is continuous. It is clear that $\varphi $ is a factor map.

Finally, if $X_V$ is minimal, then the construction associated with $\{v_{i,j}\}_{i\geq 0, 1\leq j\leq n}$ must have bounded spacer parameter, because otherwise, $1^{\mathbb {Z}}\in X_V$ and $\{1^{\mathbb {Z}}\}$ is invariant. Now it follows from our construction that the defined proper rank- $2n$ construction of W also has bounded spacer parameter, and by the implication (2) $\Rightarrow $ (1) of Corollary 5.4 (which does not require the assumption on the symbolic rank of $X_W$ ), $X_W$ is minimal.