where $v_{j}=r(f_{j})$, and $f_{1},\dots,f_{k}$ are the distinct red edges
such that $s(f_{j})=s(e_{0})$.

Therefore $\\{{\pazocal P}_{n}\\}_{n\geq 0}$ is a $\sigma$-refined sequence of
partitions of $X$. Let $(F^{\prime},D^{\prime})$ be the $l$-diagram obtained
from $(X,\sigma)$ by applying the construction 3.24. We aim to show that
$(F,D)\sim_{l}(F^{\prime},D^{\prime})$.

It is clear that the vertex sets of $(F,D)$ and of $(F^{\prime},D^{\prime})$
are identified: given $v\in F^{0,n+1}$ for $n\geq 0$, there exists a unique
blue path $(e_{0},e_{1},\dots,e_{n})$ such that $s(e_{n})=v$. Such vertex
corresponds exactly to the clopen set $Z(e_{0},\dots,e_{n})\in{\pazocal
P}_{n+1}=(F^{\prime})^{0,n+1}$. The vertex $v\in F^{0,0}$ corresponds to the
clopen $Z(v)\in{\pazocal P}_{0}=(F^{\prime})^{0,0}$. This correspondence is
clearly bijective.

Now for the edges. We distinguish between even and odd layers.

  1. (i)

For an even layer $2n\geq 0$, and given $v\in F^{0,2n},w\in F^{0,2n+1}$, there
exists a blue edge $e\in F^{1,2n}$ with source $w$ and range $v$ if and only
if $v=s(e_{2n-1}),w=s(e_{2n})$ for a unique blue path
$(e_{0},\dots,e_{2n-1},e_{2n})$, if and only if there exists a blue edge
$e(Z^{\prime},Z)\in(F^{\prime})^{1,2n}$ with source
$Z^{\prime}=Z(e_{0},\dots,e_{2n-1},e_{2n})$ and range
$Z=Z(e_{0},\dots,e_{2n-1})$.

Similarly, by the definition of $\sigma$, there exists a red edge $f\in
F^{1,2n}$ with source $w$ and range $v$ if and only if
$v=s(e_{2n-1}),w=s(e^{\prime}_{2n})$ for unique blue paths
$(e_{0},\dots,e_{2n-1}),(e^{\prime}_{0},\dots,e^{\prime}_{2n})$ satisfying
$Z(e^{\prime}_{0},\dots,e^{\prime}_{2n})\subseteq\sigma(Z(e_{0},\dots,e_{2n-1}))$,
if and only if there exists a red edge $f(Z^{\prime},Z)\in(F^{\prime})^{1,2n}$
with source $Z^{\prime}=Z(e^{\prime}_{0},\dots,e^{\prime}_{2n})$ and range
$Z=Z(e_{0},\dots,e_{2n-1})$ (note that since we are assuming that $(F,D)$ is
refined, there is at most one red edge between any two vertices of $F$). Let
us show the first of these equivalences (note that, when $n=0$, one has to
interpret $Z(e_{0},\dots,e_{2n-1})$ as $Z(v)$).

Suppose first that
$Z(e^{\prime}_{0},\dots,e^{\prime}_{2n})\subseteq\sigma(Z(e_{0},\dots,e_{2n-1}))$.
Taking

$y=(e_{0}^{\prime},\dots,e_{2n}^{\prime},e_{2n+1}^{\prime},\dots)\in
Z(e_{0}^{\prime},\dots,e_{2n}^{\prime}),$

we have that there is some $x=(e_{0},\dots,e_{2n-1},e_{2n},\dots)\in
Z(e_{0},\dots,e_{2n-1})$ such that $\sigma(x)=y$. Let
$(e_{2j},e_{2j+1},f_{2j},f_{2j+1})$ be the romb associated to the blue pair
$(e_{2j},e_{2j+1})$, for $j\geq 0$. Since $\sigma(x)=y$, we have by definition
of $\sigma$ that $(e^{\prime}_{2j+1},e^{\prime}_{2j+2},f_{2j+1},f_{2j+2})$ is
the romb associated to the red pair $(f_{2j+1},f_{2j+2})$, for all $j\geq 0$.
In particular we get that $s(f_{2n})=s(e_{2n}^{\prime})=w$ and that
$r(f_{2n})=r(e_{2n})=s(e_{2n-1})=v$. Hence we see that there is a red edge
$f\in F^{1,2n}$ from $w$ to $v$.

To show the converse, suppose there is a red edge $f_{2n}\in F^{1,2n}$ such
that $s(f_{2n})=w$ and $r(f_{2n})=v$, and let

$y=(e_{0}^{\prime},\dots,e_{2n}^{\prime},e_{2n+1}^{\prime},\dots)\in
Z(e_{0}^{\prime},\dots,e_{2n}^{\prime}).$

By property (g) in Definition 3.2, we can inductively build red pairs
$(f_{2j+1},f_{2j+2})$ for $j\geq n$, such that $f_{2j+1}\in R(f_{2j})$ and
$(e_{2j+1}^{\prime},e_{2j+2}^{\prime},f_{2j+1},f_{2j+2})$ is a romb, for all
$j\geq n$. Now, for all $j\geq n$, we can build the rombs
$(e_{2j},e_{2j+1},f_{2j},f_{2j+1})$ (see part (ii) of Proposition 3.3). Note
that $r(e_{2n})=r(f_{2n})=v=s(e_{2n-1})$, so that we can consider the point

$x=(e_{0},\dots,e_{2n-1},e_{2n},e_{2n+1},\dots)\in Z(e_{0},\dots,e_{2n-1}).$

Now we have

$\sigma(x)=(e_{0}^{\prime\prime},\dots,e_{2n}^{\prime\prime},e_{2n+1}^{\prime},e_{2n+2}^{\prime},\dots).$

In particular $(e_{0}^{\prime\prime},\dots,e_{2n}^{\prime\prime})$ is a blue
path with source $r(e_{2n+1}^{\prime})=s(e_{2n}^{\prime})=w$, so by uniqueness
of the blue paths we get that
$(e_{0}^{\prime\prime},\dots,e_{2n}^{\prime\prime})=(e_{0}^{\prime},\dots,e_{2n}^{\prime})$,
and consequently $\sigma(x)=y$, with $x\in Z(e_{0},\dots e_{2n-1})$. This
shows the desired inclusion $Z(e_{0}^{\prime},\dots
e_{2n}^{\prime})\subseteq\sigma(Z(e_{0},\dots,e_{2n-1}))$.

We have shown that there exist bijective correspondences between blue edges in
$F^{1,2n}$ and blue edges in $(F^{\prime})^{1,2n}$, and between red edges in
$F^{1,2n}$ and red edges in $(F^{\prime})^{1,2n}$.

  2. (ii)

For an odd layer $2n+1\geq 1$, and given $v\in F^{0,2n+1},w\in F^{0,2n+2}$,
there exists a blue edge $e\in F^{1,2n+1}$ with source $w$ and range $v$ if
and only if $v=s(e_{2n}),w=s(e_{2n+1})$ for a unique blue path
$(e_{0},\dots,e_{2n},e_{2n+1})$, if and only if there exists a blue edge
$e(Z^{\prime},Z)\in(F^{\prime})^{1,2n+1}$ with source
$Z^{\prime}=Z(e_{0},\dots,e_{2n},e_{2n+1})$ and range
$Z=Z(e_{0},\dots,e_{2n})$.

Similarly, by the definition of $\sigma$, there exists a red edge $f\in
F^{1,2n+1}$ with source $w$ and range $v$ if and only if
$v=s(e^{\prime}_{2n}),w=s(e_{2n+1})$ for unique blue paths
$(e_{0},\dots,e_{2n+1}),(e^{\prime}_{0},\dots,e^{\prime}_{2n})$ satisfying
$Z(e_{0},\dots,e_{2n+1})\subseteq\sigma^{-1}(Z(e^{\prime}_{0},\dots,e^{\prime}_{2n}))$,
if and only if there exists a red edge
$f(Z^{\prime\prime},Z^{\prime})\in(F^{\prime})^{1,2n+1}$ with source
$Z^{\prime\prime}=Z(e_{0},\dots,e_{2n+1})$ and range
$Z^{\prime}=Z(e^{\prime}_{0},\dots,e^{\prime}_{2n})$.

Indeed, if $\sigma(Z(e_{0},\dots,e_{2n+1}))\subseteq
Z(e^{\prime}_{0},\dots,e^{\prime}_{2n})$, then it follows directly from the
definition of $\sigma$ that there is a red edge $f\in F^{1,2n+1}$ from
$w=s(e_{2n+1})$ to $v=s(e_{2n}^{\prime})$.

Conversely, suppose that there is a red edge $f\in F^{1,2n+1}$ from
$w=s(e_{2n+1})$ to $v=s(e_{2n}^{\prime})$, and let

$x=(e_{0},\dots,e_{2n+1},e_{2n+2},\dots)\in Z(e_{0},\dots,e_{2n+1}).$

Since $w\in F^{0,2n+2}$ it follows from condition (c) in Definition 3.2 that
there is a unique red edge with source $w$, which must be equal to $f$.
Therefore, if $(e_{2n},e_{2n+1},f_{2n},f_{2n+1})$ is the romb determined by
$(e_{2n},e_{2n+1})$, then necessarily $f_{2n+1}=f$. If

$\sigma(x)=(e_{0}^{\prime\prime},\dots,e_{2n}^{\prime\prime},e_{2n+1}^{\prime\prime},\dots),$

then, by definition of $\sigma$, we have
$s(e_{2n}^{\prime})=v=r(f)=r(f_{2n+1})=r(e_{2n+1}^{\prime\prime})=s(e_{2n}^{\prime\prime})$.
By uniqueness of blue paths, we get that
$(e_{0}^{\prime\prime},\dots,e_{2n}^{\prime\prime})=(e_{0}^{\prime},\dots,e_{2n}^{\prime})$,
and therefore $\sigma(x)\in Z(e_{0}^{\prime},\dots,e_{2n}^{\prime})$, showing
that $\sigma(Z(e_{0},\dots,e_{2n+1}))\subseteq
Z(e_{0}^{\prime},\dots,e_{2n}^{\prime})$.

We have shown that there exist bijective correspondences between blue edges in
$F^{1,2n+1}$ and blue edges in $(F^{\prime})^{1,2n+1}$, and between red edges
in $F^{1,2n+1}$ and red edges in $(F^{\prime})^{1,2n+1}$.

From these one can easily deduce that $(F,D)\sim_{l}(F^{\prime},D^{\prime})$,
and so $[(F,D)]=[(F^{\prime},D^{\prime})]=\Phi(\Psi([(F,D)]))$, as we wanted
to show.

We are left to show that
$\Psi\circ\Phi=\text{id}_{\textsf{LHomeo}/\sim_{t.c.}}$. Let
$(X,\sigma)\in\textsf{LHomeo}$ and take $\\{{\pazocal P}_{n}\\}_{n\geq 0}$ to
be any sequence of $\sigma$-refined partitions of $X$. Denote by $(F,D)$ the
refined $l$-diagram constructed from $\\{{\pazocal P}_{n}\\}_{n\geq 0}$ by
applying the construction 3.24, and by $(X^{\prime},\sigma^{\prime})$ the
dynamical system in LHomeo constructed from $(F,D)$ by applying the
construction 3.34. Recall that

$X^{\prime}=\\{(e_{0},e_{1},e_{2},\dots)\mid e_{i}\in
B_{r}(e_{i}),r(e_{i+1})=s(e_{i})\text{ for all }i\geq 0\\},$

and a basis for the topology of $X^{\prime}$ is given by the cylinder sets
$Z(e_{0},e_{1},\dots,e_{n})$. We aim to show that $(X,\sigma)$ and
$(X^{\prime},\sigma^{\prime})$ are topologically conjugate.

Let us construct the homeomorphism $h:X\rightarrow X^{\prime}$ as follows.
Given $x\in X$, for each $n\geq 0$ let $Z_{n}(x)$ be the unique clopen set in
the partition ${\pazocal P}_{n}$ that contains $x$. Then $x\in\bigcap_{n\geq
0}Z_{n}(x)$, and since $\text{diam}(Z_{n}(x))$ tend to zero as
$n\rightarrow\infty$, necessarily this intersection is exactly $\\{x\\}$, that
is

$\bigcap_{n\geq 0}Z_{n}(x)=\\{x\\}.$

Let now $e_{n}(x)$ be the unique blue edge in $F^{1,n}$ such that
$r(e_{n}(x))=Z_{n}(x)$ and $s(e_{n}(x))=Z_{n+1}(x)$ (it exists since $x\in
Z_{n}(x)\cap Z_{n+1}(x)$, so necessarily $Z_{n+1}(x)\subseteq Z_{n}(x)$). We
define

$h:X\rightarrow X^{\prime},\quad x\mapsto(e_{0}(x),e_{1}(x),e_{2}(x),\dots).$

We show that $h$ is a homeomorphism, and that
$h\circ\sigma=\sigma^{\prime}\circ h$.

  1. (i)

Suppose $x,y\in X$ are such that $e_{n}(x)=e_{n}(y)$ for all $n\geq 0$. Then

$\\{x\\}=\bigcap_{n\geq 0}Z_{n}(x)=\bigcap_{n\geq 0}r(e_{n}(x))=\bigcap_{n\geq
0}r(e_{n}(y))=\bigcap_{n\geq 0}Z_{n}(y)=\\{y\\},$

i.e. $x=y$. This proves injectivity of $h$.

  2. (ii)

Let $e=(e_{0},e_{1},e_{2},\dots)\in X^{\prime}$. Putting $Z_{n}=r(e_{n})$, we
have a decreasing sequence of non-empty clopen sets $Z_{0}\supseteq
Z_{1}\supseteq Z_{2}\supseteq\cdots$. Therefore $\bigcap_{n\geq
0}Z_{n}\neq\emptyset$ by Cantor’s Intersection Theorem, and by the assumption
that $\text{diam}(Z_{n})\xrightarrow{n\rightarrow\infty}0$, it must consist of
a single point $\\{x\\}$. Then $x\in Z_{n}$ for all $n\geq 0$, so
$e_{n}(x)=e_{n}$. This proves surjectivity of $h$.

Thus $h$ is bijective. Since $X,X^{\prime}$ are compact and Hausdorff, in
order to prove that $h$ is a homeomorphism it is enough to prove continuity of
$h$.

  1. (iii)

Let $x\in h^{-1}(Z(e_{0},e_{1},\dots,e_{n}))$, so that $e_{i}(x)=e_{i}$ for
$0\leq i\leq n$. That means $x\in Z_{0}(x),\dots,Z_{n}(x),Z_{n+1}(x)$, being
$Z_{i}(x)=r(e_{i}(x))$ and $Z_{n+1}(x)=s(e_{n}(x))$. But then for any
$y\in\bigcap_{i=0}^{n+1}Z_{i}(x)=Z_{n+1}(x)$ we have that
$e_{i}(y)=e_{i}(x)=e_{i}$ for $0\leq i\leq n$, i.e. $h(y)\in
Z(e_{0},e_{1},\dots,e_{n})$. Therefore $x\in Z_{n+1}(x)\subseteq
h^{-1}(Z(e_{0},e_{1},\dots,e_{n}))$. This shows continuity of $h$.

Hence $h$ is a homeomorphism. To conclude the theorem, we need to prove that
$h$ intertwines the surjective local homeomorphisms $\sigma$ and
$\sigma^{\prime}$.

  1. (iv)

Let $x\in X$. Let also $Z_{n}(\sigma(x))\in{\pazocal P}_{n}$ be such that
$\\{\sigma(x)\\}=\bigcap_{n\geq 0}Z_{n}(\sigma(x))$, and consider the blue
edges $e_{n}:=e_{n}(\sigma(x))$, so that $r(e_{n})=Z_{n}(\sigma(x))$ and
$s(e_{n})=Z_{n+1}(\sigma(x))$. Let $\\{Z_{n}(x)\\}_{n}$ and
$\\{e_{n}(x)\\}_{n}$ be such that $\\{x\\}=\bigcap_{n\geq
0}Z_{n}(x),r(e_{n}(x))=Z_{n}(x)$ and $s(e_{n}(x))=Z_{n+1}(x)$.

For each $j\geq 0$, we complete the pair of blue edges
$(e_{2j}(x),e_{2j+1}(x))$ to a romb $(e_{2j}(x),e_{2j+1}(x),f_{2j},f_{2j+1})$.
For $j\geq 1$ we have $s(f_{2j-1})=r(f_{2j})$, so we may consider the red pair
$(f_{2j-1},f_{2j})$ and complete it to another romb
$(e_{2j-1}^{\prime},e_{2j}^{\prime},f_{2j-1},f_{2j})$. There exists then a
unique blue edge $e_{0}^{\prime}$ such that
$s(e_{0}^{\prime})=r(e_{1}^{\prime})$. Then by construction of
$\sigma^{\prime}$ we have
$\sigma^{\prime}(h(x))=(e_{0}^{\prime},e_{1}^{\prime},e_{2}^{\prime},\dots)$.
Let $Z_{n}^{\prime}:=r(e_{n}^{\prime})$ for $n\geq 0$. This is a decreasing
sequence of clopen subsets $Z_{n}^{\prime}\in{\pazocal P}_{n}$. Also, by
definition, we have

$\sigma(Z_{2j+2}(x))\subseteq Z^{\prime}_{2j+1}\subseteq\sigma(Z_{2j}(x))$

for all $j\geq 0$. In particular, $\sigma(x)\in\sigma(Z_{2j+2}(x))\subseteq
Z_{2j+1}^{\prime}$, so by uniqueness $Z_{2j+1}^{\prime}=Z_{2j+1}(\sigma(x))$.
On the other hand, note that

$Z_{2j}(\sigma(x))\cap Z_{2j}^{\prime}\supseteq Z_{2j+1}(\sigma(x))\cap
Z_{2j+1}^{\prime}=Z_{2j+1}^{\prime}\neq\emptyset$

for all $j\geq 0$. By uniqueness again, $Z_{2j}^{\prime}=Z_{2j}(\sigma(x))$.
We have thus concluded that $Z_{n}^{\prime}=Z_{n}(\sigma(x))$ for all $n\geq
0$, and so

$\sigma^{\prime}(h(x))=(e_{0}^{\prime},e_{1}^{\prime},e_{2}^{\prime},\dots)=h(\sigma(x)),$

as we wanted to show.

This concludes the proof of the theorem. ∎

##  4\. Homeomorphisms

If we restrict our attention to homeomorphisms $\varphi:X\rightarrow X$ of a
totally disconnected, compact metrizable space $X$, then by restricting the
correspondence from Theorem 3.39 we obtain a bijective correspondence between
such dynamical systems and a subfamily of the family LDiag, which we call
HDiag. In this section we describe such correspondence. We first define the
class HDiag.

######  Definition 4.1.

Let $(F,D)$ be a separated Bratteli diagram. We say that $(F,D)$ is an
$h$-diagram if the following hold:

  1. (a)

(Separation for the layers) For each $n\geq 0$ and $v\in F^{0,n}$, we have

$D_{v}=\\{B_{v},R_{v}\\}.$

The elements of $B_{v}$ will be called blue edges, and the elements of $R_{v}$
red edges.

  2. (b)

(Vertices) For each $n\geq 0$,

$F^{0,n+1}=\bigsqcup_{v\in F^{0,n}}s(B_{v})=\bigsqcup_{v\in F^{0,n}}s(R_{v}).$

Moreover, for any $v\in F^{0,n}$, we have $s(e)\neq s(e^{\prime})$ for all
distinct blue edges $e,e^{\prime}\in B_{v}$, and similarly $s(f)\neq
s(f^{\prime})$ for all distinct red edges $f,f^{\prime}\in R_{v}$.

  3. (c)

(Compatibility) For each $n\geq 0$ and $v\in F^{0,n}$, we have

$\bigsqcup_{e\in B_{v}}s(B_{s(e)})=\bigsqcup_{f\in R_{v}}s(R_{s(f)}).$

The family of $h$-diagrams will be denoted by HDiag.

######  Remarks 4.2.

  1. (1)

Condition (b) in Definition 4.1 implies that, for each $n\geq 1$ and $v\in
F^{0,n}$, there are exactly two different edges $e\in B_{r(e)},f\in R_{r(f)}$
such that $s(e)=s(f)=v$.

  2. (2)

The disjointness of both unions in condition (c) of Definition 4.1 is
automatic, due to condition (b).

The class HDiag is a subclass of the diagrams considered in Section 3. To
prove this, we need the following technical lemma, which enables us to build
rombs at any layer of an $h$-diagram $(F,D)$.

######  Lemma 4.3.

Let $v\in F^{0,n}$. Given a pair of blue edges $(e_{0},e_{1})$ such that
$r(e_{0})=v$ and $r(e_{1})=s(e_{0})$, there is a unique pair $(f_{0},f_{1})$
of red edges such that $r(f_{0})=v$, $r(f_{1})=s(f_{0})$ and
$s(f_{1})=s(e_{1})$.

Conversely, given a pair of red edges $(f_{0},f_{1})$ such that $r(f_{0})=v$
and $r(f_{1})=s(f_{0})$, there is a unique pair $(e_{0},e_{1})$ of blue edges
such that $r(e_{0})=v$, $r(e_{1})=s(e_{0})$ and $s(e_{1})=s(f_{1})$.

###### Proof.

By condition (c) in Definition 4.1 we have

$s(e_{1})\in\bigsqcup_{e\in B_{v}}s(B_{s(e)})=\bigsqcup_{f\in
R_{v}}s(R_{s(f)}),$

so there exist a pair of red edges $(f_{0},f_{1})$ such that
$s(e_{1})=s(f_{1})$, being $f_{1}\in R_{s(f_{0})}$ and $f_{0}\in R_{v}$. The
pair $(f_{0},f_{1})$ satisfies the required properties. To prove uniqueness,
assume that there exists another pair $(f^{\prime}_{0},f^{\prime}_{1})$ of red
edges such that $r(f^{\prime}_{0})=v$, $s(f^{\prime}_{0})=r(f^{\prime}_{1})$
and $s(f^{\prime}_{1})=s(e_{1})=s(f_{1})$. Condition (b) in Definition 4.1
ensures that $f^{\prime}_{1}=f_{1}$; in particular
$s(f^{\prime}_{0})=r(f^{\prime}_{1})=r(f_{1})=s(f_{0})$. Applying condition
(b) again gives $f^{\prime}_{0}=f_{0}$. That is, the pair $(f_{0},f_{1})$ is
unique.

The same arguments apply if one starts with a pair of red edges to conclude
existence and uniqueness of a pair of blue edges satisfying the desired
properties. ∎

######  Proposition 4.4.

Every $h$-diagram is an $l$-diagram.

###### Proof.

Indeed, if $(F,D)$ is an $h$-diagram, it is also an $l$-diagram of a special
kind, namely the separations for the odd layers consist only of two colors. We
need to check conditions (a)-(g) of Definition 3.2.

(a) The separations for the even layers are what they should be by definition.

(b) By Remark 4.2 (1), for each $j\geq 0$ and $v\in F^{0,2j+1}$ there is
exactly one red edge $f_{v}\in R_{r(f_{v})}$ with source $v$. Therefore here
$R(f_{v})=R_{v}$, and we have the separation

$D_{v}=\\{B_{v},R_{v}\\}=\\{B_{v},R(f_{v})\\}.$

(c) Follows directly from condition (b) in Definition 4.1.

(d) Parts (1) and (2) follow directly from condition (b) in Definition 4.1.
Also, the second part follows from the disjointness of the sets in the same
condition (b) and from the second part of that same condition.

(e) Follows directly from condition (c) in Definition 4.1.

(f)-(g) They follow from Lemma 4.3 above. ∎

Note that the equivalence relation $\sim_{l}$ in LDiag restricts to an
equivalence relation $\sim_{h}$ in HDiag. More concretely, the contraction of
an $h$-diagram is again an $h$-diagram.

######  Lemma 4.5.

The contraction $(F^{\prime},D^{\prime})$ of an $h$-diagram, as defined in
Definition 3.10, is an $h$-diagram.

###### Proof.

First observe that if $(F,D)$ is an $h$-diagram, then the contraction
$(F^{\prime},D^{\prime})$ with respect to a sequence $(m_{n})_{n\geq 0}$ in
${\mathbb{N}}\cup\\{0\\}$ with $0\leq m_{0}<m_{1}<\cdots$, $m_{0}$ even and
satisfying (CR) is given by the following data.

  1. (i)

$(F^{\prime})^{0,n}=F^{0,m_{n}}$ for $n\geq 0$.

  2. (ii)

$(F^{\prime})^{1,n}=P_{m_{n},m_{n+1}}^{B}\cup P_{m_{n},m_{n+1}}^{R}$ for
$n\geq 0$, where for a given pair $j<l$ of non-negative integers, we have

$P_{j,l}^{B}=\\{(e_{j},\dots,e_{l-1})\in P_{j,l}\mid e_{i}\in
B_{r(e_{i})},j\leq i\leq l-1\\},$ $P_{j,l}^{R}=\\{(f_{j},\dots,f_{l-1})\in
P_{j,l}\mid f_{i}\in R_{r(e_{i})},j\leq i\leq l-1\\}.$

  3. (iii)

For $v\in(F^{\prime})^{0,n}$ with $n\geq 0$,
$D_{v}^{\prime}=\\{B_{v}^{\prime},R_{v}^{\prime}\\}$ being

$B_{v}^{\prime}=\\{\overline{e}\in P_{m_{n},m_{n+1}}^{B}\mid
r(\overline{e})=v\\},\quad R_{v}^{\prime}=\\{\overline{f}\in
P_{m_{n},m_{n+1}}^{R}\mid r(\overline{f})=v\\}.$

Now the proof of the lemma follows identical arguments as in the proof of
Lemma 3.11, so we omit it. ∎

We now describe the correspondence between homeomorphisms on a totally
disconnected, compact metrizable space and $h$-diagrams. For ease of notation,
we let Homeo be the family of dynamical systems $(X,\varphi)$ being $X$ a
totally disconnected, compact metrizable space, and $\varphi:X\rightarrow X$ a
homeomorphism. Note that Homeo is a subfamily of LHomeo.

######  Theorem 4.6.

Let $(X,\varphi)\in\emph{{Homeo}}$. Then the resulting $l$-diagram $(F,D)$
obtained from $(X,\varphi)$ by applying the construction 3.24 is in fact an
$h$-diagram.

###### Proof.

Take $\\{{\pazocal P}_{n}\\}_{n\geq 0}$ to be any $\varphi$-refined sequence
of partitions of $X$. We know that $(F,D)$ so obtained is an $l$-diagram by
Lemma 3.27. We recall its construction.

For $n\geq 0$ we have $F^{0,n}={\pazocal P}_{n}$, and $F^{1,n}=B^{(n)}\cup
R^{(n)}$, where the set $B^{(n)}=\bigsqcup_{Z\in F^{0,n}}B_{Z}$ consists of
blue edges, and $R^{(n)}=\bigsqcup_{Z\in F^{0,n}}R_{Z}$ consists of red edges.
More precisely, for $Z\in F^{0,n}$,

$B_{Z}=\\{e(Z^{\prime},Z)\mid Z^{\prime}\in{\pazocal P}_{n+1}\text{ such that
}Z^{\prime}\subseteq Z\\}$

and if $j\geq 0$, then

$R_{Z}=\\{f(Z^{\prime},Z)\mid Z^{\prime}\in{\pazocal P}_{2j+1}\text{ such that
}Z^{\prime}\subseteq\varphi(Z)\\}$

for $Z\in{\pazocal P}_{2j}$, and

$R_{Z^{\prime}}=\\{f(Z^{\prime\prime},Z^{\prime})\mid
Z^{\prime\prime}\in{\pazocal P}_{2j+2}\text{ such that
}Z^{\prime\prime}\subseteq\varphi^{-1}(Z^{\prime})\\}$

for $Z^{\prime}\in{\pazocal P}_{2j+1}$.

For $j\geq 0$, the separations are given by

$D_{Z}=\\{B_{Z},R_{Z}\\}$

in case $Z\in{\pazocal P}_{2j}$, and by

$D_{Z^{\prime}}=\\{B_{Z^{\prime}}\\}\cup\\{R(f)\mid s(f)=Z^{\prime},f\in
R_{r(f)}\\}$

in case $Z^{\prime}\in{\pazocal P}_{2j+1}$.

Observe that, since $\varphi$ is a homeomorphism,

${\pazocal P}_{n}^{\varphi}=\\{\varphi(Z)\mid Z\in{\pazocal P}_{n}\\}.$

Hence for $Z\in{\pazocal P}_{2j},Z^{\prime}\in{\pazocal P}_{2j+1}$, we have
$Z^{\prime}\subseteq\varphi(Z)$ if and only if
$\varphi^{-1}(Z^{\prime})\subseteq Z$. This means precisely that there is
exactly one red edge with source $Z^{\prime}$, namely $f=f(Z^{\prime},Z)\in
R_{Z}$. Hence

$R_{Z^{\prime}}=R(f)$

and so the separation at $Z^{\prime}$ becomes

$D_{Z^{\prime}}=\\{B_{Z^{\prime}},R_{Z^{\prime}}\\}.$

This shows condition (a) in Definition 4.1. The first part of condition (b) in
Definition 4.1 follows from condition (c) in Definition 3.2 together with
Proposition 3.3 (i) (note that the disjointness of the second union in
Proposition 3.3 (i) follows in this case by the observation in the above
paragraph). The second part of condition (b) follows by the definition of the
edges in $(F,D)$, and by the observation in the paragraph above.

For condition (c) in Definition 4.1, we note that by Remark 4.2 (2) it is
enough to show that

$\bigcup_{e\in B_{Z}}s(B_{s(e)})=\bigcup_{f\in R_{Z}}s(R_{s(f)})$

for each $n\geq 0$ and $Z\in{\pazocal P}_{n}$. If $n$ is even, this follows
directly from condition (e) of Definition 3.2, so we may assume that
$n=2j+1,j\geq 0$.

Let $Z^{\prime\prime}\in s(B_{s(e)})$ for some $e\in B_{Z}$. Since ${\pazocal
P}_{2j+2}^{\varphi}=\varphi({\pazocal P}_{2j+2})$ is a partition which is
coarser than ${\pazocal P}_{2j+3}$, we can find $Z^{\prime}\in{\pazocal
P}_{2j+2}$ such that $Z^{\prime\prime}\subseteq\varphi(Z^{\prime})$. This says
that $Z^{\prime\prime}\in s(R_{Z^{\prime}})$. But $\varphi^{-1}({\pazocal
P}_{2j+1})$ is coarser than the partition ${\pazocal P}_{2j+2}$, so there
exists $Z\in{\pazocal P}_{2j+1}$ such that
$Z^{\prime}\subseteq\varphi^{-1}(Z)$. Hence $Z^{\prime\prime}=s(f^{\prime})$
with $f^{\prime}=f(Z^{\prime\prime},Z^{\prime})\in R_{s(f)}$, where
$f=f(Z^{\prime},Z)$. This proves one inclusion; the other is analogous and we
will omit it.

This concludes the proof. ∎

######  Theorem 4.7.

Let $(F,D)\in\emph{{HDiag}}$. Then the resulting dynamical system
$(X,\varphi)\in\emph{{LHomeo}}$ obtained from $(F,D)$ by applying the
construction 3.34 belongs in fact to Homeo.

###### Proof.

We recall the construction of $(X,\varphi)$. The space $X$ is defined as the
set of infinite blue paths

$X=\\{(e_{0},e_{1},e_{2},\dots)\mid e_{j}\in
B_{r(e_{j})},r(e_{j+1})=s(e_{j})\text{ for all }j\geq 0\\}$

endowed with a topology having as basis the cylinder sets
$\\{Z(e_{0},\dots,e_{n})\mid(e_{0},\dots,e_{n})\in P_{0,n}^{B},n\geq 0\\}$,
which become clopen sets. The map $\varphi:X\rightarrow X$ is defined as
follows.

Let $(e_{0},e_{1},e_{2},\dots)\in X$. For each $j\geq 0$ we complete the pair
of blue edges $(e_{2j},e_{2j+1})$ to a romb
$(e_{2j},e_{2j+1},f_{2j},f_{2j+1})$. Now, for each $j\geq 1$, we consider the
red pair $(f_{2j-1},f_{2j})$ which satisfies $s(f_{2j-1})=r(f_{2j})$, so we
can complete it to another romb
$(e_{2j-1}^{\prime},e_{2j}^{\prime},f_{2j-1},f_{2j})$. Finally, take
$e_{0}^{\prime}$ to be the unique blue edge in $F^{1,0}$ such that
$s(e_{0}^{\prime})=r(e_{1}^{\prime})$. Then

$\varphi(e_{0},e_{1},e_{2},\dots)=(e_{0}^{\prime},e_{1}^{\prime},e_{2}^{\prime},\dots).$

We already know (Lemmas 3.33 and 3.34) that $X$ is a totally disconnected,
compact metrizable space, and that $\varphi$ is a surjective local
homeomorphism. In this particular case, we can explicitly construct the
inverse of $\varphi$ by reversing the above process. Let us detail its
construction.

Take $(e_{0}^{\prime},e_{1}^{\prime},e_{2}^{\prime},\dots)\in X$. By Remarks
4.2 there exists a unique red edge $f_{0}\in R_{r(f_{0})}$ such that
$s(f_{0})=s(e_{0}^{\prime})$. By recursively applying Lemma 4.3, we construct
a sequence of red edges $f_{1},f_{2},\dots$ such that, for all $j\geq 0$,
$f_{2j+1}\in R_{s(f_{2j})}$ and the quadruple
$(e_{2j+1}^{\prime},e_{2j+2}^{\prime},f_{2j+1},f_{2j+2})$ is a romb. We use
again Lemma 4.3 to build pairs of blue edges $(e_{2j},e_{2j+1})$ such that
$(e_{2j},e_{2j+1},f_{2j},f_{2j+1})$ is a romb, for all $j\geq 0$. Define

$\varphi^{-1}(e_{0}^{\prime},e_{1}^{\prime},e_{2}^{\prime},\dots)=(e_{0},e_{1},e_{2},\dots).$

An easy inspection shows that $\varphi^{-1}$ is the inverse of $\varphi$ (the
key part is the uniqueness property in Lemma 4.3). Since $\varphi$ was already
a local homeomorphism, it must be a homeomorphism.

This proves the theorem. ∎

Putting everything together, we obtain the following bijective correspondence
between homeomorphisms and $h$-diagrams.

######  Theorem 4.8.

The constructions 3.24 and 3.34 also establish a bijective correspondence
between equivalence classes of $h$-diagrams (via isomorphism and contraction)
and topological conjugacy classes of homeomorphisms on totally disconnected,
compact metrizable spaces.

###### Proof.

Note that, by Lemma 4.5, HDiag is a saturated class in LDiag, in the sense
that $(F^{\prime},D^{\prime})\sim_{l}(F,D)$ and $(F,D)\in\textsf{HDiag}$ imply
that $(F^{\prime},D^{\prime})\in\textsf{HDiag}$. Similarly, Homeo is a
saturated class in LHomeo with respect to $\sim_{t.c.}$. Hence the result
follows directly from Theorems 3.39, 4.6 and 4.7. ∎

######  Example 4.9.

The example discussed in Examples 3.23 1) and 3.28 1) falls under the umbrella
of this section, since in this case $\varphi$ is a homeomorphism. Therefore
the $l$-diagram obtained in Examples 3.28 1) gives in fact an example of an
$h$-diagram.

##  5\. Generalizing shifts of finite type

In this section we introduce the key concept of a generalized finite shift, a
class of dynamical systems that generalizes both one-sided and two-sided
shifts of finite type. We also show that any $(X,\sigma)\in\textsf{LHomeo}$ is
an inverse limit of generalized finite shifts.

###  5.1. A quick review on shifts of finite type

It is well-known that shifts of finite type are codified by finite graphs (see
e.g. [48]). We recall the construction here.

Let $E=(E^{0},E^{1},r_{E},s_{E})$ be a finite graph, with adjacency matrix
$A=A_{E}$. We will assume that $E^{0}=\\{1,\dots,n\\}$. Here the adjacency
matrix $A_{E}=(a_{ji})$ of $E$ is the square $n\times n$ matrix such that
$a_{ji}$ is the number of edges from vertex $j$ to vertex $i$. Then we may
build the associated space

$X_{E}=\\{(e_{0},e_{1},e_{2},\dots)\mid e_{i}\in E^{1}\text{ and
}s_{E}(e_{i})=r_{E}(e_{i+1})\text{ for all }i\geq 0\\}$

with the topology generated by the cylinder sets $Z(e_{0},\dots,e_{n})$. In
this space we may define the shift map

$\sigma_{E}\colon X_{E}\to X_{E}$

by $\sigma_{E}(e_{0},e_{1},e_{2},\dots)=(e_{1},e_{2},\dots)$. Note that
$X_{E}$ is a closed invariant subset of the compact space $\prod_{n\geq
0}E^{1}$, and that it is defined by the forbidden words $ef$ for all $e,f\in
E^{1}$ such that $s(e)\neq r(f)$, so that $(X_{E},\sigma_{E})$ is a one-sided
shift of finite type. An analogous definition gives the two-sided shift, which
is a homeomorphism.

Assuming that $E$ has neither sinks nor sources (so that the matrix $A_{E}$
has no zero-rows and no zero-columns), we have that $\sigma_{E}$ is a
surjective local homeomorphism.

For $i\in E^{0}$, write $V_{i}=\\{(e_{0},e_{1},e_{2},\dots)\in X_{E}\mid
r_{E}(e_{0})=i\\}$. Let also, for $i,j\in E^{0}$, $\\{f_{ji}^{(k)}\mid 1\leq
k\leq a_{ji}\\}$ be the set of edges with initial vertex $j$ and terminal
vertex $i$. Set

$V_{ji}^{(k)}=Z(f_{ji}^{(k)})=\\{(e_{0},e_{1},e_{2},\dots)\in X_{E}\mid
e_{0}=f_{ji}^{(k)}\\}.$

Then $V_{i}=\bigsqcup_{j=1}^{n}\bigsqcup_{k=1}^{a_{ji}}V_{ji}^{(k)}$, and
$\sigma_{E}(V_{ji}^{(k)})=V_{j}$ for all $i,j,k\in E^{0}$. Observe that the
restriction of $\sigma_{E}$ to each subset $V_{ji}^{(k)}$ is injective. It
turns out that all the dynamical behavior of the one-sided shift
$(X_{E},\sigma_{E})$ is completely determined by these decompositions.

###  5.2. Generalized finite shifts

We generalize the above situation as follows.

######  Definition 5.1.

Let $(X,\sigma)\in\textsf{LHomeo}$. We say that $(X,\sigma)$ admits a
$\sigma$-cylinder decomposition if there exist

  1. (I)

a finite set $\Gamma=\bigsqcup_{i=1}^{n}\Gamma_{i}$ where each
$\Gamma_{i}\neq\emptyset$;

  2. (II)

a finite collection $\\{Z_{\gamma}\mid\gamma\in\Gamma\\}$ of non-empty compact
open, mutually disjoint subsets $Z_{\gamma}\subseteq X$;

  3. (III)

non-negative integers $\\{a_{\gamma,i}\mid\gamma\in\Gamma,1\leq i\leq n\\}$;
and

  4. (IV)

a finite collection $\\{V_{\gamma,i}^{(k)}\mid\gamma\in\Gamma,1\leq i\leq
n,1\leq k\leq a_{\gamma,i}\\}$ of non-empty compact open sets;

such that the following properties hold:

  1. (a)

$X=\bigsqcup_{\gamma\in\Gamma}Z_{\gamma}$.

  2. (b)

For each $i=1,\dots,n$, if we write
$V_{i}:=\bigsqcup_{\gamma\in\Gamma_{i}}Z_{\gamma}$, then

$V_{i}=\bigsqcup_{\gamma\in\Gamma}\bigsqcup_{k=1}^{a_{\gamma,i}}V_{\gamma,i}^{(k)}.$

  3. (c)

$\sigma|_{V_{\gamma,i}^{(k)}}$ is injective and
$\sigma(V_{\gamma,i}^{(k)})=Z_{\gamma}$ for all allowable values of
$\gamma,i,k$.

The pair ${\pazocal Z}:=(\\{Z_{\gamma}\\},\\{V_{\gamma,i}^{(k)}\\})$ will be
called a $\sigma$-cylinder decomposition of $X$.

Some immediate consequences of Definition 5.1 are the following. Since
$V_{i}\neq\emptyset$ for $1\leq i\leq n$, we have that for each $i=1,\dots,n$
there exists $\gamma\in\Gamma$ such that $a_{\gamma,i}\neq 0$. Moreover, since
$\sigma$ is surjective, we have

$X=\sigma\Big{(}\bigcup_{i=1}^{n}V_{i}\Big{)}=\bigcup_{i=1}^{n}\bigcup_{\gamma\in\Gamma}\bigcup_{k=1}^{a_{\gamma,i}}\sigma(V_{\gamma,i}^{(k)})=\bigcup_{i=1}^{n}\bigcup_{\gamma\in\Gamma}\bigcup_{k=1}^{a_{\gamma,i}}Z_{\gamma}=\bigcup_{\gamma\in\Gamma}\Big{(}\bigcup_{i=1}^{n}\bigcup_{k=1}^{a_{\gamma,i}}Z_{\gamma}\Big{)}.$

Since also $X=\bigsqcup_{\gamma\in\Gamma}Z_{\gamma}$, we have conversely that
for each $\gamma\in\Gamma$ there exists $i\in\\{1,\dots,n\\}$ such that
$a_{\gamma,i}\neq 0$. These observations show that all rows and columns of the
matrix $A:=(a_{\gamma,i})\in M_{\Gamma\times n}({\mathbb{Z}}^{+})$ are non-
zero.

Let also $I\in M_{\Gamma\times n}({\mathbb{Z}}^{+})$ be the matrix with all
entries in $\\{0,1\\}$ such that in the $i^{th}$ column it has $1$’s exactly
at the positions that belong to $\Gamma_{i}$, for $1\leq i\leq n$. The pair
$(A,I)$ will be called the pair of red-blue adjacency matrices, respectively,
of a prescribed $\sigma$-cylinder decomposition ${\pazocal Z}$ of $X$.

We consider also a notion of refined $\sigma$-cylinder decompositions of $X$.
Basically this amounts to have refinements of the two partitions
$\\{Z_{\gamma}\\}$ and $\\{V_{\gamma,i}^{(k)}\\}$ satisfying a compatibility
condition, as follows.

######  Definition 5.2.

Suppose $(X,\sigma)\in\textsf{LHomeo}$. Let ${\pazocal
Z}=(\\{Z_{\gamma}\\},\\{V_{\gamma,i}^{(k)}\\}),{\pazocal
T}=(\\{T_{\lambda}\\},\\{W_{\lambda,j}^{(l)}\\})$ be two $\sigma$-cylinder
decompositions; in more detail, suppose that

$X=\bigsqcup_{\gamma\in\Gamma}Z_{\gamma}=\bigsqcup_{\lambda\in\Lambda}T_{\lambda},\qquad\Gamma=\bigsqcup_{i=1}^{n}\Gamma_{i},\qquad\Lambda=\bigsqcup_{j=1}^{m}\Lambda_{j},$

and that, with $V_{i}=\bigsqcup_{\gamma\in\Gamma_{i}}Z_{\gamma}$ and
$W_{j}=\bigsqcup_{\lambda\in\Lambda_{j}}T_{\lambda}$, we have

$V_{i}=\bigsqcup_{\gamma\in\Gamma}\bigsqcup_{k=1}^{a_{\gamma,i}}V_{\gamma,i}^{(k)}\quad\text{
and }\quad
W_{j}=\bigsqcup_{\lambda\in\Lambda}\bigsqcup_{l=1}^{b_{\lambda,j}}W_{\lambda,j}^{(l)},$

such that $\sigma|_{V_{\gamma,i}^{(k)}}$ is injective and
$\sigma(V_{\gamma,i}^{(k)})=Z_{\gamma}$, and $\sigma|_{W_{\lambda,j}^{(l)}}$
is injective and $\sigma(W_{\lambda,j}^{(l)})=T_{\lambda}$ for all allowable
values of $\gamma,i,k$ and $\lambda,j,l$ respectively.

Then we say that ${\pazocal T}$ refines ${\pazocal Z}$ if:

  1. (a)

The partition $\\{T_{\lambda}\\}$ is a refinement of $\\{Z_{\gamma}\\}$, that
is, we can write

$Z_{\gamma}=\bigsqcup_{\lambda\in\Lambda(\gamma)}T_{\lambda},\quad\text{ with
}\Lambda=\bigsqcup_{\gamma\in\Gamma}\Lambda(\gamma).$

  2. (b)

The partition $\\{W_{\lambda,j}^{(l)}\\}$ is a refinement of
$\\{V_{\gamma,i}^{(k)}\\}$, that is, with

$\mathfrak{A}=\\{(\gamma,i,k)\mid\gamma\in\Gamma,1\leq i\leq n,1\leq k\leq
a_{\gamma,i}\\},\quad\mathfrak{B}=\\{(\lambda,j,l)\mid\lambda\in\Lambda,1\leq
j\leq m,1\leq l\leq b_{\lambda,j}\\},$

we have

$V_{\gamma,i}^{(k)}=\bigsqcup_{\eta\in\mathfrak{B}(\gamma,i,k)}W(\eta),\quad\text{
with }\mathfrak{B}=\bigsqcup_{\nu\in\mathfrak{A}}\mathfrak{B}(\nu),$

where $W(\eta)=W_{\lambda,j}^{(l)}$ whenever
$\eta=(\lambda,j,l)\in\mathfrak{B}$.

  3. (c)

The following compatibility condition is satisfied: let
$\pi_{\Lambda}\colon\mathfrak{B}\to\Lambda$ be the projection onto the first
component. Then for each $(\gamma,i,k)\in\mathfrak{A}$, $\pi_{\Lambda}$
restricts to a bijection from $\mathfrak{B}(\gamma,i,k)$ onto
$\Lambda(\gamma)$.

######  Remark 5.3.

To see the meaning of the compatibility condition in the above definition, let
us apply $\sigma$ to the decomposition
$V_{\gamma,i}^{(k)}=\bigsqcup_{\eta\in\mathfrak{B}(\gamma,i,k)}W(\eta)$ for a
given $(\gamma,i,k)\in\mathfrak{A}$. Since $\sigma|_{V_{\gamma,i}^{(k)}}$ is
injective, we have

$Z_{\gamma}=\sigma(V_{\gamma,i}^{(k)})=\bigsqcup_{\eta\in\mathfrak{B}(\gamma,i,k)}\sigma(W(\eta))=\bigsqcup_{\eta\in\mathfrak{B}(\gamma,i,k)}T_{\pi_{\Lambda}(\eta)}.$

Hence the compatibility condition (c) says that this decomposition agrees with
the decomposition of $Z_{\gamma}$ from (a).

We immediately observe transitivity of refinement, which we prove in the next
lemma.

######  Lemma 5.4.

Let ${\pazocal Z},{\pazocal T}$ and ${\pazocal S}$ be three $\sigma$-cylinder
decompositions for $(X,\sigma)$. If ${\pazocal S}$ refines ${\pazocal T}$ and
${\pazocal T}$ refines ${\pazocal Z}$, then ${\pazocal S}$ refines ${\pazocal
Z}$.

###### Proof.

Write ${\pazocal Z}=(\\{Z_{\gamma}\\},\\{V_{\gamma,i}^{(k)}\\}),{\pazocal
T}=(\\{T_{\lambda}\\},\\{W_{\lambda,j}^{(l)}\\})$ and ${\pazocal
S}=(\\{S_{\delta}\\},\\{U_{\delta,r}^{(s)}\\})$. In more detail,

$X=\bigsqcup_{\gamma\in\Gamma}Z_{\gamma}=\bigsqcup_{\lambda\in\Lambda}T_{\lambda}=\bigsqcup_{\delta\in\Delta}S_{\delta},$

where
$\Gamma=\bigsqcup_{i=1}^{n}\Gamma_{i},\Lambda=\bigsqcup_{j=1}^{m}\Lambda_{j}$
and $\Delta=\bigsqcup_{r=1}^{o}\Delta_{r}$, and that, with
$V_{i}=\bigsqcup_{\gamma\in\Gamma_{i}}Z_{\gamma},W_{j}=\bigsqcup_{\lambda\in\Lambda_{j}}T_{\lambda}$
and $U_{r}=\bigsqcup_{\delta\in\Delta_{r}}S_{\delta}$, we have

$V_{i}=\bigsqcup_{\gamma\in\Gamma}\bigsqcup_{k=1}^{a_{\gamma,i}}V_{\gamma,i}^{(k)},\quad
W_{j}=\bigsqcup_{\lambda\in\Lambda}\bigsqcup_{l=1}^{b_{\lambda,j}}W_{\lambda,j}^{(l)}\quad\text{
and }\quad
U_{r}=\bigsqcup_{\delta\in\Delta}\bigsqcup_{s=1}^{c_{\delta,r}}U_{\delta,r}^{(s)},$

such that $\sigma|_{V_{\gamma,i}^{(k)}}$ is injective and
$\sigma(V_{\gamma,i}^{(k)})=Z_{\gamma}$, $\sigma|_{W_{\lambda,j}^{(l)}}$ is
injective and $\sigma(W_{\lambda,j}^{(l)})=T_{\lambda}$, and
$\sigma|_{U_{\delta,r}^{(s)}}$ is injective and
$\sigma(U_{\delta,r}^{(s)})=S_{\delta}$ for all allowable values of
$\gamma,i,k$, of $\lambda,j,l$ and of $\delta,r,s$ respectively.

By hypothesis, we can write

$Z_{\gamma}=\bigsqcup_{\lambda\in\Lambda(\gamma)}T_{\lambda},\quad\text{ with
}\Lambda=\bigsqcup_{\gamma\in\Gamma}\Lambda(\gamma),$

and

$T_{\lambda}=\bigsqcup_{\delta\in\Delta(\lambda)}S_{\delta},\quad\text{ with
}\Delta=\bigsqcup_{\lambda\in\Lambda}\Delta(\lambda).$

Then if we write
$\Delta(\gamma):=\bigsqcup_{\lambda\in\Lambda(\gamma)}\Delta(\lambda)$ for
$\gamma\in\Gamma$, we obtain that
$\Delta=\bigsqcup_{\gamma\in\Gamma}\Delta(\gamma)$ and

$Z_{\gamma}=\bigsqcup_{\lambda\in\Lambda(\gamma)}\bigsqcup_{\delta\in\Delta(\lambda)}S_{\delta}=\bigsqcup_{\delta\in\Delta(\gamma)}S_{\delta}.$

This proves part (a) of Definition 5.2. Part (b) is proved similarly, as
follows. By hypothesis, if

$\mathfrak{A}=\\{(\gamma,i,k)\mid\gamma\in\Gamma,1\leq i\leq n,1\leq k\leq
a_{\gamma,i}\\},$ $\mathfrak{B}=\\{(\lambda,j,l)\mid\lambda\in\Lambda,1\leq
j\leq m,1\leq l\leq b_{\lambda,j}\\},$
$\mathfrak{C}=\\{(\delta,r,s)\mid\delta\in\Delta,1\leq r\leq o,1\leq s\leq
c_{\delta,r}\\},$

we have

$V_{\gamma,i}^{(k)}=\bigsqcup_{\eta\in\mathfrak{B}(\gamma,i,k)}W(\eta),\quad\text{
with }\mathfrak{B}=\bigsqcup_{\nu\in\mathfrak{A}}\mathfrak{B}(\nu),$

where $W(\eta)=W_{\lambda,j}^{(l)}$ whenever
$\eta=(\lambda,j,l)\in\mathfrak{B}$, and

$W_{\lambda,j}^{(l)}=\bigsqcup_{\rho\in\mathfrak{C}(\lambda,j,l)}U(\rho),\quad\text{
with }\mathfrak{C}=\bigsqcup_{\eta\in\mathfrak{B}}\mathfrak{C}(\eta),$

where $U(\rho)=U_{\delta,r}^{(s)}$ whenever
$\rho=(\delta,r,s)\in\mathfrak{C}$. If we define, for $\nu\in\mathfrak{A}$,
the set
$\mathfrak{C}(\nu):=\bigsqcup_{\eta\in\mathfrak{B}(\nu)}\mathfrak{C}(\eta)$,
then $\mathfrak{C}=\bigsqcup_{\nu\in\mathfrak{A}}\mathfrak{C}(\nu)$ and

$V_{\gamma,i}^{(k)}=\bigsqcup_{\eta\in\mathfrak{B}(\gamma,i,k)}\bigsqcup_{\rho\in\mathfrak{C}(\eta)}U(\rho)=\bigsqcup_{\rho\in\mathfrak{C}(\gamma,i,k)}U(\rho).$

This shows part (b), as promised. Finally it remains to prove part (c).
Consider $\pi_{\Lambda}:\mathfrak{B}\rightarrow\Lambda$ and
$\pi_{\Delta}:\mathfrak{C}\rightarrow\Delta$ the projections onto the
respective first components. By hypothesis, for each
$(\gamma,i,k)\in\mathfrak{A}$, the restriction of $\pi_{\Lambda}$ on
$\mathfrak{B}(\gamma,i,k)$ gives a bijection
$\mathfrak{B}(\gamma,i,k)\simeq\Lambda(\gamma)$ given by
$(\lambda,j,l)\mapsto\lambda$. Also, for each $(\lambda,j,l)\in\mathfrak{B}$,
the restriction of $\pi_{\Delta}$ on $\mathfrak{C}(\lambda,j,l)$ gives a
bijection $\mathfrak{C}(\lambda,j,l)\simeq\Delta(\lambda)$ given by
$(\delta,r,s)\mapsto\delta$.

So, for a fixed $(\gamma,i,k)$, we have

$\mathfrak{C}(\gamma,i,k)=\bigsqcup_{(\lambda,j,l)\in\mathfrak{B}(\gamma,i,k)}\mathfrak{C}(\lambda,j,l)$

which, under the above bijections, translates to a bijection

$\mathfrak{C}(\gamma,i,k)\simeq\bigsqcup_{\lambda\in\Lambda(\gamma)}\Delta(\lambda)=\Delta(\gamma)$

under the restriction of $\pi_{\Delta}$ on $\mathfrak{C}(\gamma,i,k)$. This
concludes the proof of the lemma. ∎

We build a finite bipartite separated graph $(E,C):=(E({\pazocal
Z}),C({\pazocal Z}))$ out of a $\sigma$-cylinder decomposition ${\pazocal Z}$
of $(X,\sigma)$ as follows. Let $E^{0}=E^{0,0}\sqcup E^{0,1}$, where
$E^{0,0}=\\{v^{1},\dots,v^{n}\\}$ and $E^{0,1}=\Gamma$. Now we set, for each
$1\leq i\leq n$,

$C_{i}:=C_{v^{i}}=\\{B_{i},R_{i}\\}$

where $B_{i}=\\{e_{\gamma,i}\mid\gamma\in\Gamma_{i}\\}$, with
$s(e_{\gamma,i})=\gamma$ and $r(e_{\gamma,i})=v^{i}$; and
$R_{i}=\\{f_{\gamma,i}^{(k)}\mid\gamma\in\Gamma,1\leq k\leq a_{\gamma,i}\\}$,
where $s(f_{\gamma,i}^{(k)})=\gamma$ and $r(f_{\gamma,i}^{(k)})=v^{i}$. Then
we have that $(E,C)$ is a finite bipartite separated graph such that there are
exactly two colors at each vertex in $E^{0,0}$, $s(e)\neq s(e^{\prime})$ for
all distinct $e\in B_{i}$ and $e^{\prime}\in B_{i^{\prime}}$, and

$E^{0,1}=\bigsqcup_{i=1}^{n}s(B_{i})=\bigcup_{i=1}^{n}s(R_{i}).$

Therefore we see that $(E,C)$ satisfies all the conditions needed to be the
first layer of an $l$-diagram.

Conversely, given an $l$-diagram $(F,D)$, we can build a $\sigma$-cylinder
decomposition for its corresponding space $(X,\sigma)\in\textsf{LHomeo}$, such
that the first layer of $(F,D)$ corresponds to the above construction applied
to this $\sigma$-cylinder decomposition. Set
$F^{0,0}=\\{v^{1},\dots,v^{n}\\}$, and then
$\Gamma=\bigsqcup_{i=1}^{n}\Gamma_{i}$, where $\Gamma_{i}=B_{i}:=B_{v^{i}}$.
Note that $\Gamma$ can be identified with the set $F^{0,1}$ by means of the
correspondence $e\leftrightarrow s(e)$. We have

$X=\bigsqcup_{e\in\Gamma}Z(e)\quad\text{ and }\quad
V_{i}:=Z(v^{i})=\bigsqcup_{e\in\Gamma_{i}}Z(e).$

It remains to build the sets $V_{e,i}^{(k)}$. For $e\in\Gamma$ and
$i\in\\{1,...,n\\}$, let $a_{e,i}$ be the number of red edges from $s(e)$ to
$v^{i}$. The set of red edges from $s(e)$ to $v^{i}$ can then be labeled as
$\\{f_{e,i}^{(k)}\mid 1\leq k\leq a_{e,i}\\}$. Fix a red edge $f_{e,i}^{(k)}$.
Then $s(f_{e,i}^{(k)})=s(e)$ and $r(f_{e,i}^{(k)})=v^{i}$. By the proof of
Lemma 3.35, for each $y\in Z(e)$ there exists a unique
$x=(e_{0},e_{1},e_{2},\dots)\in Z(v^{i})=V_{i}$ such that $\sigma(x)=y$ and
the first component $f_{0}$ of the romb $(e_{0},e_{1},f_{0},f_{1})$ determined
by $(e_{0},e_{1})$ is precisely $f_{e,i}^{(k)}$. We define $V_{e,i}^{(k)}$ to
be the set of all these pre-images $x\in Z(v^{i})$ of elements of $Z(e)$.
Observe that by definition we have that $\sigma|_{V_{e,i}^{(k)}}$ is
injective, and that $\sigma(V_{e,i}^{(k)})=Z(e)$. Moreover, we get

$V_{i}=Z(v^{i})=\bigsqcup_{e\in\Gamma}\bigsqcup_{k=1}^{a_{e,i}}V_{e,i}^{(k)}.$

Therefore the decompositions

$X=\bigsqcup_{e\in\Gamma}Z(e),\quad V_{i}=\bigsqcup_{e\in\Gamma_{i}}Z(e),\quad
V_{i}=\bigsqcup_{e\in\Gamma}\bigsqcup_{k=1}^{a_{e,i}}V_{e,i}^{(k)}$

satisfy all the necessary conditions to form a $\sigma$-cylinder decomposition
${\pazocal Z}_{0}(F,D)=(\\{Z(e)\\},\\{V_{e,i}^{(k)}\\})$ of $X$, and the
bipartite separated graph obtained from these decompositions as explained
above coincides with the first layer of $(F,D)$.

As a consequence, we have the following result.

######  Proposition 5.5.

An object $(X,\sigma)\in\emph{{LHomeo}}$ always admits a $\sigma$-cylinder
decomposition.

###### Proof.

This is immediate due to the above construction and Theorem 3.39. ∎

By the above construction, it seems that first layers of $l$-diagrams play a
special role. They deserve a name on its own right.

######  Definition 5.6.

We say that a finite bipartite separated graph $(E,C)$ is a generalized finite
shift graph if it satisfies the following properties:

  1. (a)

$C_{v}=\\{B_{v},R_{v}\\}$ for each $v\in E^{0,0}$, where $B_{v},R_{v}$ are
non-empty;

  2. (b)

$(1)\,\,E^{0,1}=\bigsqcup_{v\in E^{0,0}}s(B_{v}),\quad\text{ and also
}\quad(2)\,\,E^{0,1}=\bigcup_{v\in E^{0,0}}s(R_{v}).$

Moreover, $s(e)\neq s(e^{\prime})$ for all distinct edges $e,e^{\prime}\in
B_{v}$, for all $v\in E^{0,0}$.

We call the elements of $B:=\bigsqcup_{v\in E^{0,0}}B_{v}$ blue edges, and the
elements of $R:=\bigsqcup_{v\in E^{0,0}}R_{v}$ red edges. The red adjacency
matrix of the generalized finite shift graph $(E,C)$ is the matrix
$A=(a_{w,v})\in M_{E^{0,1}\times E^{0,0}}({\mathbb{Z}}^{+})$ such that
$a_{w,v}$ is the number of red edges from $w$ to $v$. We say that $(E,C)$ is
refined if $A$ is a $\\{0,1\\}$ matrix. We also consider the blue adjacency
matrix $I\in M_{E^{0,1}\times E^{0,0}}(\\{0,1\\})$, with

$I_{w,v}=\begin{cases}1&\text{ if }w=s(e)\text{ for (a unique) }e\in B_{v},\\\
0&\text{ otherwise}.\end{cases}$

The pair of matrices $(A,I)$ characterizes the graph $(E,C)$ completely.

It follows from the definition of $l$-diagram (Definition 3.2) that first
layers of $l$-diagrams can be characterized by the pair $(A,I)$ as defined
above.

######  Example 5.7.

The finite bipartite separated graph $(E,C)$ defined by $E^{0}=E^{0,0}\sqcup
E^{0,1}$ where

$E^{0,0}=\\{v^{1},v^{2},v^{3},v^{4}\\},\quad
E^{0,1}=\\{w^{1},w^{2},w^{3},w^{4},w^{5},w^{6}\\},$

and

$B_{v^{1}}=\\{e_{1,1},e_{2,1}\\},\quad B_{v^{2}}=\\{e_{3,2}\\},\quad
B_{v^{3}}=\\{e_{4,3}\\},\quad B_{v^{4}}=\\{e_{5,4},e_{6,4}\\},$
$R_{v^{1}}=\\{f_{1,1}\\},\quad
R_{v^{2}}=\\{f_{1,2},f_{2,2},f_{3,2}^{(1)},f_{3,2}^{(2)}\\},\quad
R_{v^{3}}=\\{f_{1,3},f_{4,3}^{(1)},f_{4,3}^{(2)},f_{5,3},f_{6,3}\\},\quad
R_{v^{4}}=\\{f_{6,4}\\},$

with $s(e_{j,i})=s(f_{j,i}^{(k)})=w^{j}$ and
$r(e_{j,i})=r(f_{j,i}^{(k)})=v^{i}$, is a generalized finite shift graph. The
graph looks as follows.

$w^{1}$$w^{2}$$w^{3}$$w^{4}$$w^{5}$$w^{6}$$v^{1}$$v^{2}$$v^{3}$$v^{4}$$E^{0,0}$$E^{1}$$E^{0,1}$
Figure 6. The generalized finite shift $(E,C)$.

Here the red-blue adjacency matrices are

$A=\begin{pmatrix}1&1&1&0\\\ 0&1&0&0\\\ 0&2&0&0\\\ 0&0&2&0\\\ 0&0&1&0\\\
0&0&1&1\end{pmatrix},\quad I=\begin{pmatrix}1&0&0&0\\\ 1&0&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\ 0&0&0&1\\\ 0&0&0&1\end{pmatrix}.$

We can now state the following result that summarizes and slightly generalizes
our previous discussion.

######  Proposition 5.8.

Let $(F,D)$ be an $l$-diagram and let $(X,\sigma)$ be the associated
surjective local homeomorphism, as defined in Subsection 3.2. Then the first
layer of $(F,D)$ is a generalized finite shift graph, and it gives rise to a
$\sigma$-cylinder decomposition for $X$.

More generally, the layers corresponding to even vertices $(F^{0,2j}\cup
F^{0,2j+1},F^{1,2j})$, $j\geq 0$, in the $l$-diagram $(F,D)$ are generalized
finite shift graphs, which are refined if $j>0$, and give rise to
$\sigma$-cylinder decompositions for $X$. For each $j\geq 0$, the pair of red-
blue adjacency matrices associated to this $\sigma$-cylinder decomposition
agrees with the pair $(A,I)$ of red and blue adjacency matrices of the
generalized finite shift graph $(F^{0,2j}\cup F^{0,2j+1},F^{1,2j})$. Moreover,
for $j^{\prime}>j$, the $\sigma$-cylinder decomposition corresponding to the
level $2j^{\prime}$ refines the $\sigma$-cylinder decomposition corresponding
to the level $2j$.

###### Proof.

By definition, it is clear that the even layers $(F^{0,2j}\cup
F^{0,2j+1},F^{1,2j})$, $j\geq 0$, are generalized finite shift graphs, which
are refined for $j>0$ by Remark 3.5 (5). Let us now describe the associated
$\sigma$-cylinder decompositions.

We have already defined the $\sigma$-cylinder decomposition for the first
layer $(F^{0,0}\cup F^{0,1},F^{1,0})$, namely ${\pazocal Z}_{0}={\pazocal
Z}_{0}(F,D)=(\\{Z(e)\\},\\{V_{e,i}^{(k)}\\})$. We now define the
$\sigma$-cylinder decomposition ${\pazocal Z}_{2j}={\pazocal Z}_{2j}(F,D)$ for
an even layer $(F^{0,2j}\cup F^{0,2j+1},F^{1,2j})$, where $j\geq 1$.

We take as a finite set for the $2j^{\text{th}}$ layer the set

$\Gamma^{2j}=P^{B}_{0,2j+1}=\bigsqcup_{\overline{e}\in
P^{B}_{0,2j}}\Gamma^{2j}_{\overline{e}},\quad\text{ with
}\quad\Gamma^{2j}_{\overline{e}}=\\{(\overline{e},e)\in P^{B}_{0,2j+1}\mid
e\in B_{s(\overline{e})}\\}.$

Now we have $X=\bigsqcup_{(\overline{e},e)\in\Gamma^{2j}}Z(\overline{e},e)$,
and we set

$V_{\overline{e}}:=\bigsqcup_{(\overline{e},e)\in\Gamma^{2j}_{\overline{e}}}Z(\overline{e},e)=\bigsqcup_{e\in
B_{s(\overline{e})}}Z(\overline{e},e)=Z(\overline{e}).$

For the construction of the remaining sets, we proceed as follows. First,
recalling that the set of vertices $F^{0,n}$ can be identified with
$P^{B}_{0,n}$ for every $n\geq 0$, we will label the red adjacency matrix of
$(F^{0,2j}\cup F^{0,2j+1},F^{1,2j})$ by the set $\Gamma^{2j}\times
P^{B}_{0,2j}$, so that, for $(\overline{e},e)\in\Gamma^{2j}$ and
$\overline{e}^{\prime}\in P^{B}_{0,2j}$,
$a_{(\overline{e},e),\overline{e}^{\prime}}=1$ if and only if there is a red
edge from $s(e)$ to $s(\overline{e}^{\prime})$, and
$a_{(\overline{e},e),\overline{e}^{\prime}}=0$ otherwise (recall from Remark
3.5 (5) that, since $j\geq 1$, the red adjacency matrix of $(F^{0,2j}\cup
F^{0,2j+1},F^{1,2j})$ is a binary matrix).

Fix $(\overline{e},e)\in\Gamma^{2j}$ and $\overline{e}^{\prime}\in
P^{B}_{0,2j}$, with $\overline{e}=(e_{0},e_{1},\dots,e_{2j-2},e_{2j-1})$, and
$\overline{e}^{\prime}=(e_{0}^{\prime},e_{1}^{\prime},\dots,e_{2j-2}^{\prime},e_{2j-1}^{\prime})$.
We now check that $a_{(\overline{e},e),\overline{e}^{\prime}}=1$ if and only
if there exists $x\in Z(\overline{e}^{\prime})$ such that $\sigma(x)\in
Z(\overline{e},e)$. Suppose first that there is some $x$ with the stated
condition. Then we can write

$x=(e_{0}^{\prime},e_{1}^{\prime},\dots,e_{2j-2}^{\prime},e_{2j-1}^{\prime},e_{2j}^{\prime},e_{2j+1}^{\prime},\dots)$

for some $e_{2j}^{\prime},e_{2j+1}^{\prime},\dots$ blue edges. For each $k\geq
0$, let $(e_{2k}^{\prime},e_{2k+1}^{\prime},f_{2k},f_{2k+1})$ be the romb
associated to the pair of blue edges $(e_{2k}^{\prime},e_{2k+1}^{\prime})$
(see Proposition 3.3 (ii)). Then by the definition of $\sigma$ (Construction
3.34), and since $\sigma(x)\in Z(\overline{e},e)$ by assumption, we have that
$(e_{2j-1},e,f_{2j-1},f_{2j})$ is the romb associated to the pair of red edges
$(f_{2j-1},f_{2j})$. Hence we have that the red edge $f_{2j}$ satisfies
$r(f_{2j})=r(e_{2j}^{\prime})=s(e_{2j-1}^{\prime})$ and also $s(f_{2j})=s(e)$,
and we see that there is a red edge from $s(e)$ to
$s(\overline{e}^{\prime})=s(e_{2j-1}^{\prime})$, and thus
$a_{(\overline{e},e),\overline{e}^{\prime}}=1$.

Conversely, suppose that there is a red edge $f_{2j}$ such that
$s(f_{2j})=s(e)$ and $r(f_{2j})=s(e_{2j-1}^{\prime})$. We will show that for
each element

$y=(e_{0},e_{1},\dots,e_{2j-1},e,e_{2j+1},e_{2j+2},\dots)\in
Z(\overline{e},e)$

there is a unique $x\in Z(\overline{e}^{\prime})$ such that $\sigma(x)=y$. The
proof of this fact resembles the proof of surjectivity in Lemma 3.35, although
some additional manipulations are needed here. Use Definition 3.2 (g) and
induction to get unique red edges $f_{n}$ for $n\geq 2j+1$ such that
$f_{2k+1}\in R(f_{2k})$ and $(e_{2k+1},e_{2k+2},f_{2k+1},f_{2k+2})$ is a romb
for each $k\geq j$. By Proposition 3.3 (ii), there are blue edges
$e^{\prime}_{n}$ for $n\geq 2j$ such that
$(e^{\prime}_{2k},e^{\prime}_{2k+1},f_{2k},f_{2k+1})$ is a romb for each
$k\geq j$. Note that $r(e_{2j}^{\prime})=r(f_{2j})=s(e^{\prime}_{2j-1})$.
Hence we may consider the element

$x=(e_{0}^{\prime},e_{1}^{\prime},\dots,e_{2j-2}^{\prime},e_{2j-1}^{\prime},e_{2j}^{\prime},e_{2j+1}^{\prime},e_{2j+2}^{\prime},\dots)\in
Z(\overline{e}^{\prime}).$

Then by the construction of $x$ and the definition of $\sigma$, we have that

$\sigma(x)=(e_{0}^{\prime\prime},e_{1}^{\prime\prime},\dots,e_{2j-2}^{\prime\prime},e_{2j-1}^{\prime\prime},e_{2j}^{\prime\prime},e_{2j+1},e_{2j+2},e_{2j+3},\dots).$

But now by uniqueness of blue edges with the same source we get that
$e=e_{2j}^{\prime\prime}$ and $e_{n}=e_{n}^{\prime\prime}$ for $0\leq n\leq
2j-1$, showing that $\sigma(x)=y$. The uniqueness of $x\in
Z(\overline{e}^{\prime})$ such that $\sigma(x)=y$ follows from the fact that
$j\geq 1$ because, following the proof of Lemma 3.35, the pre-images of $y$
are determined by the red edges departing from $s(e_{0})$, and thus $x$ will
necessarily be the pre-image of $y$ determined by the edge $f_{0}$, where
$(e_{0}^{\prime},e_{1}^{\prime},f_{0},f_{1})$ is the romb associated to the
pair of blue edges $(e_{0}^{\prime},e_{1}^{\prime})$, with $f_{1}\in
R(f_{0})$.

$e_{2j-3}$$e_{2j-2}$$e_{2j-1}$$e$$f^{\prime}_{2j-2}$$f_{2j-1}$$f_{2j}$$e^{\prime}_{2j-3}$$e^{\prime}_{2j-2}$$e^{\prime}_{2j-1}$
Figure 7. Relation between $(\overline{e},e)$ and $\overline{e}^{\prime}$ via
red edges.

Let us now continue the proof. If
$a_{(\overline{e},e),\overline{e}^{\prime}}=1$, we set

$V_{(\overline{e},e),\overline{e}^{\prime}}=\\{x\in
Z(\overline{e}^{\prime})\mid\sigma(x)\in Z(\overline{e},e)\\}.$

The set $V_{(\overline{e},e),\overline{e}^{\prime}}$ is left undefined if
$a_{(\overline{e},e),\overline{e}^{\prime}}=0$. Finally, we define

${\pazocal Z}_{2j}={\pazocal
Z}_{2j}(F,D)=(\\{Z(\overline{e},e)\mid(\overline{e},e)\in\Gamma^{2j}\\},\\{V_{(\overline{e},e),\overline{e}^{\prime}}\mid(\overline{e},e)\in\Gamma^{2j},\overline{e}^{\prime}\in
P^{B}_{0,2j}\text{ with }a_{(\overline{e},e),\overline{e}^{\prime}}=1\\}).$

Clearly conditions (a) and (b) in Definition 5.1 are satisfied. Moreover, by
what we have proven above, we have that the restriction of $\sigma$ to each
$V_{(\overline{e},e),\overline{e}^{\prime}}$, with
$a_{(\overline{e},e),\overline{e}^{\prime}}=1$, is injective, and that
$\sigma(V_{(\overline{e},e),\overline{e}^{\prime}})=Z(\overline{e},e)$.
Therefore condition (c) in Definition 5.1 is also satisfied. Hence we have
defined $\sigma$-cylinder decompositions ${\pazocal Z}_{2j}$, whose associated
red-blue adjacency matrices agree with the pair $(A,I)$ of red and blue
adjacency matrices of the generalized finite shift graph $(F^{0,2j}\cup
F^{0,2j+1},F^{1,2j})$, for each $j\geq 0$.

We now show the part on the refinement of the $\sigma$-cylinder
decompositions. By Lemma 5.4, it is enough to show it in the case where we
have two consecutive even layers $(F^{0,2j}\cup F^{0,2j+1},F^{1,2j})$ and
$(F^{0,2j+2}\cup F^{0,2j+3},F^{1,2j+2})$, $j\geq 0$.

We need to check the conditions in Definition 5.2. Note that

$\Gamma^{2j+2}=P^{B}_{2j+3}=\bigsqcup_{\overline{e}\in\Gamma^{2j}}\Gamma^{2j+2}(\overline{e}),$

with

$\Gamma^{2j+2}(\overline{e})=\\{(\overline{e},e_{2j+1},e_{2j+2})\mid
e_{2j+1}\in B_{s(\overline{e})},e_{2j+2}\in B_{s(e_{2j+1})}\\},$

and we have, for $\overline{e}\in\Gamma^{2j}$,

$Z(\overline{e})=\bigsqcup_{(\overline{e},e_{2j+1},e_{2j+2})\in\Gamma^{2j+2}(\overline{e})}Z(\overline{e},e_{2j+1},e_{2j+2}).$

This proves condition (a) in Definition 5.2.

We prove (b) and (c) first for $j\geq 1$. In this case

$\mathfrak{A}_{2j}=\\{((\overline{e},e),\overline{e}^{\prime})\mid\overline{e},\overline{e}^{\prime}\in
P^{B}_{0,2j},e\in
B_{s(\overline{e})},a_{(\overline{e},e),\overline{e}^{\prime}}=1\\},$

and we have a similar expression for $\mathfrak{A}_{2j+2}$. We can then write

$\mathfrak{A}_{2j+2}=\bigsqcup_{((\overline{e},e),\overline{e}^{\prime})\in\mathfrak{A}_{2j}}\mathfrak{A}_{2j+2}((\overline{e},e),\overline{e}^{\prime}),$

where $\mathfrak{A}_{2j+2}((\overline{e},e),\overline{e}^{\prime})$ is the set
of all elements of $\mathfrak{A}_{2j+2}$ of the form
$((\overline{e},e,e_{2j+1},e_{2j+2}),(\overline{e}^{\prime},e^{\prime}_{2j},e^{\prime}_{2j+1}))$,
for suitable edges $e_{2j+1},e_{2j+2},e^{\prime}_{2j},e_{2j+1}^{\prime}$.

With these decompositions, we obtain the equality

(5.1)
$V_{(\overline{e},e),\overline{e}^{\prime}}=\bigsqcup_{\nu\in\mathfrak{A}_{2j+2}((\overline{e},e),\overline{e}^{\prime})}V_{\nu},$

for each $((\overline{e},e),\overline{e}^{\prime})\in\mathfrak{A}_{2j}$, which
shows (b). To prove (c), one has to show that given
$((\overline{e},e),\overline{e}^{\prime})\in\mathfrak{A}_{2j}$, and given
$e_{2j+1}\in B_{s(e)}$, $e_{2j+2}\in B_{s(e_{2j+1})}$, there are unique
$e^{\prime}_{2j}\in B_{s(\overline{e}^{\prime})}$ and $e^{\prime}_{2j+1}\in
B_{s(e^{\prime}_{2j})}$ such that
$((\overline{e},e,e_{2j+1},e_{2j+2}),(\overline{e}^{\prime},e^{\prime}_{2j},e^{\prime}_{2j+1}))\in\mathfrak{A}_{2j+2}$.
The blue edges $e_{2j}^{\prime},e_{2j+1}^{\prime}$ are constructed as follows.
Let $f_{2j}$ be the unique red edge such that $s(f_{2j})=s(e)$ and
$r(f_{2j})=s(\overline{e}^{\prime})$. This edge must exist because
$a_{(\overline{e},e),\overline{e}^{\prime}}=1$. Now by Definition 3.2 (g),
there is a unique pair of red edges $(f_{2j+1},f_{2j+2})$ such that
$f_{2j+1}\in R(f_{2j})$ and $(e_{2j+1},e_{2j+2},f_{2j+1},f_{2j+2})$ is a romb.
Now use Proposition 3.3 (ii) to get a unique pair of blue edges
$(e_{2j}^{\prime},e_{2j+1}^{\prime})$ such that
$(e_{2j}^{\prime},e_{2j+1}^{\prime},f_{2j},f_{2j+1})$ is a romb. Note that
$f_{2j+2}$ is a red edge such that $s(f_{2j+2})=s(e_{2j+2})$ and
$r(f_{2j+2})=s(e_{2j+1}^{\prime})$. It follows that
$(\overline{e}^{\prime},e_{2j}^{\prime},e_{2j+1}^{\prime})$ is the unique path
in $P^{B}_{0,2j+2}$ such that
$((\overline{e},e,e_{2j+1},e_{2j+2}),(\overline{e}^{\prime},e_{2j}^{\prime},e_{2j+1}^{\prime}))\in\mathfrak{A}_{2j+2}$.

It remains to show (b) and (c) in Definition 5.2 for $j=0$. In this case, we
have the sets $V_{e_{0},i}^{(k)}$, where $e_{0}$ is a blue edge in $F^{1,0}$,
$v^{i}\in F^{0,0}$ and $1\leq k\leq a_{e_{0},i}$. Given such a triple
$(e_{0},i,k)$, the set $\mathfrak{A}_{2}(e_{0},i,k)$ is the set of all
elements
$((e_{0},e_{1},e_{2}),(e_{0}^{\prime},e_{1}^{\prime}))\in\mathfrak{A}_{2}$
such that $r(e_{0}^{\prime})=v^{i}$ and the romb associated to the pair of
blue edges $(e_{0}^{\prime},e_{1}^{\prime})$ is of the form
$(e_{0}^{\prime},e_{1}^{\prime},f_{0},f_{1})$ with $f_{0}=f_{e_{0},i}^{(k)}$.
We then have

$\mathfrak{A}_{2}=\bigsqcup_{(e_{0},i,k)\in\mathfrak{A}_{0}}\mathfrak{A}_{2}(e_{0},i,k)$

and

$V_{e_{0},i}^{(k)}=\bigsqcup_{\nu\in\mathfrak{A}_{2}(e_{0},i,k)}V_{\nu}$

for $(e_{0},i,k)\in\mathfrak{A}_{0}$. This shows (b) for $j=0$. Part (c) for
$j=0$ is proven in a similar way as in the case where $j>0$. Namely, given a
triple $(e_{0},i,k)$ as before, and given $e_{1}\in B_{s(e_{0})}$ and
$e_{2}\in B_{s(e_{1})}$, take first the unique pair of red edges
$(f_{1},f_{2})$ such that $f_{1}\in R(f_{e_{0},i}^{(k)})$ and
$(e_{1},e_{2},f_{1},f_{2})$ is a romb. Then take the unique blue pair
$(e_{0}^{\prime},e_{1}^{\prime})$ so that
$(e_{0}^{\prime},e_{1}^{\prime},f^{(k)}_{e_{0},i},f_{1})$ is a romb. Then
$((e_{0},e_{1},e_{2}),(e_{0}^{\prime},e_{1}^{\prime}))$ is the unique element
in $\mathfrak{A}_{2}(e_{0},i,k)$ projecting to $(e_{0},e_{1},e_{2})$. This
completes the proof. ∎

######  Remark 5.9.

Note the following relation between red adjacency matrices. For $j\geq 0$, let
$A_{2j}=(a_{(\overline{e},e),\overline{e}^{\prime}})\in M_{\Gamma^{2j}\times
P^{B}_{0,2j}}({\mathbb{Z}}^{+})$ be the red adjacency matrix of the
generalized finite shift graph $(F^{0,2j}\cup F^{0,2j+1},F^{1,2j})$. Let also,
for $n\geq 0$, $P_{n+1}$ be the characteristic matrix of the partitions
$\\{Z(\overline{e})\mid\overline{e}\in P^{B}_{0,n}\\}$ and
$\\{Z(\overline{e},e_{0},e_{1})\mid(\overline{e},e_{0},e_{1})\in
P^{B}_{0,n+2}\\}$, which is defined by

$(P_{n+1})_{\overline{e}^{\prime},(\overline{e},e_{0},e_{1})}=\begin{cases}1&\text{
if }\overline{e}^{\prime}=\overline{e},\\\ 0&\text{ otherwise}.\end{cases}$

Then we have the relation

$D_{2j}A_{2j}=P_{2j+2}A_{2j+2}P_{2j+1}^{T},$

where $D_{2j}$ is the diagonal $\Gamma^{2j}\times\Gamma^{2j}$ matrix having
$|P^{B}_{2j+1,2j+3}(s(\overline{e}))|$ at the position
$(\overline{e},\overline{e})$, for each $\overline{e}\in\Gamma^{2j}$.

The formula holds because for each
$(\overline{e},e_{0},e_{1})\in\Gamma^{2j+2}$ and each red edge $f_{0}$ such
that $s(f_{0})=s(\overline{e})$, there is a unique
$(\overline{e}^{\prime},e_{0}^{\prime},e_{1}^{\prime})\in P^{B}_{0,2j+2}$ such
that
$a_{(\overline{e},e_{0},e_{1}),(\overline{e}^{\prime},e_{0}^{\prime},e_{1}^{\prime})}=1$
and the romb associated to $(e_{0}^{\prime},e_{1}^{\prime})$ is of the form
$(e_{0}^{\prime},e_{1}^{\prime},f_{0},f_{1})$, where
$(e_{0},e_{1},f_{1},f_{2})$ is the romb associated to $(e_{0},e_{1})$.

$\overline{e}$$e_{0}$$e_{1}$$f_{0}$$f_{1}$$f_{2}$$\overline{e}^{\prime}$$e^{\prime}_{0}$$e^{\prime}_{1}$
Figure 8. Visual explanation of the formula
$D_{2j}A_{2j}=P_{2j+2}A_{2j+2}P_{2j+1}^{T}$.

We will now show that any generalized finite shift graph gives rise to a
dynamical system $(X,\sigma)\in\textsf{LHomeo}$. This dynamical system will be
called a generalized finite shift, and includes both one-sided and two-sided
finite shifts. For this purpose we will use the construction of the canonical
resolution of a bipartite separated graph. We first recall its definition (see
[14] and [10]).

######  Definition 5.10.

Let $(E,C)$ be any finite bipartite separated graph, and write

$C_{u}=\\{X_{1}^{u},\dots,X_{k_{u}}^{u}\\}$

for all $u\in E^{0,0}$. Then the $1$-step resolution of $(E,C)$ is the finite
bipartite separated graph denoted by $(E_{1},C^{1})$, and defined by

  1. (a)

$E_{1}^{0,0}:=E^{0,1}$ and $E_{1}^{0,1}:=\\{v(x_{1},\dots,x_{k_{u}})\mid u\in
E^{0,0},x_{j}\in X_{j}^{u}\text{ for all }1\leq j\leq k_{u}\\}$.

  2. (b)

$E_{1}^{1}:=\\{\alpha^{x_{i}}(x_{1},\dots,\widehat{x_{i}},\dots,x_{k_{u}})\mid
u\in E^{0,0},1\leq i\leq k_{u},x_{j}\in X_{j}^{u}\text{ for all }1\leq j\leq
k_{u}\\}$.

  3. (c)

The range and source for the edges in $E_{1}^{1}$ are defined by

$r_{1}(\alpha^{x_{i}}(x_{1},\dots,\widehat{x_{i}},\dots,x_{k_{u}})):=s(x_{i}),$
$s_{1}(\alpha^{x_{i}}(x_{1},\dots,\widehat{x_{i}},\dots,x_{k_{u}})):=v(x_{1},\dots,x_{k_{u}}).$

  4. (d)

$C^{1}_{v}:=\\{X(x)\mid x\in s^{-1}(v)\\}$ for $v\in E^{0,0}_{1}$, where for
$u\in E^{0,0}$, $1\leq i\leq k_{u}$, and $x_{i}\in X_{i}^{u}$,

$X(x_{i}):=\\{\alpha^{x_{i}}(x_{1},\dots,\widehat{x_{i}},\dots,x_{k_{u}})\mid
x_{j}\in X_{j}^{u}\text{ for }j\neq i\\}.$

A sequence of finite bipartite separated graphs $\\{(E_{n},C^{n})\\}_{n\geq
0}$ with $(E_{0},C^{0}):=(E,C)$ is then defined inductively by letting
$(E_{n+1},C^{n+1})$ denote the $1$-step resolution of $(E_{n},C^{n})$. The
bipartite separated graph $(E_{n},C^{n})$ is called the $n$-step resolution of
$(E,C)$. Finally $(F,D)$ is the infinite layer graph

$(F,D):=\bigcup_{n=0}^{\infty}(E_{n},C^{n}).$

It is clear by construction that $(F,D)$ is a separated Bratteli diagram,
called the canonical resolution of the finite bipartite graph $(E,C)$.

We now apply this construction to a generalized finite shift graph $(E,C)$
(see Definition 5.6).

######  Proposition 5.11.

Let $(E,C)$ be a generalized finite shift graph. Then the canonical resolution
$(F,D)$ of $(E,C)$ is an $l$-diagram.

###### Proof.

Let $(E,C)$ be a generalized finite shift graph, with
$E^{0,0}=\\{v^{1},\dots,v^{n}\\}$ and red adjacency matrix $A=(a_{w,i})$,
where $a_{w,i}$ is the number of red edges from $w\in E^{0,1}$ to $v^{i}$.

Observe that, by its natural expression as a separated Bratteli diagram, we
have $F^{0,n}=E_{n}^{0,0}$ and $F^{1,n}=E_{n}^{1}$ for all $n\geq 0$. We also
have $D_{v}=C_{v}^{n}$ for $v\in F^{0,n}$. We need to check conditions (a)-(g)
in Definition 3.2.

Note first that, by definition, property (a) for $j=0$, formulas (1) and (2)
in (d), and the second part of (d) for $j=0$ hold.

We analyze first the graph $(E_{1},C^{1})$. Observe that by Definition 5.10 we
have exactly $|s^{-1}(v)|$ elements in $D_{v}$ for each $v\in
E_{1}^{0,0}=F^{0,1}$. These are of the form

$D_{v}=\\{B_{v}\\}\cup\\{R(f_{v,j}^{(k)})\mid 1\leq j\leq n,\text{ }1\leq
k\leq a_{v,j}\\},$

where $\\{f_{v,j}^{(k)}\mid 1\leq k\leq a_{v,j}\\}$ is the set of red edges
from $v$ to $v^{j}$. For $v\in F^{0,1}$, we write

$R_{v}=\bigcup_{j=1}^{n}\bigcup_{k=1}^{a_{v,j}}R(f_{v,j}^{(k)}).$

These are the red edges at the vertices in $E^{0,1}=F^{0,1}$. Condition (b) is
therefore verified for $j=1$.

To verify condition (c) at $j=1$, observe that $F^{0,2}$ is the disjoint union
of the vertices corresponding to the different vertices $v^{i}$, for $1\leq
i\leq n$, where a vertex $w$ in $F^{0,2}$ corresponds to $v^{i}$ in case there
is a (necessarily unique) blue path going from $w$ to $v^{i}$. So it is enough
to concentrate on one such index $i$. The set of vertices corresponding to
$v^{i}$ is

$\\{v(e_{v,i},f_{w,i}^{(k)})\mid e_{v,i}\in B_{v^{i}},w\in F^{0,1},1\leq k\leq
a_{w,i}\\}.$

Here $B_{v^{i}}=\\{e_{v,i}\\}_{v\in s(B_{v^{i}})}$, with $s(e_{v,i})=v$ and
$r(e_{v,i})=v^{i}$. Clearly, for a given vertex $v(e_{v,i},f_{w,i}^{(k)})$ of
this form, there are exactly two edges with source $v(e_{v,i},f_{w,i}^{(k)})$,
namely $\alpha^{e_{v,i}}(f_{w,i}^{(k)})\in B_{v}$ and
$\alpha^{f_{w,i}^{(k)}}(e_{v,i})\in R(f_{w,i}^{(k)})\subseteq R_{w}$. Hence
condition (c) holds for $j=1$.

The validity of condition (e) for $j=0$ is clear from the above
considerations.

We have verified all the conditions up to the second layer. Now, using
induction, we suppose that, for $n>0$, all conditions hold up to the layer
$2n$, that is, suppose that all the conditions in Definition 3.2 involving
vertices from $F^{0,j}$ for $0\leq j\leq 2n$ and edges between them, hold. We
will check the conditions for the next two layers. For $v\in F^{0,2n}$ we have
$D_{v}=\\{B_{v},R_{v}\\}$ because by induction hypothesis (condition (c)),
there are exactly two edges departing from $v$, one of them a blue edge and
the other a red edge, giving rise (by Definition 5.10) to the two sets $B_{v}$
and $R_{v}$ respectively. This gives condition (a) for $j=n$. Moreover, the
elements of $B_{v}$ are of the form $\alpha^{e}(f_{1},\dots,f_{p})$, where $e$
is the unique blue edge such that $s(e)=v$, and $f_{i}\in R(g_{i})$, where
$g_{1},\dots,g_{p}$ are the different red edges such that $s(g_{i})=r(e)$. We
then have that

$s(\alpha^{e}(f_{1},\dots,f_{p}))=v(e,f_{1},\dots,f_{p}),$

so clearly we have that the sources of two distinct blue edges are distinct.
This verifies the second part of condition (d) for $j=n$. Observe that
$(F^{0,2n}\cup F^{0,2n+1},F^{1,2n})$ is again a generalized finite shift
graph, and so we can prove the same properties as before for the layers
between $2n$ and $2n+2$.

It remains to check conditions (f) and (g).

We check condition (f) for $j=n$. Suppose that we have a pair $(f_{0},f_{1})$
of red edges such that $r(f_{0})=v\in F^{0,2n-1}$ and $r(f_{1})=s(f_{0})$.
Then $f_{1}$ will be of the form

$f_{1}=\alpha^{f_{0}^{\prime}}(e_{0},g_{1},\dots,g_{p})$

where $e_{0}\in B_{v}$, $f_{0}^{\prime}\in R(h_{0})$, and $g_{i}\in R(h_{i})$
for $1\leq i\leq p$, where $\\{h_{0},h_{1},\dots,h_{p}\\}$ are the different
red edges departing from $v$. Observe that
$s(f_{0}^{\prime})=r(f_{1})=s(f_{0})$. Hence by property (c) for $j=n$ we need
to have $f_{0}^{\prime}=f_{0}$. We define
$e_{1}:=\alpha^{e_{0}}(f_{0},g_{1},\dots,g_{p})$. We have that
$r(e_{1})=s(e_{0})$, and that $e_{1}\in B_{r(e_{1})}$. Moreover,
$s(e_{1})=s(f_{1})=v(e_{0},f_{0},g_{1},\dots,g_{p})$ and
$r(e_{0})=r(f_{0})=v$, as desired. It is clear that the pair $(e_{0},e_{1})$
is the unique pair of blue edges satisfying these conditions.

We finally deal with condition (g). Suppose that we are given one of the red
edges $h_{i}$, where as before $\\{h_{0},h_{1},\dots,h_{p}\\}$ are the
different red edges departing from $v$, and we are also given a pair
$(e_{0},e_{1})$ of blue edges such that $r(e_{0})=v$ and $r(e_{1})=s(e_{0})$.
This means that $e_{1}$ is of the form

$e_{1}=\alpha^{e_{0}}(g_{0},g_{1},\dots,g_{p}),$

where $g_{j}\in R(h_{j})$ for $0\leq j\leq p$. We then define $f_{0}=g_{i}$
and

$f_{1}=\alpha^{g_{i}}(e_{0},g_{0},\dots,\hat{g_{i}},\dots,g_{p}).$

Then we get that $(e_{0},e_{1},f_{0},f_{1})$ is a romb, and that
$(f_{0},f_{1})$ is unique satisfying this property and the property that
$f_{0}\in R(h_{i})$. This shows that condition (g) is satisfied for $j=n$.

This completes the proof of the fact that $(F,D)$ is an $l$-diagram. ∎

We have observed in the above proof that the separated graphs
$(E_{2n},C^{2n})$, $n\geq 0$, are also generalized finite shift graphs. In
particular, we can talk about their red-blue adjancency matrices
$(A_{2n},I_{2n})$. A similar relation between the different matrices $A_{n}$
as the one described in Remark 5.9 can also be given. Of course, all graphs
$(E_{2n},C^{2n})$ describe exactly the same dynamical system.

######  Definition 5.12.

Let $(E,C)$ be a generalized finite shift graph, and let $(F,D)$ be its
canonical resolution, which is an $l$-diagram by Proposition 5.11. The
associated dynamical system $(X,\sigma)$, as defined in Subsection 3.2, will
be called a generalized finite shift.

The rest of this section is devoted to show how these generalized finite
shifts provide common generalizations of one-sided and two-sided shifts.

We come back to the original situation where we have
$E=(E^{0},E^{1},r_{E},s_{E})$ a finite graph with adjacency matrix $A=A_{E}$
with no zero-rows and no zero-columns, and set of vertices
$E^{0}=\\{1,\dots,n\\}$.

We can canonically build a finite bipartite separated graph
$(E^{\prime},C^{\prime})$ associated to $E$ as follows. The graph $E^{\prime}$
has set of vertices
$(E^{\prime})^{0}=(E^{\prime})^{0,0}\sqcup(E^{\prime})^{0,1}$ with
$(E^{\prime})^{0,0}$ and $(E^{\prime})^{0,1}$ disjoint copies of $E^{0}$. Let
$i\mapsto v_{i}$ be a bijection from $E^{0}$ onto $(E^{\prime})^{0,1}$, and
$i\mapsto v^{i}$ a bijection of $E^{0}$ onto $(E^{\prime})^{0,0}$. We define
$C^{\prime}_{i}=\\{B_{i},R_{i}\\}$, where $B_{i}=\\{e_{i}\\}$ with
$s(e_{i})=v_{i}$ and $r(e_{i})=v^{i}$, and where $R_{i}$ consists of a copy of
$r_{E}^{-1}(i)$. More precisely, for each $f\in E^{1}$ such that $r_{E}(f)=i$,
we define an edge $\overline{f}\in(E^{\prime})^{1}$ such that
$s(\overline{f})=s_{E}(f)_{i}$ and $r(\overline{f})=r_{E}(f)^{i}$. Then we
define $R_{i}$ as the set of such edges $\overline{f}$ with $r_{E}(f)=i$.

######  Example 5.13.

Let $E$ be the finite graph

$00$$01$$10$$11$

The associated separated graph $(E^{\prime},C^{\prime})$ is given by

$v_{00}$$v_{01}$$v_{10}$$v_{11}$$v^{00}$$v^{01}$$v^{10}$$v^{11}$

It is easy to show that the associated one-sided shift $(X_{E},\sigma_{E})$ is
precisely the subset of the set of all sequences in $\\{0,1\\}^{{\mathbb{N}}}$
where the word $000$ is forbidden. ∎

The above procedure defines a separated graph $(E^{\prime},C^{\prime})$ that,
in fact, is a generalized finite shift graph since $E$ has neither sinks nor
sources. Let $(X,\sigma)$ be the associated generalized finite shift.

######  Proposition 5.14.

The pair $(X,\sigma)$ is topologically conjugate to the one-sided shift
$(X_{E},\sigma_{E})$ as defined in Subsection 5.1.

###### Proof.

Let $(F,D)$ denote the canonical resolution of $(E^{\prime},C^{\prime})$. We
first show that the red adjacency matrix $A_{2}$ associated to the separated
graph $(F_{2},D^{2})$ is exactly the incidence matrix of $E$, and hence equal
to the adjacency matrix of $D(E)$, the dual graph of $E$. For this, we first
observe that we can identify the vertices in $F^{0,2}$ with the edges of $E$.
Indeed, each vertex in $F^{0,2}$ is of the form $v(e_{i},\overline{f})$, where
$f\in E^{1}$ with $r_{E}(f)=i$. Hence there is a bijection between $F^{0,2}$
and $E^{1}$ sending $v(e_{i},\overline{f})$ to $f$. We will identify $F^{0,2}$
with $E^{1}$ through this bijection. Now for $v_{i}\in F^{0,1}$ we have

$B_{v_{i}}=\\{\alpha^{e_{i}}(\overline{f})\mid f\in r_{E}^{-1}(i)\\},$

and

$R_{v_{i}}=\\{\alpha^{\overline{f}}(e_{j})\mid f\in
s_{E}^{-1}(i),j=r_{E}(f)\\}.$

Denoting by $f^{*}$ the edge $\alpha^{\overline{f}}(e_{j})$, we have
$r(f^{*})=v_{s_{E}(f)}$ and $s(f^{*})=f$. Further, we denote by $f^{\prime}$
the edge $\alpha^{e_{i}}(\overline{f})$, so $s(f^{\prime})=f$ and
$r(f^{\prime})=v_{r_{E}(f)}$.

Now we move to the graph $(F_{2},D^{2})$. Recall that we are identifying
$F^{0,2}$ with $E^{1}$. For $f\in E^{1}$, we have only one element in
$R(\overline{f})$, namely $R(\overline{f})=\\{f^{*}\\}$. Therefore $F^{0,3}$
can be identified with $F^{0,2}=E^{1}$ through the bijection
$v(f^{\prime},f_{1}^{*},f_{2}^{*}.\dots,f_{p}^{*})\leftrightarrow f$, where
$s_{E}^{-1}(i)=\\{f_{1},f_{2},\dots,f_{p}\\}$ and $r_{E}(f)=i$. Observe that

$r(\alpha^{f^{\prime}}(f_{1}^{*},f_{2}^{*},\dots,f_{p}^{*}))=s(f^{\prime})=f\in
F^{0,2}=E^{1}$

and
$s(\alpha^{f^{\prime}}(f_{1}^{*},f_{2}^{*},\dots,f_{p}^{*}))=v(f^{\prime},f_{1}^{*},f_{2}^{*},\dots,f_{p}^{*})$,
which we identify with $f$ through the above bijection. Hence $B_{f}$ consists
of a single edge, which connects $f\in F^{0,3}$ with $f\in F^{0,2}$.

It remains to identify the red edges in $(F_{2},D^{2})$. These are of the form

$\alpha^{g^{*}}(f^{\prime},g_{2}^{*},\dots,g_{p}^{*})$

for $\\{g,g_{2},\dots,g_{p}\\}=s_{E}^{-1}(i)$ and $r_{E}(f)=i$ for some
$i\in\\{1,\dots,n\\}$. We then have

$r(\alpha^{g^{*}}(f^{\prime},g_{2}^{*},\dots,g_{p}^{*}))=s(g^{*})=g\in
F^{0,2},\quad s(\alpha^{g^{*}}(f^{\prime},g_{2}^{*},\dots,g_{p}^{*}))=f\in
F^{0,3}$

under the above identifications. So there is at most one edge from $f\in
F^{0,3}$ to $g\in F^{0,2}$, and there is such edge if and only if
$r_{E}(f)=s_{E}(g)$. We conclude that the matrix $A_{2}$ is the incidence
matrix of the graph $E$.

More generally, it follows by induction that for each $N\geq 1$ the layer
$(F^{0,2N}\cup F^{0,2N+1},F^{1,2N})$ gives exactly the $(N+1)^{\text{th}}$
higher edge graph $E^{[N+1]}$ of $E$, which is defined as the graph having as
vertices all the paths of length $N$ in $E$, and having exactly one edge from
$e_{N}e_{N-1}\cdots e_{2}e_{1}$ to $f_{N}f_{N-1}\cdots f_{2}f_{1}$ if and only
if $e_{N}\cdots e_{2}=f_{N-1}\cdots f_{1}$, and none otherwise. The edge is
named $f_{N}e_{N}\cdots e_{2}e_{1}$ (see e.g. [48, Definition 2.3.4]).

Using these identifications it is now easy to define a conjugation between
$(X,\sigma)$ and $(X_{E},\sigma_{E})$. An element
$x=(e_{0},e_{1},e_{2},\dots)\in X_{E}$ gives rise to a finite path
$e_{N}e_{N-1}\cdots e_{1}e_{0}$ for each $N\geq 0$, which corresponds to a
unique vertex in the layer $(F^{0,2N+2}\cup F^{0,2N+3},F^{1,2N+2})$ of
$(F,D)$. By the construction of $(X,\sigma)$ given in Subsection 3.2, it is
clear that $x$ determines a unique element $\psi(x)\in X$. The map
$\psi:X_{E}\rightarrow X$ gives the desired conjugacy between
$(X_{E},\sigma_{E})$ and $(X,\sigma)$. ∎

######  Remark 5.15.

Note that one-sided shifts $(X,\sigma)$ of infinite type are not objects in
our category LHomeo, because in this case $\sigma$ is not an open map (see
[41, Theorem 1]), and therefore it is not a local homeomorphism. Hence there
is no $l$-diagram realizing a one-sided shift of infinite type. Of course, the
situation is different with the two-sided shifts, of any type, which are
always homeomorphisms, and can be concretely realized by $h$-diagrams (recall
Section 4). This has been studied in [14, 6.7–6.10], and we refer the reader
there for further details. Note in particular that, by [14, Proposition 6.8
(2)], two-sided shifts of finite type correspond to a particular class of
generalized finite shift graphs.

###  5.3. A configurational point of view

We will now obtain an explicit realization of the generalized finite shifts.
To obtain this, we rely on the work done in [10] (see also [14]) to describe
the dynamics inherent to a bipartite separated graph. We will use the notation
of [14].

######  Definition 5.16.

[14, Definition 2.11] Let $(E,C)$ be a finite bipartite separated graph, and
let $\mathbb{F}$ be the free group on $E^{1}$. Given $\xi\subseteq\mathbb{F}$
and $\alpha\in\xi$, the local configuration $\xi_{\alpha}$ of $\xi$ at
$\alpha$ is

$\xi_{\alpha}=\\{\sigma\in
E^{1}\cup(E^{1})^{-1}\mid\sigma\in\xi\cdot\alpha^{-1}\\}.$

The space $\Omega(E,C)$ is the space of all configurations
$\xi\subseteq\mathbb{F}$ such that:

  1. (a)

$1\in\xi$;

  2. (b)

$\xi$ is right convex, which means that $\xi$ is suffix closed: if
$x_{n}\cdots x_{2}x_{1}\in\xi$, where $x_{n}\cdots x_{2}x_{1}$ is a reduced
word (so that $x_{i}\in E^{1}\cup(E^{1})^{-1}$ and $x_{i}\neq x_{i+1}^{-1}$
for $1\leq i<n$), then $x_{m}\cdots x_{2}x_{1}\in\xi$ for all $1\leq m\leq n$;

  3. (c)

for each $\alpha\in\xi$, one of the following holds:

    1. (c.1)

$\xi_{\alpha}=s^{-1}(v)$ for some $v\in E^{0,1}$;

    2. (c.2)

$\xi_{\alpha}=\\{e_{X}^{-1}\mid X\in C_{v}\\}$ for some $v\in E^{0,0}$ and
$e_{X}\in X$.

In words, $\Omega(E,C)$ is the space of all the configurations
$\xi\subseteq\mathbb{F}$ containing $1$, being right convex, and whose local
configuration at each point consists of either emitting all the edges
departing from a vertex in $E^{0,1}$, or emitting the inverses of a choice of
exactly one edge $e_{X}\in X$ for each $X\in C_{v}$, $v\in E^{0,0}$. With the
induced topology from $2^{\mathbb{F}}$, $\Omega(E,C)\subseteq 2^{\mathbb{F}}$
is a totally disconnected compact metrizable space.

There is a partial action of $\mathbb{F}$ on $\Omega(E,C)$ with open compact
domains, which is defined by setting, for each
$\alpha\in\mathbb{F}\setminus\\{1\\}$, the domains as

$\Omega(E,C)_{\alpha}=\\{\xi\in\Omega(E,C)\mid\alpha^{-1}\in\xi\\}$

and the maps as

$\theta_{\alpha}:\Omega(E,C)_{\alpha^{-1}}\rightarrow\Omega(E,C)_{\alpha},\quad\theta_{\alpha}(\xi)=\xi\cdot\alpha^{-1}.$

Following [14], we will use the notation $\alpha.\xi$ to denote
$\theta_{\alpha}(\xi)$. We thus have
$\alpha.\xi=\theta_{\alpha}(\xi)=\xi\cdot\alpha^{-1}$.

For $v\in E^{0}$, the space $\Omega(E,C)_{v}$ is defined to be the space of
all configurations “starting" at vertex $v$, that is, $\Omega(E,C)_{v}$ is the
set of configurations $\xi\in\Omega(E,C)$ such that $\xi_{1}=s^{-1}(v)$ in
case $v\in E^{0,1}$, and the set of configurations $\xi\in\Omega(E,C)$ such
that $\xi_{1}=\\{e_{X}^{-1}\mid X\in C_{v}\\}$ for a choice of an edge
$e_{X}\in X$ for each $X\in C_{v}$, in case $v\in E^{0,0}$.

We will also need the notion of an $n$-ball in $\Omega(E,C)$.

######  Definition 5.17.

[14, Definition 3.4] Given $\xi\in\Omega(E,C)$ and $n\in{\mathbb{N}}$, we
define the $n$-ball $\xi^{n}$ by

$\xi^{n}=\\{\alpha\in\xi\mid|\alpha|\leq n\\}.$

Here we use the notation $|\alpha|$ to denote the length of the (reduced) word
$\alpha\in\mathbb{F}$.

Of course, the $1$-ball $\xi^{1}$ of $\xi$ is just the local configuration of
$\xi$ at $1$, together with the element $1\in\mathbb{F}$, so that
$\xi^{1}=\xi_{1}\cup\\{1\\}$. We denote by ${\pazocal B}_{n}(\Omega(E,C))$ the
collection of $n$-balls $\xi^{n}$ associated to elements $\xi\in\Omega(E,C)$.

Suppose now that $(E,C)$ is a generalized finite shift graph. In our next
result we relate the pair $(\Omega(E,C),\theta)$ with the generalized finite
shift $(X,\sigma)$ associated with $(E,C)$. Recall from Definition 5.10 the
notion of the $n$-step resolution $(E_{n},C^{n})$ of $(E,C)$.

######  Proposition 5.18.

Let $(E,C)$ be a generalized finite shift graph, let $(X,\sigma)$ be the
corresponding generalized finite shift, as in Definition 5.12, and let
$\Omega(E,C)$ be the space of configurations on $\mathbb{F}$ defined above.
Then $X$ can be identified with the subspace of configurations $\xi$ whose
local configuration $\xi_{1}$ at $1$ is of the form (c.2). Moreover, we have

$\sigma(\xi)=\theta_{a_{1}f_{0}^{-1}}(\xi),$

where $\xi_{1}=\\{e_{0}^{-1},f_{0}^{-1}\\}$, with $e_{0}\in B_{v}$, $f_{0}\in
R_{v}$, $v=r(e_{0})=r(f_{0})$, and $a_{1}$ is the unique blue edge such that
$s(a_{1})=s(f_{0})$.

###### Proof.

We will use [14, Theorem 3.22] which, for every finite bipartite separated
graph $(E,C)$, establishes a conjugacy

$\theta^{(E_{n},C^{n})}\cong(\theta^{(E,C)})^{[n]},$

where $(\theta^{(E,C)})^{[n]}$ is the $n$-ball convex subshift associated to
the convex subshift $\theta^{(E,C)}$ (see [14, Construction 3.14]). More
precisely, there is a canonical bijective correspondence

$E_{n}^{0}\to{\pazocal B}_{n}(\Omega(E,C)),\quad v\mapsto B(v)$

and a homeomorphism $\varphi_{n}\colon\Omega(E_{n},C^{n})\to\Omega(E,C)$ which
restricts to a homeomorphism

$\Omega(E_{n},C^{n})_{v}\to\\{\xi\in\Omega(E,C)\mid\xi^{n}=B(v)\\}$

for each $v\in E_{n}^{0}$. It is clear that $X$ can be identified with the
subspace of configurations $\xi$ whose local configuration $\xi_{1}$ at $1$ is
of the form (c.2), through the following correspondence. If
$x=(e_{0},e_{1},\dots)\in X$, then $\xi$ is the unique configuration such that
$\xi^{2j-1}=B(e_{0},e_{1},\dots,e_{2j-1})$, where we interpret
$(e_{0},e_{1},\dots,e_{2j-1})$ as a vertex in $E_{2j-1}^{0,1}$, for all $j\geq
1$. Note that $(e_{0},e_{1},\dots,e_{2j-1})$ can also be interpreted as a
vertex in $E_{2j}^{0,0}$, and then it correponds to $\xi^{2j}$ through the
above bijection.

Furthermore, a group isomorphism
$\Phi_{n}\colon\mathbb{F}^{[n:\Omega]}\to\mathbb{F}(E_{n}^{1})$ is established
in the proof of [14, Theorem 3.22], for all $n\geq 1$, where
$\Omega:=\Omega(E,C)$, and
$\mathbb{F}^{[n:\Omega]}:=\mathbb{F}(A^{[n:\Omega]})$ is the free group
defined in [14, Construction 3.14].

Let $\xi\in X\subseteq\Omega(E,C)$. The element $\xi$ is determined by a
sequence $(e_{0},e_{1},e_{2},\dots)$ in the Bratteli diagram $(F,D)$. Let
$(e_{2j},e_{2j+1},f_{2j},f_{2j+1})$ be the rombs associated to the pairs
$(e_{2j},e_{2j+1})$ of blue edges, for each $j\geq 0$. By definition of
$\sigma$ (Construction 3.34), we have that $\sigma(\xi)=\xi^{\prime}$, with
$\xi^{\prime}=(a_{1},e_{1}^{\prime},e_{2}^{\prime},\dots)$, where
$(e_{2j-1}^{\prime},e_{2j}^{\prime},f_{2j-1},f_{2j})$ are the rombs associated
to the pairs $(f_{2j-1},f_{2j})$ of red edges, for $j\geq 1$, and $a_{1}$ is
the unique blue edge such that $s(a_{1})=r(e_{1}^{\prime})=r(f_{1})=s(f_{0})$.

Note that, for $j\geq 1$, the edges $e_{2j-1}$ and $e_{2j}$ correspond to the
edges

$[\xi^{2j-1}\overset{e_{0}}{\longleftarrow}(e_{0}^{-1}.\,\xi)^{2j-1}]\quad\text{and}\quad[(e_{0}^{-1}.\,\xi)^{2j}\overset{e_{0}^{-1}}{\longleftarrow}\xi^{2j}]$

in $(\theta^{(E,C)})^{[2j-1]}$ and $(\theta^{(E,C)})^{[2j]}$ under the
isomorphisms $\Phi_{2j-1}$ and $\Phi_{2j}$, respectively. Similarly, the edges
$f_{2j-1}$ and $f_{2j}$ correspond to the edges

$[\xi^{2j-1}\overset{f_{0}}{\longleftarrow}(f_{0}^{-1}.\,\xi)^{2j-1}]\quad\text{and}\quad[(f_{0}^{-1}.\,\xi)^{2j}\overset{f_{0}^{-1}}{\longleftarrow}\xi^{2j}]$

in $(\theta^{(E,C)})^{[2j-1]}$ and $(\theta^{(E,C)})^{[2j]}$ under the
isomorphisms $\Phi_{2j-1}$ and $\Phi_{2j}$, respectively. It follows that the
edges $e^{\prime}_{2j-1}$ and $e^{\prime}_{2j}$ correspond to the edges

$[(a_{1}f_{0}^{-1}.\,\xi)^{2j-1}\overset{a_{1}}{\longleftarrow}(f_{0}^{-1}.\,\xi)^{2j-1}]\quad\text{and}\quad[(f_{0}^{-1}.\,\xi)^{2j}\overset{a_{1}^{-1}}{\longleftarrow}(a_{1}f_{0}^{-1}.\,\xi)^{2j}]$

in $(\theta^{(E,C)})^{[2j-1]}$ and $(\theta^{(E,C)})^{[2j]}$, respectively.
Therefore $\xi^{\prime}=(a_{1},e_{1}^{\prime},e_{2}^{\prime},\dots)$
corresponds to $(a_{1}f_{0}^{-1}).\xi=\theta_{a_{1}f_{0}^{-1}}(\xi)$, that is,
$\sigma(\xi)=\theta_{a_{1}f_{0}^{-1}}(\xi)$. ∎

With the help of the above proposition, we can give a more concrete form of a
generalized finite shift.

######  Construction 5.19.

Let $(E,C)$ be a generalized finite shift graph, and set $R=\bigsqcup_{v\in
E^{0,0}}R_{v}$. For each $f\in R$, consider a symbol $a_{f}$, and form the
alphabet $A=\\{a_{f}\mid f\in R\\}$. Moreover, let $s,r\colon A\to E^{0,0}$ be
the maps defined by $s(a_{f})=r(f)$ and $r(a_{f})=r(e)$, where $e$ is the
unique blue edge in $E^{1}$ such that $s(e)=s(f)$. Let $\mathfrak{X}$ be the
space of configurations $\xi\subseteq\mathbb{F}(A)$ on the free group
$\mathbb{F}(A)$ on $A$ such that

  1. (a)

$1\in\xi$;

  2. (b)

$\xi$ is right convex;

  3. (c)

for each $\alpha\in\xi$, the local configuration of $\xi$ at $\alpha$ is of
the following form: there exists a vertex $v\in E^{0,0}$ such that

$(\alpha.\xi)^{1}=\\{1\\}\cup\\{a_{f}\\}\cup\\{a_{f_{i}}^{-1}\mid 1\leq i\leq
n\\},$

where $f\in R_{v}$ and there is a blue edge $e\in B_{v}$ such that
$\\{f_{1},\dots,f_{n}\\}$ is the set of all red edges $g\in R$ such that
$s(g)=s(e)$.

The local homeomorphism $\tau$ on $\mathfrak{X}$ is defined by

$\tau(\xi)=\xi\cdot a_{f}^{-1},$

where $f$ is the unique red edge such that $a_{f}\in\xi^{1}$.

Observe that any configuration $\xi\in\mathfrak{X}$ contains only elements of
the form

$a_{f_{1}}^{-1}\cdots a_{f_{m}}^{-1}a_{g_{n}}\cdots a_{g_{1}}$

for $n,m\geq 0$, where $f_{i},g_{j}\in R$, $f_{m}\neq g_{n}$,
$s(f_{m})=s(g_{n})$, $r(g_{j+1})=r(e_{j})$, where $e_{j}$ is the unique blue
edge such that $s(e_{j})=s(g_{j})$ for $1\leq j\leq n-1$, and
$r(f_{i+1})=r(e_{i}^{\prime})$, where $e_{i}^{\prime}$ is the unique blue edge
such that $s(e_{i}^{\prime})=s(f_{i})$ for $1\leq i\leq m-1$.

Note that, for $\xi\in\mathfrak{X}$, and with $\\{f_{1},\dots,f_{n}\\}$ the
red edges of $E$ such that $a_{f_{i}}^{-1}\in\xi^{1}$ for all $1\leq i\leq n$,
we have $\xi\cdot a_{f_{i}}\in\mathfrak{X}$, and $\tau(\xi\cdot
a_{f_{i}})=\xi$. Hence $\tau$ is surjective, but not necessarily injective.

######  Proposition 5.20.

Let $(E,C)$ be a generalized finite shift graph. Then the dynamical systems
$(X,\sigma)$ and $(\mathfrak{X},\tau)$ are topologically conjugate.

###### Proof.

This follows from Proposition 5.18. Indeed, if $\xi\in\Omega(E,C)$ is a
configuration whose local configuration $\xi_{1}$ at $1$ is of the form (c.2),
then replacing the arrows $ef^{-1}$, where $f$ is the unique red edge in
$E^{1}$ such that $f^{-1}\in(\alpha\,.\,\xi)^{1}$, and $e$ is the unique blue
edge such that $s(e)=s(f)$, by $a_{f}$, for each $\alpha\in\xi$ of even
length, we obtain a configuration in $\mathfrak{X}$. And clearly, the process
can be reversed. This also gives the desired intertwining between $\sigma$ and
$\tau$. ∎

We now consider a concrete example of a generalized finite shift which is
neither a one-sided nor a two-sided shift.

######  Example 5.21.

Let $(E,C)$ be the generalized finite shift graph with associated red-blue
adjacency matrices

$A=\begin{pmatrix}2\\\ 1\end{pmatrix}\quad\text{and}\quad
I=\begin{pmatrix}1\\\ 1\end{pmatrix}.$

Concretely, we set $E^{0,0}=\\{v\\}$, $E^{0,1}=\\{w_{1},w_{2}\\}$,
$C_{v}=\\{B_{v},R_{v}\\}$, where $B_{v}=\\{\alpha_{0},\alpha_{1}\\}$,
$R_{v}=\\{\beta_{0},\beta_{1},\beta_{2}\\}$, with all edges having range $v$,
and with $s(\alpha_{0})=s(\beta_{0})=s(\beta_{1})=w_{1}$,
$s(\alpha_{1})=s(\beta_{2})=w_{2}$.

$w_{1}$$w_{2}$$v$$\alpha_{0}$$\beta_{0}$$\beta_{1}$$\alpha_{1}$$\beta_{2}$
Figure 9. The generalized finite shift graph $(E,C)$.

The possible local configurations are of the following form.

$\alpha_{0}$$\alpha_{1}$$\beta_{2}$$\beta_{0}$$\beta_{1}$ (a) Local
configurations of type (c.1).

$\alpha_{0}$$\alpha_{0}$$\alpha_{0}$$\alpha_{1}$$\alpha_{1}$$\alpha_{1}$$\beta_{0}$$\beta_{1}$$\beta_{2}$$\beta_{0}$$\beta_{1}$$\beta_{2}$
(b) Local configurations of type (c.2).

Figure 10. The different possible local configurations for the graph $(E,C)$.

In particular, the possible local configurations at $1$ are the ones shown in
Figure 10 (B). In Figure 11 below we give an example of the local description
of the shift map $\sigma$.

$1$$1$$\alpha_{0}$$\alpha_{0}$$\alpha_{0}$$\alpha_{0}$$\alpha_{0}$$\alpha_{1}$$\beta_{1}$$\beta_{0}$$\beta_{2}$$\beta_{0}$$\beta_{1}$$\beta_{0}$$\beta_{1}$$\beta_{0}$$\beta_{1}$$\beta_{1}$$\beta_{0}$$\alpha_{0}$$\alpha_{0}$$\alpha_{0}$$\alpha_{0}$$\alpha_{1}$$\alpha_{0}$$\beta_{0}$$\beta_{2}$$\beta_{0}$$\beta_{1}$$\beta_{0}$$\beta_{1}$$\beta_{0}$$\beta_{1}$$\beta_{1}$$\beta_{0}$$\beta_{1}$$\sigma$$\xi$$\sigma(\xi)$
Figure 11. Description of the shift map $\sigma$ on $X\subseteq\Omega(E,C)$.
Here the configuration $\xi$ is given by
$\xi=\\{1,\alpha_{0}^{-1},\beta_{1}^{-1},\beta_{0}\alpha_{0}^{-1},\beta_{1}\alpha_{0}^{-1},\beta_{0}\beta_{1}^{-1},\alpha_{0}\beta_{1}^{-1},\dots\\}$,
which has local configuration at $1$ given by
$\\{\alpha_{0}^{-1},\beta_{1}^{-1}\\}$. The shifted configuration
$\sigma(\xi)$ has local configuration at $1$ given by
$\\{\alpha_{0}^{-1},\beta_{0}^{-1}\\}$.

Taking $a=\alpha_{0}\beta_{0}^{-1}$, $b=\alpha_{0}\beta_{1}^{-1}$,
$c=\alpha_{1}\beta_{2}^{-1}$, we can further describe the generalized finite
shift $(X,\sigma)$ associated with $(E,C)$ as follows.

Let $\mathbb{F}$ be the free group on $\\{a,b,c\\}$. Then $X$ is identified as
the set $\mathfrak{X}$ of all configurations $\xi\subseteq\mathbb{F}$
satisfying the conditions (a)–(c) in Construction 5.19. In particular, the
local configuration of $\xi$ at each point is either of the form
$\\{z,a^{-1},b^{-1}\\}$ or of the form $\\{z,c^{-1}\\}$, where
$z\in\\{a,b,c\\}$. The local homeomorphism $\sigma$ is identified with $\tau$,
which consists in shifting one step in the direction pointed by the directed
edges.

$1$$1$$b$$a$$b$$a$$c$$a$$b$$a$$b$$b$$a$$b$$a$$b$$a$$c$$a$$b$$a$$b$$b$$a$$\tau$$\xi$$\tau(\xi)$
Figure 12. Description of the shift map $\tau$ on $\mathfrak{X}$. Here the
configuration $\xi$ is given by
$\xi=\\{1,b,a^{-1},b^{-1},ab,a^{-1}b,a^{-2},b^{-1}a^{-1},b^{-2},a^{-1}b^{-1},\dots\\}$,
which has local configuration at $1$ given by $\\{b,a^{-1},b^{-1}\\}$. The
shifted configuration $\sigma(\xi)$ has local configuration at $1$ given by
$\\{a,a^{-1},b^{-1}\\}$. The picture conveys the same idea of a river basin
present for instance in [30, p. 5041]. From any vertex there is only one way
forward, and this fact enables us to define the shift map.

We now establish the universal property of the dynamical system $(X,\sigma)$
associated to a generalized finite shift graph $(E,C)$. Given such a graph, we
set $B:=\bigsqcup_{v\in E^{0,0}}B_{v}$ and $R:=\bigsqcup_{v\in E^{0,0}}R_{v}$.

######  Definition 5.22.

Let $(E,C)$ be a generalized finite shift graph with red adjacency matrix
$A=(a_{e,v})\in M_{B\times E^{0,0}}({\mathbb{Z}}^{+})$, where as usual
$a_{e,v}$ is the number of red edges from $s(e)$ to $v$. Let $(Y,\rho)$ be an
object from LHomeo. An $(E,C)$-structure on $(Y,\rho)$ is given by a
$\rho$-cylinder decomposition

${\pazocal Z}=(\\{Z_{e}\mid e\in B\\},\\{W_{e,v}^{(k)}\mid e\in B,v\in
E^{0,0},1\leq k\leq a_{e,v}\\}).$

That is, we have

$Y=\bigsqcup_{e\in B}Z_{e},\qquad W_{v}:=\bigsqcup_{e\in
B_{v}}Z_{e}=\bigsqcup_{e\in B}\bigsqcup_{k=1}^{a_{e,v}}W_{e,v}^{(k)},$

$\rho|_{W_{e,v}^{(k)}}$ is injective, and $\rho(W_{e,v}^{(k)})=Z_{e}$ for all
allowable indices $e,v,k$.

We say that an $(E,C)$-structure ${\pazocal
Z}=(\\{Z_{e}\\},\\{W_{e,v}^{(k)}\\})$ on $(Y,\rho)$ is universal if for any
other $(E,C)$-structure $\tilde{{\pazocal
Z}}=(\\{\tilde{Z}_{e}\\},\\{\tilde{W}_{e,v}^{(k)}\\})$ on
$(\tilde{Y},\tilde{\rho})\in\textsf{LHomeo}$, there is a unique
$(E,C)$-equivariant continuous map $\psi\colon\tilde{Y}\to Y$. Here
$(E,C)$-equivariant means that $\psi(\tilde{Z}_{e})\subseteq Z_{e}$ and
$\psi(\tilde{W}_{e,v}^{(k)})\subseteq W_{e,v}^{(k)}$ for all allowable indices
$e,v,k$, and that $\psi\circ\tilde{\rho}=\rho\circ\psi$.

By Propositions 5.11 and 5.8, every generalized finite shift $(X,\sigma)$,
associated to a generalized finite shift graph $(E,C)$, has a natural
$(E,C)$-structure. It turns out that this structure is universal.

######  Theorem 5.23.

Let $(X,\sigma)$ be the generalized finite shift associated to the generalized
finite shift graph $(E,C)$. Then the natural $(E,C)$-structure on $X$ is
universal.

###### Proof.

The result follows from the universality of the $(E,C)$-dynamical system
$\Omega(E,C)$ ([10, Corollary 6.11]). We thus recall the definition of this
dynamical system. We will identify $X$ as a subspace of $\Omega(E,C)$
throughout the proof, as in Proposition 5.18.

The $(E,C)$-dynamical system on $\Omega(E,C)$ consists of the collection of
open compact subsets $\Omega(E,C)_{v}$, for $v\in E^{0}$, and the open compact
subsets $H_{e}$ and $H_{f}$, for $e\in B$ and $f\in R$, where

$H_{e}=\\{\xi\in\Omega(E,C)\mid e^{-1}\in\xi_{1}\\},\qquad
H_{f}=\\{\xi\in\Omega(E,C)\mid f^{-1}\in\xi_{1}\\},$

together with the homeomorphisms $\theta_{e}\colon\Omega(E,C)_{s(e)}\to H_{e}$
and $\theta_{f}\colon\Omega(E,C)_{s(f)}\to H_{f}$, which we defined just after
Definition 5.16. Note that for each $v\in E^{0,0}$, we have
$\Omega(E,C)_{v}=\bigsqcup_{e\in B_{v}}H_{e}=\bigsqcup_{f\in
R_{v}}H_{f}\subseteq X$.

Let us also identify the natural $(E,C)$-structure on $(X,\sigma)$, adopting
the picture of $(X,\sigma)$ from Proposition 5.18. We have that the space
corresponding to $e\in B$ is just $H_{e}=Z(e)$, and the space corresponding to
$f\in R$ is just $H_{f}$. Of course we have $X=\bigsqcup_{v\in
E^{0,0}}\Omega(E,C)_{v}$ and $\Omega(E,C)_{v}=\bigsqcup_{e\in B_{v}}H_{e}$ for
each $v\in E^{0,0}$. We also have

$\Omega(E,C)_{v}=\bigsqcup_{f\in R_{v}}H_{f}=\bigsqcup_{e\in
B}\bigsqcup_{k=1}^{a_{e,v}}H_{e,v}^{(k)},$

where $H_{e,v}^{(k)}:=H_{f_{e,v}^{(k)}}$, since for each $f\in R_{v}$ there is
a unique $e\in B$ such that $s(e)=s(f)$, and then there is a unique $1\leq
k\leq a_{e,v}$ such that $f=f_{e,v}^{(k)}$. By the proof of Proposition 5.18,
the space $H_{e,v}^{(k)}$ agrees with the space $V_{e,v}^{(k)}$ defined just
before Proposition 5.5, namely the space of all $x=(e_{0},e_{1},\dots)\in X$
such that the romb corresponding to the pair of blue edges $(e_{0},e_{1})$ is
of the form $(e_{0},e_{1},f_{0},f_{1})$ with $f_{0}=f_{e,v}^{(k)}$. Hence, the
$(E,C)$-structure ${\pazocal H}:=(\\{H_{e}\\},\\{H_{e,v}^{(k)}\\})$ on
$(X,\sigma)$ agrees with the previously defined $\sigma$-cylinder
decomposition ${\pazocal Z}_{0}(F,D)$, where $(F,D)$ is the canonical
resolution of $(E,C)$.

Now let ${\pazocal Z}=(\\{Z_{e}\\},\\{W_{e,v}^{(k)}\\})$ be an
$(E,C)$-structure on $(Y,\rho)\in\textsf{LHomeo}$. We need to show that there
is a unique $(E,C)$-equivariant continuous map $\psi\colon Y\to X$. We first
build an $(E,C)$-dynamical system

$(\\{\overline{Y}_{v}\mid v\in E^{0}\\},\\{\overline{H}_{x}\mid x\in
E^{1}\\},\\{\overline{\theta}_{x}\mid x\in E^{1}\\})$

on $\overline{Y}=Y^{0}\sqcup Y^{1}$, where $Y^{0}$ and $Y^{1}$ are disjoint
copies of $Y$. For $i=0,1$ we denote by $y^{i}$ the canonical image of $y\in
Y$ in $Y^{i}$, and by $N^{i}$ the canonical image of a subset $N$ of $Y$ in
$Y^{i}$, with $W^{(k,i)}_{e,v}:=(W^{(k)}_{e,v})^{i}$.

First we set $\overline{Y}_{v}=\bigsqcup_{e\in B_{v}}Z_{e}^{0}$ for $v\in
E^{0,0}$, and $\overline{Y}_{w}=Z_{e}^{1}$ for $w\in E^{0,1}$, where $e$ is
the unique blue edge such that $s(e)=w$. Now we set
$\overline{H}_{e}=Z^{0}_{e}$ for $e\in B$, and
$\overline{H}_{f}=W_{e,v}^{(k,0)}$ for $f\in R$ with $r(f)=v$, where $e\in B$
is the unique blue edge such that $s(e)=s(f)$, and $k$ is the unique index in
$\\{1,\dots,a_{e,v}\\}$ such that $f=f_{e,v}^{(k)}$. Finally we define

$\overline{\theta}_{e}\colon\overline{Y}_{s(e)}\to\overline{H}_{e},\qquad\overline{\theta}_{e}(z^{1})=z^{0},\text{
for }z\in Z_{e},$

and

$\overline{\theta}_{f}\colon\overline{Y}_{s(f)}\to\overline{H}_{f},\qquad\overline{\theta}_{f}(z^{1})=\Big{(}\rho|_{W_{e,v}^{(k)}}\Big{)}^{-1}(z)^{0},\text{
for }z\in Z_{e},\text{ where }f=f_{e,v}^{(k)}.$

By [10, Corollary 6.11], there exists a unique $(E,C)$-equivariant continuous
map $\overline{\psi}\colon\overline{Y}\to\Omega(E,C)$. This means that
$\overline{\psi}(\overline{Y}_{v})\subseteq\Omega(E,C)_{v}$ for all $v\in
E^{0}$, $\overline{\psi}(\overline{H}_{x})\subseteq H_{x}$ for all $x\in
E^{1}$, and that
$\overline{\psi}(\overline{\theta}_{x}(y))=\theta_{x}(\overline{\psi}(y))$ for
all $y\in\overline{Y}_{s(x)}$, $x\in E^{1}$. Note that, for $y^{0}\in Y^{0}$,
we have $\overline{\psi}(y^{0})\in\Omega(E,C)_{v}$ for some $v\in E^{0,0}$,
and hence $\overline{\psi}(y^{0})\in X$. Thus the map

$\psi\colon Y\to X,\quad\psi(y)=\overline{\psi}(y^{0})$

is well-defined. We have

$\psi(Z_{e})=\overline{\psi}(Z_{e}^{0})=\overline{\psi}(\overline{H}_{e})\subseteq
H_{e},\qquad\psi(W_{e,v}^{(k)})=\overline{\psi}(W_{e,v}^{(k,0)})=\overline{\psi}(\overline{H}_{f_{e,v}^{(k)}})\subseteq
H_{e,v}^{(k)}.$

We finally check that $\psi\circ\rho=\sigma\circ\psi$. For this it is enough
to check that
$\psi\circ\rho|_{W_{e,v}^{(k)}}=\sigma|_{H_{e,v}^{(k)}}\circ\psi$ for all
$e\in B$, $v\in E^{0,0}$ and $1\leq k\leq a_{e,v}$. For $z\in Z_{e}$, we
compute

$\displaystyle\psi((\rho|_{W_{e,v}^{(k)}})^{-1}(z))$
$\displaystyle=\overline{\psi}((\rho|_{W_{ev}^{(k)}})^{-1}(z)^{0})$
$\displaystyle=\overline{\psi}(\overline{\theta}_{f_{e,v}^{(k)}}(z^{1}))$
$\displaystyle=\theta_{f_{e,v}^{(k)}}(\overline{\psi}(z^{1}))$
$\displaystyle=\theta_{f_{e,v}^{(k)}}(\overline{\psi}(\overline{\theta}_{e}^{-1}(z^{0})))$
$\displaystyle=\theta_{f_{e,v}^{(k)}}\theta_{e}^{-1}(\overline{\psi}(z^{0}))=(\sigma|_{H^{(k)}_{e,v}})^{-1}(\psi(z)),$

where we have used Proposition 5.18 in the last equality. Hence $\psi$ is an
$(E,C)$-equivariant continuous map from $Y$ to $X$. Uniqueness of $\psi$
follows from uniqueness of $\overline{\psi}$. Indeed, the universality of the
$(E,C)$-dynamical system $\Omega(E,C)$ turns out to be equivalent to the
universality of the natural $(E,C)$-structure on $X$. ∎

######  Remark 5.24.

For a dynamical system $(Y,\rho)\in\textsf{LHomeo}$ with an $(E,C)$-structure
$(\\{\overline{H}_{e}\\}_{e\in B},\\{\overline{H}_{f}\\}_{f\in R})$, the
unique $(E,C)$-equivariant map $\psi\colon Y\to X$ from Theorem 5.23 can be
explicitly described. Indeed, observe that we have decompositions

$Y=\bigsqcup_{e\in B}\overline{H}_{e}=\bigsqcup_{f\in R}\overline{H}_{f}$

and homeomorphisms

$\rho_{f}\colon\overline{H}_{f}\to\overline{H}_{e},$

for each $f\in R$, where $e$ is the unique blue edge such that $s(e)=s(f)$.
Then, for $y\in Y$, $\psi(y)$ consists of the set of elements

$a_{f_{1}}^{-1}\cdots a_{f_{m}}^{-1}a_{g_{n}}\cdots
a_{g_{1}}\in\mathbb{F}(A),$

in the notation of Construction 5.19, such that

$y\in\text{Dom}(\rho_{f_{1}}^{-1}\circ\cdots\circ\rho_{f_{m}}^{-1}\circ\rho_{g_{n}}\circ\cdots\circ\rho_{g_{1}}),$

where the above is a composition of partial maps (see e.g. [32, Chapter 2]).

We end this section by proving that any $(X,\sigma)\in\textsf{LHomeo}$ can be
expressed as an inverse limit of a sequence of generalized finite shifts.

Let $(X,\sigma)\in\textsf{LHomeo}$, and fix an $l$-diagram $(F,D)$
corresponding to $(X,\sigma)$ under the bijection established in Theorem 3.39.
For each $i\geq 0$, write $E(i):=(F^{0,2i}\cup F^{0,2i+1},F^{1,2i})$ and let
$C(i)$ be the partition of $E(i)^{1}$ inherited from $D$. Then, by Proposition
5.8, $(E(i),C(i))$ is a generalized finite shift graph. We let
$(X_{i},\sigma_{i})$ be the generalized finite shift (Definition 5.12)
associated to the canonical resolution $(F(i),D(i))$ of the generalized finite
shift graph $(E(i),C(i))$. Moreover, by Proposition 5.8, $(X,\sigma)$ admits
an $(E(i),C(i))$-structure for each $i\geq 0$. It follows from Theorem 5.23
that there is a unique continuous $(E(i),C(i))$-equivariant map
$\psi_{i}\colon(X,\sigma)\to(X_{i},\sigma_{i})$ for each $i\geq 0$, where we
consider the natural $(E(i),C(i))$-structure on $(X_{i},\sigma_{i})$.

######  Proposition 5.25.

The space $(X_{i+1},\sigma_{i+1})$ admits an $(E(i),C(i))$-structure, in such
a way that the natural $(E(i+1),C(i+1))$-structure on $(X_{i+1},\sigma_{i+1})$
is a refinement (in the sense of Definition 5.2) of the mentioned
$(E(i),C(i))$-structure.

In particular, the map $\psi_{i}$ factors through $\psi_{i+1}$.

###### Proof.

Let $(G,H)$ be the Bratteli diagram whose first two layers consist of the
layers $E(i)=(F^{0,2i}\cup F^{0,2i+1},F^{1,2i})$ and $(F^{0,2i+1}\cup
F^{0,2i+2},F^{1,2i+1})$, respectively, with the separations inherited from
$D$, and whose other layers coincide with the layers of $(F(i+1),D(i+1))$,
moved down in two units, so that for instance $G^{0,k}=F(i+1)^{0,k-2}$ for all
$k\geq 2$. Using Proposition 5.11 it is easy to show that $(G,H)$ is an
$l$-diagram. By Theorem 3.36, we have that the dynamical system associated to
the $l$-diagram $(G,H)$ is conjugate to $(X_{i+1},\sigma_{i+1})$. In
particular, we see from Proposition 5.8 that $(X_{i+1},\sigma_{i+1})$ admits
an $(E(i),C(i))$-structure, and moreover the natural
$(E(i+1),C(i+1))$-structure on $(X_{i+1},\sigma_{i+1})$ is a refinement of the
$(E(i),C(i))$-structure.

Now Theorem 5.23 provides us with a unique equivariant continuous map

$\psi_{i,i+1}\colon X_{i+1}\to X_{i},$

with respect to the respective $(E(i),C(i))$-structures. Since the three maps
$\psi_{i}$, $\psi_{i+1}$ and $\psi_{i,i+1}$ preserve the
$(E(i),C(i))$-structure, we get by uniqueness that
$\psi_{i}=\psi_{i,i+1}\circ\psi_{i+1}$. This concludes the proof of the
proposition. ∎

We therefore have an inverse system $\\{(X_{i},\sigma_{i}),\psi_{i,j}\\}$ in
LHomeo, and hence we get a dynamical system $(\varprojlim
X_{i},\tilde{\sigma})$ together with an equivariant continuous map

$\psi\colon(X,\sigma)\to(\varprojlim
X_{i},\tilde{\sigma}),\quad\psi(x)=(\psi_{0}(x),\psi_{1}(x),\dots).$

######  Theorem 5.26.

With the notation as above, we have that the natural equivariant map

$\psi\colon X\to\varprojlim X_{i}$

is a homeomorphism.

###### Proof.

Let $x=(e_{0},e_{1},e_{2},\dots)\in X$, where all $e_{i}\in F^{1,i}$ are blue
edges. Since $\psi_{i}$ is $(E(i),C(i))$-equivariant, we have

$\psi_{i}(x)=(e_{2i},\tilde{e}_{i,1},\tilde{e}_{i,2},\dots),$

where $\tilde{e}_{i,j}\in F(i)^{1,j}$ for $j\geq 1$. It follows that if
$x=(e_{i}),x^{\prime}=(e^{\prime}_{i})\in X$ and $\psi(x)=\psi(x^{\prime})$,
then $e_{2i}=e^{\prime}_{2i}$ for all $i\geq 0$. But then
$e_{2i+1}=e^{\prime}_{2i+1}$ for all $i\geq 0$, since there is a unique blue
edge in $F$ joining $r(e_{2i+2})$ with $s(e_{2i})$. It follows that $\psi$ is
injective.

We now show that $\psi$ is surjective. Let
$(x_{0},x_{1},x_{2},\dots)\in\varprojlim X_{i}$. Write

$x_{i}=(e_{2i},\tilde{e}_{i,1},\tilde{e}_{i,2},\dots),$

where $e_{2i}\in F^{1,2i}$ and $\tilde{e}_{i,j}\in F(i)^{1,j}$ for all $j\geq
1$. Since $\psi_{i,i+1}(x_{i+1})=x_{i}$ and all the maps $\psi_{i,i+1}$ are
$(E(i),C(i))$-equivariant, it follows by a simple induction that there is a
unique blue path

$x=(e_{0},e_{1},e_{2},\dots,e_{2i},e_{2i+1},e_{2i+2},\dots)\in X.$

We show that $\psi(x)=(x_{0},x_{1},x_{2},\dots)$. For this we need the
description of the maps $\psi_{j}$ given in Remark 5.24. We will concentrate
in showing that $\psi_{0}(x)=x_{0}$. Consider some $i\geq 0$. Then the first
$2i$ coordinates of $\psi_{0}(x)$ are determined by the set of all elements

$a_{f_{1}}^{-1}\cdots a_{f_{m}}^{-1}a_{g_{n}}\cdots a_{g_{1}}\in\mathbb{F}(A)$

where $f_{1},\dots,f_{m},g_{1},\dots,g_{n}$ are red edges in $E(0)$ satisfying
the requirements given in Construction 5.19, such that

$x\in\text{Dom}(\theta_{f_{1}}^{-1}\circ\cdots\circ\theta_{f_{m}}^{-1}\circ\theta_{g_{n}}\circ\cdots\circ\theta_{g_{1}})$

and with $n+m\leq i$. Here $\\{\theta_{f}\\}$ is the set of partial maps as in
Remark 5.24 associated to the $(E(0),C(0))$-structure on $X$. Now it follows
from the construction of $\sigma$ (given in Construction 3.34) and from the
proof of surjectivity of $\sigma$ (given in Lemma 3.35) that this information
is contained in the blue path $(e_{0},e_{1},\dots,e_{2i-1})$, which is the
initial segment of $x$ of length $2i$. Note that, by the same argument as
before and the definition of the maps $\psi_{j,j+1}$, we see that, for $y\in
X_{i}$, the first $2i$ components of $\psi_{0,i}(y)$ are determined by the
unique blue path $(e_{0}^{\prime},e_{1}^{\prime},\dots,e^{\prime}_{2i-1})$
joining $r(y)\in F^{0,{2i}}$ with a vertex in $F^{0,0}$ in the graph $F$,
because the $(E(0),C(0))$-structure on $X_{i}$ is determined by the canonical
$(E(0),C(0))$-structure on the $l$-diagram obtained by topping $(F(i),D(i))$
with the first $2i$ layers of $(F,D)$. Since $\psi_{i}(x)=(e_{2i},\dots)$ and
$x_{i}=(e_{2i},\dots)$, that unique blue path is precisely
$(e_{0},e_{1},\dots,e_{2i-1})$ for both $\psi_{i}(x)$ and $x_{i}$, and thus
the first $2i$ components of $\psi_{0,i}(\psi_{i}(x))$ agree with the first
$2i$ components of $\psi_{0,i}(x_{i})$. But
$\psi_{0,i}(\psi_{i}(x))=\psi_{0}(x)$ and $\psi_{0,i}(x_{i})=x_{0}$, hence the
first $2i$ coordinates of $\psi_{0}(x)$ must agree with the first $2i$
coordinates of $x_{0}$. Since this holds for all $i\geq 0$, we get that
$\psi_{0}(x)=x_{0}$.

In the same way we may prove that $\psi_{n}(x)=x_{n}$ for all $n\geq 1$,
completing the proof. ∎

##  6\. Approximating algebras associated to a dynamical system

Let $(K,\ast)$ be a field endowed with an involution $\ast$. In this section
we analyze both the Steinberg algebra $A_{K}({\pazocal G})$ and the
$C^{*}$-algebra $C^{*}({\pazocal G})$ of the Deaconu-Renault groupoid
${\pazocal G}={\pazocal G}(X,\sigma)$ associated to an object
$(X,\sigma)\in\textsf{LHomeo}$. We will describe approximating processes of
such algebras by means of the corresponding algebras associated with the
sequence of generalized finite shifts $(X_{i},\sigma_{i})$ described at the
end of Section 5.

###  6.1. The general approximation construction

Fix $(X,\sigma)\in\textsf{LHomeo}$ and a $\sigma$-refined sequence of
partitions $\\{{\pazocal P}_{n}\\}_{n\geq 0}$ of $X$. Let $(F,D)$ be the
$l$-diagram associated to $\\{{\pazocal P}_{n}\\}_{n\geq 0}$, see Construction
3.24. We adopt the terminology and notation introduced in Section 5.

For each $n\geq 0$ we define bipartite finite separated graphs $(F_{n},D^{n})$
as follows:

$F_{n}^{0}:=F_{n}^{0,0}\sqcup F_{n}^{0,1}\text{ with
}F_{n}^{0,0}:=F^{0,n}\text{ and }F_{n}^{0,1}:=F^{0,n+1},$
$F_{n}^{1}:=F^{1,n},$

and $D_{v}^{n}:=D_{v}$ for $v\in F_{n}^{0,0}$. Note that
$r(F_{n}^{1})=F_{n}^{0,0}$ and $s(F_{n}^{1})=F_{n}^{0,1}$. With this notation,
the generalized finite shift graphs $(E(n),C(n))$ introduced in Section 5 are
precisely $(F_{2n},C^{2n})$.

For notational convenience, we write $L_{n}:=L_{K}(F_{n},D^{n})$, the Leavitt
path algebra of the separated graph $(F_{n},D^{n})$. Also $p_{n}$ will denote
the projection in $L_{n}$ on $F_{n}^{0,0}$, so that

$p_{n}=\sum_{v\in F_{n}^{0,0}}v.$

######  Lemma 6.1.

Let $j\geq 0$ be an integer.

  1. $i)$

For any vertex $v\in F_{2j}^{0,0}$,

$\textbf{S}(v):=\sum_{e\in B_{v}}\sum_{e^{\prime}\in
B_{s(e)}}s(e^{\prime})=\sum_{f\in R_{v}}\sum_{f^{\prime}\in
R(f)}s(f^{\prime})\in L_{2j+1}.$

  2. $ii)$

For any vertex $v\in F_{2j+1}^{0,0}$ and any $g\in R_{r(g)}$ such that
$s(g)=v$,

$\textbf{S}(v):=\sum_{e\in B_{v}}\sum_{e^{\prime}\in
B_{s(e)}}s(e^{\prime})=\sum_{f\in R(g)}\sum_{f^{\prime}\in
R_{s(f)}}s(f^{\prime})\in L_{2j+2}.$

###### Proof.

Note that in both $i)$ and $ii)$ the sum $\sum_{e\in B_{v}}\sum_{e^{\prime}\in
B_{s(e)}}s(e^{\prime})$ is a sum of distinct vertices in $F^{n+2}$, where $n$
is either $2j$ or $2j+1$.

  1. i)

The set of vertices $s(e^{\prime})$ in the above expression is in bijective
correspondence with the set of pairs $(e,e^{\prime})$, where $e\in B_{v}$ and
$e^{\prime}\in B_{s(e)}$. By Proposition 3.3 (ii), the set of pairs
$(e,e^{\prime})$ where $e\in B_{v}$ and $e^{\prime}\in B_{s(e)}$ is in
bijective correspondence with the set of pairs $(f,f^{\prime})$ where $f\in
R_{v}$, $f^{\prime}\in R(f)$, and $(f,f^{\prime})$ corresponds to
$(e,e^{\prime})$ if and only if $s(f^{\prime})=s(e^{\prime})$. The result
follows.

  2. ii)

The argument is similar to the one given for the first part, taking into
account conditions (f) and (g) in the definition of $l$-diagram (Definition
3.2).∎

######  Remark 6.2.

Note that each $e\in s^{-1}(v)$ determines a unique set $X(e)\in D_{v}$. This
notation is compatible with the notation introduced in Definition 5.10. With
this notation, the formulas in Lemma 6.1 take a uniform shape:

$\textbf{S}(v)=\sum_{e\in X}\sum_{e^{\prime}\in X(e)}s(e^{\prime})$

for any $X\in D_{v}$, $v\in F^{0}$.

######  Proposition 6.3.

There exist unital $*$-homomorphisms $\widetilde{\phi}_{n}\colon
L_{n}\rightarrow L_{n+1}$ such that $\widetilde{\phi}_{n}(p_{n})=p_{n+2}$ and
$\widetilde{\phi}_{n}(p_{n+1})=p_{n+1}$.

###### Proof.

We define $\widetilde{\phi}_{n}:L_{n}\rightarrow L_{n+1}$ by the rules

$\widetilde{\phi}_{n}(\alpha)=\begin{cases}w&\text{ if }\alpha=w\in
F_{n}^{0,1};\\\ \textbf{S}(v)&\text{ if }\alpha=v\in F_{n}^{0,0};\\\
\sum_{\overline{e}\in B_{s(e)}}\overline{e}^{*}&\text{ if }\alpha=e\in
F_{n}^{1}\text{ is a blue edge};\\\ \sum_{\overline{f}\in
R(f)}\overline{f}^{*}&\text{ if }n=2j\text{ and }\alpha=f\in F_{n}^{1}\text{
is a red edge};\\\ \sum_{\overline{f}\in R_{s(f)}}\overline{f}^{*}&\text{ if
}n=2j+1\text{ and }\alpha=f\in F_{n}^{1}\text{ is a red edge}.\end{cases}$

We must check that the defining relations for $L_{n}$ in Definition 2.1 are
satisfied. The relations (V), (E) are easily verified. For (SCK1) take first
$e,f\in B_{v}$, where $v\in F_{n}^{0,0}=F^{0,n}$. We compute

$\begin{split}\widetilde{\phi}_{n}(e)^{*}\widetilde{\phi}_{n}(f)=\Big{(}\sum_{\overline{e}\in
B_{s(e)}}\overline{e}\Big{)}\Big{(}\sum_{\overline{f}\in
B_{s(f)}}\overline{f}^{*}\Big{)}=\sum_{\overline{e}\in
B_{s(e)}}\sum_{\overline{f}\in
B_{s(f)}}\overline{e}\overline{f}^{*}.\end{split}$

The product $\overline{e}\overline{f}^{*}$ is zero unless
$s(\overline{e})=s(\overline{f})$. Since the edges are blue, this implies that
$\overline{e}=\overline{f}$ (see Definition 3.2). In particular,
$s(e)=r(\overline{e})=r(\overline{f})=s(f)$, and again we deduce that $e=f$.
Hence

$\widetilde{\phi}_{n}(e)^{*}\widetilde{\phi}_{n}(f)=\delta_{e,f}\Big{(}\sum_{\overline{e}\in
B_{s(e)}}\overline{e}\overline{e}^{*}\Big{)}=\delta_{e,f}s(e)=\widetilde{\phi}_{n}(\delta_{e,f}s(e)),$

where we have used the relation (SCK2) inside $L_{n+1}$.

Assume now that $n=2j$ for $j\geq 0$ and that $e,f\in R_{v}$ for $v\in
F_{2j}^{0,0}$. Then

$\begin{split}\widetilde{\phi}_{n}(e)^{*}\widetilde{\phi}_{n}(f)=\sum_{\overline{e}\in
R(e)}\sum_{\overline{f}\in R(f)}\overline{e}\overline{f}^{*}.\end{split}$

Again, the product $\overline{e}\overline{f}^{*}$ is zero unless
$s(\overline{e})=s(\overline{f})$. We claim that in that case we necessarily
have $(e,\overline{e})=(f,\overline{f})$. This follows from Proposition 3.3
(ii). Indeed, a first application of this proposition gives that there is a
unique pair $(e_{0},e_{1})$ of blue edges such that $r(e_{0})=r(e)=v$,
$r(e_{1})=s(e_{0})$ and $s(e_{1})=s(\overline{e})$. But now, since we are
assuming that $s(\overline{e})=s(\overline{f})$, we have that
$(e,\overline{e})$ and $(f,\overline{f})$ are two pairs of red edges with
$\overline{e}\in R(e)$ and $\overline{f}\in R(f)$ which correspond to the same
pair $(e_{0},e_{1})$ of blue edges, so the uniqueness part of Proposition 3.3
(ii) gives that $(e,\overline{e})=(f,\overline{f})$, as desired. Hence

$\widetilde{\phi}_{n}(e)^{*}\widetilde{\phi}_{n}(f)=\delta_{e,f}\Big{(}\sum_{\overline{e}\in
R(e)}\overline{e}\overline{e}^{*}\Big{)}=\delta_{e,f}s(e)=\widetilde{\phi}_{n}(\delta_{e,f}s(e)),$

where again we have used the relation (SCK2) inside $L_{n+1}$.

Assume finally that $n=2j+1$ for $j\geq 0$ and that $e,f\in R(g)$ for $g\in
R_{r(g)}$ such that $s(g)\in F_{2j+1}^{0,0}$. Then we have

$\begin{split}\widetilde{\phi}_{n}(e)^{*}\widetilde{\phi}_{n}(f)=\sum_{\overline{e}\in
R_{s(e)}}\sum_{\overline{f}\in
R_{s(f)}}\overline{e}\overline{f}^{*}.\end{split}$

Again, the product $\overline{e}\overline{f}^{*}$ is zero unless
$s(\overline{e})=s(\overline{f})$. We claim that, as in the previous case, we
necessarily have $(e,\overline{e})=(f,\overline{f})$. The argument is similar
to the above, now using conditions (f) and (g) in Definition 3.2. Namely, by
(f) there is a unique blue pair $(e_{0},e_{1})$ such that
$(e_{0},e_{1},e,\overline{e})$ is a romb. Now by the uniqueness part of (g)
(and since $e,f\in R(g)$), we deduce that $(e,\overline{e})=(f,\overline{f})$.
Hence

$\widetilde{\phi}_{n}(e)^{*}\widetilde{\phi}_{n}(f)=\delta_{e,f}\Big{(}\sum_{\overline{e}\in
R_{s(e)}}\overline{e}\overline{e}^{*}\Big{)}=\delta_{e,f}s(e)=\widetilde{\phi}_{n}(\delta_{e,f}s(e))$

again by the relation (SCK2) inside $L_{n+1}$.

For (SCK2) note that, for $e\in B_{v}$, $v\in F_{n}^{0,0}$, we have

$\widetilde{\phi}_{n}(e)\widetilde{\phi}_{n}(e)^{*}=\sum_{f,g\in
B_{s(e)}}f^{*}g=\sum_{f\in B_{s(e)}}s(f),$

so that

$\sum_{e\in
B_{v}}\widetilde{\phi}_{n}(e)\widetilde{\phi}_{n}(e)^{*}=\sum_{e\in
B_{v}}\sum_{f\in B_{s(e)}}s(f)=\textbf{S}(v)=\widetilde{\phi}_{n}(v)$

because of Lemma 6.1. Similarly, we get that (SCK2) is also preserved for red
edges.

This analysis gives the well-definiteness of the maps
$\widetilde{\phi}_{n}\colon L_{n}\rightarrow L_{n+1}$. Now note that

$\widetilde{\phi}(p_{n+1})=\sum_{w\in F_{n}^{0,1}}w=p_{n+1}\quad\text{ and
}\quad\widetilde{\phi}_{n}(p_{n})=\sum_{v\in
F_{n}^{0,0}}\textbf{S}(v)=\sum_{w\in F_{n+1}^{0,1}}w=p_{n+2}.$

The proposition is now proved. ∎

For $m>n$, denote by $\widetilde{\phi}_{n,m}\colon L_{n}\rightarrow L_{m}$ the
composition $\widetilde{\phi}_{m-1}\circ\cdots\circ\widetilde{\phi}_{n}$. Set

$L_{\infty}:=\varinjlim(L_{n},\widetilde{\phi}_{n,m}),$

with canonical maps $\widetilde{\phi}_{n,\infty}\colon L_{n}\to L_{\infty}$.

We will need the following observations.

######  Lemma 6.4.

With the above notation, and with $k=m-n$, the following hold.

  1. i)

If $v_{1},\dots,v_{l}$ are mutually distinct vertices in $F_{n}$, then each of
$\widetilde{\phi}_{n,m}(v_{i})$ is a sum of mutually distinct vertices in
$F_{m}$, and moreover the vertices appearing in the decompositions of two
different vertices $v_{i}$ and $v_{j}$ are also mutually distinct.

  2. ii)

If $f\in F_{n}^{1}$, then $\widetilde{\phi}_{n,m}(f)$ is a sum of edges $g\in
F_{m}^{1}$ whose sources are mutually distinct if $k$ is even, and is a sum of
“ghost” edges $g^{*}$ for edges $g\in F_{m}^{1}$ whose sources are mutually
distinct if $k$ is odd. Moreover, for each $w\in F_{m}^{0,0}$, all the edges
$g\in F_{m}^{1}$ appearing in the above sum with range $w$ belong to the same
set $Z\in D_{w}$.

###### Proof.

The easy proof of $i)$ is left to the reader. For $ii)$, we proceed by
induction on $k$. If $k=1$ then $\widetilde{\phi}_{n}(f)=\sum_{\overline{f}\in
X(f)}\overline{f}^{*}$, where we use the notation introduced in Remark 6.2.
Since $r(\overline{f})=r(\overline{f}^{\prime})$ for all
$\overline{f},\overline{f}^{\prime}\in X(f)$, the result follows from the fact
that, except possibly at the first layer, distinct blue (respectively,
distinct red) edges with the same range must have distinct sources (see
Definition 3.2 and Remark 3.5 (5)).

Now suppose that the result holds for $k\geq 1$. Assume first that $k$ is
even. Then we have that $\widetilde{\phi}_{n,n+k}(f)=\sum_{g\in L}g$, where
$L$ is a set of edges in $F_{n+k}=F_{m}$ whose sources are mutually distinct,
and such that all members of $L$ with the same range belong to the same
element of the partition $D$. Using the notation introduced in Remark 6.2, we
have

$\widetilde{\phi}_{n,n+k+1}(f)=\sum_{g\in
L}\widetilde{\phi}_{n+k}(g)=\sum_{g\in L}\sum_{\overline{g}\in
X(g)}\overline{g}^{*},$

where $X(g)\in D_{s(g)}$. Let $w\in F_{m}^{0}$ and let $X^{w}\in D_{w}$ be the
unique set in the partition $D_{w}$ to which all elements $g$ of $L$ with
range $w$ belong. Using Lemma 6.1 (and Remark 6.2), we have

$\sum_{\\{g\in L\mid r(g)=w\\}}\sum_{\overline{g}\in
X(g)}s(\overline{g})\leq\sum_{g\in X^{w}}\sum_{\overline{g}\in
X(g)}s(\overline{g})=\textbf{S}(w).$

Since in addition $\textbf{S}(w)\cdot\textbf{S}(w^{\prime})=0$ for distinct
$w,w^{\prime}\in F_{m}^{0,0}$, we conclude that
$\\{s(\overline{g})\mid\overline{g}\in X(g),g\in L\\}$ is a set of mutually
distinct vertices in $F_{m+1}^{0,1}$. Moreover, if
$r(\overline{g})=r(\overline{h})$ for $\overline{g}\in X(g)$ and
$\overline{h}\in X(h)$, with $g,h\in L$, then we get

$s(g)=r(\overline{g})=r(\overline{h})=s(h)$

and by induction hypothesis we conclude that $g=h$, so that
$\overline{g},\overline{h}\in X(g)=X(h)$, as wanted.

The case where $k$ is odd is treated in the same way. ∎

We now recall some basic definitions, see [32, Definition 12.9].

######  Definition 6.5.

A set $F$ of partial isometries of a $*$-algebra $A$ is said to be tame if for
all two elements $u,u^{\prime}\in U$, where $U$ is the multiplicative
$*$-subsemigroup of $A$ generated by $F$, we have that $e(u)$ and
$e(u^{\prime})$ are commuting elements of $A$, where $e(v)=vv^{*}$ for $v\in
U$. Note that if $F$ is a tame set of partial isometries, then all elements of
$U$ are indeed partial isometries, and therefore the elements $e(u)$, with
$u\in U$, are mutually commuting projections in $A$.

A $*$-algebra $A$ is said to be tame if it is generated as a $*$-algebra by a
tame set of partial isometries. If $A$ is a $*$-algebra generated by a subset
$F$ of partial isometries, there is a universal tame $*$-algebra
$A^{\text{ab}}$ associated to $F$. By definition, there is a surjective
$*$-homomorphism $\pi\colon A\to A^{\text{ab}}$ so that $\pi(F)$ is a tame set
of partial isometries generating $A^{\text{ab}}$, and such that for any
$*$-homomorphism $\psi\colon A\to B$ of $A$ to a $*$-algebra $B$ such that
$\psi(F)$ is a tame set of partial isometries in $B$, there is a unique
$*$-homomorphism $\overline{\psi}\colon A^{\text{ab}}\to B$ such that
$\psi=\overline{\psi}\circ\pi$. Indeed, we have $A^{\text{ab}}=A/J$, where $J$
is the ideal of $A$ generated by all the commutators $[e(u),e(u^{\prime})]$,
where $u,u^{\prime}\in U$.

If $(E,C)$ is a separated graph, then the universal tame $*$-algebra
associated to the generating set $E^{0}\sqcup E^{1}$ of partial isometries of
$L_{K}(E,C)$ is denoted by $L^{{\rm ab}}_{K}(E,C)$. In addition, we denote by
${\pazocal O}(E,C)$ the $C^{*}$-algebra $C^{*}(E,C)/{\pazocal J}$, where
${\pazocal J}$ is the closed ideal of $C^{*}(E,C)$ generated by all the
commutators $[e(u),e(u^{\prime})]$, for $u,u^{\prime}\in U$, where $U$ is the
multiplicative $*$-subsemigroup of $C^{*}(E,C)$ generated by $E^{0}\cup
E^{1}$. Clearly ${\pazocal O}(E,C)$ is universal with respect to
$*$-homomorphisms from $C^{*}(E,C)$ to a tame $C^{*}$-algebra.

The following lemma will be crucial for our main result. We continue with the
notation introduced at the beginning of this section.

######  Lemma 6.6.

Let $U_{n}$ be the multiplicative $*$-subsemigroup of
$L_{n}=L_{K}(F_{n},D^{n})$ generated by the set $F_{n}^{1}$. Let $\omega$ be a
word in $U_{n}$ of length $k$. Then
$\widetilde{\phi}_{n,n+k}(\omega\omega^{*})$ is a sum of distinct vertices of
$F_{n+k}$.

###### Proof.

We proceed by induction on $k$. If $k=1$, then either $\omega=e$ or
$\omega=e^{*}$ for some $e\in F_{n}^{1}$. In the first case $\omega=e$, we
have

$\widetilde{\phi}_{n}(\omega\omega^{*})=\widetilde{\phi}_{n}(ee^{*})=\Big{(}\sum_{\overline{e}\in
X(e)}\overline{e}^{*}\Big{)}\Big{(}\sum_{\overline{e}^{\prime}\in
X(e)}\overline{e}^{\prime}\Big{)}=\sum_{\overline{e}\in X(e)}s(\overline{e}),$

which is a sum of mutually distinct vertices of $F_{n+1}$. In the second case
$\omega=e^{*}$, we have

$\widetilde{\phi}_{n}(\omega\omega^{*})=\widetilde{\phi}_{n}(e^{*}e)=\widetilde{\phi}_{n}(s(e))=s(e).$

The result follows for $k=1$.

Suppose now that the result is true for words of length $k$ and let $\omega\in
U_{n}$ be a word of length $k+1$. Write $\omega=x\omega_{0}$, where
$\omega_{0}$ is a word of length $k$ and $x$ is a word of length $1$ (so that
$x=f$ or $x=f^{*}$ for $f\in F_{n}^{1}$). By induction hypothesis we can write

$\widetilde{\phi}_{n,n+k}(\omega_{0}\omega_{0}^{*})=\sum_{i=1}^{t}v_{i}$

for mutually distinct vertices $v_{i}$ of $F_{n+k}$. We now use Lemma 6.4
(ii). Suppose first that $\widetilde{\phi}_{n,n+k}(x)=\sum_{g\in L}g^{*}$,
where $L$ is a family of edges in $F_{n+k}$ whose sources are mutually
distinct, such that, for each $w\in F_{n+k}^{0,0}$, all the edges $g\in L$
with range $w$ belong to the same set $Z\in D_{w}$. We then have

$\displaystyle\widetilde{\phi}_{n,n+k}(\omega\omega^{*})$
$\displaystyle=\widetilde{\phi}_{n,n+k}(x)\widetilde{\phi}_{n,n+k}(\omega_{0}\omega_{0}^{*})\widetilde{\phi}_{n,n+k}(x)^{*}$
$\displaystyle=\Big{(}\sum_{g\in
L}g^{*}\Big{)}\Big{(}\sum_{i=1}^{t}v_{i}\Big{)}\Big{(}\sum_{h\in L}h\Big{)}$
$\displaystyle=\sum_{i=1}^{t}\sum_{\\{g,h\in L\mid r(g)=r(h)=v_{i}\\}}g^{*}h$
$\displaystyle=\sum_{i=1}^{t}\sum_{\\{g\in L\mid r(g)=v_{i}\\}}s(g),$

where for the last equality we have used the fact that any two edges $g,h\in
L$ with the same range $w$ belong to the same element of $D_{w}$. Now since
the sources of edges in $L$ are mutually distinct, we see that
$\widetilde{\phi}_{n,n+k}(\omega\omega^{*})$ is a sum of mutually distinct
vertices in $F_{n+k}^{0,1}$. Since
$\widetilde{\phi}_{n,n+k+1}(\omega\omega^{*})=\widetilde{\phi}_{n+k}(\widetilde{\phi}_{n,n+k}(\omega\omega^{*}))$,
we obtain the result in this case by Lemma 6.4 (i).

Suppose now that $\widetilde{\phi}_{n,n+k}(x)$ is a sum of edges in $F_{n+k}$.
Then necessarily

$\widetilde{\phi}_{n,n+k+1}(x)=\widetilde{\phi}_{n+k}(\widetilde{\phi}_{n,n+k}(x))$

will be of the form $\sum_{g\in L}g^{*}$, where $L$ is a family of edges in
$F_{n+k+1}$ whose sources are mutually distinct, such that, for each $w\in
F_{n+k+1}^{0,0}$, all the edges $g\in L$ with range $w$ belong to the same set
$Z\in D_{w}$. Since $\widetilde{\phi}_{n,n+k+1}(\omega_{0}\omega_{0}^{*})$ is
a sum of mutually distinct vertices by induction hypothesis and by Lemma 6.4
(i), the same argument as before gives us the result also in this case. ∎

We are now in a position to show that the algebra $L_{\infty}=\varinjlim
L_{n}$ is tame, and compute it by means of the universal tame $*$-algebras of
$L_{n}$ with respect to the sets $F_{n}$.

######  Theorem 6.7.

The $*$-algebra $L_{\infty}$ is tame, with generating set of partial
isometries
$F_{\infty}^{1}:=\bigcup_{n=0}^{\infty}\widetilde{\phi}_{n,\infty}(F_{n}^{1})$.
Moreover, the natural $*$-homomorphisms $\widetilde{\phi}_{n,\infty}\colon
L_{n}\to L_{\infty}$ factor through $L^{{\rm ab}}_{n}:=L^{{\rm
ab}}_{K}(F_{n},D^{n})$ with homomorphisms $\overline{\phi}_{n,\infty}\colon
L^{{\rm ab}}_{n}\to L_{\infty}$, and we have

$L_{\infty}=\bigcup_{n=0}^{\infty}\widetilde{\phi}_{n,\infty}(L_{n})=\bigcup_{n=0}^{\infty}\overline{\phi}_{n,\infty}(L^{{\rm
ab}}_{n}).$

###### Proof.

Let $n,n^{\prime}$ be non-negative integers, and let $\omega,\omega^{\prime}$
be words in $U_{n}$ and $U_{n^{\prime}}$ respectively. By Lemmas 6.6 and 6.4
(i), there exists $m\geq n,n^{\prime}$ such that both
$\widetilde{\phi}_{n,m}(e(\omega))$ and
$\widetilde{\phi}_{n^{\prime},m}(e(\omega^{\prime}))$ are finite sums of
mutually distinct vertices of $F_{m}$. Therefore they are commuting elements
of $L_{m}$. It follows that
$F_{\infty}^{1}=\bigcup_{n=0}^{\infty}\widetilde{\phi}_{n,\infty}(F_{n}^{1})$
is a tame set of partial isometries of $L_{\infty}$. By the universal property
of $L^{{\rm ab}}_{n}$, $\widetilde{\phi}_{n,\infty}$ factors (uniquely)
through a $*$-homomorphism $\overline{\phi}_{n,\infty}\colon L^{{\rm
ab}}_{n}\to L_{\infty}$, and obviously
$\widetilde{\phi}_{n,\infty}(L_{n})=\overline{\phi}_{n,\infty}(L^{{\rm
ab}}_{n})$. This completes the proof. ∎

Let $\widetilde{\phi}_{n}\colon L_{n}\to L_{n+1}$ be the $*$-homomorphism
defined in Proposition 6.3. Observe that

$\widetilde{\phi}_{n+1}\circ\widetilde{\phi}_{n}(p_{n})=\widetilde{\phi}_{n+1}\Big{(}\sum_{w\in
F_{n+1}^{0,1}}w\Big{)}=p_{n+2}\in L_{n+2}.$

From now on we will write ${\pazocal A}_{n}=p_{n}L_{n}p_{n}$, and we will
denote by $\phi_{n}\colon{\pazocal A}_{2n}\rightarrow{\pazocal A}_{2n+2}$ the
restriction to ${\pazocal A}_{2n}$ of the composition
$\widetilde{\phi}_{2n+1}\circ\widetilde{\phi}_{2n}$. We consider the inductive
limit ${\pazocal A}_{\infty}:=\varinjlim({\pazocal A}_{2n},\phi_{n})$, with
canonical maps $\phi_{n,\infty}\colon{\pazocal A}_{2n}\rightarrow{\pazocal
A}_{\infty}$. For $m>n$, we set
$\phi_{n,m}:=\phi_{m-1}\circ\cdots\circ\phi_{n}\colon{\pazocal
A}_{2n}\to{\pazocal A}_{2m}$.

######  Lemma 6.8.

We have a natural $*$-isomorphism

${\pazocal A}_{\infty}\cong pL_{\infty}p,$

where $p=\widetilde{\phi}_{2n,\infty}(p_{2n})$ for all $n\geq 0$.

###### Proof.

We have obvious compatible $*$-homomorphisms
$\widetilde{\phi}_{2n,\infty}\colon{\pazocal A}_{2n}\rightarrow pL_{\infty}p$,
and hence we get a $*$-homomorphism

$\lambda\colon{\pazocal A}_{\infty}\to pL_{\infty}p$

defined by $\lambda(\phi_{n,\infty}(a))=\widetilde{\phi}_{2n,\infty}(a)$ for
$a\in{\pazocal A}_{2n}$. It is straightforward to check that $\lambda$ is a
$*$-isomorphism. ∎

Our aim for the rest of this subsection is to find a $*$-isomorphism
$A_{K}({\pazocal G}(X,\sigma))\cong{\pazocal A}_{\infty}$, thus providing a
graph-theoretic model for the Steinberg algebra $A_{K}({\pazocal
G}(X,\sigma))$ of the Deaconu-Renault groupoid ${\pazocal G}(X,\sigma)$
associated to $(X,\sigma)$. We will also deal with the corresponding
$C^{*}$-algebras.

To start with, we define $*$-homomorphisms $\varphi_{n}\colon{\pazocal
A}_{2n}\rightarrow A_{K}({\pazocal G}(X,\sigma))$. For this, we use another
presentation for ${\pazocal A}_{2n}$, given by [5, Proposition 2.8]. We warn
the reader about the different convention used for sources and ranges in [5].

######  Definition 6.9.

[5, Definition 2.7] Let $(E,C)$ be a finite bipartite separated graph with
$E^{0}=E^{0,0}\sqcup E^{0,1}$, $s(E^{1})=E^{0,1}$ and $r(E^{1})=E^{0,0}$. Let
$LV_{K}(E,C)$ be the universal $*$-algebra with generators $P_{V}\sqcup T$
where $P_{V}=\\{p_{v}\mid v\in E^{0,0}\\}$ and $T=\\{\tau_{(e,f)}\mid e,f\in
E^{1},s(e)=s(f)\\}$, and subject to the relations

  1. (V’)

$p_{v}p_{w}=\delta_{v,w}p_{v}$, and $p_{v}=p_{v}^{*}$ for $v,w\in E^{0,0}$;

  2. (T)

$\tau_{(e,f)}^{*}=\tau_{(f,e)}$ for $e,f\in E^{1}$ with $s(e)=s(f)$;

  3. (E’)

$\tau_{(e,f)}\cdot p_{r(f)}=p_{r(e)}\cdot\tau_{(e,f)}=\tau_{(e,f)}$ for
$e,f\in E^{1}$ with $s(e)=s(f)$;

  4. (SCK1’)

$\tau_{(e,f)}\cdot\tau_{(g,h)}=\delta_{f,g}\tau_{(e,h)}$ for $e,f,g,h\in
E^{1}$ with $s(e)=s(f)$, $s(g)=s(h)$ and $f,g\in Y$ belonging to the same set
$Y\in C$;

  5. (SCK2’)

$p_{v}=\sum_{e\in Y}\tau_{(e,e)}$ for all $v\in E^{0,0}$ and all $Y\in C_{v}$.

We denote by $LV^{{\rm ab}}_{K}(E,C)$ the abelianized Leavitt path algebra of
$LV_{K}(E,C)$ with respect to its generating family $T=\\{\tau_{(e,f)}\mid
e,f\in E^{1},s(e)=s(f)\\}$ of partial isometries.

######  Proposition 6.10.

[5, Proposition 2.8] Let $(E,C)$ be a finite bipartite separated graph with
$E^{0}=E^{0,0}\sqcup E^{0,1}$, $s(E^{1})=E^{0,1}$ and $r(E^{1})=E^{0,0}$, and
set $V=\sum_{v\in E^{0,0}}v\in L_{K}(E,C)$. Then there is a unique
$*$-isomorphism

$\varphi_{V}\colon LV_{K}(E,C)\to VL_{K}(E,C)V$

such that $\varphi_{V}(p_{v})=v$ for all $v\in E^{0,0}$ and
$\varphi_{V}(\tau_{(e,f)})=ef^{*}$ for all $e,f\in E^{1}$ with $s(e)=s(f)$.

Let us now define the $*$-homomorphisms $\varphi_{n}\colon{\pazocal
A}_{2n}\rightarrow A_{K}({\pazocal G}(X,\sigma))$. We use the presentation of
${\pazocal A}_{2n}=p_{2n}L_{K}(F_{2n},D^{2n})p_{2n}\cong
LV_{K}(F_{2n},D^{2n})$ provided by Definition 6.9 (see Proposition 6.10
above). Recall that the vertices of $F_{2n}^{0,0}$ correspond to the sets in
the partition ${\pazocal P}_{2n}$ of $X$, and similarly, the vertices in
$F_{2n}^{0,1}$ correspond to the sets in the partition ${\pazocal P}_{2n+1}$
of $X$.

Given $p_{v}$, with $v\in F_{2n}^{0,0}$, there is a unique clopen subset
$U\in{\pazocal P}_{2n}$ such that $v=U$. Define

$\varphi_{n}(p_{v})=\varphi_{n}(p_{U})=\mathds{1}_{U}\in A_{K}({\pazocal
G}(X,\sigma)),$

where we identify $Z(U,0,0,U)$ with $U$. Now we want to define the images of
elements of the form $\tau_{(e,f)}$, where $e$ and $f$ are edges in $F_{2n}$
such that $s(e)=s(f)$. We write $s(e)=s(f)=W\in{\pazocal P}_{2n+1}$,
$r(e)=U\in{\pazocal P}_{2n}$ and $r(f)=V\in{\pazocal P}_{2n}$. We need to
distinguish various cases.

  1. (a)

$e,f$ are both blue edges. By definition, we have then that $W\subseteq U$ and
$W\subseteq V$. In this case, define

$\varphi_{n}(\tau_{(e,f)})=\mathds{1}_{W}\in A_{K}({\pazocal G}(X,\sigma)),$

where we identify $Z(W,0,0,W)$ with $W$.

  2. (b)

$e$ and $f$ are both red edges. In this case we have $W\subseteq\sigma(U)$ and
$W\subseteq\sigma(V)$. Define

$\varphi_{n}(\tau_{(e,f)})=\mathds{1}_{Z((\sigma|_{U})^{-1}(W),1,1,(\sigma|_{V})^{-1}(W))}\in
A_{K}({\pazocal G}(X,\sigma)).$

Observe that all sets $Z$ in the partitions ${\pazocal P}_{m}$ satisfy the
property that the restriction of $\sigma$ to $Z$ is injective. Hence
$Z((\sigma|_{U})^{-1}(W),1,1,(\sigma|_{V})^{-1}(W))$ is well-defined.

  3. (c)

$e$ is a red edge and $f$ is a blue edge. In this case we have
$W\subseteq\sigma(U)$ and $W\subseteq V$. Define

$\varphi_{n}(\tau_{(e,f)})=\mathds{1}_{Z((\sigma|_{U})^{-1}(W),1,0,W)}\in
A_{K}({\pazocal G}(X,\sigma)).$

  4. (d)

$e$ is a blue edge and $f$ is a red edge. Then $W\subseteq U$ and
$W\subseteq\sigma(V)$. Define

$\varphi_{n}(\tau_{(e,f)})=\mathds{1}_{Z(W,0,1,(\sigma|_{V})^{-1}(W))}\in
A_{K}({\pazocal G}(X,\sigma)).$

It is straightforward to check that the defining relations from Definition 6.9
are preserved, and hence we obtain well-defined $*$-homomorphisms

$\varphi_{n}\colon{\pazocal A}_{2n}\to A_{K}({\pazocal G}(X,\sigma))$

for all $n\geq 0$.

Observe that the algebras $L_{2n}=L_{K}(F_{2n},D^{2n})$ are
${\mathbb{Z}}$-graded for all $n\geq 0$, with grading determined by setting
$d(v)=0$ for each vertex $v\in F_{2n}^{0}$, $d(e)=0$ if $e\in F_{2n}^{1}$ is a
blue edge, and $d(f)=1$ if $f\in F_{2n}^{1}$ is a red edge. For each $n\geq
0$, ${\pazocal A}_{2n}$ is a graded $*$-subalgebra of $L_{2n}$, and the maps
$\phi_{n}\colon{\pazocal A}_{2n}\to{\pazocal A}_{2n+2}$ are graded
$*$-homomorphisms, so we get the structure of a ${\mathbb{Z}}$-graded
$*$-algebra in ${\pazocal A}_{\infty}$.

######  Proposition 6.11.

The morphisms $\varphi_{n}\colon{\pazocal A}_{2n}\to A_{K}({\pazocal
G}(X,\sigma))$ are graded and compatible with the maps
$\phi_{n}\colon{\pazocal A}_{2n}\to{\pazocal A}_{2n+2}$ from the inductive
system $({\pazocal A}_{2n},\phi_{n})$. Therefore they define a well-defined
graded $*$-homomorphism

$\varphi\colon{\pazocal A}_{\infty}\to A_{K}({\pazocal G}(X,\sigma)).$

###### Proof.

It is clear from the definitions that the maps $\varphi_{n}$ are graded
$*$-homomorphisms for all $n\geq 0$, so it only remains to show compatibility
with the maps $\phi_{n}$.

First, if $v\in F_{2n}^{0,0}$, then $v=U$ for a unique $U\in{\pazocal
P}_{2n}$, and $\phi_{n}(v)=\textbf{S}(v)$ corresponds to the decomposition

$U=\bigsqcup_{\\{\overline{U}\in{\pazocal P}_{2n+2}\mid\overline{U}\subseteq
U\\}}\overline{U}$

of $U$ as a disjoint union of sets from ${\pazocal P}_{2n+2}$. Hence we obtain
that $(\varphi_{n+1}\circ\phi_{n})(p_{v})=\varphi_{n}(p_{v})$.

There are four cases to consider concerning the elements $\tau_{(e,f)}$, all