# A classification of nonexpansive Bratteli-Vershik systems

Karl Petersen  Department of Mathematics, CB 3250 Phillips Hall, University of
North Carolina, Chapel Hill, NC 27599 USA<EMAIL_ADDRESS>and  Sandi
Shields  College of Charleston, 66 George St., Charleston, SC 29424-0001 USA
<EMAIL_ADDRESS>
###### Abstract.

We study simple, properly ordered nonexpansive Bratteli-Vershik ($BV$)
systems. Correcting a mistake in an earlier paper, we redefine the classes
standard nonexpansive ($SNE$) and strong standard nonexpansive ($SSNE$). We
define also the classes of very well timed and well timed systems, their
opposing classes of untimed and very untimed systems (which feature, as
subclasses of “Case (2)”, in the work of Downarowicz and Maass as well as
Hoynes on expansiveness of $BV$ systems of finite topological rank), and
several related classes according to the existence of indistinguishable pairs
(of some “depth”) and their synchronization (“common cuts”). We establish some
properties of these types of systems and some relations among them. We provide
several relevant examples, including a problematic one that is conjugate to a
well timed system while also (vacuously) in the classes “Case (2)”. We prove
that the class of all simple, properly ordered nonexpansive $BV$ systems is
the disjoint union of the ones conjugate to well timed systems and those
conjugate to untimed systems, thereby showing that nonexpansiveness in $BV$
systems arises in one of two mutually exclusive ways.

###### Key words and phrases:

Bratteli-Vershik system, expansiveness, odometer

###### 2020 Mathematics Subject Classification:

37B10, 37B02, 28D05

##  1\. Introduction

Bratteli-Vershik ($BV$) systems present visually and combinatorially the
hierarchical mechanisms that drive measure-preserving and topological
dynamical systems. Vershik [Vershik1981Uniform, Vershik1981Markov] and Herman-
Putnam-Skau [HPS1992] showed that every measure-preserving system and every
minimal homeomorphism on the Cantor set is isomorphic (in the appropriate
sense) to a $BV$ system. As with any system, it is useful, when possible, to
code the dynamics as a subshift, so that the methods of symbolic dynamics and
formal languages can be brought into action. This is possible exactly when the
system is expansive (in the topological setting), or essentially expansive (in
the measure-preserving setting; see [AFP]). There has been considerable
progress on the question of when $BV$ systems are expansive or not, or when a
sequence of morphisms is recognizable: [dm2008,
BezuglyiKwiatkowskiMedynets2009, FPS2017, AFP, Berthe2017, FPS2020], for
example. Here we explore the reasons why a $BV$ system might be nonexpansive.
We focus on $BV$ systems that are simple (so the topological dynamical system
is minimal) and properly ordered (there is a unique minimal path and a unique
maximal path), but not necessarily of finite topological rank (conjugate to
one with a bounded number of vertices per level).

Many nonexpansive systems are conjugate to odometers, which can be represented
by diagrams of bounded width, or seem to be similar to the Gjerde-Johansen
example [GJ2000]*Figure 4, which has unbounded width and is not conjugate to
an odometer. For brevity we will refer to this system, which motivated much of
this investigation, as the GJ example. In an earlier paper [FPS2017],
abstracting key properties of this example, the authors (including us)
proposed a class of systems (standard nonexpansive, $SNE$) as a model for how
nonexpansivesness can arise in $BV$ systems. Proposition 5.4 of [FPS2017]
asserted that every standard nonexpansive system, according to the definition
given there, has unbounded width and cannot be conjugate to an odometer.
Example 3.7 below shows that neither of these properties is guaranteed under
the old definition of $SNE$; that definition was flawed, being based on the
assumption that two paths with the same sequence of ordinal edge labels had to
move simultaneously to new vertices. We cannot even guarantee that $SNE$
systems, as originally defined, are nonexpansive. Here we correct those
mistakes by providing a new definition of $SNE$ (Definition 3.4), as well as
its invariant version $SSNE$ (Definition 3.5). Nonexpansiveness now follows
from Proposition 3.8, and Proposition 3.11 and Theorem 3.12 show that any
system satisfying the new definition of $SNE$ is not conjugate to any odometer
and has unbounded width.

In Section 4 we study the GJ example in detail, to prove that it is standard
nonexpansive and to show how a slight modification ruins this property. For
the latter, we use the technique of splitting $j$-symbols from [dm2008] and
prove that the $2$-coding of every path in the GJ example is aperiodic
(Observation 4.4) and the related fact that a certain factor system gives rise
to a recognizable family of morphisms (Observation 4.5).

We define two types of systems, well timed or untimed, as well as their
subclasses very well timed and very untimed. The GJ example is very well
timed. The untimed systems are related to what Downarowicz and Maass called
“Case (2)” in [dm2008]. Theorem 5.15 shows that the family of all nonexpansive
systems is the disjoint union of the class of systems which are conjugate to
some well timed system and the class of those conjugate to some untimed
system.

Hoynes [Hoynes2017] used a slightly different “Case (2)”, and both [dm2008]
and [Hoynes2017] used telescoping to arrive at stronger properties. Definition
5.3 states the definitions of these properties and a few of the others that
would arise from permuting or negating the quantifiers concerning depth and
cuts. Proposition 5.7 clarifies what happens to depth and cuts under
telescoping. Working towards the proof of Theorem 5.15, we characterize in
Proposition 5.10 the systems that are conjugate to some well timed system as
those that are “weakly well timed”. We establish relations among these various
types of systems. Example 5.6 shows that odometers can be either untimed but
not very untimed, or very untimed, depending on how they are presented.
Example 5.13 is a system that is both weakly well timed and (vacuously)
satisfies the property in each “Case (2)”; this indicates that, as written,
the proofs of the main theorem of [dm2008, Hoynes2017] are slightly
incomplete.

In Section 6 we describe the possible diagrams for bounded width very untimed
systems: they are all “kite shaped”, like the example in Figure 2. In the
final section we mention several questions suggested by the foregoing, for
example whether there are any untimed systems that have unbounded width and
whether every well timed system is conjugate to a system that is in $SNE$.

######  Acknowledgment.

We thank Sarah Bailey Frick for valuable contributions during the first half
of this project.

##  2\. Some definitions and notation

We deal with Bratteli-Vershik ($BV$) systems $(X,T)$. Each system is built on
an ordered Bratteli diagram, which is a countably infinite, directed, graded,
graph. For each $n=0,1,2,\dots$ there is a finite nonempty set of vertices
$V_{n}$. $V_{0}$ consists of a single vertex, called the “root”. The set of
edges is the disjoint union of finite nonempty sets $E_{n},n\geq 0$, with
$E_{n}$ denoting the set of edges with source in $V_{n}$ and target in
$V_{n+1}$. We assume that every vertex has at least one outgoing edge, and
every vertex other than the root has at least one incoming edge. There can be
multiple edges between pairs of vertices. The space $X$ is the set of infinite
paths (sequences $x=x_{0}x_{1}\dots$, each $x_{i}$ denoting an edge from level
$i$ to level $i+1$) starting at the root at level $i=0$. For a path $x$ we
denote by $v_{i}(x)$ the vertex of the path at level $i$. If $i<j$, we will
say that level $j$ is after or later than level $i$, which is earlier or
before level $j$. Since diagrams are often drawn with later levels below
earlier ones, we may sometimes say that level $j$ is below level $i$ if $i<j$
and use up and down to refer to relative positions in such diagrams. $X$ is a
compact metric space when we specify that two paths have distance $1/2^{n}$ if
they agree from levels $0$ to $n$ and disagree leaving level $n$. To avoid
degenerate situations we assume that $X$ is homeomorphic to the Cantor set.

The edges entering each vertex are totally ordered, and this yields a partial
order on the set of infinite paths as follows. Two paths $x$ and $y$ are
comparable in case they are cofinal: there is a smallest $N>0$ such that
$x_{n}=y_{n}$ for all $n\geq N$. In this case $v_{N}(x)=v_{N}(y)$, and
$x_{N-1}\neq y_{N-1}$; we agree that $x<y$ if $x_{N-1}<y_{N-1}$, and $x>y$ if
not. The set of minimal paths, meaning those all of whose edges are minimal
into all of their vertices, will be denoted by $X_{\min}$, and similarly the
set of maximal paths will be denoted by $X_{\max}$. The Vershik map $T$ is
defined from the set of nonmaximal paths to the set of nonminimal paths by
mapping each path $x$ to its successor, the smallest $y>x$. We assume that the
diagrams are properly ordered, which means that there is a unique minimal path
and a unique maximal path. This implies that the Vershik map is perfectly
ordered: it extends to a homeomorphism $T$ on the set $X$ of all infinite
paths from the root by mapping the unique maximal path to the unique minimal
path. (See [BezuglyiKwiatkowskiYassawi2014, BezuglyiYassawi2016]).

The system is called minimal if every orbit is dense, and it is called simple
if it has a telescoping for which there are complete connections between
adjacent levels. It can be proved that a properly ordered system is minimal if
and only if it is simple.

We will say that two systems are conjugate if there is an equivariant
homeomorphism from one to the other that sends minimal points to minimal
points. The relation of diagram equivalence is the one generated by graph
isomorphism and telescoping. Herman-Putnam-Skau [HPS1992]*Section 4 (see also
[GPS1995]*p. 70 and p. 72, Theorem 3.6) characterized conjugacy of minimal
pointed topological dynamical systems defined by properly ordered (called
there “essentially simple”) $BV$ diagrams (with unique minimal points), as
follows: two systems are conjugate if and only if their diagrams are
equivalent, and this happens if and only if there exists a diagram $Z$ that
telescopes on odd levels to a telescoping of one of the diagrams and on even
levels to a telescoping of the other.

We emphasize that in the following, unless stated otherwise, every system is a
$BV$ system defined by a properly ordered, simple diagram, and every conjugacy
is an equivariant homeomorphism between two such systems that sends the unique
minimal point in one system to the unique minimal point in the other (but we
may include reminders about these hypotheses anyway). For convenience we may
use the same symbol, such as $X$ or $Y$, to denote a diagram as well as the
system that it defines, and we may also use the same symbol, such as $T$, for
Vershik maps on different systems, or suppress it entirely. By “pair” we mean
a pair of distinct elements, unless stated otherwise. The width of a level is
the number of vertices at that level. Recall that in [dm2008] the topological
rank of a system $(X,T)$ is defined to be the minimum among all $BV$ systems
$(Y,T)$ conjugate to $(X,T)$ of the supremum of the widths of the levels of
$(Y,T)$. For further terminology and background, see [HPS1992, GPS1995,
Durand2010, FPS2017] and the references cited there.

For each $k\geq 1$ denote by $A_{k}$ the finite alphabet whose elements are
the finite paths (segments, strings of edges) from the root to level $k$. For
each $a\in A_{k}$, the set $[a]=\\{x\in X:x_{0}\dots x_{k-1}=a\\}$ is a clopen
cylinder set, and $\mathcal{P}_{k}=\\{[a]:a\in A_{k}\\}$ is a partition of $X$
into clopen sets. The map $\pi_{k}:X\to A_{k}^{\mathbb{Z}}$ is defined by
$(\pi_{k}x)_{n}=a$ if and only if $T^{n}x\in[a]$. The doubly infinite sequence
$\pi_{k}x$ is called the $k$-coding of $x$. Let $\Sigma_{k}=\pi_{k}X$, and
denote by $\sigma$ the shift transformation on $A_{k}^{\mathbb{Z}}$. Then
$\pi_{k}:(X,T)\to(\Sigma_{k},\sigma)$ is a factor map: it is continuous, onto,
and it commutes with the transformations.

######  Definition 2.1.

For each vertex $v$ at level $k\geq 1$, denote by $P(v)$ the set of paths from
the root to $v$, define the dimension of the vertex $v$ to be $\dim v=|P(v)|$,
and concatenate the elements of $P(v)$, in their lexicographic order (defined
by the edge ordering), as $p_{1}\dots p_{\dim v}$ (so that
$[p_{j+1}]=T[p_{j}]$ for $j=1,\dots,\dim v-1$). We call the string $p_{1}\dots
p_{\dim v}$ on symbols from the alphabet $A_{k}$ the $k$-basic block at $v$
and denote it by $B_{k}(v)$. Each path from the root to level $k$ determines,
by truncation, for each $i<k$ a unique path from the root to level $i$, so
there is a natural factor map $A_{k}\to A_{i}$ which converts $B_{k}(v)$ to a
string, which we denote by $B_{i}(v)$ and call the $i$-basic block at $v$, of
the same length on the alphabet $A_{i}$.

######  Definition 2.2.

The coding by vertices at level $j<n$ of a vertex $w$ at level $n$, denoted by
$C_{j}(w)$, is defined as follows. List in their order in the diagram the
paths entering $w$ from vertices at level $j$ as $\\{p_{1},\dots,p_{m}\\}$ and
denote the source of $p_{i}$ by $v_{i},i=1,\dots,m$. Then $C_{j}(w)=v_{1}\dots
v_{m}$. (For $j=n-1$, this is the “morphism read on $\mathcal{V}_{n}$” in
[Durand2010]*p. 342).

######  Definition 2.3.

We say that a $BV$ system is nonexpansive (abbreviated $NE$) if for every
$k\geq 1$ there exists a pair of paths with the same $k$-coding (by finite
paths from level $0$ to level $k$).

Thus a system is expansive if and only if there is a $k\geq 1$ such that the
map $\pi_{k}:(X,T)\to(\Sigma_{k},\sigma)$ is injective (and hence a
conjugacy).

######  Definition 2.4.

We say two paths $x$ and $x^{\prime}$ are depth $k\geq 0$ if they have the
same $k$-coding but not the same $(k+1)$-coding.

If two paths have the same $k$-coding, then they agree from the root to level
$k$.

For any pair of (distinct) paths $x$ and $x^{\prime}$ there exists a $j$ such
that $x$ and $x^{\prime}$ do not have the same $j$-coding. Hence, telescoping
can be used to convert a nonexpansive $BV$ system into one with the property
that for every $k\geq 1$ there exists a depth $k$ pair of paths.

######  Definition 2.5.

We say two paths $x$ and $x^{\prime}$ have a (common) $j$ cut if there is an
integer $m$ such that the initial segments of $T^{m}x$ and $T^{m}x^{\prime}$
are minimal into level $j$. Then we say that the pair has a cut at time $m$.

We say that two paths differ at level $j$ if they follow different edges into
level $j$. Note that if two paths $x$ and $x^{\prime}$ differ at level $k+1$
and have a $k+1$ cut, then $m$ can be chosen so that $T^{m}x$ and
$T^{m}x^{\prime}$ are minimal into distinct vertices at level $k+1$. (For
detailed explanation, see top of page 7 in [Hoynes2017]).

##  3\. Standard nonexpansive systems

######  Definition 3.1.

Let $k\geq 1$ and $n>k$. We say that two vertices $v,w$ at level $n$ are
$k$-equivalent, and write $v\sim_{k}w$, if they are strongly uniformly ordered
with respect to level $k$, meaning that the set of paths from vertices at
level $k$ to $v$ is order isomorphic with the set of paths from vertices at
level $k$ to $w$. Equivalently, the $k$-basic blocks at $v$ and $w$ are
identical. (In [FPS2017] a uniformly ordered level $n+1$ was defined to be a
level all of whose vertices had $n$-basic blocks that were powers of a single
block on the alphabet of paths from the root to level $n$.)

Any pair of vertices that is $(k+1)$-equivalent is also $k$-equivalent. (If
$v$ and $w$ at level $n>k+1$ have equal basic blocks in terms of the paths
from the root to level $k+1$, when these blocks are rewritten in terms of
paths to level $k$ the results will still be identical.)

######  Definition 3.2.

We say that two paths $x$ and $x^{\prime}$ are $k$-equivalent at level $n$ if
we have $v_{n}(x)\sim_{k}v_{n}(x^{\prime})$ and moreover their paths from the
root to level $n$ have the same ordinal path label. We say $x$ and
$x^{\prime}$ are $k$-equivalent, and write $x\sim_{k}x^{\prime}$, if they
agree from the root to level $k$ and, for all $n>k$, $x$ and $x^{\prime}$ are
$k$-equivalent at level $n$.

######  Remark 3.3.

Recall that for a path $x$ and $n\geq k\geq 1$, the $k$-basic block
$B_{k}(v_{n}(x))$ lists in their assigned order the truncations to levels from
$0$ to $k$ of paths from the root to $v_{n}(x)$, say as $q_{1}\dots
q_{\dim(v_{n}(x))}$ (see Definition 2.1). Suppose that $i$ is the index in
$[1,\dim(v_{n}(x))]$ for which $q_{i}$ is the initial segment of $x$ from the
root to level $n$. This information can be conveyed by writing out the string
$B_{k}(v_{n}(x))$ with a “dot” immediately preceding $q_{i}$. For example, if
$x$ follows only minimal edges to level $n$, then $B_{k}(v_{n}(x))=.q_{1}\dots
q_{\dim(v_{n}(x))}$. Two paths $x,x^{\prime}$ are $k$-equivalent at level
$n>k$ if $B_{k}(v_{n}(x))=B_{k}(v_{n}(x^{\prime}))$ and, informally, these two
basic blocks have the “dot” in the same place.

######  Definition 3.4.

We say that a nonexpansive $BV$ system is standard nonexpansive (SNE) if for
every $k\geq 1$ there is a pair of $k$-equivalent paths.

######  Definition 3.5.

We say that two paths $x$ and $x^{\prime}$ are $k$-same, and write
$x\approx_{k}x^{\prime}$, if $T^{j}x\sim_{k}T^{j}x^{\prime}$ for all
$j\in\mathbb{Z}$. We say that a system is strong standard nonexpansive (SSNE)
if for every $k\geq 1$ there is a pair of distinct $k$-same paths.

######  Remark 3.6.

The paper [FPS2017] introduced a different definition of standard
nonexpansive: For every $k$ there should exist a pair of paths that agree from
the root to a level $n>k$, have the same sequence of edge labels, and have the
same $k$-basic blocks at all levels after $k$. In the new definition (3.4) we
replace the requirement that the sequences of edge labels be the same with the
stronger requirement that the two paths always have the “dot” in the same
place of their equal $k$-basic blocks.
Proposition 5.4 of [FPS2017] asserted that every standard nonexpansive system,
according to the definition given there, has unbounded width. The following
example shows that this is not correct. In Theorem 3.12 below we show that the
new definition of $SNE$ given here does suffice to guarantee unbounded width.

######  Example 3.7.

This example satisfies the old definition of $SNE$, but not the new one. It
has bounded width and is conjugate to an odometer (since after telescoping to
the levels with exactly two vertices, all levels are uniformly ordered, see
[FPS2017]); so it is a counterexample to Prop 5.4 of [FPS2017] and the
assertion in its proof that any system conjugate to an odometer cannot have
pairs of paths with the same edge labels.
In this example (see Figure 1) the dark path and the dotted path are identical
from the root to the vertex labeled $b$ at level $3$, where they diverge. They
continue below with the same sequence of ordinal edge labels. At level $1$
(not shown) there are two vertices, $u$ and $v$, and at level $2$ there are
four vertices, with codings by vertices of the previous level
$u^{2},u^{3},uv^{2},v^{2}u$. The vertices $u$ and $v$ connect to $a$ and $b$
via edges in the order that leads to identical codings by vertices
$u^{2}uvvu^{3}$ at $a$ and $u^{3}vvuu^{2}$ at $b$. At all levels $2$ and
later, the paths have identical codings by vertices on the symbols $u$ and
$v$, but the “dots” for the two paths are in different places after level $4$.
After level $0$, the diagram repeats levels and edges with period two, so it
has bounded width.
To find such pairs of paths for values of $k$ larger than $1$, use the
periodicity of the diagram. For example, to deal with $1$ replaced by $3$ (and
hence also by $2$), we may form a pair of paths that is identical from the
root to the vertex labeled $B$ at level $5$, where it splits into two paths
following the same sequence of edge labels as before ($22121212\dots$). These
paths always enter vertices with the same codings by level $3$ vertices $a,b$,
but not with the dot in the same place (after level $6$). Then for $3$
replaced by $5$ (and hence also $4$), take two paths that follow identical
edges from the root to the vertex $D$ at level $7$, where they split and
follow the same sequence of edges $22121\dots$. These paths always enter
vertices with the same codings by level $5$ vertices $A,B$, and hence also by
the vertices at level $4$, but not with the dots in the same place. Etc.

$a=u^{2}uvvu^{3}$$b=u^{3}vvuu^{2}$$a^{2}$$a^{3}$$a.bb$$b.ba$$A=a^{2}a.bba^{3}$$B=a^{3}b.baa^{2}$$A^{2}$$A^{3}$$ABB$$BBA$$C$$D$$C^{2}$$C^{3}$$CDD$$DDC$$2$$2$$2$$2$$1$$1$$2$$2$$1$$1$
Figure 1. Part of a diagram whose system satisfies the old definition of $SNE$
but not the new one

The strict synchronizing structure of $k$-equivalent pairs of paths guarantees
that they cannot be separated by their $k$-codings.

######  Proposition 3.8.

In any system, if two paths are $k$-equivalent at infinitely many levels, then
they have the same $k$-codings.

###### Proof.

Suppose that $x,x^{\prime}$ is a pair of distinct paths for which there is an
infinite increasing sequence $(n_{j})$ such that for each $j$ the paths
$x,x^{\prime}$ are $k$-equivalent at level $n_{j}$. Their (identical)
$k$-basic blocks at their $n_{j}$-level vertices
$v_{n_{j}}(x),v_{n_{j}}(x^{\prime})$ have lengths increasing to infinity,
$j\geq 1$. If the lengths of the segments of the blocks both to the left and
right of their dots increase unboundedly, $x$ and $x^{\prime}$ will have
identical $k$-codings. Suppose that the segments to one side of the dot, for
example the left, stay bounded. Then there are $m\in[0,\infty)$ and $j_{0}$
such that the segment to the left has length $m$ at all levels $n_{j}$ for
$j\geq j_{0}$. At these levels $T^{-m}x,T^{-m}x^{\prime}$ are minimal paths
from the root to the vertices $v_{n_{j}}(x),v_{n_{j}}(x^{\prime})$. Since
$n_{j}$ is arbitrarily large, $T^{-m}x$ and $T^{-m}x^{\prime}$ consist
entirely of minimal edges, so both equal the minimal path $x_{\min}$,
contradicting our assumption that the paths are distinct. Analogously, using
uniqueness of the maximal path, the segments of the basic blocks to the right
cannot stay bounded. ∎

######  Corollary 3.9.

Standard nonexpansive implies nonexpansive.

######  Proposition 3.10.

If two paths $x$ and $x^{\prime}$ are $k$-equivalent, then for every $n>k$
they have an $n$ cut, and there is a $j\geq k$ such that they are depth $j$.
Hence, in any system, if two $k$-equivalent paths are depth $n\geq k$, then
they have an $n+1$ cut.

###### Proof.

For any path and any $n>k$, the position of the dot in the $k$-basic block at
level $n$ represents the smallest number of applications of $T^{-1}$ applied
to that path necessary to produce a path that is minimal into level $n$. Since
$x$ and $x^{\prime}$ are $k$-equivalent, the dot is in the same position for
their $k$-basic blocks at level $n$, so the paths $x$ and $x^{\prime}$ have an
$n$ cut.

Given paths $x$ and $x^{\prime}$ in a system that are $k$-equivalent, by
Proposition 3.8 they have the same $k$-codings. There exists a largest $j\geq
k$ such that for all $m\in\mathbb{Z}$, $T^{m}x$ and $T^{m}x^{\prime}$ agree to
level $j$. It follows that $x$ and $x^{\prime}$ are depth $j$. ∎

Note that the converse of each statement in Proposition 3.10 is not true: it
can happen for a depth $k$ pair and some $n>k$ that the $k$-basic block at
$v_{n}(x)$ is a proper prefix of the $k$-basic block at $v_{n}(x^{\prime})$,
and then the paths would not be $k$-equivalent but could still have an $n$
cut.

The following Proposition will be extended in Theorem 5.14.

######  Proposition 3.11.

No $SNE$ system can be conjugate to any odometer.

###### Proof.

Suppose that $(X,T)$ is a system that is conjugate to an odometer. By
[FPS2017]*Theorem 5.3 it has a telescoping that has infinitely many uniformly
ordered levels. We claim that for $k=1$ no pair in the telescoped system can
be $k$-equivalent, so the telescoped system in fact cannot be $SNE$.

Suppose that $x,x^{\prime}$ is a $1$-equivalent pair in the telescoped system.
Since $x\neq x^{\prime}$ but their first edges are equal,
$x_{0}=x^{\prime}_{0}$, the pair first disagrees at some level $j>1$, that is
to say, they follow different edges from level $j-1$ to level $j$. If they
enter the same vertex at level $j$ along these different edges, they cannot
have their dots in the same place. So the paths are at different vertices at
level $j$, and then, by induction, they must be at different vertices at each
level $n$ for all $n\geq j$.

Look at the first uniformly ordered level $n>j$. At the vertices $v_{n}(x)$
and $v_{n}(x^{\prime})$, because of $1$-equivalence we have $(n-1)$-basic
blocks of the same length, and then because of uniform order these
$(n-1)$-basic blocks are equal. Since $x,x^{\prime}$ are at different vertices
at level $n-1$, their dots are at different places in the $(n-1)$-basic blocks
at $v_{n}(x),v_{n}(x^{\prime})$. When these $(n-1)$-basic blocks are expanded
to $1$-basic blocks, the dots for $x,x^{\prime}$ will be at different places,
contradicting $1$-equivalence of $x$ and $x^{\prime}$. ∎

The following Theorem follows directly from the preceding Proposition combined
with the main result of [dm2008], but we give a direct proof here in order to
re-establish Proposition 5.4 of [FPS2017] in our context, with the new
definition of $SNE$.

######  Theorem 3.12.

SNE implies unbounded width.

###### Proof.

Starting with $j_{0}=1$, using Proposition 3.10, we can find two paths
$x^{(0)},y^{(0)}$ that are $j_{0}$-equivalent and follow different edges into
some level $j_{1}>j_{0}$. As above, because they are $j_{0}$-equivalent, they
must pass through different vertices at level $j_{1}$, and hence they pass
through different $j_{0}$-equivalent vertices for all $j\geq j_{1}$.

If they pass through $j_{1}$-equivalent vertices at infinitely many levels,
they must have the dot in the same place in their basic blocks at those
levels, because when the $j_{1}$-basic blocks are expanded to $j_{0}$-basic
blocks, they are the same and have the dot in the same place, so the dot had
to be in the same place for the $j_{1}$-basic blocks. Then $x^{(0)}$ and
$y^{(0)}$ would be $j_{1}$-equivalent at these infinitely many levels, and so
by Proposition 3.8 would have the same $j_{1}$-coding, but they do not. Thus
for all large enough $n$, $x^{(0)}$ and $y^{(0)}$ pass through vertices that
are $j_{0}$-equivalent and not $j_{1}$-equivalent.

Continuing, given $K\geq 1$, for each $k=0,\dots,K-1$ we can find $N$,
integers $j_{0}<j_{1}<\dots<j_{K}$, and pairs $x^{(k)},y^{(k)}$ that for each
$n\geq N$ and $k=0,\dots,K-1$ pass through vertices that are
$j_{k}$-equivalent and not $j_{k+1}$-equivalent:

(3.1) $v_{n}(x^{(k)})\sim_{j_{k}}v_{n}(y^{(k)})\quad\text{ while }\quad
v_{n}(x^{(k)})\nsim_{j_{k+1}}v_{n}(y^{(k)}).$

But these pairs of vertices at any level $n\geq N$ are all different, because
not $j_{k}$-equivalent implies not $j_{k+1}$-equivalent. If there are at most
$R$ vertices at every level and $K>R^{2}$, this is a contradiction. ∎

##  4\. Modifing the GJ example to ruin the property $SNE$

We will show that the example of Gjerde and Johansen [GJ2000]*Figure 4, which
has unbounded width and is not conjugate to any odometer, is $SNE$ (even
$SSNE$) by specifying for any $k$ exactly which pairs of paths are
$k$-equivalent. We will also determine for any $k$ which pairs are depth $k$.

Denote the system in the GJ example by $X$. Given $j>0$ there are $2j$
vertices in the diagram at level $j$. For any $i\leq 2j$, let $v(j,i)$ denote
the $i$’th vertex from the left (beginning with $i=1$) in level $j$. For
$i=1,2,\dots$ and $j>i$, paths that at level $j$ pass only through vertices
$v(j,2i)$ or $v(j,2i+1)$ will be said to constitute the $i$’th Morse component
$MC(i)$ of the diagram. In particular, $MC(i)$ begins at level $i+1$.

######  Observation 4.1.

(1) For any $i\geq 1$, at any level of $MC(i)$ the $i$-basic blocks are the
same (at both vertices), as can be verified by writing them out. It can also
be verified that the $(i+1)$-basic blocks at the vertices of $MC(i)$ differ
after level $i+1$.

(2) In the diagram for $X$, each vertex is connected to every vertex at the
previous level by exactly one edge, so all vertices at any level have the same
dimension. From this it can be argued inductively that for every $j\geq 1$ two
paths have the same ordinal path label from the root into level $j$ precisely
when they have the same sequence of edge labels into level $j$.

######  Proposition 4.2.

. The following statements hold for the GJ example, $X$.
(1) If two edges at level $n\geq 2$ have the same label, then they have
different targets. If they also have different sources, then they are
contained in $MC(j)$ for some $j<n.$
(2) A pair of paths $x$, $x^{\prime}$ with the same sequence of edge labels
first differ at level $j\geq 1$ if and only if they enter a Morse component at
level $j$ and remain in that Morse component at all levels $j$ and higher,
never again meeting the same vertex.
(3) If two paths have the same sequence of edge labels and first differ at
level $j$, then for some $k<j$ they enter $MC(k)$ at level $j$ and are
$k$-equivalent and not $(k+1)$-equivalent. Conversely, if two paths are
$k$-equivalent and not $(k+1)$-equivalent, then they have the same sequence of
edge labels and enter $MC(k)$ at the level where they first differ.
(4) For any $k\geq 2$, a pair of paths is depth $k$ if and only if those paths
are $k$-equivalent but not $(k+1)$-equivalent. (Equivalently, in view of (3),
two paths are depth $k$ if and only they have the same sequence of edge labels
and enter $MC(k)$ at the level where they first differ.)

###### Proof.

Proof of (1): Fix $n\geq 2$. All edges with source $v(n,1)$ are labeled 1 and
all edges with source $v(n,2n)$ are labeled $2n$. For all $j=1,...,n-1$, all
edges with source $v(n,2j)$ are labeled $2j$, with the exception of the single
edge between $v(n,2j)$ and $v(n+1,2j+1)$, which is labeled $2j+1$. Likewise,
all edges with source $v(n,2j+1)$ are labeled $2j+1$, with the exception of
the single edge between $v(n,2j+1)$ and $v(n+1,2j+1)$, which is labeled $2j$.
If two edges between levels $n$ and $n+1$ have the same label $1$ or $2n$,
they must have the same source (and different targets). If two edges between
levels $n$ and $n+1$ have the same label and different sources, then for some
$j=1,...,n-1$ one of these edges has source $v(n,2j)$, the other has source
$v(n,2j+1)$, and the two edges have different targets and lie in $MC(j)$.

Proof of (2): Two paths with the same sequence of edge labels cannot first
differ into level $1$, because the edges out of $v(1,1)$ all have label $1$,
while the edges out of $v(1,2)$ all have label $2$; so the paths first differ
into level $j\geq 2$, arriving at different vertices at level $j$. Leaving
level $j$, the two paths traverse edges with the same label and different
sources. By part (1), both of these edges are in the same Morse component and
have different targets. Hence, both paths also traverse different edges with
the same label and different sources into level $j+2$. By repeated
applications of part (1) we get that the two paths enter a Morse component at
level $j$ and remain in that Morse component at all levels $j$ and higher.

Conversely, if a pair of paths $x$, $x^{\prime}$ with the same sequence of
edge labels enters a Morse component at level $j$ and remains in that Morse
component at all levels $j$ and higher, then $x$ and $x^{\prime}$ first differ
at level $j$. This is because, from above, the pair cannot differ before
entering a Morse component, so they meet different vertices at a first level
$n>j$. But since two edges in a Morse component with the same label have
different sources, $n=j$. In fact, $x$ and $x^{\prime}$ do not ever meet the
same vertex in that Morse component, since the edges entering any vertex have
distinct labels.

Proof of (3): If paths $x$ and $x^{\prime}$ have the same sequence of edge
labels, then by (2) $x$ and $x^{\prime}$ enter a Morse component $MC(k)$ at
the first level $j>k$ where they differ, and they will meet distinct vertices
in $MC(k)$ at all levels $j$ and higher. Hence, by Observation 4.1 (1), they
will have the same $k$-basic blocks at levels $j$ and higher, whereas their
$(k+1)$-basic blocks at level $j+1$ will differ. By Observation 4.1 (2), they
will have the same ordinal path label from the root into all levels. Thus the
dot will be in the same place in their identical $k$-basic blocks at all
levels $k+1$ and higher: the pair will be $k$-equivalent but not
$(k+1)$-equivalent.

Conversely, we show that any pair of $k$-equivalent paths $x$, $x^{\prime}$
that are not $(k+1)$-equivalent enter $MC(k)$ at the first level where they
differ and have the same sequence of edge labels. By the definition of
$k$-equivalence, such paths agree to level $k$ and their $k$-basic blocks at
levels $k+1$ and higher are the same with the dot in the same place. In
particular, $x$ and $x^{\prime}$ have the same ordinal path label from level
$k$ into any higher level and they agree to level $k$, so they have the same
ordinal path label from the root into any level. Hence, by Observation 4.1
(2), they have the same sequence of edge labels. Denote by $j$ the first level
at which $x$ and $x^{\prime}$ differ (so that $j>k$). Then, by (2), for some
$i<j$ both paths enter $MC(i)$ at level $j$ and are contained in that
component at all higher levels. By the preceding paragraph, $x$ and
$x^{\prime}$ are $i$-equivalent but not $(i+1)$-equivalent. Therefore, $i=k$.

Proof of (4): Suppose that $k\geq 2$ and $x$, $x^{\prime}$ is a depth $k$
pair. Then $x$ and $x^{\prime}$ have the same $k$-coding and therefore the
same $2$-coding. As shown later in Example 4.6, this means that $x$,
$x^{\prime}$ are $2$-equivalent, in other words, at all levels after level
$2$, they have the same $2$-basic block with the dot in the same place. At any
level $n\geq k$, the $k$-basic block $B_{k}(v)$ at any vertex $v$ factors via
the $1$-block code mentioned in Definition 2.1 onto $B_{2}(v)$, and the result
has the same length with the dot in the same position. It follows that the
$k$-basic blocks at $v_{n}(x)$ and $v_{n}(x^{\prime})$ have equal length and
the dot in the same position. Since $x$ and $x^{\prime}$ have the same
$k$-coding, then their $k$-basic blocks at all levels must be the same.
Therefore, $x$ and $x^{\prime}$ are $k$-equivalent. Furthermore, $x$ and
$x^{\prime}$ are not $(k+1)$-equivalent, since at some time in their orbits
they follow different paths into level $(k+1)$ (by definition of depth $k$),
whereas by Proposition 3.8 $(k+1)$-equivalent paths must have the same
$(k+1)$-coding. Hence, for all $k\geq 2$, any depth $k$ pair is $k$-equivalent
but not $(k+1)$-equivalent.

Conversely, for all $k\geq 2$, any pair of paths that is $k$-equivalent has
the same $k$-coding by Proposition 3.8. If that pair is not
$(k+1)$-equivalent, then, it is not $i$-equivalent for any $i>k$. As just
shown, this means it cannot be depth $i$ for any $i>k$. Therefore that pair is
depth $k$. ∎

######  Corollary 4.3.

The GJ example is standard nonexpansive

###### Proof.

By Part (4) of the Proposition, given $k\geq 2$ two paths with the same
sequence of edge labels that enter $MC(k)$ at the level where they first
differ are depth $k$. For example, the path $x$ that passes through the
vertices $v(1,j)$ for all $j,1\leq j\leq k$, then $v(j,2k)$ for all $j>k$ and
the path $y$ that passes through the vertices $v(1,j)$ for $1\leq j\leq k$,
then $v(j,2k+1)$ for all $j>k$, are depth $k$. ∎

The following observations are related to Example 4.6.

######  Observation 4.4.

In the GJ example, the $2$-coding of every path is aperiodic (also sometimes
called “nonperiodic”).

###### Proof.

We will show that the forward coding of the minimal path $x_{\min}$ by
vertices at level $2$, $(v_{2}(T^{j}x),j\geq 0)$, is aperiodic. This implies
that the $2$-coding of $x_{\min}$ is aperiodic, and hence, since the orbit of
$x_{\min}$ is dense, the $2$-coding of every path in $X$ is aperiodic. Fix a
large $n\geq 3$. The idea is to reduce $C_{2}(v(n,1))$ to a long initial block
of the famous Prouhet-Thue-Morse sequence, which is known to be aperiodic.
For $n\geq 2$ we have

(4.1) $C_{n-1}(v(n,1))=v(n-1,1)v(n-1,2)v(n-1,3)\dots v(n-1,2n-2),$

but note that for $n\geq 4$

(4.2) $C_{n-2}(v(n-1,3))=v(n-2,1)v(n-2,3)v(n-2,2)v(n-2,)\dots v(n-2,2n-4).$

This switch in order of adjacent symbols occurs at every level $n-1\geq 3$.
Working towards the expansion of $C_{2}(v(n,1))$ as a string on symbols
$v(2,m),m=1,2,3,4$, in Equation 4.1 replace each $v(n-1,i)$ by
$C_{n-2}(v(n-1,i))$, then in the result (which is $C_{n-2}(v(n,1))$) replace
each $v(n-2,i)$ by $C_{n-3}(v(n-2,i))$, etc., until we finally arrive at
$C_{2}(v(n,1))$. Note that, reading from left to right in any
$C_{j}(v(n,i)),2\leq j<n$, from time to time the symbols $v(j,2)$ and $v(j,3)$
switch order.
We will now repeat this process of repeatedly expanding $C_{n-1}(v(n,1))$,
deliberately losing some information at each step, to produce for each
$j=n-1,\dots,2$ a string $\tilde{C}_{j}$ on the alphabet
$\\{v(j,2),v(j,3),0_{j}\\}$. In Equation 4.1, replace each $v(n-1,i)$ for $i$
not equal to $2$ or $3$ by $0_{n-1}$, arriving at a block $\tilde{C}_{n-1}$ on
the alphabet $\\{v(n-1,2),v(n-1,3),0_{n-1}\\}$.
Then in $\tilde{C}_{n-1}$ replace each $v(n-1,2)$ by $C_{n-2}(v(n-1,2)$, each
$v(n-1,3)$ by $C_{n-2}(v(n-1,3))$, each $0_{n-1}$ by $(0_{n-2})^{2n-4}$, and
finally each $v(n-2,i)$ for $i$ not equal to $2$ or $3$ by $0_{n-2}$, arriving
at a block $\tilde{C}_{n-2}$ on the alphabet
$\\{v(n-2,2),v(n-2,3),0_{n-2}\\}$. Noting that $|C_{n-2}(v(n-1,i))|=2n-4$ for
all $i$, we have formed a $1$-block factor $\tilde{C}_{n-2}$ of
$C_{n-2}(v(n,1))$ on the alphabet $\\{v(n-2,2),v(n-2,3),0_{n-2}\\}$.
Continue analogously until in the end we have a block $\tilde{C}_{2}$ on the
alphabet $\\{v(2,2),v(2,3),0_{2}\\}$ which is a symbol-by-symbol ($1$-block)
factor of $C_{2}(v(n,1))$.
If the forward coding of $x_{\min}$ by vertices at level $2$ were periodic,
then there would be a nonempty block $P$ on the symbols $v(2,m)\,(m=1,2,3,4)$,
and a (possibly empty) prefix $Q$ of $P$, such that for large enough $n$ we
would have $C_{2}(v(n,1))=P^{k}Q$ for some $k\geq 2$. Then $\tilde{C}_{2}$
would have the form $\tilde{P}^{k}\tilde{Q}$ with $\tilde{P},\tilde{Q}$
$1$-block images of $P,Q$.
But note that for each $j=n-1,\dots,3$ a symbol $v(j,2)$ in $\tilde{C}_{j}$
expands to a block $0_{j-1}v(j-1,2)v(j-1,3)0_{j-1}^{2j-5}$ in
$\tilde{C}_{j-1}$, while a symbol $v(j,3)$ in $\tilde{C}_{j}$ expands to a
block $0_{j-1}v(j-1,3)v(j-1,2)0_{j-1}^{2j-5}$ in $\tilde{C}_{j-1}$. Thus if we
ignore the symbol $0_{j-1}$ in each $\tilde{C}_{j}$ we see substrings on
$\\{a=v(\cdot,2),b=v(\cdot,3)\\}$ that expand according to the Prouhet-Thue-
Morse (PTM) substitution $a\to ab,b\to ba$.
If $\tilde{C}_{2}$ were of the form $\tilde{P}^{k}\tilde{Q}$ as above, with
$n$ large enough that $k\geq 2$, then deleting the symbol $0_{2}$ from
$\tilde{P}$ and $\tilde{Q}$ would present a long initial block of the PTM
sequence in the form $P_{0}^{k}Q_{0}$ with $P_{0},Q_{0}$ blocks on
$\\{a,b\\}$, and $|P_{0}|\geq 1$ (because $\tilde{C}_{2}$ contains symbols
other than $0_{2}$). But the PTM sequence $abba\,baab\dots$ is aperiodic, in
fact it cannot begin with $BB$ for any block $B$. Therefore the forward coding
of $x_{\min}$ by vertices at level $2$ is not periodic, and hence the
$2$-coding of $x_{\min}$ is not periodic. ∎

######  Observation 4.5.

In the GJ example, identifying at every level $j>2$ the vertices other than
$v(j,3)$ produces a sequence of morphisms that forms a recognizable family.

###### Proof.

Denote by $\mathcal{V}_{j}$ the set of vertices in our diagram at level $j$.
For each $j\geq 2$ define an alphabet $A_{j}=\\{D_{j},E_{j}\\}$ and a (many-
to-one) map $\phi_{j}:\mathcal{V}_{j}\to A_{j}$ as follows. Assign to $v(j,3)$
the symbol $D_{j}=\phi_{j}(v(j,3))$, and to each $v(j,i)$ for $i\neq 3$ assign
the symbol $E_{j}=\phi_{j}(v(j,i))$. When we expand each symbol (vertex)
$v(j,i)$ to $C_{j-1}(v(j,i))$ the effect on the $\phi_{j}$-images
($D_{j},E_{j},D_{j-1},E_{j-1}$) at both levels produces the morphism

(4.3) ${E_{j}\to E_{j-1}^{2}D_{j-1}E_{j-1}^{2j-5},\qquad D_{j}\to
E_{j-1}D_{j-1}E_{j-1}^{2j-4}}$

for all $j\geq 3$, in the sense of concatenation of blocks.
Denoting as usual by $A_{j}^{+}$ the set of nonempty words on the alphabet
$A_{j}$, Equation (4.3) defines a sequence of morphisms $\tau_{j}:A_{j}\to
A_{j-1}^{+}$, as (for example) in [Berthe2017]. By keeping track of the
position of $D_{j}$ in codings of paths by images of vertices at level $j$
under $\phi_{j}$ one can prove directly that the sequence is recognizable, in
the sense that every sequence on the alphabet $A_{j}$ has at most one
desubstitution, or factorization, on the alphabet $A_{j+1}$. To see this,
suppose that we are given a bisequence on the alphabet $A_{j}$. When we see a
block $F(j,q)=D_{j}E_{j}^{q}D_{j}$, since the block $D_{j+1}D_{j+1}$ does not
appear in the coding of any path by the images of vertices at level $j+1$, it
must be the case that $F(j,q)$ is a subblock, in a uniquely determined
position, of

(4.4) $\displaystyle
E_{j+1}E_{j+1}=E_{j}E_{j}D_{j}E_{j}^{2j-3}\,\,E_{j}E_{j}D_{j}E_{j}^{2j-3}\quad\text{
if }q=2j-1,$ $\displaystyle
E_{j+1}D_{j+1}=E_{j}E_{j}D_{j}E_{j}^{2j-3}\,\,E_{j}D_{j}E_{j}^{2j-2}\quad\text{
if }q=2j-2,\text{ or }$ $\displaystyle
D_{j+1}E_{j+1}=E_{j}D_{j}E_{j}^{2j-2}\,\,E_{j}E_{j}D_{j}E_{j}^{2j-3}\quad\text{
if }q=2j.$

Thus the coding of an orbit by $D_{2},E_{2}$ determines its coding by
$D_{j},E_{j}$ for all $j>2$. This also follows from [Berthe2017]*Theorem 5.1,
since each of these alphabets has only two elements and each infinite
bisequence generated by the sequence of morphisms is aperiodic, by the
arguments in the proof of Observation 4.4. ∎

Note that the natural sequence of morphisms for the diagram defined by the
coding of vertices at each level $j$ beyond some level $j_{0}$ by the vertices
at level $j-1$ cannot be recognizable, because the GJ system is nonexpansive.
(The coding of any path by vertices at some level would then determine the
codings at all subsequent levels and hence the entire path.) Moreover, the BV
system corresponding to the sequence of morphisms $\tau_{j}$ is of finite
topological rank (as well as expansive). In fact it has topological rank $2$:
it cannot be $1$ (conjugate to an odometer) by [FPS2017]*Theorem 5.3 and
Observation 4.4, since for every $n>1$ the $n$-coding of the orbit of any path
is not periodic.

We show now how the GJ example can be modified to spoil the strict
requirements of the definition of $SSNE$ to produce an example of a
nonexpansive system that is not even $SNE$ and is not conjugate to any
odometer. (By [dm2008] any such example necessarily has unbounded width.)
Further, this example is very well timed (see Definition 5.2).

In [dm2008], Downarowicz and Maass introduced a handy way to visualize paths
and their orbits in a $BV$ system by means of 3-sidedly infinite arrays of
$j$-symbols. Every vertex $v$ at level $j\geq 0$ has an associated $j$-symbol
which is also labeled $v$. The $j$-symbol is a finite rectangular matrix with
$j+1$ rows consisting of subrectangles ($i$-symbols for $0\leq i\leq j$), as
described in [dm2008]*p. 741. An array represents the entire orbit of a path
and its $k$-codings for all $k\geq 0$. The path itself is indicated by an
arrow pointing to the left edge of its time $0$ rectangle at level $0$, and
this arrow indicates, by extending it vertically downward, all the other
rectangles (vertices) through which the path passes at levels $n>0$. We call
this extension the vertical that corresponds to the path. Further development
and use of these arrays can be seen in [BezuglyiKwiatkowskiMedynets2009] and
[Hoynes2017].

The family of arrays determined by a diagram has a type of consistency called
“agreeable” in [BezuglyiKwiatkowskiMedynets2009, Def. 3.2]: each $j$-symbol
with a fixed name $v$ has for its first $j$ rows the same concatenation of
$(j-1)$-symbols. Conversely, given an agreeable family of arrays, we can
construct its unique associated $BV$ diagram (which may or may not be properly
ordered). For each vertex $v$ at level $j$, the names, in order, of the
$(j-1)$-symbols comprising the first $j$ rows of its associated $j$-symbol
specify the edges connecting certain vertices at level $j-1$ to it in a
certain order, possibly with repeats. This concatenation of $(j-1)$-symbols
determines the $(j-1)$-basic block $B_{j-1}(v)$ at the vertex v, since it
lists in order the paths from the root to $v$. The Vershik map $T$ on the
diagram corresponds to sliding each array one “notch”, i.e. level-$0$
rectangle width, to the right. Thus there is a natural correspondence between
$BV$ systems, their diagrams, and agreeable families of arrays; in the
following we deal with them interchangeably.

######  Example 4.6.

Denote the system in the GJ example [GJ2000]*Figure 4, p. 1699 by $X$.
In their proof in [dm2008], Downarowicz and Maass make use of a modification
of the Bratteli diagram and hence of all arrays representing orbits of the
system. See [dm2008, pp. 743–744, Figure 4] and [Hoynes2017, p. 204]). We will
use a version of their splitting technique to ensure that for every $i\geq 1$
and every $j\geq i+1$ the two vertices at level $j$ in the $i$’th Morse
component have different $1$-basic blocks.
For every $j\geq 2$ and every $i<j$, we replace the $j$-symbol $v_{i}=v(j,2i)$
with two symbols $v_{i}^{\prime}$ and $v_{i}^{\prime\prime}$ so that
$|B_{1}(v_{i}^{\prime})|=|B_{1}(v(j-1,1))|$ and the concatenation of
$B_{1}(v_{i}^{\prime})$ and $B_{1}(v_{i}^{\prime\prime})$ is $B_{1}(v_{i})$.
In particular, $B_{1}(v_{i}^{\prime})$ is a proper prefix of $B_{1}(v_{i})$.
In the DM arrays, every occurrence of the $j$-symbol with label $v_{i}$ is
then replaced by two $j$-symbols labeled $v_{i}^{\prime}$ and
$v_{i}^{\prime\prime}$ respectively. We leave the $j$-symbols for all other
vertices at level $j$ unchanged, so their $1$-basic blocks are the same in the
new diagram as they are in $X$. The result is that the left vertices in every
Morse component of $X$ have been split into two vertices, whereas the right
vertices remain intact.
Note that for every $j\geq 2$, the modifications at level $j$ extend vertical
bars (i.e rectangle boundaries) at level $j-1$ in the original diagram for $X$
by only one level. None of these bars is extended further by a modification at
level $j+1$. Hence, no new infinitely long vertical lines consisting entirely
of rectangle boundaries are created. This means that after all modifications,
the system represented by the diagram remains in the class of properly ordered
systems. Moreover, every orbit still meets every vertex, so the system is
simple. Also note that since this new system is conjugate to $X$, it is not
conjugate to any odometer.
The new system has the property that at any level $j\geq 2$ and $i=1,...,j-1$,
the $1$-basic block at the right vertex $v(j,2i+1)$ of the i’th Morse
component through level $j$ is longer than the $1$-basic blocks at the two new
vertices. Hence, the right vertex cannot be $1$-equivalent to either of these
new vertices. We claim that for every $k\geq 1$ paths that were $k$-equivalent
in the old diagram are no longer $k$-equivalent in the new diagram. This is
because $k$-equivalent paths in the old diagram are also $1$-equivalent in
that diagram. By Proposition 4.2, any pair of paths that are $1$-equivalent in
the original diagram are contained in a Morse component at all levels after
which they first differ, so they can no longer be $1$-equivalent in the new
diagram. Then since $k$-equivalence implies $1$-equivalence, the paths are no
longer $k$-equivalent in the new diagram.
The $1$-coding of every path in $X$ is periodic with period length 2. However,
at every level $j>2$ the $2$-basic block at $v(j,3)$ differs from the
$2$-basic blocks at at all other vertices, and as a result the $2$-coding is
not periodic (see Observation 4.4). We will exploit this property of $2$-basic
blocks to show that no new $2$-equivalent paths are created by our
modifications. It then follows that the new system has no $2$-equivalent
paths, hence is not $SNE$.
By Prop 3.8, any $2$-equivalent pair in the new system has the same new
$2$-coding. In fact, it is the image of a pair in $X$ that has the same old
$2$-coding. This is because there is an invertible mapping between the old and
the new codings. Specifically, in both the old and the new system each path in
$X$ is represented by a vertical in its array. Row 2 of the array displays the
$2$-coding of the path and the placement of the dot is specified by the
vertical. The splitting of rectangles and relabeling at level 2 is reversible
and does not change the position of the vertical. So a pair of $2$-equivalent
paths in the new diagram is the image of a pair in the diagram for $X$ with
the same $2$-coding. We now show that any such pair in $X$ is $2$-equivalent.
Let $x$, $x^{\prime}$ be paths in the diagram for $X$ with the same
$2$-coding. Using the preceding observations about $D_{j}$ and $E_{j}$ (see
Observation 4.5), we argue inductively that for every $j\geq 3$, any time the
orbit of $x$ is minimal from the root to $v(j,3)$, the orbit of $x^{\prime}$
is as well (and vice versa). Then because all $j$-basic blocks at level $j$
have the same length ($\dim v(j,1)$), any time the orbit of one of these paths
changes vertices at level $j$, the other one does as well. Since $x$ and
$x^{\prime}$ have the same 2-coding, it follows that they must be
$2$-equivalent at level $j$ (i.e not only do they have the same $2$-basic
block at level $j$ but their dot is in the same place).
First note that, since $x$ and $x^{\prime}$ have the same $2$-coding, their
orbits must meet $v(2,3)$ always at the same time; in particular, any time the
orbit of one of these paths is minimal from the root into $v(2,3)$ the orbit
of the other is as well.
Next fix $j\geq 2$ and assume that the orbits of $x$ and $x^{\prime}$ are
minimal into $v(j,3)$ always at the same time. Since the number of paths is
the same from the root into any vertex at level $j$, these orbits must be
minimal into each vertex at level $j$ always at the same time.
Now suppose that for some $m$, $T^{m}x$ is minimal from the root into
$v(j+1,3)$, so that $\phi_{j+1}(v_{j+1}(T^{m}x))=D_{j+1}$ and
$T^{m+\dim(v(j,1))}x$ is minimal to $v(j,3)$, which maps under $\phi_{j}$ to
$D_{j}$. We claim that since the orbits of $x$ and $x^{\prime}$ always hit
$v(j,3)$ at the same time, we cannot have $v_{j+1}(T^{m}x^{\prime})=v(j+1,i)$
for some $i\neq 3$, i.e. we cannot have
$\phi_{j+1}(v_{j+1}(T^{m}x^{\prime}))=E_{j+1}$ rather than $D_{j+1}$. This
follows from looking at the next vertices at level $j+1$ hit by the orbits of
$x$ and $x^{\prime}$. Since the block $v(j+1,3)v(j+1,3)$ cannot appear in the
coding of any path by vertices at level $j+1$, the orbit of $x$ next hits
$v(j+1,i)$ for some $i\neq 3$, which has image $E_{j+1}$, while the orbit of
$x^{\prime}$ next hits a vertex with image either $E_{j+1}$ or $D_{j+1}$. As
seen in the proof of Observation 4.5, in the block $D_{j+1}E_{j+1}$ (on
symbols $D_{j},E_{j}$) in the coding of the orbit of $x$ (by images of
vertices under $\phi_{j}$), consecutive appearances of $D_{j}$ are separated
by a distance $2j$, while in the two possible blocks $E_{j+1}E_{j+1}$ and
$E_{j+1}D_{j+1}$ in the coding of the orbit of $x^{\prime}$ the consecutive
appearances of $D_{j}$ are separated by distance either $2j-1$ or $2j-2$.
Therefore $v_{j+1}(T^{m}x^{\prime})=v(j+1,3)$.
Since $T^{m+\dim(v(j,1))}x$ is minimal to $v(j,3)$ and the orbits of $x$ and
$x^{\prime}$ always hit $v(j,3)$ at the same time, we must have that
$T^{m+\dim(v(j,1))}x^{\prime}$ is also minimal to $v(j,3)$. Since the dot for
both $\phi_{j}(T^{m+\dim(v(j,1))}x)$ and
$\phi_{j}(T^{m+\dim(v(j,1))}x^{\prime})$ is at the beginning of an appearance
of $D_{j}$, and $D_{j}$ appears only once in each $\phi_{j}(v(j+1,3))$,
applying $T^{-\dim((v(j,1))}$ shows that both $T^{m}x$ and $T^{m}x^{\prime}$
are minimal from the root into $v(j+1,3)$.

##  5\. Well timed and untimed systems

In this section we define several classes of nonexpansive $BV$ systems
($W,W_{0},DM2$, $H2,U,U_{0},U_{1}$, and $U_{2}$) according to various
possibilities for the existence of pairs of paths with cuts. Recall that all
systems under consideration are nonexpansive, properly ordered, and simple. It
will turn out that every such system is conjugate to a system in exactly one
of $W$ (“well timed”) or $U$ (“untimed”).

######  Definition 5.1.

For any class $S$ of systems, we will denote the class of systems conjugate to
some system in $S$ by $CS$:

(5.1) $CS=\\{X:\text{there is }Y\in S\text{ such that }X\text{ is conjugate to
}Y\\}.$

We denote the class of systems not conjugate to any system in $S$ by $NCS$,
and the class of systems not in $S$ by $\neg S$. Note that it is not true that
if $X,Y\in CS$ then $X$ must be conjugate to $Y$.

In their proof that every bounded width system is either expansive or
conjugate to an odometer, Downarowicz and Maass [dm2008] considered a class of
systems that they called Case (2), which we denote here by $DM2$. In
[Hoynes2017] Hoynes’ Case (2) is a slightly weaker condition, which we call
$H2$, apparently still sufficient for the proof to succeed. Downarowicz and
Maass as well as Hoynes actually used stronger properties, opposite to well
timed, which here we call very untimed ($U_{0}$) and $U_{2}$. Here is a list
of some relevant classes of systems, obtained by varying quantifiers.

######  Definition 5.2.

We say that a depth $k$ pair of paths has long cuts if for every $n>k$ the
pair has an $n$ cut.

######  Definition 5.3.

We define the following classes of systems:

(1) We say that a system is well timed if for every $k\geq 1$ for every $j>k$
there is a depth $k$ pair with a $j$ cut. We denote the class of well timed
systems by $W$.

(2) We say that a system is very well timed if for every $k\geq 1$ there
exists a depth $k$ pair with long cuts. We denote the class of very well timed
systems by $W_{0}$.

(3) $DM2$: For infinitely many $k$ there is a $j(k)>k$ such that no depth $k$
pair has a $j(k)$ cut. The smallest such $j(k)$ is called a $k$ cutoff.

(4) $H2$: For infinitely many $k$ for every depth $k$ pair $x,x^{\prime}$
there is a $j(k,x,x^{\prime})>k$ such that $x,x^{\prime}$ has no
$j(k,x,x^{\prime})$ cut. The smallest such $j(k,x,x^{\prime})$ could be called
a $k$ pair cutoff for the pair $x,x^{\prime}$.

(5) $U$ (untimed): For every $k$ there is a $k$ cutoff. (I.e., for every
$k\geq 1$ there is a $j(k)>k$ such that no depth $k$ pair has a $j(k)$ cut.)

(6) $U_{0}$ (very untimed): For every $k$ no depth $k$ pair has a $k+1$ cut.
(I.e., for every $k$, $k+1$ is a $k$ cutoff).

(7) $U_{2}$: For every $k$ there is a depth $k$ pair with no $k+1$ cut.

(8) $U_{1}$: For infinitely many $k$ there is a depth $k$ pair $x,x^{\prime}$
and there is a $j(k,x,x^{\prime})>k$ such that the pair $x,x^{\prime}$ has no
$j(k,x,x^{\prime})$ cut. (I.e., for infinitely many $k$ there is a depth $k$
pair with a pair cutoff.)

Recall that the GJ example is $SNE$ and is very well timed. The modified GJ
example (Example 4.6) is not $SNE$ but it is very well timed, because the only
changes we made to the DM arrays were to add additional vertical bars, which
will not destroy the existing cuts.

######  Remark 5.4.

In [dm2008] the proof of the main theorem in Case (2) begins by telescoping
any system in $DM2$ so that the result is in $U_{0}$. Hoynes
[Hoynes2017]*Remark 4.4 does not see why this should always be possible, but
suggests that changing the universal quantifier to existential, i.e. replacing
$DM2$ with $U_{1}$, does allow one to telescope such a system to one in
$U_{2}$, and that should be enough to let the proof proceed.
The proof in [Hoynes2017] assumes $H2$, telescopes so that for every $k\geq 1$
there is a depth $k$ pair, and then telescopes to obtain a system in $U_{2}$.
The proof of Sublemma 4.1, though, applies $H2$ to possibly ineligible pairs
$y_{i},y_{i^{\prime}}$, because $H2$ is not closed under telescoping. Because
the telescoped system is in $U_{2}$, for every $i$ there is a pair
$x_{i},x_{i}^{\prime}$ with no $i+1$ cut. But the pair $y_{i},y_{i^{\prime}}$
of the proof could be depth $i^{\prime}$ and without a cutoff, if it were the
image under the telescoping of a pair of a depth other than one of the
infinitely many “good” $k$ in the definition of $H2$ (see Proposition 5.7).
Indeed, in Example 5.13 we present a system that is in both classes $DM2$
(vacuously) and $WW$ (see Definition 5.8, below) and for which telescoping to
any strictly increasing sequence of levels takes it out of the class $H2$ (and
hence out of $DM2$). Such a system cannot be telescoped into $U_{0}$, because
then it would be in both $CU$ and $WW=CW$ (see Proposition 5.10), but by
Theorem 5.14 these classes are disjoint.
We think that Sublemma 4.1 of [Hoynes2017] can be proved as follows. Assuming
that $H2$ is satisfied nonvacuously, let $i_{0}$ and all the other $i$’s
mentioned in the argument be good $k$’s according to the definition of $H2$.
They may not fill up an interval in the integers, but given $L$ one can pick a
sequence of them of length $L$ and proceed to write the same argument, being
careful to choose the pairs $x_{i},x_{i}^{\prime}$ so that their cutoffs
$j(i,i^{\prime})$ interleave the levels with good $k$’s.

######  Remark 5.5.

(1) $U_{0}\subset U\subset DM2\subset H2$, and $U_{2}\subset U_{1}$.
(2) Each of the classes $W,W_{0},U,U_{0}$ is closed under telescoping. We will
later (in Remark 5.12) provide a proof of this for $W$ and $W_{0}$. To see
that the very untimed property persists under telescoping, note that if in a
telescoping to levels $\\{n_{l},l\geq 0\\}$ of a very untimed system we found
a depth $j$ pair with a $j+1$ cut, that pair would correspond to a pair in the
original system of some depth $k\geq n_{j}$ with an $n_{j+1}$ cut, and hence
with a $k+1$ cut—cf. Proposition 5.7, (3) and (4).

######  Example 5.6.

Every odometer presented with one vertex at every level is in $U_{0}$ (very
untimed). To see this, suppose that $x,x^{\prime}$ are depth $k\geq 1$, so
that at some time $m$ in their orbits they follow different edges from level
$k$ to level $k+1$, i.e. $(T^{m}x)_{k}\neq(T^{m}x^{\prime})_{k}$.
Because $(T^{j}x)_{k}$ and $(T^{j}x^{\prime})_{k}$, $j\in\mathbb{Z}$, follow
the same periodic sequence of edges as $j$ varies, we have that, for all
$j\in\mathbb{Z}$, $(T^{j}x)_{k}\neq(T^{j}x^{\prime})_{k}$. In particular, $x$
and $x^{\prime}$ cannot have a $k+1$ cut, since this would require that for
some $j$ they both follow the unique minimal edge from level $k$ to level
$k+1$.
Note that in this example for every $k\geq 1$ there is a depth $k$ pair, but
that is not a requirement in the definition of $U_{0}$.
We used here a general principle that applies in any system: If level $k+1$ is
strongly uniformly ordered with respect to level $k$ (see Definition 3.1) and
$x,x^{\prime}$ follow edges with different ordinal labels from level $k$ to
level $k+1$, so do $T^{j}x$ and $T^{j}x^{\prime}$ for all $j\in\mathbb{Z}$.

On the other hand, every odometer can be presented (up to conjugacy) with at
least two vertices per level after the root and all levels strongly uniformly
ordered. We claim that for such a system for every $k\geq 1$ there is a depth
$k$ pair with a $k+1$ cut, but no depth $k$ pair can have a $k+2$ cut. Thus
every such system is in $U\setminus U_{0}$, with a $k$ cutoff of $k+2$ for
every $k\geq 1$.
At each level $n\geq 1$, the system has vertices $v(n,1),\dots,v(n,q_{n})$ for
some $q_{n}\geq 2$. Edges with source $v(n,i)$ have ordinal label $i$.
Given $k\geq 1$, let $x$ be a path that is minimal from the root to
$v(k+1,1)$, and let $x^{\prime}$ be a path that is minimal from the root to
$v(k,1)$ at level $k$ and then follows the edge (labeled $1$) to $v(k+1,2)$ at
level $k+1$. Because $T^{j}x,T^{j}x^{\prime}$ are at strongly uniformly
ordered vertices at level $k+1$ for all $j\in\mathbb{Z}$, they follow edges
with the same ordinal label from level $k$ to level $k+1$. Thus $x,x^{\prime}$
have the same $k$-coding and hence the pair is depth $k$. The paths
$x,x^{\prime}$ also follow minimal edges to level $k+1$, so they have a $k+1$
cut. Thus the system is not in $U_{0}$.
We show now that if $x,x^{\prime}$ is a depth $k$ pair, then it cannot have a
$k+2$ cut, so the system is in $U$, with $k$ cutoff equal to $k+2$. For
suppose that $x,x^{\prime}$ is a depth $k$ pair. Applying a power of $T$ if
necessary, we may assume that these paths follow different edges from level
$k$ to level $k+1$. Because level $k+1$ is strongly uniformly ordered with
respect to level $k$, for all $j\in\mathbb{Z}$ the paths $T^{j}x$ and
$T^{j}x^{\prime}$ follow different edges from level $k$ to level $k+1$, and
hence they are at different vertices at level $k+1$:

(5.2) $v_{k+1}(T^{j}x)\neq v_{k+1}(T^{j}x^{\prime}).$

Thus the edges downward from these vertices to level $k+2$ always have
different ordinal labels, precluding existence of a $k+2$ cut.

We aim to show that $CW\subset NCU$ and $NCW\subset CU$, so that the family of
simple, perfectly ordered nonexpansive $BV$ systems is the disjoint union of
$CW$ and $CU$;

(5.3) ${W_{0}\subset W\subset CW=NCU\subset NCU_{0}.}$

For this purpose we need to know how pairs of some depth and cuts in a system
relate to those in a telescoping of that system. If $\tilde{X}$ is a
telescoping of $X$, we will call $X$ a lift of $\tilde{X}$. The following
Proposition says, informally, that pairs of some depth in one of $X,\tilde{X}$
telescope (lift) to pairs of a related depth in the other, as do cuts for
those pairs. One consequence is that nonexistence of cutoffs is preserved
under telescoping and lifts.

######  Proposition 5.7.

Let $(X,T)$ be a system and $(\tilde{X},\tilde{T})$ be another system obtained
by telescoping $X$. The following statements hold for every $k\geq 1:$
(1) There exists an $\tilde{i}(k)\leq k$ such that the image of any depth $k$
pair in $X$ under the telescoping is depth $\tilde{i}(k)$ in $\tilde{X}$.
Furthermore, $\tilde{i}(k)\rightarrow\infty$ as $k\rightarrow\infty$.
(2) For all sufficiently large $j$ there exists $\tilde{J}(j)\leq j$ such that
if a depth $k$ pair in $X$ has a $j$ cut then the depth $\tilde{i}(k)$ image
of that pair in $\tilde{X}$ has a $\tilde{J}(j)$ cut. Furthermore,
$\tilde{J}(j)\rightarrow\infty$ as $j\rightarrow\infty$.
(3) For every depth $k$ pair $\tilde{x}^{(k)},\tilde{y}^{(k)}$ in $\tilde{X}$
there exists an $i(\tilde{x}^{(k)},\tilde{y}^{(k)})\geq k$ such that
$\tilde{x}^{(k)},\tilde{y}^{(k)}$ is the image under the telescoping of a
depth $i(\tilde{x}^{(k)},\tilde{y}^{(k)})$ pair in $X$.
(4) For any $j>k$ there exists a $J(j)\geq j$ such that if
$\tilde{x}^{(k)},\tilde{y}^{(k)}$ is a depth $k$ pair in $\tilde{X}$ with a
$j$ cut, then that pair is the image under the telescoping of a depth
$i(\tilde{x}^{(k)},\tilde{y}^{(k)})$ pair in $X$ with a $J(j)$ cut.

###### Proof.

Suppose that $\tilde{X}$ is a telescoping of $X$ to levels $(n_{l},l\geq 0)$
(with the root being at level $n_{0}=0$).

Proof of (1) and (2):
If $k<n_{1}$, then after telescoping from the root to level $n_{1}$, the image
of any pair of paths that is depth $k$ in $(X,T)$ is a pair of paths with
different 1-codings in $(\tilde{X},\tilde{T})$, i.e., $\tilde{i}(k)=0$. So
assume $k\geq n_{1}.$ In other words, there exists $l_{k}\geq 1$ such that
$n_{l_{k}}\leq k<k+1\leq n_{l_{k}+1}$. In particular, $l_{k}\to\infty$ as
$k\to\infty$.

Let $x^{(k)},y^{(k)}$ be a depth $k$ pair of paths in $X$. Since these paths
agree to level $k$ and differ at level $k+1$, they have the same $n_{l_{k}}$
coding, but not the same $n_{l_{k}+1}$ coding. This means row $n_{l_{k}}$ in
the array of $j$-symbols for $x^{(k)}$ is identical to row $n_{l_{k}}$ in the
array for $y^{(k)}$ and is met in the same position by the verticals (see the
discussion preceding Example 4.6) for both paths, whereas the same is not true
for row $n_{l_{k}+1}$. For every $l\geq 1$, the telescoping removes all rows
of the arrays between rows $n_{l}$ and $n_{l+1}$, so that what used to be row
$n_{l}$ becomes row $l$. Afterwards, row $l_{k}$ in the array for the image of
$x^{(k)}$ is identical to row $l_{k}$ in the array for the image of $y^{(k)}$
and is met in the same position by the verticals for both paths, whereas the
same is not true for row $l_{k+1}$. Hence, the image of the pair
$x^{(k)},y^{(k)}$ under the telescoping is depth $l_{k}$. Letting
$\tilde{i}(k)=l_{k}$, (1) is proved.

Suppose that $j>k$ and the paths $x^{(k)}$ and $y^{(k)}$ have a $j$ cut. This
cut appears as a pair of vertical segments in the arrays for the two paths
that begin in the same position in row 0, end in row $j$, and consist entirely
of rectangle boundaries. Find $l_{j}$ such that $n_{l_{j}}\leq j<n_{l_{j}+1}$.
After removing the rows for the telescoping, these vertical segments in the
arrays for $x^{(k)}$ and $y^{(k)}$ extend from level $0$ to level $l_{j}$ and
still consist entirely of rectangle boundaries. Hence both represent minimal
paths in $\tilde{X}$ from the root to level $l_{j}$. It follows that the image
of $x^{(k)},y^{(k)}$ has an $l_{j}$ cut. Letting $\tilde{J}(j)=l_{j}$, (2) is
proved.

Proof of (3) and (4):
Now let $\tilde{x}^{(k)},\tilde{y}^{(k)}$ be a depth $k$ pair of paths in
$\tilde{X}$. This means row $k$ in the array for $\tilde{x}^{(k)}$ is
identical to row $k$ in the array for $\tilde{y}^{(k)}$ and is met in the same
position by the verticals for both paths, whereas the same is not true for row
$k+1$. In the original diagram, there is a pair of paths whose image under the
telescoping is $\tilde{x}^{(k)},\tilde{y}^{(k)}$. If we restore the rows in
their respective arrays that were removed by the telescoping, rows $k$ and
$k+1$ in the arrays for $\tilde{x}^{(k)}$ and $\tilde{y}^{(k)}$ become rows
$n_{k}$ and $n_{k+1}$. Hence, for some
$i(\tilde{x}^{(k)},\tilde{y}^{(k)})\in[n_{k},n_{k+1})$, it must be the case
that these arrays are now the same from row $0$ to row
$i(\tilde{x}^{(k)},\tilde{y}^{(k)})$ with the vertical in the same position,
and that the same is not true for row $i(\tilde{x}^{(k)},\tilde{y}^{(k)})+1$.
It follows that $\tilde{x}^{(k)},\tilde{y}^{(k)}$ is the image of a depth
$i(\tilde{x}^{(k)},\tilde{y}^{(k)})$ pair of paths in $X$ under the
telescoping. This proves (3).

For (4), it is important to note that when we reinsert rows into an array for
$\tilde{X}$ that were removed during the telescoping to get an array for some
path in $X$, any vertical segment bounding a rectangle in row $l\geq 1$ of the
former becomes a rectangle boundary in row $n_{l}$ of the new array. Moreover,
the vertical line containing this rectangle boundary at level $n_{l}$ must
include rectangle boundaries from row $n_{l}$ all the way up to row 0. So any
$j$ cut for a depth $k$ pair in $\tilde{X}$ just becomes longer when we
reinsert into their respective arrays rows that were removed by the
telescoping. Specifically, any $j$ cut for a depth $k$ pair of paths
$\tilde{x}^{(k)},\tilde{y}^{(k)}$ in $\tilde{X}$ corresponds to an $n_{j}$ cut
for the preimage of $\tilde{x}^{(k)},\tilde{y}^{(k)}$ before the telescoping.
Letting $J(j)=n_{j}$, (4) is proved. ∎

Some of the systems that we shall encounter while proving our main results
will have the property encapsulated in the following definition.

######  Definition 5.8.

We say that a system is _weakly well timed_ if for infinitely many $k$ for
every $j>k$ there is a depth $k$ pair with a $j$ cut. $WW$ denotes the class
of systems with this property.

By definition, $W\subset\neg DM2\subset WW$. We will want to know what happens
to the well timed and $WW$ properties under microscoping and telescoping.

It is not necessarily the case that if $(\tilde{X},\tilde{T})$ is a
telescoping of $(X,T)$ and is well timed, then $(X,T)$ is well timed. For
example, suppose we telescope $X$ to even levels. It could happen that $(X,T)$
has no pairs with odd depth, and yet for every $k\geq 1$ and every $j>k$ there
exists a depth $k$ pair in $(\tilde{X},\tilde{T})$ with a $j$ cut whose lift
to $(X,T)$ is depth $2k$. In this case, $(\tilde{X},\tilde{T})$ is well timed
while $(X,T)$ is not.

######  Lemma 5.9.

Let $(X,T)$ be a system and $(\tilde{X},\tilde{T})$ be another system obtained
by telescoping $X$ to levels $(n_{l},l\geq 0)$.
(1) If $(X,T)\in W$, then $(\tilde{X},\tilde{T})\in W$. Likewise, if $(X,T)\in
W_{0}$, then $(\tilde{X},\tilde{T})\in W_{0}$.
(2) If $(X,T)\in WW$, then $(\tilde{X},\tilde{T})\in WW$.
(3) If $(\tilde{X},\tilde{T})\in WW$, then $(X,T)\in WW$.

###### Proof.

Proof of (1):
Suppose $(X,T)$ is well timed. Fix $k\geq 1$ and let $j>k$. There exists in
$X$ a depth $n_{k}$ pair with an $n_{j}$ cut. As shown in the proof of
Proposition 5.7, parts (1) and (2), after the telescoping this yields a depth
$k$ pair in $\tilde{X}$ with a $j$ cut.

If $(X,T)$ is very well timed, then for every $k\geq 1$ there exists a depth
$n_{k}$ pair with an $n_{j}$ cut for all $j>k$. The above argument shows that
this yields a depth $k$ pair with long cuts after the telescoping.

Proof of (2):
Suppose $(X,T)\in WW$. In other words, there exists an increasing sequence
$(m_{k})$ such that for every $k$ and every $j>m_{k}$ there is a depth $m_{k}$
pair in $(X,T)$ with a $j$ cut.

By Proposition 5.7 (1), for every $k\geq 1$ there exists $\tilde{i}(m_{k})\leq
m_{k}$ such that the image of any depth $m_{k}$ pair in $X$ under the
telescoping is depth $\tilde{i}(m_{k})$ in $\tilde{X}$. Let
$\tilde{m}_{k}=\tilde{i}(m_{k})$. Since $\tilde{i}(k)\rightarrow\infty$ as
$j\rightarrow\infty$, the sequence $(\tilde{m}_{k})$ is increasing.

Given $k\geq 1$ and $j>m_{k}$, find in $(X,T)$ a depth $m_{k}$ pair with a $j$
cut. By Proposition 5.7 (2), there exists a $\tilde{J}(j)\leq j$ such that the
depth $\tilde{m}_{k}$ image of this pair in $(\tilde{X},\tilde{T})$ has a
$\tilde{J}(j)$ cut. Since $\tilde{J}(j)\rightarrow\infty$ as
$j\rightarrow\infty$, it follows that $(\tilde{X},\tilde{T})\in WW$.

Proof of (3):
Suppose that $(\tilde{X},\tilde{T})\in WW$. There exists an increasing
sequence $(\tilde{m}_{k})$ such that for every $k$ and every $j>\tilde{m}_{k}$
there is a depth $\tilde{m}_{k}$ pair
$\tilde{x}^{({\tilde{m}_{k}})},\tilde{y}^{({\tilde{m}_{k}})}$ in $\tilde{X}$
with a $j$ cut.

Fix $k\geq 1$. As we vary $j$, the pair
$\tilde{x}^{({\tilde{m}_{k}})},\tilde{y}^{({\tilde{m}_{k}})}$ may change, and
hence $i(\tilde{x}^{({\tilde{m}_{k}})},\tilde{y}^{({\tilde{m}_{k}})})$ may
vary. However, it was shown in the proof of Proposition 5.7 (3) that for every
depth $\tilde{m}_{k}$ pair
$\tilde{x}^{({\tilde{m}_{k}})},y^{({\tilde{m}_{k}})}$, we have
$n_{\tilde{m}_{k}}\leq
i(\tilde{x}^{({\tilde{m}_{k}})},\tilde{y}^{({\tilde{m}_{k}})})<n_{\tilde{m}_{k}+1}$.
Hence, there exists $m_{k}\in[n_{\tilde{m}_{k}},n_{\tilde{m}_{k+1}})$ such
that for infinitely many $j>\tilde{m}_{k}$, the corresponding pair
$\tilde{x}^{({\tilde{m}_{k}})},\tilde{y}^{({\tilde{m}_{k}})}$ lifts to a depth
$m_{k}$ pair, which by Proposition 5.7 (4) has a $J(j)>j$ cut. Since
$J(j)\rightarrow\infty$ as $j\rightarrow\infty$ (and the $m_{k}$ are all
distinct), it follows that $(X,T)\in WW$. ∎

######  Proposition 5.10.

For a system $(X,T)$ the following statements are equivalent:
(1) $(X,T)$ is weakly well timed.
(2) $(X,T)$ has a telescoping that is well timed.
(3) $(X,T)$ is conjugate to a well timed system.
Thus $WW=CW$.

###### Proof.

We show first that (1) implies (2). Suppose there are infinitely many $k$ for
which for every $j>k$ there exists a depth $k$ pair in $X$ with a $j$ cut. In
other words, there exists an increasing sequence $(m_{k})$ such that for all
$k$ and for every $j>m_{k}$ there is a depth $m_{k}$ pair in $X$ with a $j$
cut. Let $\tilde{X}$ be the telescoping of $X$ to levels $(m_{k})$.

Given $k\geq 1$ and $j>m_{k}$, find in $(X,T)$ a depth $m_{k}$ pair with a $j$
cut. By Proposition 5.7 (1) and (2), the image of this pair is depth $k$ with
a $\tilde{J}(j)$ cut. Since $\tilde{J}(j)\rightarrow\infty$ as
$j\rightarrow\infty$, it follows that $\tilde{X}$ is well timed.

That (2) implies (3) is clear, since conjugacy of $BV$ systems is the
equivalence relation that corresponds to the one for diagrams that is
generated by telescoping and isomorphism [HPS1992, Section 4]. So if $(X,T)$
has a telescoping that is well timed then it is conjugate to that well timed
system.

To prove that (3) implies (1), suppose that $(X,T)$ is conjugate to a well
timed system $(Y,S)$. As remarked above, then there are a system $Z$ and
telescopings $\tilde{X}$ of $X$ and $\tilde{Y}$ of $Y$ such that $Z$
telescopes on even levels $0,2,4,\dots$ to $\tilde{X}$ and on odd levels
$0,1,3,\dots$ to $\tilde{Y}$. By Lemma 5.9 (1), any telescoping of a well
timed system is also well timed. So we may assume that $\tilde{Y}=Y$.

Since $W\subset WW$, $Y$ is also weakly well timed. Hence by Lemma 5.9 (3),
$Z\in WW$. Next consider the telescoping of $Z$ to $\tilde{X}$. By Lemma 5.9
(2), $(\tilde{X},\tilde{T})\in WW$.

Applying part (3) of the lemma again, we conclude that $(X,T)\in WW$. ∎

With appropriate adjustments the foregoing Proposition and its proof adapt to
very well timed systems.

######  Proposition 5.11.

For a system $(X,T)$ the following statements are equivalent:
(1) There are infinitely many $k$ for which there exists a depth $k$ pair with
long cuts.
(2) $(X,T)$ has a telescoping that is very well timed.
(3) $(X,T)$ is conjugate to a very well timed system.

######  Remark 5.12.

Part (3) of Proposition 5.11 shows that parts (1) and (2) persist under
telescoping.

######  Example 5.13.

As promised in Remark 5.4, we show that $WW\cap DM2\neq\emptyset$. We modify
the diagram for the GJ example $X$ by changing every other (even-numbered)
Morse component so that all its vertices are uniformly ordered: for all $j\geq
4$ and all even $i$ such that $1\leq i<j-1$ we change the ordering at
$v(j,2i+1)$ so that it is left to right. Let $Y$ denote the new diagram and
system, with the same vertices $v(j,i)$ and “Morse components” $MC(k)$ as in
$X$. Then for every odd $k$, there is still a depth $k$ pair with long cuts
(these are just the same as they were in the GJ example, see Proposition 4.2
and Corollary 4.3), so the new system is in $WW$ .
(In more detail, as in Corollary 4.3, let $x$ and $x^{\prime}$ be paths that
agree to level $k+1$ and pass through $v(k+1,2k)$ and $v(k+1,2k+1)$,
respectively, and such that the ordinal edge label for every edge in $x$ and
in $x^{\prime}$ after level $k+1$ is $2k$. In other words, $x$ and
$x^{\prime}$ enter the Morse component $MC(k)$ at its top and then follow its
two sides all the way down. By Proposition 4.2, $x$ and $x^{\prime}$ are depth
$k$. It is easily verified that for every $j>k$, there exists a time $m<0$
such that $T^{m}x$ and $T^{m}x^{\prime}$ agree to level $j-1$ and are minimal
from the root into $v(j,2k)$ and $v(j,2k+1)$, respectively. Therefore $x$ and
$x^{\prime}$ are depth $k$ and have long cuts.)
We show now that for all even $k$ there are no longer any depth $k$ pairs, so
the new system is (vacuously) in $DM2$ (and hence in $H2$). The idea is that,
as in $X$, the only candidates for depth $k$ pairs are paths down $MC(k)$, but
the changed edge orders cause these pairs to have different $2$-codings, so
they have become depth $1$.
All changes to $X$ were at levels $4$ and higher. Moreover, for every $j\geq
2$, the coding of $v(j,3)$ by vertices at level $j-1$ remains the same in $Y$
as it was in $X$; and at vertices in $Y$ where this coding is different than
it was in $X$, it is now the same as the coding at $v(j,1)$. Hence, if as in
Observation 4.5 we expand each symbol (vertex) $v(j,i)$ to $C_{j-1}(v(j,i))$,
the effect on the $\phi_{j}$-images (strings on $D_{j},E_{j},D_{j-1},E_{j-1}$)
produces the same morphisms as before. The argument made in Example 4.6 that
paths in $X$ with the same $2$-coding are $2$-equivalent can then be applied
to show the same holds true in $Y$. Hence, the proof of Proposition 4.2 (4)
still works to show that for any $k\geq 2$ any depth $k$ pair in the new
system represented by $Y$ is $k$-equivalent but not $(k+1)$-equivalent.
Now let $x$, $x^{\prime}$ be a pair of paths in $Y$ that for some $k\geq 2$ is
depth $k$ in the corresponding system. Since Observation 4.1 (2) still holds
for $Y$, we can argue as in the proof of Proposition 4.2 (3) that $x$ and
$x^{\prime}$ in $Y$ have the same sequence of edge labels, enter $MC(k)$ at
the level at which they first differ and are contained in $MC(k)$ at all
subsequent levels. At any level of $Y$ only pairs of edges inside one of the
odd Morse components in $Y$ can have the same label and distinct sources.
Hence, $k$ must be odd, and so there are no pairs of paths with even depth in
the new system.
Telescoping the new diagram to a strictly increasing sequence of levels
$j_{n}$ will produce a system that is not in $H2$ (or $DM2$). This is because
given any $n$ there are an odd integer $k\in[j_{n},j_{n+1})$ and a depth $k$
pair with long cuts. After the telescoping this pair will be depth $n$ (by the
proof of Proposition 5.7 (1)) and will still have long cuts (by Proposition
5.7 (2)), so the telescoped system is in $W_{0}$. By definition, $W_{0}\cap
H2=\emptyset$, so the class $H2$ is not closed under telescoping.
Remark 5.4 mentioned that $H2$ not being closed under telescoping impacts the
proofs in [dm2008, Hoynes2017]. If in some system for every odd $k$ there were
a depth $k$ pair with a cutoff (so the system is nonvacuously in $H2$, unlike
our example), and for every even $k$ there were a depth $k$ pair with long
cuts (so the system is also in $WW$), the same argument as above would show
that telescoping to any strictly increasing sequence of levels produces a
system that is not in $H2$. If the levels were chosen to produce a system in
$U_{2}$, the result would be in $U_{2}\setminus H2$. The proofs in [dm2008,
Hoynes2017] could be clarified by ruling out this case or dealing with it,
maybe along the lines we suggest in Remark 5.4.

The following Theorem subsumes Proposition 3.11.

######  Theorem 5.14.

No system that is conjugate to a well timed system can also be conjugate to an
untimed system: $CW\subset NCU$.

###### Proof.

Suppose that $X$ is a system that is conjugate to a well timed system and is
also conjugate to an untimed system $Y$. As mentioned above, by [HPS1992,
Theorem 4.7], then there are a system $Z$ and telescopings $\tilde{X}$ of $X$
and $\tilde{Y}$ of $Y$ such that $Z$ telescopes on even levels $0,2,4,\dots$
to $\tilde{X}$ and on odd levels $0,1,3,\dots$ to $\tilde{Y}$.

By Proposition 5.10, $X\in WW$. Repeated application of Lemma 5.9 gives
$\tilde{X}\in WW$, then $Z\in WW$, then $\tilde{Y}\in WW$, and finally $Y\in
WW$. This is a contradiction, since by definition no system can be both
untimed and weakly well timed. ∎

######  Theorem 5.15.

The family of nonexpansive systems is the disjoint union of those conjugate to
well timed systems and those conjugate to untimed systems:

(5.4) $NE=CW\sqcup CU.$

###### Proof.

By Theorem 5.14, $CW\subset NCU$, so it remains only to show that $NCW\subset
CU$. By Proposition 5.10, if $(X,T)\in NCW$, then there is a $k_{0}$ such that
for all $k\geq k_{0}$ there is a $k$ cutoff $j(k)>k$: no depth $k$ pair has a
$j(k)$ cut. Let us telescope to levels $n_{l},l\geq 0$, with $n_{1}>k_{0}$ to
produce the system $(\tilde{X},\tilde{T})$. Suppose that $\tilde{k}\geq 1$ and
$\tilde{x},\tilde{y}$ is a depth $\tilde{k}$ pair in $\tilde{X}$ with a
$\tilde{j}$ cut. Proposition 5.7 (3) and (4) tell us that then
$\tilde{x},\tilde{y}$ is the image under the telescoping of a pair $x,y$ in
$X$ of depth $i(\tilde{x},\tilde{y})\in[n_{\tilde{k}},n_{\tilde{k}+1})$ with a
$J(\tilde{j})=n_{\tilde{j}}$ cut. Every $i\in[n_{\tilde{k}},n_{\tilde{k}+1})$
has a cutoff $j(i)$ in $X$, in particular
$J(\tilde{j})<j(i(\tilde{x},\tilde{y}))$. Since

(5.5) $\tilde{j}\leq
J(\tilde{j})<j(i(\tilde{x},\tilde{y}))\leq\max\\{j(i):i\in[n_{\tilde{k}},n_{\tilde{k}+1})\\},$

$\tilde{j}$ is bounded, over all pairs in $\tilde{X}$ of depth $\tilde{k}$.
Thus in $\tilde{X}$ for every $\tilde{k}$ there is a $\tilde{k}$ cutoff. This
shows that telescoping past $k_{0}$ produces a system in $U$, and hence
$(X,T)\in CU$. ∎

##  6\. Describing untimed systems

Now we take a few steps towards determining exactly which $BV$ systems are
very untimed. If a system $(X,T)$ is very untimed and not conjugate to an
odometer, then, by [dm2008], it must have infinite “topological rank”. At the
moment we do not have an example of a very untimed system that has unbounded
width.

######  Lemma 6.1.

If in a $BV$ diagram level $n+1$ is uniformly ordered and has more than one
vertex, then in the $BV$ system there is a depth $n$ pair of paths with an
$n+1$ cut.

###### Proof.

Choose two paths $x,x^{\prime}$ that are minimal into distinct vertices at
level $n+1$. Since level $n+1$ is uniformly ordered, the $n$-basic block at
each vertex at level $n+1$ is periodic with shortest repeating block $P$. (By
this we mean that for each vertex $v$ at level $n+1$, there is an integer
$n_{v}$ such that the $n$-basic block at $v$ is $P^{n_{v}}$). Thus the
$n$-factor is a rotation on $|P|$ points, and the $n$-coding of every path is
the two-sided sequence $P^{\infty}$, with a choice of the center position.
Since $x$ and $x^{\prime}$ are minimal into level $n+1$, their “dot” is at the
beginning of an explicit appearance of $P$ in the $n$-basic blocks at
$v_{n+1}(x)$ and $v_{n+1}(x^{\prime})$ respectively, so $x,x^{\prime}$ have
the same $n$-coding. Hence $x$ and $x^{\prime}$ are depth $n$ with an $n+1$
cut. ∎

######  Proposition 6.2.

Every bounded width very untimed system has only finitely many levels with
more than one vertex.

###### Proof.

By [dm2008], bounded width and nonexpansive implies conjugate to an odometer.
By [FPS2017] (and also mentioned in [GJ2000]) there is a telescoping that has
infinitely many uniformly ordered levels. Let $n+1$ ($n>0$ ) be a uniformly
ordered level in the telescoped system, which is still very untimed (see
Remark 5.5, (2)). Because of the Lemma and because no depth $n$ pair can have
an $n+1$ cut, this level $n+1$ must have just one vertex. Returning to the
original (not telescoped) system, the level $m$ corresponding to level $n+1$
in the telescoped system has just one vertex. Any level that follows a level
with just one vertex is uniformly ordered, and so, by the Lemma, level $m+1$
in the original system must also have just one vertex. Thus every level in the
original system from level $m+1$ on has just one vertex. ∎

######  Example 6.3.

So one way to produce very untimed systems is to begin with levels with more
than one vertex with edges ordered so that all the edges downward from each
vertex have different ordinal labels. Exiting these levels downward there can
be no cuts. Then continue forever with single-vertex levels connected by
multiple edges. Any such system is conjugate to an odometer, upon telescoping
from the root to the first single-vertex level.

######  Definition 6.4.

A $BV$ system is deterministic if for every $n\geq 1$ the edges downward from
each vertex at level $n$ have different ordinal labels.

The following Proposition provides more detail about the possible form of
deterministic diagrams such as the one in Example 6.3.

######  Proposition 6.5.

Let $(X,T)$ be a deterministic system. For each $n\geq 1$ let $k_{n}$ denote
the number of vertices at level $n$. Then $k_{1}\geq k_{2}\geq\dots$, and
eventually all $k_{n}=1$. (So the diagram has the form of the previous
Example.)

###### Proof.

If $k_{n}<k_{n+1}$ for some $n\geq 1$, then the label $1$ must be repeated on
edges between levels $n$ and $n+1$, so $k_{n}\geq k_{n+1}$ for all $n$.

If eventually all $k_{n}=k$, we claim that there would be $k$ minimal paths,
and so we must have $k=1$. This is because from each low (late) enough level
$n+1$ there are $k$ minimal edges (labeled $1$) upward to $k$ different
vertices at level $n$ (because each vertex at level $n$ has a single edge
downward that is labeled $1$), and hence $k$ minimal paths from level $n+1$ up
to the root. Looking at larger and larger values of $n$, we can see $k$
distinct paths that follow only minimal edges. ∎

######  Example 6.6.

Not every bounded width very untimed system must be deterministic, as Figure 2
shows. The small numbers are ordinal edge labels, and the strings in
parentheses are $1$-basic blocks. In this example the only possible cuts are
for pairs of paths among $x=0auA...$ (thick edges), $x^{\prime}=0avB\dots$
(dashed edges) and $x^{\prime\prime}=0bwB\dots$ (or pairs in the orbits of
such pairs). While $x$ and $x^{\prime}$ have a $2$ cut, they are not depth
$1$; $x$ and $x^{\prime\prime}$ have $2$ and $3$ cuts, but they are not depth
$1$ or $2$; and $x^{\prime}$,$x^{\prime\prime}$ have a $2$ cut, but they are
not depth $1$. In fact there are no depth $1$ or depth $2$ pairs.

$0$$a$$b$$c$$u\,(ab)$$1$$2$$1$$2$$1$$2$$1$$2$$2$$1$$1$$2$$1$$1$$v\,(ab)$$w\,(bc)$$A\,(abbc)$$B\,(bcab)$$AB$
Figure 2. A nondeterministic bounded width very untimed system

##  7\. Conclusion and questions

The family of simple, properly ordered Bratteli-Vershik systems is the
disjoint union of the expansive systems, the systems conjugate to well timed
systems, and the systems conjugate to untimed systems:

(7.1) $BV=E\sqcup CW\sqcup CU.$

The foregoing suggests several questions:

1\. Is there an example of an untimed (or even very untimed) system that has
unbounded width? Is there an untimed system that has infinite topological
rank, equivalently is not conjugate to any odometer? Is the class $U$ closed
under conjugacy?

2\. Is every well timed system conjugate to an $SSNE$ (or $SNE$) system? If
so, one could regularize nonexpansiveness by modifying any member of this
class $W$ of well timed $NE$ systems so that it becomes $SSNE$.

3\. There are many questions about the relations among the classes of systems
that we have defined, and others that could be defined. Exactly which classes
can overlap, exactly what inclusions are there among them, exactly which of
these inclusions are strict, etc.? Is every odometer (by which we here mean
all levels strictly uniformly ordered, cf. Example 5.6) in $U$? Is every
system conjugate to an odometer in $U$? Is $SNE$ contained in $W$? Can $H2$
and $W$ intersect? (We know $DM2\cap W=\emptyset$.) Is $U_{1}\cap
W\neq\emptyset$? Is $U_{2}\cap W\neq\emptyset$?

## References