# Subshifts of Finite Symbolic Rank

Su Gao  School of Mathematical Sciences and LPMC, Nankai University, Tianjin
300071, P.R. China<EMAIL_ADDRESS>and  Ruiwen Li  School of Mathematical
Sciences and LPMC, Nankai University, Tianjin 300071, P.R. China
<EMAIL_ADDRESS>
###### Abstract.

The definition of subshifts of finite symbolic rank is motivated by the finite
rank measure-preserving transformations which have been extensively studied in
ergodic theory. In this paper we study subshifts of finite symbolic rank as
essentially minimal Cantor systems. We show that minimal subshifts of finite
symbolic rank have finite topological rank, and conversely, every minimal
Cantor system of finite topological rank is either an odometer or conjugate to
a minimal subshift of finite symbolic rank. We characterize the class of all
minimal Cantor systems conjugate to a rank-$1$ subshift and show that it is
dense but not generic in the Polish space of all minimal Cantor systems.
Within some different Polish coding spaces of subshifts we also show that the
rank-1 subshifts are dense but not generic. Finally we study topological
factors of minimal subshifts of finite symbolic rank. We show that every
infinite odometer and every irrational rotation is the maximal equicontinuous
factor of a minimal subshift of symbolic rank $2$, and that a subshift factor
of a minimal subshift of finite symbolic rank has finite symbolic rank.

###### 2020 Mathematics Subject Classification:

Primary 37B10; Secondary 54H15

The first author acknowledges the partial support of his research by the
National Natural Science Foundation of China (NSFC) grants 12250710128 and
12271263.

##  1\. Introduction

This paper is a contribution to the study of symbolic and topological
dynamical systems, but the main notion studied here originated from ergodic
theory.

One of the main sources of examples and counterexamples in ergodic theory has
been the measure-preserving transformations constructed from a cutting-and-
stacking process. The first such example was given by Chacón in [10] more than
half a century ago, and since then there has been a large literature devoted
to the study of measure-theoretic properties of these transformations. In
fact, much of the work concentrated on the so-called rank-one transformations,
where there is only one stack in every step of the cutting-and-stacking
process. This is partially because the class of rank-one transformations forms
a dense $G_{\delta}$ set in (the Polish space of) the class of all measure-
preserving transformations (c.f., e.g. [23] and [24]), and thus behaviors of
the rank-one transformations capture the generic behaviors of all measure-
preserving transformations. On the other hand, measure-preserving
transformations of arbitrary finite rank (where there is a uniform finite
bound on the number of stacks used in every step of the cutting-and-stacking
process, first defined by Ornstein–Rudolph–Weiss in [39]) have also been
extensively studied. In particular, they are known to have different models,
some of which are of geometric nature, and some symbolic. Ferenczi [23] gave
an excellent survey over a quarter of a century ago.

Meanwhile, there has also been an effort to develop an analogous theory of
topological dynamics for Cantor systems of finite topological rank. The
starting point was [33] by Herman–Putnam–Skau, in which essentially minimal
Cantor systems were described by their Bratteli–Vershik representations
through a nested sequence of Kakutani–Rohlin partitions. These were seen to be
analogous/parallel to the cutting-and-stacking processes used to build
measure-preserving transformations. In the time period between [33] and [14]
by Downarowicz–Maass, many special kinds of Cantor systems were studied
through their Bratteli–Vershik representations. These systems included
odometers, substitution subshifts, linear recurrent subshifts, symbolic
codings of interval exchange maps, etc. Along with these studies, there had
been a number of attempts to produce a good definition of topological rank for
Cantor systems. In the end, the natural notion of finite rank Bratteli diagram
(where there is a uniform bound on the number of nodes on all of its levels)
was chosen to be the connotation of finite topological rank. The supporting
evidence was abundant. First, all the special kinds of systems mentioned above
have finite rank Bratteli diagrams. More importantly, in [14] the authors
considered general Cantor minimal systems with finite rank Bratteli diagrams
and proved that they are either equicontinuous or else expansive and hence
conjugate to subshifts. This important dichotomy theorem and its proof
suggested that the assumption of finite rank Bratteli diagram has far-reaching
consequences. Shortly after, the terminology of finite topological rank
started to be used (by Durand in [16]), and since then the Cantor minimal
systems of finite topological rank have been more extensively studied in this
generality (see Durand [16], Bressaud–Durand–Maass [9],
Bezuglyi–Kwiatkowski–Medynets–Solomyak [7], Donoso–Durand–Maass–Petite [13],
Durand–Perrin [17] and Golestani–Hosseini [25]).

At the same time, motivated by the symbolic definition of measure-preserving
transformations of finite rank, Ferenczi [22] introduced a notion of
$\mathcal{S}$-adic subshift. An $\mathcal{S}$-adic subshift is defined from a
substitution process with infinitely many levels, and it has finite alphabet
rank if there is a bound on the numbers of letters used on all of the levels.
This notion of rank, as well as its interplay with the notion of finite
topological rank, have been studied in Durand [15], Berthé–Delecroix [5],
Leroy [38], Berthé–Steiner–Thuswaldner–Solomyak [6],
Donoso–Durand–Maass–Petite [13], Espinoza [19] and [20] etc. In particular, it
has been shown in [13] that every minimal Cantor system of finite topological
rank is either an odometer or conjugate to an $\mathcal{S}$-adic subshift of
finite alphabet rank. Conversely, every $\mathcal{S}$-adic subshift of finite
alphabet rank has finite topological rank.

In this paper we consider a notion of symbolic rank for subshifts which is
more directly motivated by the symbolic definition of finite rank measure-
preserving transformations. In some sense, a subshift of finite symbolic rank
is simply a finite rank measure-preserving transformation without the measure.
Rank-one subshifts in this sense have been studied in the literature. For
example, they are known to have zero topological entropy, and Bourgain [8]
proved Sarnak’s conjecture for minimal rank-one subshifts. Other topological
properties of rank-one subshifts have been considered in
Adams–Ferenczi–Petersen [1], Danilenko [11], El Abdalaoui–Lemańczyk–de la Rue
[18], Etedadialiabadi–Gao [21], Gao–Hill [25] and [26], Gao–Ziegler [27] and
[28], etc.

A systematic study of subshifts of finite symbolic rank started in [29]. Among
other things, it was proved that they all have zero topological entropy. Much
of [29] focused on the combinatorial properties of infinite words that
generate subshifts of finite symbolic rank, and not so much on the topological
properties of the subshifts themselves. In particular, one of the main
questions left unsolved was how the symbolic rank relates to the other more
well-established notions of rank for various Cantor systems.

In this paper we prove a number of results on the topological properties of
subshifts of finite symbolic rank. The subshifts we consider all have alphabet
$\\{0,1\\}$, and therefore they are subshifts of $2^{\mathbb{Z}}$. We show
that any minimal subshift of finite symbolic rank has finite topological rank
(Theorem 6.7) and conversely, any Cantor minimal system of finite topological
rank is either an odometer or conjugate to a minimal subshift of finite
symbolic rank (Theorem 6.9). Thus the notion of finite symbolic rank is
essentially the same as the notion of finite topological rank, and by previous
results ([13]), it is also essentially the same as the notion of finite
alphabet rank.

Although we do not solve any open problem about systems of finite topological
rank in this paper, it is our hope that the relatively new notion of finite
symbolic rank will provide a new perspective and an alternative approach to
the future study of these systems. For example, one of the key open problems
about minimal Cantor systems is their classification up to topogical
conjugacy. In [25] the authors have described a complete classification of
subshifts of symbolic rank one. This gives hope that further progress can be
made by considering subshifts of higher symbolic ranks. Another example of a
major open problem is Sarnak’s conjecture for topological dynamical systems of
zero topological entropy. Bourgain [8] gave a proof for all minimal subshifts
of symbolic rank one, and in [21] Sarnak’s conjecture was confirmed for a
class of subshifts of symbolic rank one which is essentially minimal but not
minimal. Note that by results of [2] or our Proposition 6.11 below, subshifts
of symbolic rank one can have arbitrarily high topological rank or alphabet
rank. Hence the results of [25], [8] and [21] cannot be covererd by results
about any fixed topological rank or alphabet rank.

In this paper we also consider various classes of Cantor systems and
characterize their descriptive complexity. In particular, the class of all
Cantor systems can be coded by the Polish space $\mbox{\rm Aut}(\mathcal{C})$
(this is defined and discussed in Subsection 2.3), and we show that the
classes of all essentially minimal Cantor systems, minimal Cantor systems, as
well as those whose topological rank has a fixed bound all form $G_{\delta}$
subspaces, and hence are Polish (Section 3). On the other hand, the class of
all minimal Cantor systems conjugate to a rank-$1$ subshift is dense but not
generic in the Polish space of all minimal Cantor systems (Proposition 7.1).
Additionally we consider two more Polish spaces of subshifts as done in [40]
and show that the class of minimal subshifts conjugate to a rank-1 subshift is
dense in these spaces but is not generic in either of them. This is in
contrast with the situation in the measure-theoretic setting. Nevertheless,
together with the results of [40], our results show that the class of minimal
subshifts conjugate to one of symbolic rank $\leq 2$ is generic in both of
these Polish coding spaces of subshifts.

We also consider topological factors of minimal subshifts of finite symbolic
rank (Section 8). We improve Theorem 6.9 by showing that a minimal subshift of
finite topological rank $\geq 2$ must be of finite symbolic rank itself
(Corollary 8.4), and is not just conjugate to a subshift of finite symbolic
rank as guaranteed by Theorem 6.9. However, the symbolic rank of the subshift
might be much greater than the one to which it is conjugate. We show that any
infinite odometer and any irrational rotation is the maximal equicontinuous
factor of a minimal subshift of symbolic rank $2$, which is in contrast with
known results about rank-$1$ subshifts.

The rest of the paper is organized as follows. In Section 2 we give the
preliminaries on descriptive set theory, topological dynamical systems,
(essentially) minimal Cantor systems, ordered Bratteli diagrams,
Kakutani–Rohlin partitions, subshifts, and what it means for a subshift to
have finite symbolic rank. In Section 3 we compute the descriptive complexity
of the classes of essentially minimal Cantor systems and those with
topological rank $\leq n$ for some $n\geq 1$, by giving some topological
characterizations of these classes within the Polish space of all Cantor
systems. In Section 4 we give a topological characterization of all minimal
Cantor systems conjugate to a rank-$1$ subshift. In Section 5 we characterize
minimal subshifts of finite symbolic rank as exactly those admitting a proper
finite rank construction with bounded spacer parameter. This will be a basic
tool in the study of minimal subshifts of finite symbolic rank. Section 6 is
the main section of this paper, in which we prove the main theorems (Theorems
6.7 and 6.9) which clarify the relationship between the notions of symbolic
rank and topological rank. We give some examples to show that our results are
in some sense optimal. We also prove a result connecting the notion of finite
alphabet rank for $\mathcal{S}$-adic subshifts with the notion of finite
symbolic rank. This gives an alternative proof of Theorem 6.9 via the main
result of [13]. In Section 7 we consider the density and the genericity of the
class of all minimal subshifts conjugate to a rank-$1$ subshift in various
Polish coding spaces of Cantor systems and subshifts. Finally, in Section 8 we
consider topological factors of minimal subshifts of finite symbolic rank.

Acknowledgments. We thank Fabien Durand, Felipe García-Ramos, Samuel Petite,
and Todor Tsankov for useful discussions on the topics of this paper.
Particularly, we thank Samuel Petite for pointing out reference [19] to us and
Felipe García-Ramos for bringing reference [40] to our attention. We also
thank the anonymous referee for useful suggestions which improved the paper
and for pointing out reference [12].

##  2\. Preliminaries

###  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 [35].

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

$d_{H}(A,B)=\max\left\\{\sup\\{d(x,B)\,:\,x\in A\\},\sup\\{d(y,A)\,:\,y\in
B\\}\right\\}.$

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 iff $Y$ 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{array}[]{rcl}{\bf\Sigma}^{0}_{1}&=&\mbox{ the collection of all open
subsets of $X$}\\\ {\bf\Pi}^{0}_{1}&=&\mbox{ the collection of closed subsets
of $X$}\\\
{\bf\Sigma}^{0}_{\alpha}&=&\left\\{\bigcup_{n\in\mathbb{N}}A_{n}\,:\,A_{n}\in{\bf\Pi}^{0}_{\beta_{n}}\mbox{
for some $\beta_{n}<\alpha$}\right\\}\\\
{\bf\Pi}^{0}_{\alpha}&=&\left\\{X\setminus
A\,:\,A\in{\bf\Sigma}^{0}_{\alpha}\right\\}\end{array}$

We also define
${\bf\Delta}^{0}_{\alpha}={\bf\Sigma}^{0}_{\alpha}\cap{\bf\Pi}^{0}_{\alpha}$.
Thus ${\bf\Delta}^{0}_{1}$ is the collection of all clopen subsets of $X$.
With this notation,
$\bigcup_{\alpha<\omega_{1}}{\bf\Sigma}^{0}_{\alpha}=\bigcup_{\alpha<\omega_{1}}{\bf\Pi}^{0}_{\alpha}=\bigcup_{\alpha<\omega_{1}}{\bf\Delta}^{0}_{\alpha}$
is the collection of all Borel subsets of $X$. The collection of all
$G_{\delta}$ subsets of $X$ is exactly ${\bf\Pi}^{0}_{2}$.

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. $A$ 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 nonmeager in $X$ if it is not meager in $X$; $A$ 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. [4] and [37]. 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
$\rho(T^{n}x,T^{n}y)<\epsilon$. 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), i.e., 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 $\\{T^{k}x\,:\,k\in\mathbb{Z}\\}$. If $A$ is a clopen subset of
$X$, the return times of $x$ to $A$ is defined as $\mbox{\rm
Ret}_{A}(x)=\\{n\in\mathbb{Z}\,:\,T^{n}x\in A\\}$. We regard $\mbox{\rm
Ret}_{A}(x)$ 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. $T$ is minimal if every orbit is dense,
i.e., for all $x\in X$, $\\{T^{k}x\,:\,k\in\mathbb{Z}\\}$ 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 $\mathcal{C}$, i.e., for
$x,y\in\mathcal{C}$, if $x\neq y$ then

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

Let

$\mbox{\rm Aut}(\mathcal{C})=\\{T\,:\,\mbox{$T$ is a homeomorphism from
$\mathcal{C}$ to $\mathcal{C}$}\\}$

be equipped with the compact-open topology, or equivalently the supnorm
metric, i.e., for $T,S\in\mbox{\rm Aut}(\mathcal{C})$,

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

Then $\mbox{\rm Aut}(\mathcal{C})$ is a Polish space (c.f., e.g. [36]). Let
$M(\mathcal{C})$ be the set of all minimal homeomorphisms of $\mathcal{C}$.
Then for a $T\in\mbox{\rm Aut}(\mathcal{C})$, $T\in M(\mathcal{C})$ iff for
all nonempty clopen $U\subseteq\mathcal{C}$, there is $N\in\mathbb{N}$ such
that $\mathcal{C}=\bigcup_{-N\leq n\leq N}T^{n}U$. This characterization
implies that $M(\mathcal{C})$ is a $G_{\delta}$ subset of $\mbox{\rm
Aut}(\mathcal{C})$, and hence $M(\mathcal{C})$ is also a Polish space.
$M(\mathcal{C})$ 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, i.e., a
nonempty 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 [33], [30]
and [14]. Some notations are from [16] and [13]. Recall that a Bratteli
diagram is an infinite graph $(V,E)$ with the following properties:

  * •

The vertex set $V$ is decomposed into pairwise disjoint nonempty 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 nonempty 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 $e^{\prime}$ are
$\preceq$-comparable if and only if $\mathsf{r}(e)=\mathsf{r}(e^{\prime})$.

A finite or infinite path in a Bratteli diagram $(V,E)$ is a sequence
$(e_{1},e_{2},\dots)$ 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 $p=(e_{n+1},\dots,e_{m})\in E_{n,m}$, 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 $p\preceq^{\prime}q$ for
$p=(e_{n+1},\dots,e_{m}),q=(f_{n+1},\dots,f_{m})\in E_{n,m}$ 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 $(V^{\prime},E^{\prime})$ 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 $(V^{\prime},E^{\prime},\preceq^{\prime})$ with
the order $\preceq^{\prime}$ 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 $(V^{\prime},E^{\prime})$ 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

$X_{B}=\\{(e_{n})_{n\geq 1}\,:\,e_{n}\in
E_{n},\mathsf{r}(e_{n})=\mathsf{s}(e_{n+1})\mbox{ for all $n\geq 1$}\\}.$

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 $x_{\scriptsize\rm
max}=(e_{n})_{n\geq 1},x_{\scriptsize\rm min}=(f_{n})_{n\geq 1}\in X_{B}$ such
that for every $n\geq 1$, $e_{n}$ is a $\preceq$-maximal element and $f_{n}$
is a $\preceq$-minimal element. $B=(V,E,\preceq)$ 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 $\lambda_{B}:X_{B}\to X_{B}$ as follows:
$\lambda_{B}(x_{\scriptsize\rm max})=x_{\scriptsize\rm min}$; if
$(e_{n})_{n\geq 1}\in X_{B}$ and $(e_{n})_{n\geq 1}\neq x_{\scriptsize\rm
max}$, then let

$\lambda_{B}((e_{1},e_{2},\dots,e_{k},e_{k+1},\dots))=(f_{1},f_{2},\dots,f_{k},e_{k+1},\dots),$

where $k$ is the least such that $e_{k}$ is not $\preceq$-maximal, $f_{k}$ is
the $\preceq$-successor of $e_{k}$, and $(f_{1},\dots,f_{k-1})$ 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 $(X_{B},\lambda_{B})$ is
an essentially minimal Cantor system ([33]), 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 $(X_{B},\lambda_{B})$
is a minimal Cantor system ([30]). If two simple ordered Bratteli diagrams are
equivalent, then the Bratteli–Vershik systems generated by them are conjugate,
with the conjugacy map sending $x_{\scriptsize\rm min}$ to $x_{\scriptsize\rm
min}$.

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 $\mbox{\rm
rank}_{\scriptsize\rm top}(X,T)$. Such Bratteli diagrams have been called
finite rank Bratteli diagrams in the literature, but [16] 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 [33]
and [30], with some notations from [13].

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

$\mathcal{P}=\\{T^{j}B(k)\,:\,1\leq k\leq d,\,0\leq j<h(k)\\}$

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

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

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
$\mathcal{P}(k)=\\{T^{j}B(k)\,:\,0\leq j<h(k)\\}$ is the $k$-th 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.

######  Lemma 2.1 ([33], Lemma 4.1).

Let $(X,T)$ be an essentially minimal Cantor system, $Y$ be the unique minimal
set, $y\in Y$ and $Z$ be a clopen subset of $X$ containing $y$, and let
$\mathcal{Q}$ be a finite partition of $X$ into clopen sets. Then there is a
Kakutani–Rohlin partition $\mathcal{P}$ such that $y\in B(\mathcal{P})=Z$ and
$\mathcal{P}$ refines $\mathcal{Q}$, i.e., every element of $\mathcal{Q}$ is a
union of elements of $\mathcal{P}$.

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 $T^{n}x\in Z$, 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 $h(1),\dots,h(d)$ enumerate all $h>0$
where $A_{h}\neq\varnothing$. Then $\\{T_{j}A_{h(k)}\,:\,1\leq k\leq d,\,0\leq
j<h(k)\\}$ is a Kakutani–Rohlin partition with base $Z$.

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

######  Theorem 2.2 ([33], Theorem 4.2).

For any essentially minimal Cantor system $(X,T)$ and $x$ in the unique
minimal set, there exist

  * •

positive integers $d_{n}$ for $n\geq 0$, with $d_{0}=1$,

  * •

positive integers $h_{n}(k)$ for $n\geq 0$ and $0\leq k<d_{n}$, with
$h_{0}(1)=1$,

  * •

Kakutani–Rohlin partitions $\mathcal{P}_{n}$ for $n\geq 0$, where

$\mathcal{P}_{n}=\\{T^{j}B_{n}(k)\,:\,1\leq k\leq d_{n},\,0\leq
j<h_{n}(k)\\},$

with $B_{0}(1)=X$,

such that for all $n\geq 0$,

  1. (1)

each $\mathcal{P}_{n+1}$ refines $\mathcal{P}_{n}$,

  2. (2)

$B(\mathcal{P}_{n+1})\subseteq B(\mathcal{P}_{n})$,

  3. (3)

$\bigcap_{n}B(\mathcal{P}_{n})=\\{x\\}$,

  4. (4)

$\bigcup_{n}\mathcal{P}_{n}$ generates the topology of $X$.

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
[33]. For each $n\geq 0$, let

$V_{n}=\\{\mathcal{P}_{n}(k)\,:\,1\leq k\leq d_{n}\\}.$

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 $T^{j}B_{n}(k)\subseteq\bigcup_{0\leq
i<h_{n}(\ell)}T^{i}B_{n-1}(\ell)$. Then, if $e_{j_{1}},\dots,e_{j_{m}}$ 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 $e_{j}\preceq
e_{j^{\prime}}$ iff $j\leq j^{\prime}$. It was proved in [33] 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 $x_{\scriptsize\rm min}$ 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 [33] that the equivalence class of the ordered
Bratteli diagram does not depend on the choice of the Kakutani–Rohlin
partitions in the procedure, i.e., all ordered Bratteli diagrams obtained
through this procedure are equivalent.

Conversely, if $B=(V,E,\preceq)$ is an essentially simple ordered Bratteli
diagram and $(X_{B},\lambda_{B})$ is the Bratteli–Vershik system generated by
$B$, then there is a nested system of Kakutani–Rohlin partitions for
$(X_{B},\lambda_{B})$ and $x_{\scriptsize\rm min}$ such that the ordered
Bratteli diagram $B^{\prime}$ 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 [25] and [29].

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
$\\{0,1,\dots,N-1\\}$ 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

$\sigma(x)(n)=x(n+1).$

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.

######  Lemma 2.3.

An infinite subshift is not equicontinuous. In particular, it is not conjugate
to any infinite odometer.

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

$X_{V}=\\{x\in 2^{\mathbb{Z}}\,:\,\mbox{ every finite subword of $x$ is a
subword of $V$}\\}.$

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 nonempty. 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 iff $V$ is periodic,
i.e., 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 [25]
and [29], 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 ([23]). We first define (symbolic) rank-1
subshifts, which are also called Ferenczi subshifts in [2] to honor the fact
that Ferenczi popularized the concept in [23].

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 nonnegative
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:

$\begin{array}[]{rcl}v_{0}&=&0,\\\ v_{n+1}&=&v_{n}1^{s_{n,1}}v_{n}\cdots
v_{n}1^{s_{n,r_{n}-1}}v_{n}.\end{array}$

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 $(v_{1},\dots,v_{k+1})$ of elements of $S$ and
a sequence $(s_{1},\dots,s_{k})$ of nonnegative integers for $k\geq 1$ such
that

$w=v_{1}1^{s_{1}}v_{2}\cdots v_{k}1^{s_{k}}v_{k+1}.$

The sequence $(s_{1},\dots,s_{k})$ is called the spacer parameter of the
building; it is bounded by $M$ if $s_{1},\dots,s_{k}\leq M$. We say that every
word of $S$ is used in this building if $\\{v_{1},\dots,v_{k+1}\\}=S$. When
there is a building of $w$ from $S$, we also say that $w$ is built from $S$;
when the building consists of $(v_{1},\dots,v_{k+1})$ and
$(s_{1},\dots,s_{k})$, 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 $S_{i}\triangleq\\{v_{i,1},\dots,v_{i,n_{i}}\\}$
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
$i$-th 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

  * •

$v_{i+1,j}$,

  * •

$u_{1},\dots,u_{k+1}\in S_{i}$, where $(u_{1},\dots,u_{k+1})$ and
$(a_{1},\dots,a_{k})$ give the building of $v_{i+1,j}$ from $S_{i}$,

  * •

all expected subwords of $u_{1},\dots,u_{k+1}\in S_{i}$.

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 immediate 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
$i<i^{\prime}$, for all $1\leq j^{\prime}\leq n_{i^{\prime}}$ we obtain a
building of $v_{i^{\prime},j^{\prime}}$ from $S_{i}$ by composing the
buildings of elements of $S_{\iota}$ from $S_{\iota-1}$ for all
$i+1\leq\iota\leq i^{\prime}$. 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 $\mbox{\rm rank}_{\scriptsize\rm symb}(X)=\mbox{\rm
rank}_{\scriptsize\rm symb}(X,\sigma)$.

By definition, if $\mbox{\rm rank}_{\scriptsize\rm symb}(X)=n$ 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:

$v_{0}=0;\ v_{n+1}=v_{n}v_{n}1v_{n}.$

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 Subsection 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 ([29]). 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.

##  3\. Some computations of descriptive complexity

In this section we compute the descriptive complexity of various classes of
Cantor systems. We first show that the class of all essentially minimal Cantor
systems is a $G_{\delta}$ subset of $\mbox{\rm Aut}(\mathcal{C})$. Then we
give a characterization of all essentially minimal Cantor systems with bounded
topological rank. As a consequence, we show that for each $n\geq 1$, the class
of all essentially minimal Cantor systems of topological rank $\leq n$ is a
$G_{\delta}$ subset of $\mbox{\rm Aut}(\mathcal{C})$. This implies that for
each $n\geq 1$, the class of all minimal Cantor systems of topological rank
$\leq n$ is a $G_{\delta}$ subset of $M(\mathcal{C})$.

We first give a characterization of essential minimality for a Cantor system
$(X,T)$. We say a subset $A$ of $X$ has the finite covering property if there
is some $N\in\mathbb{N}$ such that $\bigcup_{-N\leq n\leq N}T^{n}A=X$.

######  Proposition 3.1.

Let $(X,T)$ be a Cantor system and let $\rho\leq 1$ be a compatible metric on
$X$. Then the following are equivalent:

  1. (1)

$(X,T)$ is essentially minimal.

  2. (2)

For any clopen set $A$ of $X$, if $A$ has the finite covering property then
there is a clopen subset $B$ of $A$ with the finite covering property such
that $\mbox{\rm diam}(B)\leq\mbox{\rm diam}(A)/2$.

###### Proof.

First assume $(X,T)$ is essentially minimal. Suppose $A$ is a clopen subset of
$X$ with the finite covering property, i.e. for some $N\in\mathbb{N}$ we have
$\bigcup_{-N\leq n\leq N}T^{n}A=X$. Let $x$ be an arbitrary element of the
unique minimal set of $X$. Then for some $-N\leq n\leq N$, $x\in T^{n}A$,
where $T^{n}A$ is still clopen. Let $Y$ be a clopen subset of $T^{n}A$
containing $x$ such that $\mbox{\rm diam}(T^{-n}Y)\leq\mbox{\rm diam}(A)/2$.
By Theorem 1.1 of [33], $\bigcup_{k\in\mathbb{Z}}T^{k}Y=X$. By the compactness
of $X$, $Y$ has the finite covering property. Let $B=T^{-n}Y$. Then
$B\subseteq A$, $\mbox{\rm diam}(B)\leq\mbox{\rm diam}(A)/2$ and $B$ also has
the finite covering property.

Conversely, assume (2) holds. Starting with $A_{0}=X$ and repeatedly applying
(2), we obtain a decreasing sequence $\\{A_{n}\\}_{n\geq 0}$ of clopen subsets
of $X$ such that $\mbox{\rm diam}(A_{n})\leq 2^{-n}$ and each $A_{n}$ has the
finite covering property. Let $x$ be the unique element of $\bigcap_{n}A_{n}$.
Then any clopen subset $B$ of $X$ containing $x$ has the finite covering
property. By Theorem 1.1 of [33], $(X,T)$ is essentially minimal. ∎

Let $E(\mathcal{C})$ be the set of all essentially minimal homeomorphisms of
$\mathcal{C}$.

######  Corollary 3.2.

$E(\mathcal{C})$ is a $G_{\delta}$ subset of $\mbox{\rm Aut}(\mathcal{C})$,
hence is a Polish space.

###### Proof.

Note that for any clopen subset $A$ of $\mathcal{C}$, $A$ has the finite
covering property for $(\mathcal{C},T)$ is an open condition for
$T\in\mbox{\rm Aut}(\mathcal{C})$. Thus condition (2) of Proposition 3.1 gives
a $G_{\delta}$ condition for $T\in\mbox{\rm Aut}(\mathcal{C})$. ∎

We next give a characterization of essentially minimal Cantor systems of
bounded topological rank.

######  Theorem 3.3.

Let $(X,T)$ be an essentially minimal Cantor system, $\rho\leq 1$ be a
compatible complete metric on $X$, and $n\geq 1$. The following are
equivalent:

  1. (1)

$(X,T)$ has topological rank $\leq n$.

  2. (2)

There exists $x\in X$ such that for all $\epsilon>0$, there is a
Kakutani–Rohlin partition $\mathcal{P}$ with no more than $n$ many towers such
that $\mbox{\rm diam}(A)<\epsilon$ for all $A\in\mathcal{P}$, $\mbox{\rm
diam}(B(\mathcal{P}))<\epsilon$, and $x\in B(\mathcal{P})$.

  3. (3)

For any clopen subset $Z$ of $X$ with the finite covering property, for any
finite partition $\mathcal{Q}$ into clopen sets, there is a Kakutani–Rohlin
partition $\mathcal{P}$ with no more than $n$ many towers such that
$B(\mathcal{P})\subseteq Z$, $\mbox{\rm diam}(B(\mathcal{P}))\leq\mbox{\rm
diam}(Z)/2$ and $\mathcal{P}$ refines $\mathcal{Q}$.

###### Proof.

We first show (1)$\Rightarrow$(2). Suppose $(X,T)$ has topological rank $\leq
n$. Then there is an essentially simple ordered Bratteli diagram
$B=(V,E,\preceq)$ such that $(X,T)$ is conjugate to the Bratteli–Vershik
system $(X_{B},\lambda_{B})$ generated by $B$, and for all $k\geq 1$,
$|V_{k}|\leq n$. It suffices to verify that (2) holds for
$(X_{B},\lambda_{B})$. Let $y=x_{\scriptsize\rm min}$. Each level $V_{k}$ of
$B$ gives rise to a Kakutani–Rohlin partition $\mathcal{P}_{k}$, where each
set in $\mathcal{P}_{k}$ corresponds to a path from $V_{0}$ to a vertex in
$V_{k}$, and $B(\mathcal{P}_{k})$ consists of all the basic open sets
correspondent to the minimal paths from $V_{0}$ to each vertex of $V_{k}$.
Since $y$ is the unique minimal infinite path, we have
$\bigcap_{k}B(\mathcal{P}_{k})=\\{y\\}$. Let $\eta\leq 1$ be the standard
metric on $X_{B}$. Let $\epsilon>0$. Then there is a large enough $k$ such
that all the minimal paths from $V_{0}$ to the vertices of $V_{k}$ agree on
the first $k^{\prime}<k$ many edges where $2^{-k^{\prime}}<\epsilon$. This
implies that $\mbox{\rm diam}(B(\mathcal{P}_{k}))\leq
2^{-k^{\prime}}<\epsilon$. Also, for all $A\in\mathcal{P}_{k}$, $\mbox{\rm
diam}(A)\leq 2^{-k}<2^{-k^{\prime}}<\epsilon$. $\mathcal{P}_{k}$ has
$|V_{k}|\leq n$ many towers. Since $y\in B(\mathcal{P}_{k})$, we have that
$\mathcal{P}_{k}$ witnesses (2) for $(X_{B},\lambda_{B})$.

Next we show (2)$\Rightarrow$(3). Suppose (2) holds for $x\in X$. We note that
for any $n\in\mathbb{Z}$, (2) holds also for $T^{n}x$. This is because, the
property described in (2) is invariant under topological conjugacy, and
$T^{n}:(X,T)\to(X,T)$ is a topological conjugacy sending $x$ to $T^{n}x$. Let
$Z$ be a clopen subset of $X$ with the finite covering property. Without loss
of generality assume $\dim(Z)>0$. Then $Z$ meets every orbit in $X$, and
therefore there is $x\in Z$ such that the property in (2) holds. Let
$\mathcal{Q}$ be a finite partition of $X$ into clopen sets. Let $\delta>0$ be
the infimum of $d(y,z)$ where $y,z$ are from different elements of
$\mathcal{Q}$. Let $\xi>0$ be such that $x\in\\{y\in
X\,:\,\rho(x,y)<\xi\\}\subseteq Z$ and $\mbox{\rm diam}(Z)>2\xi$. Let
$\epsilon=\min\\{\delta,\xi\\}>0$. Let $\mathcal{P}$ be a Kakutani–Rohlin
partition with no more than $n$ many towers such that $\mbox{\rm
diam}(A)<\epsilon$ for all $A\in\mathcal{P}$, $\mbox{\rm
diam}(B(\mathcal{P}))<\epsilon$, and $x\in B(\mathcal{P})$. Then
$B(\mathcal{P})\subseteq\\{y\in X\,:\,\rho(x,y)<\xi\\}\subseteq Z$, $\mbox{\rm
diam}(B(\mathcal{P}))<\xi<\mbox{\rm diam}(Z)/2$, and $\mathcal{P}$ refines
$\mathcal{Q}$ because for any $A\in\mathcal{P}$, $\mbox{\rm diam}(A)<\delta$.

Finally we prove (3)$\Rightarrow$(1). Assume $(X,T)$ is essentially minimal
and (3) holds. Note that the base of any Kakutani–Rohlin partition has the
finite covering property. By applying (3) repeatedly, we obtain a system of
Kakutani–Rohlin partitions $\\{\mathcal{P}_{k}\\}_{k\geq 0}$ so that
$\mathcal{P}_{0}=\\{X\\}$, each $\mathcal{P}_{k+1}$ refines $\mathcal{P}_{k}$,
$B(\mathcal{P}_{k+1})\subseteq B(\mathcal{P}_{k})$, $\mbox{\rm
diam}(B(\mathcal{P}_{k+1}))\leq\mbox{\rm diam}(B(\mathcal{P}_{k}))/2$, each
$\mathcal{P}_{k}$ consists of no more than $n$ many towers, and
$\bigcup_{k}\mathcal{P}_{k}$ generates the topology of $X$. Let $x$ be the
unique element of $\bigcap_{k}B(\mathcal{P}_{k})$. Then any clopen subset of
$X$ containing $x$ has the finite covering property. By Theorem 1.1 of [33],
$x$ is in the unique minimal set of $X$. Now $\\{\mathcal{P}_{k}\\}_{k\geq 0}$
is a nested system of Kakutani–Rohlin partitions in the sense of Theorem 2.2,
which gives rise to an ordered Bratteli diagram for $(X,T)$ with each level
consisting of no more than $n$ vertices. Thus $\mbox{\rm rank}_{\scriptsize\rm
top}(X,T)\leq n$. ∎

######  Corollary 3.4.

For any $n\geq 1$, the set of all essentially minimal $T\in\mbox{\rm
Aut}(\mathcal{C})$ with topological rank $\leq n$ is a $G_{\delta}$ subset of
$E(\mathcal{C})$. Similarly, for any $n\geq 1$, the set of all minimal
$T\in\mbox{\rm Aut}(\mathcal{C})$ with topological rank $\leq n$ is a
$G_{\delta}$ subset of $M(\mathcal{C})$.

###### Proof.

This follows immediately from clause (3) of Theorem 3.3. ∎

We also have the following immediate corollary regarding the descriptive
complexity of (essentially) minimal Cantor systems with finite topological
rank.

######  Corollary 3.5.

The set of all essentially minimal $T\in\mbox{\rm Aut}(\mathcal{C})$ with
finite topological rank is a ${\bf\Sigma}^{0}_{3}$ subset of $E(\mathcal{C})$.
Similarly, the set of all minimal $T\in\mbox{\rm Aut}(\mathcal{C})$ with
finite topological rank is a ${\bf\Sigma}^{0}_{3}$ subset of $M(\mathcal{C})$.

Here we remark that the proof of Theorem 3.3 implies that in clauses (2) and
(3) of Theorem 3.3 one may replace “no more than $n$ many towers” with
“exactly $n$ many towers.” Similar proofs also give that, if the system
$(X,T)$ considered is minimal, then in clause (2) of Theorem 3.3 one may
replace “there exists $x\in X$” with either “for nonmeager many $x\in X$” or
“for comeager many $x\in X$.”

Finally we note that the set of all infinite odometers form a dense
$G_{\delta}$ in $M(\mathcal{C})$.

######  Proposition 3.6.

The set of all infinite odometers is a dense $G_{\delta}$ in the space of all
minimal Cantor systems.

###### Proof.

Since the set of all infinite odometers is just the set of all minimal Cantor
systems of topological rank $1$, it is a $G_{\delta}$ in the space of all
minimal Cantor systems by Corollary 3.4. We only verify that it is dense. Let
$(X,T)$ be a minimal Cantor system and suppose $\mathcal{P}$ is a clopen
partition of $X$. We only need to define an infinite odometer $S$ on $X$ such
that $SZ=TZ$ for all $Z\in\mathcal{P}$. Consider $\tilde{T}=T^{-1}$. Then
$(X,\tilde{T})$ is again a mininal Cantor system. If we define an infinite
odometer $\tilde{S}$ on $X$ such that $\tilde{S}^{-1}Z=\tilde{T}^{-1}Z$ for
all $Z\in\mathcal{P}$, then $S=\tilde{S}^{-1}$ is again an infinite odometer,
and $SZ=TZ$ holds for all $Z\in\mathcal{P}$. Thus we focus on $(X,\tilde{T})$
in the rest of this proof.

By Lemma 2.1 $\mathcal{P}$ can be refined by a Kakutani–Rohlin partition for
$\tilde{T}$. Therefore, without loss of generality, we may assume that
$\mathcal{P}$ itself is a Kakutani–Rohlin partition. Suppose
$\mathcal{P}=\\{\tilde{T}^{j}B(k)\,:\,1\leq k\leq d,\,0\leq j<h(k)\\}$.

We define a directed graph $G=(V,E)$, where $V=\\{v_{1},\cdots,v_{d}\\}$ has
$d$ vertices, and for any $1\leq k,k^{\prime}\leq d$, there is a directed edge
$e\in E$ from $v_{k}$ to $v_{k^{\prime}}$ iff
$B(k^{\prime})\cap\tilde{T}^{h(k)}B(k)\neq\varnothing$. It follows from the
minimality of $\tilde{T}$ that $G=(V,E)$ is strongly connected, i.e, there is
a directed path from any vertex to any other vertex. Now fix a finite sequence
$p=(e_{1},\cdots,e_{m})$ of edges in $G$ such that
$(e_{1},\cdots,e_{m},e_{1})$ is a directed path and
$\\{e_{1},\cdots,e_{m}\\}=E$. Then $p$ is a directed cycle in $G$.

Consider an edge $e\in E$, say $e$ is from $v_{k}$ to $v_{k^{\prime}}$. Let
$n_{e}$ be the number of times $e$ appears in $p$. Let
$A_{e}=B(k)\cap\tilde{T}^{-h(k)}(B(k^{\prime}))$. Then $A_{e}$ is a clopen set
in $X$. Let $\\{A_{e,1},\dots,A_{e,n_{e}}\\}$ be a partition of $A_{e}$ into
$n_{e}$ many clopen subsets of $X$. If $e$ appears in $p$ as
$e_{i_{1}},\dots,e_{i_{n_{e}}}$, we associate with each $e_{i_{j}}$ the set
$A_{e,j}$ for $1\leq j\leq n_{e}$.

Thus we have obtained disjoint nonempty clopen sets $C_{1},\cdots,C_{m}$ such
that $\mathcal{Q}\triangleq\\{C_{1},\cdots,C_{m}\\}$ is a partition of
$B(\mathcal{P})$, and for any $1\leq i\leq m$, if $e_{i}$ is an edge from
$v_{k}$ to $v_{k^{\prime}}$ then $C_{i}\subseteq B(k)$ and
$\tilde{T}^{h(k)}C_{i}\subseteq B(k^{\prime})$. We define an odometer
$\tilde{S}\,:\,X\to X$ such that for any $1\leq i\leq m$, if $e_{i}$ is an
edge from $v_{k}$ to $v_{k^{\prime}}$ then $\tilde{S}^{h(k)}C_{i}=C_{i+1}$
(with $C_{m+1}=C_{1}$) and $\tilde{S}^{j}C_{i}=\tilde{T}^{j}C_{i}$ for $1\leq
j<h(k)$. In fact $\mathcal{Q}$ is a Kakutani–Rohlin partition for $\tilde{S}$
(with one tower), and $\tilde{S}$ is defined by recursive refinements starting
with $\mathcal{Q}$. It is now clear that $\tilde{S}^{-1}Z=\tilde{T}^{-1}Z$ for
all $Z\in\mathcal{P}$ as desired. ∎

##  4\. A characterization of minimal rank-$1$ subshifts

In this section we give an explicit topological characterization for all
minimal Cantor systems which are conjugate to infinite rank-$1$ subshifts. In
contrast to the results in Section 3, the descriptive complexity of this
characterization will be on a higher level than $G_{\delta}$.

Define

$\mathcal{Z}=\left\\{x\in 2^{\mathbb{Z}}\,:\,\forall n\ \exists m>n\
x(m)=0\mbox{ and }\forall n\ \exists m>n\ x(-m)=0\right\\}.$

Then $\mathcal{Z}$ is a $\sigma$-invariant dense $G_{\delta}$ subset of
$2^{\mathbb{Z}}$.

For a bi-infinite word $x\in\mathcal{Z}$ and a finite word $v\in\mathcal{F}$,
we say that $x$ is built from $v$ if $\sigma^{n}(x)$ can be written in the
form

$\sigma^{n}(x)=\cdots v1^{s_{-2}}v1^{s_{-1}}v1^{s_{0}}\cdot
v1^{s_{1}}v1^{s_{2}}\cdots$

for an bi-infinite sequence $(\cdots,s_{-2},s_{-1},s_{0},s_{1},s_{2},\cdots)$
of nonnegative integers and for some $n\in\Z$. For finite words
$u,v\in\mathcal{F}$, we say that $u$ is built from $v$ if there are
nonnegative integers $s_{1},\dots,s_{k}$ for $k\geq 1$ such that

$u=v1^{s_{1}}v\cdots v1^{s_{k}}v.$

The demonstrated occurrences of $v$ in $u$ are called expected occurrences.

######  Lemma 4.1.

Let $x\in\mathcal{Z}$ and $v\in\mathcal{F}$. Then the following are
equivalent:

  1. (i)

$x$ is built from $v$.

  2. (ii)

For all $m\in\mathbb{N}$ there exists a finite word $u$ such that
$x\\!\upharpoonright\\![-m,m]$ is a subword of $u$ and $u$ is built from $v$.

###### Proof.

The implication (i)$\Rightarrow$(ii) is immediate. We show
(ii)$\Rightarrow$(i). Let $n$ be the number of $0$s in $v$, i.e., $n$ is the
number of distinct occurrences of $0$ in $v$. Let $m_{0}$ be large enough such
that $x\\!\upharpoonright\\![-m_{0},m_{0}]$ contains at least $n$ many $0$s.
Let $k_{1}<\dots<k_{n}\in[-m_{0},m_{0}]$ be such that $x(k_{i})=0$ for all
$1\leq i\leq n$ and that if $k_{1}\leq k\leq k_{n}$ is such that $x(k)=0$ then
$k=k_{i}$ for some $1\leq i\leq n$. By (ii), for each $m\geq m_{0}$ there is a
finite word $u$ such that $x\\!\upharpoonright\\![-m,m]$ is a subword of $u$
and $u$ is built from $v$. Exactly one of $k_{1},\dots,k_{n}$ corresponds to a
starting position of an expected occurrence of $v$ in $u$. We denote this
value of $k\in\\{k_{1},\dots,k_{n}\\}$ as $k(m)$. Let
$k_{\infty}\in\\{k_{1},\dots,k_{n}\\}$ be such that for infinitely many $m\geq
m_{0}$, $k(m)=k_{\infty}$. Let $M_{\infty}$ be the infinite set such that for
all $m\in M_{\infty}$, $k(m)=k_{\infty}$. Then $v$ occurs in $x$ starting at
position $k_{\infty}$. We claim that for all $k>k_{\infty}$ such that $x(k)=0$
and there are a multiple of $n$ many $0$s from $k_{\infty}$ to $k-1$, $v$
occurs in $x$ starting at position $k$. This is because, fixing such a $k$ and
letting $m\in M_{\infty}$ with $m\geq k+|v|$, there is a finite word $u$ such
that $x\\!\upharpoonright\\![-m,m]$ is a subword of $u$ and $u$ is built from
$v$; since the occurrence of $v$ starting at position $k_{\infty}$ corresponds
to an expected occurrence of $v$ in $u$, it follows that there is another
expected occurrence of $v$ in $u$ starting at the position corresponding to
$k$, and so $v$ occurs in $x$ starting at position $k$. By a similar argument
we can also prove a claim that for all $k<k_{\infty}$ such that $x(k)=0$ and
there are a multiple of $n$ many $0$s from $k$ to $k_{\infty}-1$, $v$ occurs
in $x$ starting at position $k$. Putting these two claims together, we
conclude that $x$ is built from $v$. ∎

Lemma 4.1 implies immediately that for any $v\in\mathcal{F}$, the set of all
$x\in\mathcal{Z}$ such that $x$ is built from $v$ is closed in $\mathcal{Z}$.

Let $(X,T)$ be a Cantor system and let $A$ be a clopen subset of $X$. Define
$\mathcal{B}_{T}(A)$ to be the smallest Boolean algebra $\mathcal{B}$ of
subsets of $X$ such that $T^{n}A\in\mathcal{B}$ for all $n\in\mathbb{Z}$. We
say that $(T,A)$ is generating if $\mathcal{B}_{T}(A)$ contains all clopen
subsets of $X$.

######  Theorem 4.2.

Let $(X,T)$ be a minimal Cantor system and $x_{0}\in X$. Then the following
are equivalent:

  1. (1)

$(X,T)$ is conjugate to a (infinite) rank-$1$ subshift.

  2. (2)

There is a clopen subset $A$ of $X$ such that $(T,A)$ is generating and for
all $n\in\mathbb{N}$ there is a $v\in\mathcal{F}$ satisfying:

     * •

$|v|\geq n$ and $\mbox{\rm Ret}_{A}(x_{0})$ is built from $v$, and

     * •

for any $u\in\mathcal{F}$ such that $|u|\geq|v|$ and $\mbox{\rm
Ret}_{A}(x_{0})$ is built from $u$, there exists $u^{\prime}\in\mathcal{F}$
such that $|u^{\prime}|\leq|u|+|v|$, $u^{\prime}$ is built from $v$, and $u$
is an initial segment of $u^{\prime}$.

###### Proof.

Clause (2) is apparently conjugacy invariant, thus to see (1)$\Rightarrow$(2),
we may assume $V$ is a rank-$1$ word, $X=X_{V}$ is an infinite minimal
rank-$1$ subshift, and $T=\sigma$. Let $A=\\{x\in X\,:\,x(0)=1\\}$. Then
$(T,A)$ is generating, and $\mbox{\rm Ret}_{A}(x_{0})=x_{0}$. The set of all
finite words $v$ such that $V$ is built from $v$ is a subset of the set of all
finite words $v$ such that $x_{0}$ is built from $v$. Now given any
$n\in\mathbb{N}$, let $v\in\mathcal{F}$ be such that $V$ is built
fundamentally from $v$ (see Definition 2.13 of [25]). Then by Proposition 2.16
of [25], for any $u\in\mathcal{F}$ such that $|u|\geq|v|$ and $V$ is built
from $u$, $u$ is built from $v$. This proves (2) by Proposition 2.36 of [25].

Conversely, assume $A$ is a clopen subset of $X$ witnessing (2). Since $(T,A)$
is generating, the map $\mbox{\rm Ret}_{A}:X\to 2^{\mathbb{Z}}$ is a
homeomorphic embedding such that $\mbox{\rm Ret}_{A}\circ
T=\sigma\circ\mbox{\rm Ret}_{A}$. Thus $\mbox{\rm Ret}_{A}(X)$ is a minimal
subshift, and $\mbox{\rm Ret}_{A}$ is a conjugacy map. By repeatedly applying
(2), we obtain an infinite sequence of finite words $\\{v_{n}\\}_{n\geq 0}$ in
$\mathcal{F}$ such that $\mbox{\rm Ret}_{A}(x_{0})$ is built from each $v_{n}$
and for all $n\geq 0$, $v_{n}$ is an initial segment of $v_{n+1}$ and
$v_{n+1}$ is an initial segment of some $u$ which is built from $v_{n}$. This
allows us to define an infinite word $V=\lim_{n}v_{n}$. By definition, $V$ is
a rank-$1$ word. To finish the proof it suffices to verify that $\mbox{\rm
Ret}_{A}(X)=X_{V}$. By the minimality of $\mbox{\rm Ret}_{A}(X)$, for any
$y\in\mbox{\rm Ret}_{A}(X)$, the set of all finite subwords of $y$ coincides
with the set of all finite subwords of $\mbox{\rm Ret}_{A}(x_{0})$. On the
other hand, our assumption guarantees that the set of all finite subwords of
$\mbox{\rm Ret}_{A}(x_{0})$ coincides with the set of all finite subwords of
$V$. Thus $\mbox{\rm Ret}_{A}(X)=X_{V}$ and $X$ is conjugate to $X_{V}$, a
rank-$1$ subshift. ∎

The apparent descriptive complexity given by clause (2) of the above theorem
is ${\bf\Sigma}^{0}_{5}$, which is significantly more complex than
$G_{\delta}$.

##  5\. Proper finite rank constructions

The following is a basic property regarding symbolic rank-$n$ constructions.

######  Proposition 5.1.

Let $n\geq 1$. Suppose $\\{T_{i}\\}_{i\geq 0}$ is a sequence of finite subsets
of $\mathcal{F}$ such that $T_{0}=\\{0\\}$ and for all $i\geq 0$, $|T_{i}|\leq
n$ and each element of $T_{i+1}$ is built from $T_{i}$. Then there is a
rank-$n$ construction with associated rank-$n$ generating sequence
$\\{v_{i,j}\\}_{i\geq 0,1\leq j\leq n_{i}}$ such that for all $i\geq 0$,
$v_{i,1},\dots,v_{i,n_{i}}\in T_{i}$.

###### Proof.

For each $i\geq 0$ and $v\in T_{i+1}$, fix a building of $v$ from $T_{i}$.
Define a binary relation $R$ on $\bigcup_{i\geq 0}T_{i}$ by $R(u,v)$ if for
some $i\geq 0$, $u\in T_{i}$, $v\in T_{i+1}$, and the building of $v$ from
$T_{i}$ starts with $u$. Let $<$ be the transitive closure of $R$. Then $<$ is
a (strict) partial order on $\bigcup_{i\geq 0}T_{i}$. We inductively define an
infinite $R$-chain of words $\\{u_{i}\\}_{i\geq 0}$, i.e., $u_{i}\in T_{i}$
and $R(u_{i},u_{i+1})$ for all $i\geq 0$. Let $u_{0}=0$. Note that there are
infinitely many words $u\in\bigcup_{i\geq 0}T_{i}$ such that $u_{0}<u$ (in
fact $u_{0}<u$ for all $u\in\bigcup_{i\geq 0}T_{i}$). In general, assume
$u_{i}$ has been defined such that there are infinitely many
$w\in\bigcup_{i\geq 0}T_{i}$ with $u_{i}<w$. In particular the set
$W=\\{w\in\bigcup_{j\geq i+2}T_{j}\,:\,u_{i}<w\\}$ is infinite. Note that for
each $w\in W$ there is a $u_{w}\in T_{i+1}$ such that $R(u_{i},u_{w})$ and
$u_{w}<w$. Since $T_{i+1}$ is finite, there is a $v\in T_{i+1}$ such that for
infinitely many $w\in W$, $u_{w}=v$. Let $u_{i+1}=v$. Then there are
infinitely many $w\in\bigcup_{i\geq 0}T_{i}$ such that $u_{i+1}<w$. This
finishes the inductive construction.

Now define $v_{i,j}$ for each $i\geq 0$ so that $v_{i,1}=u_{i}$ and
$\\{v_{i,1},\dots,v_{i,n_{i}}\\}=T_{i}$, where $n_{i}=|T_{i}|$. With the fixed
buildings, this gives a rank-$n$ construction as required. ∎

Next we characterize the rank-$n$ subshifts which have proper rank-$n$
constructions. We use $1^{\mathbb{Z}}$ to denote the element $x\in
2^{\mathbb{Z}}$ where $x(k)=1$ for all $k\in\mathbb{Z}$.

######  Theorem 5.2.

Let $n\geq 1$ and let $X$ be a subshift of symbolic rank $n$. The following
are equivalent:

  1. (1)

There exists a rank-$n$ word $V$ such that $X=X_{V}$, and $V$ has a proper
rank-$n$ construction.

  2. (2)

For any rank-$n$ word $V$ such that $X=X_{V}$, $V$ has a proper rank-$n$
construction.

  3. (3)

For any $x\in X$ such that $x\neq 1^{\mathbb{Z}}$, the orbit of $x$ is dense
in $X$.

###### Proof.

We first show (1)$\Rightarrow$(3). Suppose $V$ is a rank-$n$ word such that
$X=X_{V}$, and $V$ has a proper rank-$n$ construction with associated rank-$n$
generating sequence $\\{v_{i,j}\\}_{i\geq 0,1\leq j\leq n}$. For each $i\geq
0$, define $a_{i}=\max_{1\leq j\leq n}|v_{i,j}|$. Let $x\in X_{V}$ and assume
$x\neq 1^{\mathbb{Z}}$. There exists an $m\in\mathbb{Z}$ so that $x(m)=0$. Fix
an $i\geq 0$ and consider the finite word
$u=x\\!\upharpoonright\\![m-a_{i+1},m+a_{i+1}]$. By the definition of $X_{V}$,
$u$ is a subword of $V$. Since $V$ is built from $S_{i+1}$, by considering the
length of $u$ we get that there is $1\leq j_{0}\leq n$ such that
$v_{i+1,j_{0}}$ is a subword of $u$. By the properness of the rank-$n$
construction, $v_{i,1}$ is a subword of $v_{i+1,j_{0}}$, and hence a subword
of $x$. This implies that the orbit of $x$ is dense in $X_{V}$.

Next we show (3)$\Rightarrow$(2). Let $V$ be a rank-$n$ word such that
$X=X_{V}$. Suppose for any $x\in X_{V}$ such that $x\neq 1^{\mathbb{Z}}$, the
orbit of $x$ is dense in $X_{V}$. We fix a rank-$n$ construction with
associated rank-$n$ generating sequence $\\{v_{i,j}\\}_{i\geq 0,1\leq j\leq
n_{i}}$, where $V=\lim_{i}v_{i,1}$. Since $V$ is a rank-$n$ word, it does not
have a rank-$(n-1)$ construction; by telescoping if necessary, we may assume
that $n_{i}=n$ for all $i\geq 0$. Also, without loss of generality, we assume
that for all $i\geq 0$, all finite words in $S_{i}$ are expected subwords of
$V$. In particular, if $i<i^{\prime}$, then every finite word in $S_{i}$ is an
expected subword of some word in $S_{i^{\prime}}$.

Next we claim that for any $i_{0}>0$ and $1\leq j_{0}\leq n$, there exists
$i>i_{0}$ such that for any $1\leq j\leq n$, $v_{i_{0},j_{0}}$ is an expected
subword of $v_{i,j}$. Assume not; then for any $i>i_{0}$ there is $1\leq j\leq
n$ such that $v_{i_{0},j_{0}}$ is not an expected subword of $v_{i,j}$. We
define a sequence $\\{T_{k}\\}_{k\geq 0}$ of finite subsets of $\mathcal{F}$
as follows. Let $T_{0}=\\{0\\}$. For $k>0$, let $T_{k}$ be the set of all
$v_{i_{0}+k,j}$ for $1\leq j\leq n$ such that $v_{i_{0},j_{0}}$ is not an
expected subword of $v_{i_{0}+k,j}$. Then for all $k\geq 0$, $T_{k}\subseteq
S_{i_{0}+k}$ and so $|T_{k}|\leq n-1$. Also, for all $k\geq 0$, each element
of $T_{k+1}$ is built from $T_{k}$. By Proposition 5.1, there is a
rank-$(n-1)$ construction with associated rank-$(n-1)$ generating sequence
$\\{w_{k,\ell}\\}_{k\geq 0,1\leq\ell\leq m_{k}}$ such that for all $k\geq 0$
and $1\leq\ell\leq m_{k}$, $w_{k,\ell}\in T_{k}\subseteq S_{i_{0}+k}$. Let
$W=\lim_{k}w_{k,1}$. Then every finite subword of $W$ is a subword of $V$.
Hence $X_{W}$ is an closed invariant subset of $X_{V}$. It is clear from the
construction of $W$ that there is $x\in X_{W}$ such that $x\neq
1^{\mathbb{Z}}$. Since the orbit of $x$ is dense in $X_{V}$, we get that
$X_{W}=X_{V}$. This contradicts our assumption that $\mbox{\rm
rank}_{\scriptsize\rm symb}(X_{V})=n$.

Using the claim, and by telescoping, we obtain a proper rank-$n$ construction
for $V$.

Finally, (2)$\Rightarrow$(1) is immediate. ∎

Note that the implications (1)$\Rightarrow$(3) in the above theorem do not
require that $X$ is of symbolic rank $n$.

######  Corollary 5.3.

Let $n\geq 1$ and $X$ be an infinite subshift of symbolic rank $\leq n$.
Suppose $X=X_{V}$ and $V$ has a proper rank-$n$ construction. Then
$(X,\sigma)$ is an essentially minimal Cantor system. In particular, there is
$k\in\mathbb{N}$ such that $0^{k}$ is not a subword of $V$.

###### Proof.

Since $X=X_{V}$ where $V$ has a proper rank-$n$ construction, $V$ is
recurrent. Since $X_{V}$ is infinite, it is a Cantor set. Now if
$1^{\mathbb{Z}}\not\in X$, then by Theorem 5.2 (3) $X$ is minimal; if
$1^{\mathbb{Z}}\in X$ then $\\{1^{\mathbb{Z}}\\}$ is invariant and by Theorem
5.2 (3) it is the unique minimal set in $X$. Thus $(X,\sigma)$ is an
essentially minimal Cantor system. In either case, $0^{\mathbb{Z}}\not\in
X_{V}$, thus there is $k\in\mathbb{N}$ such that $0^{k}$ is not a subword of
$V$. ∎

Note that any rank-$1$ construction is proper, and thus any infinite rank-$1$
subshift is an essentially minimal Cantor system.

######  Corollary 5.4.

Let $n\geq 1$ and let $X$ be an infinite subshift of symbolic rank $n$. Then
the following are equivalent:

  1. (1)

$X$ is minimal.

  2. (2)

There exists a rank-$n$ word $V$ such that $X=X_{V}$, and $V$ has a proper
rank-$n$ construction with bounded spacer parameter.

  3. (3)

For any rank-$n$ word $V$ such that $X=X_{V}$, $V$ has a proper rank-$n$
construction with bounded spacer parameter.

###### Proof.

To see (1)$\Rightarrow$(3), suppose $X$ is minimal. Then
$1^{\mathbb{Z}}\not\in X$ and clause (3) of Theorem 5.2 holds. By Theorem 5.2,
for any rank-$n$ word $V$ such that $X=X_{V}$, $V$ has a proper rank-$n$
construction with associate rank-$n$ generating sequence $\\{v_{i,j}\\}_{i\geq
0,1\leq j\leq n}$. Without loss of generality, we may assume that every word
in this sequence is an expected subword of $V$. We claim that this given
proper rank-$n$ construction has bounded spacer parameter. Otherwise there are
arbitrarily large $k$ with $1^{k}$ as a subword of $V$, and then
$1^{\mathbb{Z}}\in X_{V}=X$, a contradiction.

The implication (3)$\Rightarrow$(2) is immediate.

Finally, we prove (2)$\Rightarrow$(1). Suppose $V$ is a rank-$n$ word such
that $X=X_{V}$, and $V$ has a proper rank-$n$ construction with bounded spacer
parameter. Then $1^{\mathbb{Z}}\not\in X_{V}$, and by Theorem 5.2 $X$ is
minimal. ∎

Again, we remark that the implications (2)$\Rightarrow$(1) of the above
corollary do not require that $X$ be a subshift of symbolic rank $n$.

##  6\. Finite symbolic rank and finite topological rank

In this section we prove that minimal subshifts of finite symbolic rank have
finite topological rank, and conversely, any minimal Cantor system of finite
topological rank is either an odometer or conjugate to a subshift of finite
symbolic rank. Together with previous results ([14] and [13]), our results
show that the three notions of finite rank for minimal expansive Cantor
systems all coincide with each other.

###  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
$v_{i,1},\dots,v_{i,n}$ 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)

$E_{0}=\\{(k,j)\,:\,x(k)=0\mbox{ and }j=1\\}$; and

  4. (iv)

for each $(k,j)\in E_{i}$, there is exactly one $(k^{\prime},j^{\prime})\in
E_{i+1}$ such that $k^{\prime}\leq k$ and $k^{\prime}+|v_{i+1,j^{\prime}}|\geq
k+|v_{i,j}|$.

If every $x\in X_{V}$ has a unique reading, we say that $\\{v_{i,j}\\}_{i\geq
0,1\leq j\leq m}$ 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 (Proposition
2.29 of [25]).

######  Lemma 6.1.

Let $n\geq 1$, $\\{v_{i,j}\\}_{i\geq 0,1\leq j\leq n}$ be a rank-$n$
generating sequence, and $V=\lim_{i}v_{i,1}$. Then any $x\in X_{V}$ has a
reading.

###### Proof.

We fix a rank-$n$ construction of $V$ with associated rank-$n$ generating
sequence $\\{v_{i,j}\\}_{i\geq 0,1\leq j\leq n}$. Without loss of generality
we may assume that every $v_{i,j}$ is an expected subword of $V$, and that for
each $i\geq 1$, the words $v_{i,1},\dots,v_{i,n}$ are distinct. For each
$i\geq 0$, 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 $b_{i+1}\geq 2b_{i}$ for all $i\geq 0$.

We consider several cases. Case 1: $x=1^{\mathbb{Z}}$. In this case a unique
reading is given by $E_{i}=\varnothing$ for all $i\geq 0$.

Case 2: There exists $k_{0}\in\mathbb{Z}$ such that $x(k_{0})=0$ and $x(k)=1$
for all $k<k_{0}$. First fix any $i\geq 0$. Define
$u_{i}=x\\!\upharpoonright\\![k_{0}-a_{i},k_{0}+a_{i}]$. Since $u_{i}$ is a
subword of $V$, $V$ is built from $\\{v_{i,1},\dots,v_{i,n}\\}$, and
$a_{i}\geq|v_{i,j}|$ for all $1\leq j\leq n$, we have that for some $1\leq
j_{0}\leq n$, $k_{0}$ is the starting position of an occurrence of
$v_{i,j_{0}}$ in $x$. Now following the rank-$n$ construction of $V$, by an
induction on $t=i,i-1,\dots,0$, we define collections $E^{i}_{t}$ for $t\leq
i$ as follows. First let $E^{i}_{i}=\\{(k_{0},j_{0})\\}$. Suppose now
$E^{i}_{t}$ has been defined, which is a collection of some pairs $(k,j)$,
where $k$ is the starting position of an occurrence of $v_{t,j}$ in $x$.
Assume that for $(k_{1},j_{1}),(k_{2},j_{2})\in E^{i}_{t}$ with $k_{1}<k_{2}$,
we have $k_{1}+|v_{t,j_{1}}|\leq k_{2}$. Now for each $(k,j)\in E^{i}_{t}$,
the building of $v_{t,j}$ from $\\{v_{t-1,1},\dots,v_{t-1,n}\\}$ in the fixed
rank-$n$ construction gives rise to pairs $(k^{\prime},j^{\prime})$, where
$v_{t-1,j^{\prime}}$ occurs at position $k^{\prime}$ and this occurrence
corresponds to the occurrence of $v_{t-1,j^{\prime}}$ in the building of
$v_{t,j}$ as an expected subword. We put all such $(k^{\prime},j^{\prime})$ in
$E^{i}_{t-1}$. It is clear that for
$(k_{1}^{\prime},j_{1}^{\prime}),(k_{2}^{\prime},j_{2}^{\prime})\in
E^{i}_{t-1}$ with $k_{1}^{\prime}<k_{2}^{\prime}$, we have
$k_{1}^{\prime}+|v_{t-1,j_{1}^{\prime}}|\leq k_{2}^{\prime}$. It is also clear
that for each $(k^{\prime},j^{\prime})\in E^{i}_{t-1}$, there is exactly one
$(k,j)\in E^{i}_{t}$ such that $k\leq k^{\prime}$ and $k+|v_{t,j}|\geq
k^{\prime}+|v_{t-1,j^{\prime}}|$. Finally, we note that
$E^{i}_{0}=\\{(k,j)\,:\,x(k)=0,k_{0}\leq k\leq k_{0}+|v_{i,j}|-1,\mbox{ and
}j=1\\}$. This finishes the definition of $E^{i}_{t}$ for $t\leq i$. We have
that for all $k_{0}\leq k\leq k_{0}+b_{i}+1$, $(k,1)\in E^{i}_{0}$ iff
$x(k)=0$.

For $i\geq 0$, define
$e_{i}\in\\{0,1\\}^{\mathbb{N}\times\mathbb{Z}\times\\{1,\cdots,n\\}}$ by
letting $e_{i}(t,k,j)=1$ iff $t\leq i$ and $(k,j)\in E^{i}_{t}$. Since
$\\{0,1\\}^{\mathbb{N}\times\mathbb{Z}\times\\{1,\cdots,n\\}}$ is compact,
there exists an accumulation point $e$ of $\\{e_{i}\\}_{i\geq 0}$. For each
$t\geq 0$, define $E_{t}=\\{(k,j)\,:\,e(t,k,j)=1\\}$. Since
$\\{b_{i}\\}_{i\geq 0}$ is strictly increasing, we conclude that for all
$k\geq k_{0}$, $(k,1)\in E_{0}$ iff $x(k)=0$. The other properties of a
reading are also easily verified. Thus $\\{E_{t}\\}_{t\geq 0}$ is a reading of
$x$.

Case 3: There exists $k_{0}\in\mathbb{Z}$ such that $x(k_{0})=0$ and $x(k)=1$
for all $k>k_{0}$. This case is similar to Case 2.

Case 4: For any $k\in\mathbb{Z}$ there are $k_{1}<k<k_{2}$ such that
$x(k_{1})=x(k_{2})=0$. Let $k_{0}$ be an integer satisfying $x(k_{0})=0$. For
$i\geq 0$, let $\ell_{i,1}$ be the $(2a_{i}+1)$th natural number such that
$x(k_{0}+\ell_{i,1})=0$; let $\ell_{i,2}$ be the $(2a_{i}+1)$th natural number
such that $x(k_{0}-\ell_{i,2})=0$. Define
$u_{i}=x\\!\upharpoonright\\![k_{0}-\ell_{i,2},k_{0}+\ell_{i,1}]$. Then
$u_{i}$ is a subword of $V$. Since $V$ is built from
$\\{v_{i,1},\dots,v_{i,n}\\}$, by the definition of $a_{i}$, there exist
$m_{i}<k_{0}$ and a subword $w_{i}$ of $V$ such that

  * •

$w_{i}$ is of the form $v_{i,j_{1}}1^{s_{1}}v_{i,j_{2}}1^{s_{2}}v_{i,j_{3}}$,
where $1\leq j_{1},j_{2},j_{3}\leq n$ and $s_{1},s_{2}\geq 0$,

  * •

$m_{i}$ is the starting position of an occurrence of $w_{i}$ in $x$, and

  * •

$m_{i}+|v_{i,j_{1}}1^{s_{1}}|\leq k_{0}\leq
m_{i}+|v_{i,j_{1}}1^{s_{1}}v_{i,j_{2}}|-1$.

Now we proceed as in the proof of Case 2 to define $E^{i}_{t}$ for all $t\leq
i$ and finally obtain a reading $\\{E_{t}\\}_{t\geq 0}$ of $x$ by compactness.
∎

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 $1\leq j\leq n$, $v_{i,j}$ is not of the form

$\alpha 1^{s_{1}}v_{i,j_{1}}1^{s_{2}}v_{i,j_{2}}\cdots
v_{i,j_{k-1}}1^{s_{k}}\beta$

where $k\geq 1$, $\alpha$ is a nonempty suffix of some $v_{i,j_{k}}$, and
$\beta$ is a nonempty 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.

######  Lemma 6.2.

Consider a good rank-$n$ construction with associate rank-$n$ generating
sequence $\\{v_{i,j}\\}_{i\geq 0,1\leq j\leq n}$. Then $\\{v_{i,j}\\}_{i\geq
0,1\leq j\leq n}$ has unique readability.

###### Proof.

Let $x\in X_{V}$. Let $\\{E_{i}\\}_{i\geq 0}$ be a reading of $x$, which
exists by Lemma 6.1. By the definition of a reading, we have that for each
$i\geq 0$, $E_{i}$ gives a way in which $x$ is built from
$\\{v_{i,1},\dots,v_{i,n}\\}$. Now suppose $\\{E_{i}^{\prime}\\}_{i\geq 0}$ is
another reading of $x$, and suppose $i\geq 0$ is the smallest such that
$E_{i}\neq E^{\prime}_{i}$. Without loss of generality, let $(k,j)\in
E_{i}\setminus E^{\prime}_{i}$.

Consider two cases. Case 1: there is $j^{\prime}\neq j$ such that
$(k,j^{\prime})\in E^{\prime}_{i}$. In this case without loss of generality
assume $|v_{i,j}|>|v_{i,j^{\prime}}|$. Then $v_{i,j}$ can be written in the
form $v_{i,j^{\prime}}1^{s_{1}}v_{i,j_{1}}\cdots
v_{i,j_{\ell}}1^{s_{\ell+1}}\beta$ for some $\ell\geq 0$ and nonempty $\beta$
where $\beta$ is a prefix of some $v_{i,j_{\ell+1}}$. This contradicts the
assumption that our rank-$n$ construction is good.

Case 2: there is no $j^{\prime}$ such that $(k,j^{\prime})\in E^{\prime}_{i}$.
By the definition of a reading, there is a unique $(k^{\prime},j^{\prime})\in
E^{\prime}_{i}$ where $k^{\prime}<k$ such that $k\leq
k^{\prime}+|v_{i,j^{\prime}}|$. If $k^{\prime}+|v_{i,j^{\prime}}|\leq
k+|v_{i,j}|$ then $v_{i,j}$ can be written in the form $\alpha
1^{s_{1}}v_{i,j_{1}}\cdots v_{i,j_{\ell}}1^{s_{\ell+1}}\beta$, contradicting
the assumption that our rank-$n$ construction is good. If
$k^{\prime}+|v_{i,j^{\prime}}|>k+|v_{i,j}|$ then $v_{i,j^{\prime}}$ can be
written in the form $\alpha 1^{s_{1}}v_{i,j_{1}}\cdots
v_{i,j_{\ell}}1^{s_{\ell+1}}\beta$, again contradicting our assumption. ∎

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

######  Lemma 6.3.

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

$\\{x\in X_{V}\,:\,\mbox{ there is an expected occurrence of $v_{i,j}$ at
position $k$}\\}$

is clopen in $X_{V}$.

###### Proof.

Let $E_{i,j,k}$ denote the set in question. Then $x\in E_{i,j,k}$ iff for any
$0\leq t\leq|v_{i,j}|-1$, $x(k+t)=v_{i,j}(t)$ and for any $1\leq
j^{\prime}\leq n$ and $k^{\prime}\leq k$, if $j^{\prime}\neq j$ and
$k^{\prime}+|v_{i,j^{\prime}}|\geq k+|v_{i,j}|$, then there is $0\leq
s\leq|v_{i,j^{\prime}}|-1$ such that $x(k^{\prime}+s)\neq
v_{i,j^{\prime}}(s)$. This implies that $E_{i,j,k}$ is clopen. ∎

######  Proposition 6.4.

Let $n\geq 1$ and $X$ be an infinite subshift of symbolic rank $\leq n$.
Suppose $X=X_{V}$ and $V$ has a proper rank-$n$ construction. Then there
exists a word $W$ with a good rank-$2n$ construction such that $X$ is a factor
of $X_{W}$. Moreover, if in addition $X$ is minimal, then $W$ can be chosen so
that $X_{W}$ is minimal.

###### Proof.

Fix a proper rank-$n$ construction of $V$ with associated rank-$n$ generating
sequence $\\{v_{i,j}\\}_{i\geq 0,1\leq j\leq n}$. Without loss of generality
assume $v_{i,1},\dots,v_{i,n}$ 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 $i\geq 0$, 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

$w_{1,q}=\left\\{\begin{array}[]{ll}v_{i_{1},q}&\mbox{ if $1\leq q\leq n$,}\\\
\\\ 010^{|v_{i_{1},q-n}|-4}10&\mbox{ if $n+1\leq q\leq
2n$.}\end{array}\right.$

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

$m_{p}=\left\lceil\frac{a_{i_{p}+1}}{b_{i_{p}}}\right\rceil.$

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 $1\leq j\leq n$, $v_{i_{p+1},j}$ in the form

(1) $v_{i_{p},j_{1}}1^{s_{1}}\cdots 1^{s_{\ell}}v_{i_{p},j_{\ell+1}}$

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 (1): for any $1\leq t\leq\ell+2-2m_{p}$,
$\\{j_{t},\cdots,j_{t+2m_{p}-1}\\}=\\{1,\cdots,n\\}$. This is because,

(2) $u\triangleq v_{i_{p},j_{t}}1^{s_{t}}\cdots v_{i_{p},j_{t+2m_{p}-1}}$

consists of $2m_{p}$ many consecutive expected occurrences of subwords of
$v_{i_{p+1},j}$ of the form $v_{i_{p},j^{\prime}}$; since $v_{i_{p+1},j}$ is
built from $\\{v_{i_{p}+1,1},\dots,v_{i_{p}+1,n}\\}$, by our definition of
$m_{p}$, $u$ must contain some expected occurrence of
$v_{i_{p}+1,j^{\prime}}$, where $1\leq j^{\prime}\leq n$, as a subword. Hence
(*) 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 (1).
For $1\leq t\leq\ell+1$, define

$\phi(t)=\left\\{\begin{array}[]{ll}j_{t}+n&\mbox{ if $2\leq t\leq 2m_{p}+j+1$
or $\ell-2m_{p}-j+1\leq t\leq\ell$,}\\\ j_{t}&\mbox{
otherwise}\end{array}\right.$

and

$\psi(t)=\left\\{\begin{array}[]{ll}j_{t}+n&\mbox{ if $2\leq t\leq
2m_{p}+j+n+1$ or $\ell-2m_{p}-j-n+1\leq t\leq\ell$,}\\\ j_{t}&\mbox{
otherwise.}\end{array}\right.$

Then define

$w_{p+1,j}=w_{p,\phi(1)}1^{s_{1}}\cdots 1^{s_{\ell}}w_{p,\phi(\ell+1)}$

and

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

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

We verify that the construction defined is proper, i.e., for all $p\geq 1$ and
$1\leq q\leq 2n$, all words in $\\{w_{p,1},\dots,w_{p,2n}\\}$ 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 (*),
$\\{\phi(t_{0}),\dots,\phi(t_{0}+2m_{p}-1)\\}=\\{j_{t_{0}},\dots,j_{t+2m_{p}-1}\\}=\\{1,\dots,n\\}$.
Thus all words in $\\{w_{p,1},\dots,w_{p,n}\\}$ are used in the building of
$w_{p+1,q}$. On the other hand, for $2\leq t\leq 2m_{p}+1$, $\phi(t)=j_{t}+n$.
By property (*) again,
$\\{\phi(2),\dots,\phi(2m_{p}+1)\\}=\\{j_{2}+n,\dots,j_{2m_{p}+1}+n\\}=\\{n+1,\dots,2n\\}$.
Hence all words in $\\{w_{p,n+1},\dots,w_{p,2n}\\}$ 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) $\alpha 1^{r_{1}}w_{p,q_{1}}1^{r_{2}}w_{p,q_{2}}\cdots
w_{p,q_{d-1}}1^{r_{d}}\beta$

where $d\geq 1$, $\alpha$ is a nonempty suffix of some $w_{p,q_{d}}$ and
$\beta$ is a nonempty prefix of some $w_{p,q_{d+1}}$, and

  * •

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

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

First suppose $p=2$. We observe that $w_{2,q}$ can be written as
$u_{1}yu_{2}$, where $0^{k_{0}}$ is not a subword of $y$, $y$ begins and ends
with 0, every word of $\\{v_{i_{1},1},\dots,v_{i_{1},n}\\}$ occurs at least 3
different times in $y$, and both $u_{1}$ and $u_{2}$ are of the form

(4) $\alpha 01^{s_{1}}010^{h_{1}}101^{s_{2}}010^{h_{2}}10\cdots
010^{h_{2m_{p}+q}}101^{s_{2m_{p}+q+1}}0\beta$

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 (1),
$y$ can be taken to contain subwords of the form (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 (2) contains a distinct
occurrence of each word in $\\{v_{i_{1},1},\dots,v_{i_{1},n}\\}$. Note that
for $q^{\prime}\neq q$, $w_{2,q^{\prime}}$ does not have a subword of the form
(4), hence $w_{2,q}$ is not a subword of $w_{2,q^{\prime}}$. Now suppose
$w_{2,q}$ can be written in the form of (3), then by the above observation,
there are $1\leq q_{1},q_{2}\leq 2n$, a nonempty suffix $y_{1}$ of
$w_{2,q_{1}}$, and a nonempty prefix $y_{2}$ of $w_{2,q_{2}}$, such that
$w_{2,q}=y_{1}1^{s}y_{2}$ for some nonnegative 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
$3$ different occurrences of each word in
$\\{v_{i_{1},1},\dots,v_{i_{1},n}\\}$, while $z$ does not have this property,
a contradiction. Next suppose $q_{1}\neq q$. 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,
contradicting the assumption that $y_{1}$ is nonempty. The case $q_{2}\neq q$
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
$u_{1}yu_{2}$, where $u_{1}$ and $u_{2}$ are of the form

(5) $w_{p,q_{1}}1^{s_{1}}w_{p,q_{2}}1^{s_{2}}\cdots
w_{p,q_{2m_{p}+q+1}}1^{s_{2m_{p}+q+1}}w_{p,q_{2m_{p}+q+2}}$

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
$q\neq q^{\prime}$ then $w_{p+1,q^{\prime}}$ does not contain a subword of the
form (5), hence $w_{p+1,q}$ is not a subword of $w_{p+1,q^{\prime}}$. Next
assume $w_{p+1,q}$ can be written in the form (3) with $p+1$ replacing $p$.
Then by the above observation, there are $1\leq q_{1},q_{2}\leq 2n$, a
nonempty suffix $y_{1}$ of $w_{p+1,q_{1}}$, and a nonempty prefix $y_{2}$ of
$w_{p+1,q_{2}}$ such that $w_{p+1,q}=y_{1}1^{s}y_{2}$ for some nonnegative
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 $3$ different occurrences
of words in $\\{w_{p,1},\dots,w_{p,n}\\}$, a contradiction. Next suppose
$q_{1}\neq q$. 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, contradicting the
assumption that $y_{1}$ is nonempty. The case $q_{2}\neq q$ 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 $k^{\prime}\leq k$, then let
$\varphi(x)(k)=v_{i_{2},j}(k-k^{\prime})$; 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. ∎

The following is a corollary to the proof of Proposition 6.4.

######  Proposition 6.5.

Let $n\geq 1$ and $X$ be an infinite subshift of symbolic rank $\leq n$.
Suppose $X=X_{V}$ and $V$ has a proper rank-$n$ construction
$\\{v_{i,j}\\}_{i\geq 0,1\leq j\leq n}$ which has unique readability. Then for
any $i\geq 0$, $1\leq j\leq n$, and $k\in\mathbb{Z}$, the set

$\\{x\in X\,:\,\mbox{ there is an expected occurrence of $v_{i,j}$ in $x$ at
position $k$}\\}$

is clopen in $X$.

###### Proof.

Let $W$ be the infinite word with a good rank-$2n$ construction with
associated rank-$2n$ generating sequence $\\{w^{\prime}_{p,q}\\}_{p\geq
0,1\leq q\leq 2n}$ and let $\varphi:X_{W}\to X_{V}$ be the factor map both
given in the proof of Proposition 6.4. Given $i\geq 0$, $1\leq j\leq n$, and
$k\in\mathbb{Z}$, let $E_{i,j,k}$ denote the set in question. It suffices to
show that $E_{i,j,k}$ is clopen for all $i=i_{p}$ for some $p>1$.

Suppose $i=i_{p}$ for some $p>1$. Note that a reading of $y\in X_{W}$ can
determine a reading of $\varphi(y)$. Thus $\varphi^{-1}(E_{i,j,k})$ consists
exactly of those $y\in X_{W}$ such that there is an expected occurrence of
$w^{\prime}_{p,j}$ or $w^{\prime}_{p,j+n}$ in $y$ at position $k$. By our
construction, $\varphi^{-1}(E_{i,j,k})$ is easily seen to be clopen. Thus
$E_{i,j,k}$ is clopen. ∎

We are now ready to compute the topological rank of a minimal Cantor system if
it has a good construction.

######  Proposition 6.6.

Let $n\geq 1$. Let $X$ be an infinite minimal subshift of symbolic rank $\leq
n$. Suppose $X=X_{V}$ and $V$ has a good rank-$n$ construction. Then $X$ has
finite topological rank.

###### Proof.

Fix a good rank-$n$ construction with associated rank-$n$ generating sequence
$\\{v_{i,j}\\}_{i\geq 0,1\leq j\leq n}$. Let $\ell\in\mathbb{N}$ be such that
$1^{\ell}$ is not a subword of $V$. Note that for each $i\geq 0$, there are at
least 4 distinct expected occurrences of $v_{i,1}$ in $v_{i+3,1}$. We let
$k_{i}$ be the starting position of the second expected occurrence of
$v_{i,1}$ in $v_{i+3,1}$. Let $x_{0}$ be the unique element of
$2^{\mathbb{Z}}$ such that for all $i\geq 0$, there exists an occurrence of
$v_{3i,1}$ which starts at the position $-\sum_{0\leq i^{\prime}\leq
i-1}k_{3i^{\prime}}$. Then every finite subword of $x_{0}$ is a subword of
$v_{3i,1}$ for some $i\geq 0$, and thus $x_{0}\in X_{V}$.

Now for every $m\geq 2$, let $A_{m}$ be the set of all $x\in X_{V}$ such that
there is an expected occurrence of $v_{3m,1}$ in $x$ starting at the position
$-\sum_{0\leq i\leq m-1}k_{3i}$, which is the second expected occurrence of
$v_{3m,1}$ in an expected occurrence of $v_{3m+3,j}$ in $x$ for some $1\leq
j\leq n$. By Lemma 6.3 each $A_{m}$ is clopen in $X_{V}$. By definition
$x_{0}\in A_{m}$.

Now consider the canonical Kakutani-Rohlin partition $\mathcal{P}$ with base
$A_{m}$ defined in the remark after Lemma 2.1. The number of towers in
$\mathcal{P}$ corresponds to the number of different $h>0$ such that $h$ is
the smallest positive integer with $\sigma^{h}(x)\in A_{m}$ for some $x\in
A_{m}$. Suppose $x\in A_{m}$ and let $1\leq j\leq n$ be the integer such that
an expected occurrence of $v_{3m+3,j}$ in $x$ contains the position $0$.
Suppose $h$ is the smallest positive integer with $\sigma^{h}(x)\in A_{m}$.
Then there is an expected occurrence of $v_{3m+3,j^{\prime}}$ in $x$ for some
$1\leq j^{\prime}\leq n$ such that the second expected occurrence of
$v_{3m,1}$ in this occurrence of $v_{3m+3,j^{\prime}}$ starts exactly at
$h-\sum_{0\leq i\leq m-1}k_{3i}$. By the minimality of $h$, we get that the
expected occurrence of $v_{3m+3,j}$ and this expected occurrence of
$v_{3m+3,j^{\prime}}$ are only separated by some $1^{s}$. Conversely, the
expected occurrence of some $v_{3m+3,j^{\prime}}$ immediately to the right of
the expected occurrence of $v_{3m+3,j}$ determines the smallest $h$ such that
$\sigma^{h}(x)\in A_{m}$. Therefore, for $1\leq j,j^{\prime}\leq n$ and $0\leq
s\leq\ell$, if we let $B_{j,s,j^{\prime}}$ be the set of all $x\in X_{V}$ such
that there is an expected occurrence of $v_{3m,1}$ in $x$ starting at the
position $-\sum_{0\leq i\leq m-1}k_{3i}$, which is the second expected
occurrence of $v_{3m,1}$ in an expected occurrence of $v_{3m+3,j}$ in $x$, and
this expected occurrence of $v_{3m+3,j}$ is followed by $1^{s}$ and an
expected occurrence of $v_{3m+3,j^{\prime}}$ in $x$, we know that
$\\{B_{j,s,j^{\prime}}:1\leq j,j^{\prime}\leq n;0\leq s\leq\ell\\}$ is a
clopen partition of $A_{m}$ and this partition refines
$\mathcal{P}\upharpoonright A_{m}$. In summary, we obtain a new
Kakutani–Rohlin partition $\mathcal{P}^{\prime}$ whose base is still $A_{m}$,
and if $B_{j,s,j^{\prime}}\neq\varnothing$, then
$B_{j,s,j^{\prime}}\in\mathcal{P}^{\prime}$. $\mathcal{P}^{\prime}$ has at
most $n^{2}\ell$ towers.

Finally note that the diameter of $A_{m}$ is at most $2^{-|v_{3m-3,1}|}$ since
for any $x\in A_{m}$,
$x\\!\upharpoonright\\![-|v_{3m-3,1}|,|v_{3m-3,1}|]=x_{0}\\!\upharpoonright\\![-|v_{3m-3,1}|,|v_{3m-3,1}|]$
and is thus completely fixed. Similarly, every clopen set in
$\mathcal{P}^{\prime}$ has a diameter at most $2^{-|v_{3m-3,1}|}$. By Theorem
3.3 (2), $X_{V}$ has topological rank at most $n^{2}\ell$. ∎

######  Theorem 6.7.

Let $X$ be an infinite minimal subshift of finite symbolic rank. Then $X$ has
finite topological rank.

###### Proof.

By Corollary 5.4 $X=X_{V}$ where $V$ has a proper rank-$n$ construction for
some $n\geq 1$. By Proposition 6.4 there is a word $W$ which has a good
rank-$2n$ construction such that $X_{V}$ is a factor of $X_{W}$ and $X_{W}$ is
minimal. By Proposition 6.6, $X_{W}$ has finite topological rank. Thus $X_{V}$
has finite topological rank by the main theorem (Theorem 1.1) of [31]. ∎

In [31] the authors showed that if a minimal Cantor system $(Y,S)$ is a factor
of a minimal Cantor system $(X,T)$ of finite topological rank, then $\mbox{\rm
rank}_{\scriptsize\rm top}(Y,S)\leq 3\mbox{\rm rank}_{\scriptsize\rm
top}(X,T)$. In [19] Corollary 4.8 this is improved to $\mbox{\rm
rank}_{\scriptsize\rm top}(Y,S)\leq\mbox{\rm rank}_{\scriptsize\rm top}(X,T)$.
Combining these with our results, we can state the following quantitative
result.

######  Corollary 6.8.

Let $X_{V}$ be an infinite minimal subshift of finite symbolic rank. Then

$\mbox{\rm rank}_{\scriptsize\rm top}(X_{V},\sigma)\leq 4(M+1)(\mbox{\rm
rank}_{\scriptsize\rm symb}(X_{V}))^{2}$

where $M$ is a bound for the spacer parameter of any proper construction of
$V$.

###  6.2. From finite topological rank to finite symbolic rank

It was proved in [14] that every minimal Cantor system of finite topological
rank is either an odometer or a subshift on a finite alphabet. We will show
that in case it is a subshift it is conjugate to a subshift of finite symbolic
rank.

We use the following notation from [31] (with a slight modification) in this
subsection. Let $B=(V,E,\preceq)$ be an ordered Bratteli diagram. For each
$i\geq 1$, let $V_{i}^{*}$ denote the set of all words on the alphabet
$V_{i}$, and define a map $\eta_{i+1}:V_{i+1}\to V_{i}^{*}$ as follows. For
$v\in V_{i+1}$, enumerate all edges $e\in E_{i+1}$ with $\mathsf{r}(e)=v$ in
the $\preceq$-increasing order as $e_{1},\dots,e_{k}$, and define

$\eta_{i+1}(v)=\mathsf{s}(e_{1})\cdots\mathsf{s}(e_{k}).$

We also define $\eta_{1}:V_{1}\to E_{1}^{*}$, where $E_{1}^{*}$ is the set of
all words on the alphabet $E_{1}$. For $v\in V_{1}$, enumerate all $e\in
E_{1}$ with $\mathsf{r}(e)=v$ in the $\preceq$-increasing order as $e_{1}\dots
e_{k}$, and define

$\eta_{1}(v)=e_{1}\cdots e_{k}.$

######  Theorem 6.9.

Every minimal Cantor system $(X,T)$ of finite topological rank is an odometer
or is conjugate to a minimal subshift $X_{V}$ of finite symbolic rank.
Moreover, if $(X,T)$ is not an odometer, then $\mbox{\rm rank}_{\scriptsize\rm
symb}(X_{V})\leq\mbox{\rm rank}_{\scriptsize\rm top}(X,T)$.

###### Proof.

We just need to show that for every simple ordered Bratteli diagram
$B=(W,E,\preceq)$ where $|W_{i}|\leq n$ for all $i\geq 1$, if the Bratteli-
Vershik system $(X_{B},\lambda_{B})$ generated by $B$ is not an odometer, then
it is conjugate to $X_{V}$ for a word $V$ which has a rank-$n$ construction.
By telescoping if necessary, we assume without loss of generality that the
following properties hold for $B$:

  1. (1)

for each $i\geq 0$, $w\in W_{i}$ and $w^{\prime}\in W_{i+1}$, there is an edge
$e\in E_{i+1}$ with $\mathsf{s}(e)=w$ and $\mathsf{r}(e)=w^{\prime}$;

  2. (2)

for each $i\geq 1$, $|W_{i}|\geq 2$;

  3. (3)

for each $i\geq 1$, there are vertices $w^{i}_{\min}$ and $w^{i}_{\max}$ in
$W_{i}$ such that for every $w\in W_{i+1}$, $\eta_{i+1}(w)$ starts with
$w^{i}_{min}$ and ends with $w^{i}_{max}$;

  4. (4)

for each $w\in W_{1}$, $|\eta_{1}(w)|\gg n$;

  5. (5)

for any $x,y\in X_{B}$, if $x\neq y$, then there exists $w\in W_{1}$ such that
$\mbox{\rm Ret}_{A_{w}}(x)\neq\mbox{\rm Ret}_{A_{w}}(y)$, where $A_{w}$
denotes the union of the Kakutani–Rohlin tower determined by $w$.

For (3), we consider the unique $x_{\scriptsize\rm min}$ and
$x_{\scriptsize\rm max}$. Fix an $i\geq 1$. Let $w^{i}_{\min}\in W_{i}$ be the
vertex in $W_{i}$ which $x_{\scriptsize\rm min}$ passes through and
$w^{i}_{\max}\in W_{i}$ be the vertex in $W_{i}$ which $x_{\scriptsize\rm
max}$ passes through. Then by the uniqueness of $x_{\scriptsize\rm min}$,
there is an $i_{0}>i$ such that for all $i^{\prime}\geq i_{0}$ and $w\in
W_{i^{\prime}}$ the minimal path between $v_{0}$ and $w$ passes through
$w^{i}_{\min}$. Similarly, there is an $i_{1}$ such that for all
$i^{\prime}\geq i_{1}$ and $w\in W_{i^{\prime}}$, the maximal path between
$v_{0}$ and $w$ passes through $w^{i}_{\max}$. Now we get (3) by telescoping.

For (5) we use the main theorem of [14], which guarantees that $(X,T)$ is a
subshift on a finite alphabet. In particular there is a finite partition
$\mathcal{P}$ of $X$ into clopen sets such that the smallest Boolean algebra
containing elements of $\mathcal{P}$ and closed under $T$ and $T^{-1}$
contains all clopen subsets of $X$. Now we also have that $(X,T)$ is conjugate
to $(X_{B},\lambda_{B})$. Thus there is also a finite partition $\mathcal{Q}$
of $X_{B}$ into clopen sets such that the smallest Boolean algebra containing
elements of $\mathcal{Q}$ and closed under $\lambda_{B}$ and
$\lambda_{B}^{-1}$ contains all clopen subsets of $X_{B}$. Hence there is
$i\geq 1$ such that every element of $\mathcal{Q}$ is the union of the basic
open sets given by the paths from $v_{0}$ to some elements of $W_{i}$. Let $F$
be the set of all paths from $v_{0}$ to an element of $W_{i}$. For each $p\in
F$ let $N_{p}$ denote the basic open set of $X_{B}$ given by $p$. Then for all
$x,y\in X_{B}$ with $x\neq y$, there is $p\in F$ such that $\mbox{\rm
Ret}_{N_{p}}(x)\neq\mbox{\rm Ret}_{N_{p}}(y)$. Now for any $w\in W_{i}$, let
$A_{w}$ denote the union of the Kakutani–Rohlin tower determined by $w$, then
$A_{w}$ is the clopen set given by all paths from $v_{0}$ to $w$. We claim
that for all $x,y\in X_{B}$ with $x\neq y$, there is $w\in W_{i}$ such that
$\mbox{\rm Ret}_{A_{w}}(x)\neq\mbox{\rm Ret}_{A_{w}}(y)$. For this, note that
for any $w\in W_{i}$, if we enumerate all paths from $v_{0}$ to $w$ in the
$\preceq^{\prime}$-increasing order as $p_{1},\dots,p_{k}$, then for any $x\in
X_{B}$, $1\leq j\leq k$ and $m\in\mathbb{Z}$, $m\in\mbox{\rm
Ret}_{N_{p_{j}}}(x)$ iff $m-ak-j+1,\dots,m\in\mbox{\rm Ret}_{A_{w}}(x)$ and
$m-ak-j\notin\mbox{\rm Ret}_{A_{w}}(x)$ for a natural number $a$. Thus if
$x\neq y\in X_{B}$ and $p$ is a path from $v_{0}$ to $w$ such that $\mbox{\rm
Ret}_{N_{p}}(x)\neq\mbox{\rm Ret}_{N_{p}}(y)$, then $\mbox{\rm
Ret}_{A_{w}}(x)\neq\mbox{\rm Ret}_{A_{w}}(y)$. Now (5) follows by telescoping.

For each $i\geq 1$, enumerate the elements of $W_{i}$ as
$w_{i,1},w_{i,2},\cdots,w_{i,n_{i}}$, where $2\leq n_{i}\leq n$, so that
$w_{i,1}=w^{i}_{\min}$. Define

$v_{1,j}=0(01)^{j}0^{|\eta_{1}(w_{1,j})|-2n-4j-2}(10)^{j+n}0$

for $1\leq j\leq n_{1}$. For $i\geq 2$, assume $v_{i-1,j}$ have been defined
for all $1\leq j\leq n_{i-1}$. Then we define

$v_{i,j}=v_{i-1,j_{1}}v_{i-1,j_{2}}\cdots v_{i-1,j_{k}}$

if

$\eta_{i}(w_{i,j})=w_{i-1,j_{1}}w_{i-1,j_{2}}\cdots w_{i-1,j_{k}}.$

It is clear that this defines a rank-$n$ construction. Let
$V=\lim_{i}v_{i,1}$.

We note that $V$ has a proper rank-$m$ construction for some $m\leq n$. In
fact, let $m\geq 2$ be the smallest such that $n_{i}=m$ for infinitely many
$i\geq 2$. Let $\\{i_{k}\\}_{k\geq 0}$ enumerate this infinite set of indices.
Then by telescoping with respect to $\\{i_{k}\\}_{k\geq 0}$, we obtain a
proper rank-$m$ construction for $V$. We note in addition that the rank-$m$
generating sequence associated to this construction is a subsequence of the
rank-$n$ generating sequence $\\{v_{i,j}\\}_{i\geq 0,1\leq j\leq n_{i}}$, and
therefore has bounded spacer parameter. Thus by Corollary 5.4 $X_{V}$ is
minimal.

We claim that for any $1\leq j\leq n_{1}$, $v_{1,j}$ is not of the form

$\alpha 1^{s_{1}}v_{1,j_{1}}1^{s_{2}}\cdots v_{1,j_{k-1}}1^{s_{k}}\beta$

where $k>0$, $\alpha$ is a nonempty suffix of some $v_{1,j_{k}}$, and $\beta$
is a nonempty prefix of some $v_{1,j_{k+1}}$. This follows easily from the
observation that $v_{1,j}$ has a prefix of the form $00(10)^{j}00$ and a
suffix of the form $00(01)^{j+n}00$, and for $1\leq j^{\prime}\leq n_{1}$
where $j^{\prime}\neq j$, $v_{1,j^{\prime}}$ does not contain either of these
words as a subword.

Let $Y=(2^{W_{1}})^{\mathbb{Z}}$ and view it as a shift over the alphabet
$2^{W_{1}}$. Define $\theta:X_{B}\to Y$ by

$\theta(x)(k)(w)=\mbox{\rm Ret}_{A_{w}}(x)(k).$

Then $\theta$ is clearly continuous. It is easy to check that
$\theta\circ\lambda_{B}=\sigma\circ\theta$. By (5), $\theta$ is injective.
Thus $\theta$ is a conjugacy map between $(X_{B},\lambda_{B})$ and
$\theta(X_{B})$, which is a subshift of $Y$.

Finally, we verify that $X_{V}$ is conjugate to $\theta(X_{B})$. For this we
define $\varphi:X_{V}\to(2^{W_{1}})^{\mathbb{Z}}$ by letting
$\varphi(z)(k)(w_{1,j})=1$ iff there is $k^{\prime}\leq k$ with
$k^{\prime}+|v_{1,j}|-1\geq k$ such that $v_{1,j}$ occurs in $z$ starting at
position $k^{\prime}$. $\varphi$ is well defined because of the above claim.
It is clear that $\varphi$ is continuous and injective, and
$\varphi\circ\sigma=\sigma\circ\varphi$. Thus $\varphi$ is a conjugacy map
between $X_{V}$ and $\varphi(X_{V})$. To complete our proof, it suffices to
show $\theta(X_{B})=\varphi(X_{V})$. Consider a $y\in X_{V}$ such that
$y\\!\upharpoonright\\![0,\infty)=V$. Then by our definitions of $\theta$ and
$\varphi$, and particularly because $|v_{i,j}|=|\eta(w_{1,j})|$ for all $1\leq
j\leq n_{1}$, we have $\theta(x_{\scriptsize\rm
min})\\!\upharpoonright\\!{[0,\infty)}=\varphi(y)\\!\upharpoonright\\![0,\infty)$.
By the shift-invariance and the compactness of $\theta(X_{B})$ and
$\varphi(X_{V})$, we get $\theta(X_{B})\cap\varphi(X_{V})\neq\varnothing$. By
the minimality of $\theta(X_{V})$ and $\varphi(X_{B})$, we have
$\theta(X_{B})=\varphi(X_{V})$ as required. ∎

The consideration of the shift $Y$ in the above proof is motivated by the work
in [34] (the construction before Theorem 3.4 in [34]).

###  6.3. Some examples

In this subsection we give some examples to demonstrate that the results in
the preceding subsections are optimal. We first show that a non-minimal
rank-$1$ subshift need not have finite topological rank.

######  Proposition 6.10.

There exists a rank-$1$ word $V$ such that $X_{V}$ is not minimal and $X_{V}$
is not of finite topological rank.

###### Proof.

For any $n\geq 0$, let $r_{n}\geq 2n+5$ and $s_{n,1},\dots,s_{n,r_{n}-1}$ be
nonnegative integers satisfying the following:

  1. (i)

$s_{n,1}=3n+1$, and $s_{n,r_{n}-1}=3n+2$;

  2. (ii)

for all $1<i<r_{n}-1$, $s_{n,i}=3m$ for some $0\leq m\leq n+1$;

  3. (iii)

for any $1\leq m\leq n+1$, there exist $1<i<r_{n}-1$ such that
$s_{n,i}=s_{n,i+1}=3m$.

Then as usual, define $v_{0}=0$ and
$v_{n+1}=v_{n}1^{s_{n,1}}v_{n}1^{s_{n,2}}\cdots 1^{s_{n,r_{n}-1}}v_{n}$
inductively, and let $V=\lim_{n}v_{n,1}$.

We note that for any $n\geq 1$, $0\leq m\leq n+1$ and $u$ a nonempty prefix of
$v_{n}$, $01^{3m}u$ is not a suffix of $v_{n}$.

Toward a contradiction, assume $X_{V}$ has topological rank $K\geq 1$. Fix a
positive integer $N\geq 1$. Then by Theorem 3.3 there is a Kakutani–Rohlin
partition $\mathcal{P}$ of $X_{V}$ with the following properties:

  1. (a)

$\mathcal{P}$ has $K$ many towers, with bases $B_{1},\dots,B_{K}$;

  2. (b)

$1^{\mathbb{Z}}\in B(\mathcal{P})=\bigcup_{1\leq k\leq K}B_{k}$ and $\mbox{\rm
diam}(B(\mathcal{P}))<2^{-N-2}$;

  3. (c)

$\mbox{\rm diam}(A)<2^{-N-2}$ for all $A\in\mathcal{P}$.

Since every $A\in\mathcal{P}$ is clopen, there exists $N^{\prime}>N+4$ such
that for every $A\in\mathcal{P}$, there exists
$U_{A}\subseteq\\{0,1\\}^{2N^{\prime}+1}$ with

$A=\\{x\in X_{V}\,:\,x\\!\upharpoonright\\![-N^{\prime},N^{\prime}]\in
U_{A}\\}.$

Let $n\gg N^{\prime}+3N$.

Fix any $1\leq m\leq n+1$. Let $x\in X_{V}$ be such that
$v_{n}1^{3m}v_{n}1^{3m}v_{n}$ occurs in $x$ at position $-|v_{n}|$, and each
of the three demonstrated occurrences of $v_{n}$ is expected. Then from the
definition of $N^{\prime}$, and because $|v_{n}|\geq 2^{n}\gg n\gg
N^{\prime}$, we have that $\sigma^{|v_{n}|+3m}(x)$ and $x$ belong to the same
set in the partition $\mathcal{P}$. Thus there is $0<j\leq|v_{n}|+3m$ such
that $\sigma^{j}(x)\in B(\mathcal{P})$. Let $t=3m-j$. Then $-|v_{n}|\leq
t<3m$, $y\triangleq\sigma^{3m-t}(x)\in B_{k}$ for some $1\leq k\leq K$,
$v_{n}1^{3m}v_{n}1^{3m}v_{n}$ occurs in $y$ at position $t-|v_{n}|-3m$, and
every $z\in X_{V}$ with an occurrence of $v_{n}1^{3m}v_{n}1^{3m}v_{n}$ at
position $t-|v_{n}|-3m$ is in $B_{k}$. Let $t_{m}$ be the least such $t$ and
$k_{m}$ be the corresponding $k$. We note the following two properties of the
element $y$. First, there is an occurrence of $v_{n}1^{3m}v_{n}$ in $y$ at
position $t_{m}$. Second, because of the minimality of $t_{m}$, we have that
for any $0\leq j\leq t_{m}+|v_{n}|$, $\sigma^{j}(y)\in\sigma^{j}(B_{k_{m}})$
and $\sigma^{j}(B_{k_{m}})\cap B(\mathcal{P})=\varnothing$ when $j\neq 0$, and
so $\sigma^{j}(B_{k_{m}})$ is an element of $\mathcal{P}$ (it is one of the
sets in the $k_{m}$-th tower of $\mathcal{P}$).

We claim that for any $1\leq m_{1},m_{2}\leq\lfloor N/3\rfloor$ with
$m_{1}\neq m_{2}$, we must have $k_{m_{1}}\neq k_{m_{2}}$. Toward a
contradiction, assume $k\triangleq k_{m_{1}}=k_{m_{2}}$. Without loss of
generality assume $t_{m_{1}}\leq t_{m_{2}}$.

Consider first the case $t_{m_{1}}<t_{m_{2}}$. Then we have a subclaim that
$t_{m_{2}}\geq t_{m_{1}}+3m_{1}$. To see this, let $y_{1}\in B_{k}$ be an
element with an occurrence of $v_{n}1^{3m_{1}}v_{n}$ at position $t_{m_{1}}$
as above, and similarly $y_{2}\in B_{k}$ be an element with an occurrence of
$v_{n}1^{3m_{2}}v_{n}$ at $t_{2}$. Since $\mbox{\rm diam}(B_{k})<2^{-N-2}$,
$y_{1}\\!\upharpoonright\\![-N,N]=y_{2}\\!\upharpoonright\\![-N,N]$. Also,
since for all $0\leq j\leq t_{m_{2}}+|v_{n}|$, $\sigma^{j}(B_{k})$ is an
element of $\mathcal{P}$, which has diameter $<2^{-N-2}$, we have that
$y_{1}\\!\upharpoonright\\![-N+j,j+N]=y_{2}\\!\upharpoonright\\![-N+j,j+N]$
for all $0\leq j\leq t_{m_{2}}+|v_{n}|$. In particular
$y_{2}(t_{m_{2}}+|v_{n}|-1)=0=y_{1}(t_{m_{2}}+|v_{n}|-1)$. Since
$t_{m_{2}}+|v_{n}|-1\geq t_{m_{1}}+|v_{n}|$ and $y_{1}$ has an occurrence of
$1^{3m_{1}}$ at $t_{m_{1}}+|v_{n}|$, we must have $t_{m_{2}}+|v_{n}|-1\geq
t_{m_{1}}+|v_{n}|+3m_{1}-1$, or $t_{m_{2}}\geq t_{m_{1}}+3m_{1}$ as in the
subclaim. Note that our argument above gives that

$y_{1}\\!\upharpoonright\\![t_{m_{1}}+|v_{n}|-1,t_{m_{2}}+|v_{n}|-1]=y_{2}\\!\upharpoonright\\![t_{m_{1}}+|v_{n}|-1,t_{m_{2}}+|v_{n}|-1].$

Since $t_{m_{2}}\geq t_{m_{1}}+3m_{1}$, the left-hand side is a word of the
form $01^{3m_{1}}u$ where $u$ is a nonempty prefix of $v_{n}$. But the right-
hand side is a suffix of $v_{n}$. This contradicts our construction of
$v_{n}$.

Thus $t_{m_{1}}=t_{m_{2}}$. Denote $t\triangleq t_{m_{1}}=t_{m_{2}}$. Without
loss of generality assume $m_{1}<m_{2}$. By the above argument we again have
$y_{1}\\!\upharpoonright\\![-N+t+|v_{n}|,t+|v_{n}|+N]=y_{2}\\!\upharpoonright\\![-N+t+|v_{n}|,t+|v_{n}|+N]$.
Since $3m_{1}<3m_{2}\leq N$, we have in particular
$y_{1}\\!\upharpoonright\\![t+|v_{n}|,t+|v_{n}|+3m_{2}-1]=y_{2}\\!\upharpoonright\\![t+|v_{n}|,t+|v_{n}|+3m_{2}-1]$.
But the left-hand side is of the form $1^{3m_{1}}u$ where $u$ is a nonempty
prefix of $v_{n}$, while the right-hand side is $1^{3m_{2}}$, a contradiction.

This finishes our proof of the claim that whenever $1\leq m_{1}\neq
m_{2}\leq\lfloor N/3\rfloor$, we have $k_{m_{1}}\neq k_{m_{2}}$. It follows
from the claim that $K\geq\lfloor N/3\rfloor$. This contradicts the
arbitrariness of $N$. ∎

The next examples show that the topological rank is not bounded by a function
of the symbolic rank alone, thus the extra parameter as in Corollary 6.8 is
necessary. The proposition is also a consequence of [2, Corollary 4.9].

######  Proposition 6.11.

For any $N>1$, there is a minimal rank-$1$ subshift whose topological rank is
at least $N$.

###### Proof.

Fix $p\geq 2N$ and $q\gg N$. Define $v_{0}=0$ and

$v_{n+1}=(v_{n}1)^{q}v_{n}1^{a_{n,1}}v_{n}1^{a_{n,2}}\cdots
1^{a_{n,p}}v_{n}(1^{N+2}v_{n})^{q},$

where $a_{n,1},\dots,a_{n,p}$ are nonnegative integers satisfying the
following:

  1. (i)

for any $1\leq i\leq p$, $2\leq a_{n,i}\leq N+1$;

  2. (ii)

for any $2\leq m\leq N+1$, there is $1\leq i\leq p$ such that
$a_{n,i}=a_{n,i+1}=m$.

Let $V=\lim_{n}v_{n}$.

By an easy induction we have that for all $n\geq 1$ and $1\leq m\leq N+1$, if
$u$ is a nonempty prefix of $v_{n}$, then $01^{m}u$ is not a suffix of
$v_{n}$.

Consider a Kakutani–Rohlin partition $\mathcal{P}$ of $X_{V}$ such that

  1. (a)

$\mathcal{P}$ has $K$ many towers, with bases $B_{1},\cdots,B_{K}$;

  2. (b)

$\mbox{\rm diam}(B(\mathcal{P}))<2^{-N-4}$;

  3. (c)

$\mbox{\rm diam}(A)<2^{-N-4}$ for all $A\in\mathcal{P}$.

Since every $A\in\mathcal{P}$ is clopen, there exists $N^{\prime}>N+6$ such
that for every $A\in\mathcal{P}$, there exists
$U_{A}\subset\\{0,1\\}^{2N^{\prime}+1}$ with

$A=\\{x\in X_{V}\,:\,x\\!\upharpoonright\\![-N^{\prime},N^{\prime}]\in
U_{A}\\}.$

Let $n\gg N^{\prime}+3N$. Similar to the proof of Proposition 6.10, we can
define, for each $2\leq m\leq N+1$, numbers $t_{m}$ where $-|v_{n}|\leq
t_{m}\leq m$, $k_{m}$ where $1\leq k_{m}\leq K$, and an element $y\in
B_{k_{m}}$ such that $v_{n}1^{m}v_{n}$ occurs in $y$ at position $t_{m}$ and
for all $0\leq 1\leq t_{m}+|v_{n}|$, $\sigma^{q}(B_{k_{m}})$ is an element of
$\mathcal{P}$.

As in the proof of Proposition 6.10, we have that for all $2\leq
m_{1},m_{2}\leq N+1$, if $m_{1}\neq m_{2}$, then $k_{m_{1}}\neq k_{m_{2}}$.
This implies that $K\geq N$. ∎

###  6.4. From finite alphabet rank to finite symbolic rank

In this subsection we explore some connections between subshifts of finite
symbolic rank and $\mathcal{S}$-adic subshifts of finite alphabet rank
considered by various authors, e.g. [6] and [13].

We first recall the basic definition of $\mathcal{S}$-adic subshifts and
related notions following [13]. For a finite alphabet $A$, let $A^{*}$ be the
set of all finite words on $A$. If $A,B$ are finite alphabets, a morphism
$\tau:A^{*}\to B^{*}$ is a map satisfying that $\tau(\varnothing)=\varnothing$
and for all $u,v\in A^{*}$, $\tau(uv)=\tau(u)\tau(v)$. A directive sequence is
a sequence of morphisms $\boldsymbol{\tau}=(\tau_{n}\,:\,A_{n+1}^{*}\to
A_{n}^{*})_{n\geq 0}$. For $0\leq n<N$, denote by $\tau_{[n,N)}$ the morphism
$\tau_{n}\circ\tau_{n+1}\circ\cdots\circ\tau_{N-1}$. For any $n\geq 0$, define

$L^{(n)}(\boldsymbol{\tau})=\\{w\in A_{n}^{*}\,:\,\mbox{$w$ occurs in
$\tau_{[n,N)}(a)$ for some $a\in A_{N}$ and $N>n$}\\}$

and

$X^{(n)}_{\boldsymbol{\tau}}=\\{x\in A_{n}^{\mathbb{Z}}\,:\,\mbox{every finite
subword of $x$ is a subword of some $w\in L^{(n)}(\boldsymbol{\tau})$}\\}.$

$X^{(n)}_{\boldsymbol{\tau}}$ is a subshift on the alphabet $A_{n}$, and we
denote the shift map by $\sigma$. Now let
$X_{\boldsymbol{\tau}}=X^{(0)}_{\bf\tau}$. Then
$(X_{\boldsymbol{\tau}},\sigma)$ is the $\mathcal{S}$-adic subshift generated
by the directive sequence $\boldsymbol{\tau}$. The alphabet rank of
$\boldsymbol{\tau}$ is defined as

$\mbox{AR}(\boldsymbol{\tau})=\liminf_{n\to\infty}|A_{n}|$

and the alphabet rank of a subshift $(X,\sigma)$ as

$\mbox{AR}(X)=\inf\\{\mbox{AR}(\boldsymbol{\tau})\,:\,X_{\boldsymbol{\tau}}=X\\}.$

As a convention, $\inf\varnothing=+\infty$.

There is a similar notion of telescoping for directive sequence
$\boldsymbol{\tau}$ which does not change the $\mathcal{S}$-adic subshift
generated by $\boldsymbol{\tau}$.

An $\mathcal{S}$-adic subshift $X_{\boldsymbol{\tau}}$ is primitive if for any
$n\geq 0$ there exists $N>n$ such that $\tau_{[n,N)}(a)$ contains all letters
in $A_{n}$ for all $a\in A_{N}$.

If $\tau:A^{*}\to B^{*}$ is a morphism, $x\in B^{\mathbb{Z}}$, and $Y\subseteq
A^{\mathbb{Z}}$ is a subshift, then a $\tau$-representation of $x$ in $Y$ is a
pair $(k,y)\in\mathbb{Z}\times Y$ such that $x=\sigma^{k}(\tau(y))$. Moreover,
$(k,y)$ is a centered $\tau$-representation if $0\leq k<|\tau(y(0))|$ in
addition. Now $\tau$ is recognizable in $Y$ if each $x\in B^{\mathbb{Z}}$ has
at most one centered $\tau$-representation in $Y$, and a directive sequence
$\boldsymbol{\tau}=(\tau_{n}:A_{n+1}^{*}\to A_{n}^{*})_{n\geq 0}$ is
recognizable if for each $n\geq 0$, $\tau_{n}$ is recognizable in
$X_{\boldsymbol{\tau}}^{(n+1)}$. An $\mathcal{S}$-adic subshift
$X_{\boldsymbol{\tau}}$ is recognizable if $\boldsymbol{\tau}$ is
recognizable.

######  Theorem 6.12.

Let $X_{\boldsymbol{\tau}}$ be a primitive, recognizable $\mathcal{S}$-adic
subshift of finite alphabet rank $K$. Then $(X_{\boldsymbol{\tau}},\sigma)$ is
conjugate to a subshift of finite symbolic rank $\leq K$. Moreover, there
exists a proper rank-$K$ construction for a uniquely readable rank-$K$
generating sequence $\\{v_{i,j}\\}_{i\geq 0,1\leq j\leq K}$ such that
$(X_{\boldsymbol{\tau}},\sigma)$ is conjugate to $(X_{V},\sigma)$, where
$V=\lim_{i}v_{i,1}$.

###### Proof.

This is similar to the proof of Theorem 6.9. By telescoping if necessary, we
assume without loss of generality that the following properties hold for
$\boldsymbol{\tau}$:

  1. (1)

for each $i\geq 0$, $a\in A_{i}$ and $b\in A_{i+1}$, $\tau_{i}(b)$ contains
the letter $a$;

  2. (2)

for each $i\geq 1$, $|A_{i}|=K$;

  3. (3)

for each $a\in A_{1}$, $|\tau_{0}(a)|\gg K$.

Since each $A_{i}$ is finite, a finite splitting argument similar to the proof
of Proposition 5.1 shows that we can enumerate each $A_{i}$ as
$a_{i,1},\dots,a_{i,n_{i}}$ such that $n_{i}=K$ for all $i>1$ and for each
$i\geq 0$, $\tau_{i}(a_{i+1,1})$ starts with $a_{i,1}$. Now, as in the proof
of Theorem 6.9, define

$v_{1,j}=0(01)^{j}0^{|\tau_{0}(a_{1,j})|-2n-4j-2}(10)^{j+n}0$

for $1\leq j\leq K$. For $i\geq 1$ and $1\leq j\leq K$, if

$\tau_{i}(a_{i+1,j})=a_{i,j_{1}}a_{i,j_{2}}\cdots a_{i,j_{k}},$

then let

$v_{i+1,j}=v_{i,j_{1}}v_{i,j_{2}}\cdots v_{i,j_{k}}.$

This gives a proper rank-$K$ construction for $V=\lim_{i}v_{i,1}$.

Clearly the recognizability of $\boldsymbol{\tau}$, together with our
definition of $v_{1,j}$, imply the unique readability of $\\{v_{i,j}\\}_{i\geq
0,1\leq j\leq K}$. Now $(X_{\boldsymbol{\tau}},\sigma)$ and $(X_{V},\sigma)$
are conjugate by the substitution $\tau_{0}(a_{1,j})\mapsto v_{1,j}$. ∎

With Theorem 6.12, our Theorem 6.9 becomes a consequence of the main theorem
of [13] which states that every minimal Cantor system of finite topologival
rank is either an odometer or conjugate to a primitive, recognizable
$\mathcal{S}$-adic subshift of finite alphabet rank.

##  7\. Density and genericity of subshifts of finite symbolic rank

It is known that the set of rank-$1$ measure-preserving transformations is a
dense $G_{\delta}$ subset of the Polish space of all measure-preserving
transformations (see [23]). Here in the topological setting, we show that the
situation is different. In fact we consider various different spaces of Cantor
systems and subshifts and show that the class of all rank-$1$ subshifts is
dense in all but one of them but generic in none. On the other hand, we note
that subshifts of symbolic rank $2$ are generic in the spaces for all
transitive and totally transitive subshifts.

We start with the coding space for all minimal Cantor systems.

######  Proposition 7.1.

The set of all minimal Cantor systems conjugate to a rank-$1$ subshift is
dense but not generic in the space of all minimal Cantor systems.

###### Proof.

By Proposition 3.6 and Lemma 2.3, the set of all subshifts is meager, and not
generic, in the space of all minimal Cantor systems. For the density, in view
of Proposition 3.6, it suffices to show that infinite minimal rank-$1$
subshifts can approximate any infinite odometer. To be precise, we need to
show that for all $k\geq 2$ there is an infinite rank-$1$ subshift $X_{V}$ and
a clopen subset $A$ of $X_{V}$ such that $\sigma^{k}(A)=A$ and
$\\{A,\sigma(A),\dots,\sigma^{k-1}(A)\\}$ form a partition of $X_{V}$.

Fix $k\geq 2$. We define the following Chacón-like rank-$1$ generating
sequence:

$\begin{array}[]{rcl}v_{0}&=&0\\\ v_{1}&=&0^{2k}1^{k}0^{k}\\\
v_{n+1}&=&v_{n}v_{n}1^{k}v_{n}\mbox{ for $n\geq 1$.}\end{array}$

Let $V=\lim_{n}v_{n}$. Then $X_{V}$ is infinite. Let $A$ be the set of all
$x\in X_{V}$ such that

$x\\!\upharpoonright\\![0,3k-1]\in\\{0^{3k},0^{2k}1^{k},0^{k}1^{k}0^{k},1^{k}0^{k}1^{k},1^{k}0^{2k}\\}.$

Then $A$ is a clopen subset of $X_{V}$ with the required property. ∎

To investigate the density and the genericity of subshifts of finite symbolic
rank, we consider some spaces of subshifts as defined in [40]. First, let
$\mathcal{S}_{2}$ be the space of all $\sigma$-invariant closed subsets of
$2^{\mathbb{Z}}$. $\mathcal{S}_{2}$ is a $G_{\delta}$ subspace of
$K(2^{\mathbb{Z}})$, and hence is a Polish space. The Hausdorff metric on
$\mathcal{S}_{2}$ is equivalent to the following metric which is easier to
work with in our setting. For $X\in\mathcal{S}_{2}$ and integer $n\geq 0$, let
$L_{n}(X)$ be the set of all finite words of length $n$ which occurs in some
element of $X$. Let $L(X)=\bigcup_{n}L_{n}(X)$. For $X,Y\in\mathcal{S}_{2}$,
let

$d_{L}(X,Y)=2^{-\inf\\{n\,:\,L_{n}(X)\neq L_{n}(Y)\\}}.$

However, $\mathcal{S}_{2}$ is not a perfect space; in particular, the finite
subshifts are isolated points in this space. Thus, following [40] we consider
the following perfect subspace, which in particular includes all infinite
rank-$1$ subshifts. Let $\mathcal{S}^{\prime}_{2}$ be the subspace of all
elements of $\mathcal{S}_{2}$ which are not isolated (the notation is inspired
by the Cantor–Bendixson derivative; see [35]). $\mathcal{S}^{\prime}_{2}$ is a
perfect subspace of $\mathcal{S}_{2}$, and hence a Polish space.

Recall that a Cantor system $(X,T)$ is (point) transitive if there exists
$x\in X$ such that the orbit of $x$ is dense in $X$; it is totally (point)
transitive if for all integer $n\geq 1$ there exists $x\in X$ such that
$\\{T^{nk}x\,:\,k\in\mathbb{Z}\\}$ is dense in $X$.

Let $\mathcal{T}^{\prime}_{2}$ be the subspace of all transitive subshifts in
$\mathcal{S}^{\prime}_{2}$. Let $\overline{\mathcal{T}^{\prime}_{2}}$ be the
closure of $\mathcal{T}^{\prime}_{2}$ in $\mathcal{S}^{\prime}_{2}$. Moreover,
let $\mathcal{TT}^{\prime}_{2}$ be the subspace of all totally transitive
subshifts in $\mathcal{S}^{\prime}_{2}$. Let
$\overline{\mathcal{TT}^{\prime}_{2}}$ be the closure of
$\mathcal{TT}^{\prime}_{2}$ in $\mathcal{S}^{\prime}_{2}$. Then
$\overline{\mathcal{TT}^{\prime}_{2}}\subseteq\overline{\mathcal{T}^{\prime}_{2}}$
are both closed subspaces of $\mathcal{S}^{\prime}_{2}$, hence are Polish
spaces, and the metric $d_{L}$ remains a compatible metric on these subspaces.

The following theorem shows that minimal rank-$1$ subshifts can approximate
infinite minimal subshifts of topological rank $2$ in the sense of $d_{L}$.

######  Theorem 7.2.

Let $n\geq 1$ and let $(X,\sigma)$ be an infinite minimal subshift of
topological rank $2$. Then there exists an infinite minimal subshift
$(Y,\sigma)$ such that $L_{n}(X)=L_{n}(Y)$ and $(Y,\sigma)$ is conjugate to
$(X_{V},\sigma)$ for some infinite rank-1 word $V$. Moreover, if $(X,\sigma)$
is totally transitive, then we can find $(Y,\sigma)$ which is also totally
transitive.

###### Proof.

By the main theorem of [13], $(X,\sigma)$ is conjugate to a primitive,
recognizable $\mathcal{S}$-adic subshift of alphabet rank 2. By the proof of
Theorem 6.12, there exists a proper rank-$2$ construction for an infinite word
$W$ with the following properties:

  1. (1)

the associated rank-$2$ generating sequence $\\{w_{i,j}\\}_{i\geq 0,1\leq
j\leq 2}$ has unique readability;

  2. (2)

for all $i\geq 1$ and $j=1,2$, the spacer parameter in the building of
$w_{i+1,j}$ from $\\{w_{i,1},w_{i,2}\\}$ is bounded by 0;

  3. (3)

$(X_{W},\sigma)$ is conjugate to $(X,\sigma)$.

Let $f$ be a conjugacy map from $(X_{W},\sigma)$ to $(X,\sigma)$.

Fix $n_{1}\geq 1$ such that for any $x,y\in X_{W}$ and $k\in\mathbb{Z}$,
whenever
$x\\!\upharpoonright\\!{[k-n_{1},k+n_{1}]}=y\\!\upharpoonright\\!{[k-n_{1},k+n_{1}]}$,
we have $f(x)(k)=f(y)(k)$. For any $v\in L(X_{W})$, if $|v|>2n_{1}$ and for
some $x\in X_{W}$ and $k\in\mathbb{Z}$, we have
$x\\!\upharpoonright\\!{[k,k+|v|-1]}=v$, then define
$\Phi(v)=f(x)\\!\upharpoonright\\!{[k+n_{1},k+|v|-n_{1}-1]}$. Clearly
$\Phi(v)$ is well defined and does not depend on the choice of $x$.

For any finite or infinite word $u$ and $m\leq|u|$, let $L_{m}(u)$ denote the
set of all subwords of $u$ of length $n$.

Let $i_{0}\geq 1$ be sufficiently large such that for $j=1,2$,
$|w_{i_{0},j}|>2n+4n_{1}$ and $L_{n+2n_{1}}(W)=L_{n+2n_{1}}(w_{i_{0},j})$.
Since $X$ is infinite, $W$ is aperiodic, and it follows that there is
$j_{0}\in\\{1,2\\}$ such that for any $j=1,2$, both
$w_{i_{0},j_{0}}w_{i_{0},j}\in L(W)$ and $w_{i_{0},j}w_{i_{0},j_{0}}\in L(W)$.
For the same reason, there exists $i_{1}>i_{0}$ such that
$\Phi({w_{i_{1},1}})$ does not have a period $t$ for any
$t\leq|w_{i_{0},1}|+|w_{i_{0},2}|$, i.e., there are $0\leq
a<a+kt<|\Phi(w_{i_{1},1})|$ such that
$\Phi(w_{i_{1},1})(a)\neq\Phi(w_{i_{1},1})(a+kt)$.

Define a rank-1 generating sequence by letting

$v_{1}=w_{i_{0},j_{0}}w_{i_{1},1}w_{i_{0},j_{0}}$

and for any $i\geq 1$,

$v_{i+1}=v_{i}v_{i}1^{|w_{i_{0},j_{0}}|}v_{i}.$

As usual, let $V=\lim_{i}v_{i}$. Then $V$ is a minimal aperiodic infinite
rank-1 word.

Define a map $g$ from $X_{V}$ to $2^{\mathbb{Z}}$ as follows. For $x\in
X_{V}$, if $k$ is a part of an expected occurrence of $v_{1}$ in $x$, then set
$g(x)(k)=x(k)$; if not, let $k^{\prime}$ be the starting position of the next
expected occurrence of $v_{1}$ in $x$, and set
$g(x)(k)=w_{i_{0},j_{0}}(|w_{i_{0},j_{0}}|+k-k^{\prime})$. Let $Z=g(X_{V})$.
Then $(Z,\sigma)$ is a subshift and $g$ is a factor map. By our definition,
$L_{n+2n_{1}}(Z)=L_{n+2n_{1}}(W)=L_{n+2n_{1}}(X_{W})$.

For $x\in Z$ and $k\in\mathbb{Z}$, define
$h(x)(k)=\Phi({x\\!\upharpoonright\\!{[k-n_{1},k+n_{1}]}})$. Let $Y=h(Z)$.
Then $(Y,\sigma)$ is a subshift and $h$ is a factor map. By our definition,
$L_{n}(Y)=L_{n}(X)$. It also follows that there exists $y\in Y$ such that $y$
does not have a period $t$ for any $t\leq|w_{i_{0},1}|+|w_{i_{0},2}|$.

By Theorem 1.5 of [27], the maximal equicontinuous factor of $X_{V}$ is a
finite cycle of length $p$, where $p$ is the maximum such that for
sufficiently large $i$, $p$ divides both $|v_{i}|$ and
$|v_{i}|+|w_{i_{0},j_{0}}|$. It follows that $p$ is a factor of
$|w_{i_{0},j_{0}}|$. However, since $Y$, a factor of $X_{V}$, contains an
element which does not have a period $t$ for any $t\leq|w_{i_{0},j_{0}}|$, we
conclude that $Y$ is an infinite set. By the main theorem of [28], any
nontrivial factor of $X_{V}$ is conjugate to $X_{V}$. Thus $Y$ is conjugate to
$X_{V}$. This finishes the proof of the main conclusion of the theorem.

Suppose $(X,\sigma)$ is totally transitive. We define $W$,
$\\{w_{i,j}\\}_{i\geq 0,1\leq j\leq 2}$, $n_{1}$, $i_{0}$, and $i_{1}$ as
before. We claim that $|w_{i_{0},1}|$ and $|w_{i_{0},2}|$ are relatively
prime. To see this, let $a=\gcd(|w_{i_{0},1}|,|w_{i_{0},2}|)$ and assume
$a>1$. Then by property (2), the set of all $x\in X_{W}$ such that there
exists an expected occurrence of $w_{i_{0},1}$ or $w_{i_{0},2}$ starting at
some multiple of $a$ is a clopen, $\sigma^{a}$-invariant, proper subset of
$X$, contradicting the assumption that $(X,\sigma)$ is totally transitive.

Let $p=|w_{i_{0},j_{0}}|$ and $q=|w_{i_{0},3-j_{0}}|$. Since $p,q$ are
relatively prime, we can find a positive integer $m$ such that
$|w_{i_{0},j_{0}}w_{i_{1},1}|+(m+1)p$ and $p-q$ are relatively prime. We
inductively define a rank-1 generating sequence as follows. First let

$v_{1}=w_{i_{0},j_{0}}w_{i_{1},1}(w_{i_{0},j_{0}})^{m}.$

For $i\geq 1$, if $v_{i}$ has been defined such that $|v_{i}|+p$ and
$|v_{i}|+q$ are relatively prime, then let $v_{i+1}$ be defined to satisfy the
following properties:

  1. (i)

$v_{i+1}$ is built from $v_{i}$ and the spacer parameters are only selected
from $\\{p,q\\}$;

  2. (ii)

for any $0\leq j<i$, there exist $k_{1}<k_{2}$ such that $k_{2}-k_{1}-j$ is a
multiple of $i$, and $k_{1},k_{2}$ are the starting positions of expected
occurrences of $v_{i}$ in $v_{i+1}$;

  3. (iii)

$|v_{i+1}|+p$ and $|v_{i+1}|+q$ are relatively prime.

Let $V=\lim_{i}v_{i}$. Then $V$ is a minimal aperiodic infinite rank-$1$ word.
By (ii), $(X_{V},\sigma)$ is totally transitive. The rest of the argument is
identical to the above proof. ∎

######  Corollary 7.3.