# A correspondence between surjective local homeomorphisms and a family of
separated graphs

Pere Ara  Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de
Barcelona, 08193 Cerdanyola del Vallès (Barcelona), Spain, and  Centre de
Recerca Matemàtica, Edifici Cc, Campus de Bellaterra, 08193 Cerdanyola del
Vallès (Barcelona), Spain<EMAIL_ADDRESS>and  Joan Claramunt  Departamento
de Matemáticas, Universidad Carlos III de Madrid, Av. de la Universidad 30,
28911 Leganés (Madrid), Spain<EMAIL_ADDRESS>
###### Abstract.

We present a graph-theoretic model for dynamical systems $(X,\sigma)$ given by
a surjective local homeomorphism $\sigma$ on a totally disconnected compact
metrizable space $X$. In order to make the dynamics appear explicitly in the
graph, we use two-colored Bratteli separated graphs as the graphs used to
encode the information. In fact, our construction gives a bijective
correspondence between such dynamical systems and a subclass of separated
graphs which we call l-diagrams. This construction generalizes the well-known
shifts of finite type, and leads naturally to the definition of a generalized
finite shift. It turns out that any dynamical system $(X,\sigma)$ of our
interest is the inverse limit of a sequence of generalized finite shifts. We
also present a detailed study of the corresponding Steinberg and $C^{*}$
algebras associated with the dynamical system $(X,\sigma)$, and we use the
above approximation of $(X,\sigma)$ to write these algebras as colimits of the
associated algebras of the corresponding generalized finite shifts, which we
call generalized finite shift algebras.

###### Key words and phrases:

local homeomorphism, Bratteli diagram, separated graph, groupoid,
$C^{*}$-algebra, Steinberg algebra

###### 2020 Mathematics Subject Classification:

37B10, 46L55, 16S88

Both authors were partially supported by the project
PID2020-113047GB-I00/AEI/10.13039/501100011033. The first-named author was
also partially supported by the Spanish State Research Agency, through the
Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence
in R$\&$D (CEX2020-001084-M). The second named author was also partially
supported by the ERC Starting Grant “Limits of Structures in Algebra and
Combinatorics” No. 805495.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Separated graphs
    2. 2.2 Groupoids and associated algebras
    3. 2.3 Dynamical systems
  3. 3 The correspondence between local homeomorphisms and $l$-diagrams
    1. 3.1 From LHomeo to LDiag
    2. 3.2 From LDiag to LHomeo
    3. 3.3 Equivalence between both constructions
  4. 4 Homeomorphisms
  5. 5 Generalizing shifts of finite type
    1. 5.1 A quick review on shifts of finite type
    2. 5.2 Generalized finite shifts
    3. 5.3 A configurational point of view
  6. 6 Approximating algebras associated to a dynamical system
    1. 6.1 The general approximation construction
    2. 6.2 Functoriality properties
    3. 6.3 Algebras as colimits of generalized finite shift algebras

##  1\. Introduction

Dynamical systems consisting of a pair $(X,f)$, where $X$ is a topological
space and $f\colon X\to X$ is a continuous map, have attracted a great deal of
attention along the years. In particular, an important line of investigation
consists of approximating a dynamical system by means of other dynamical
systems defined by combinatorial data. In many occasions, invariants
associated to the original dynamical system can be computed using these
combinatorial approximations. In this respect, symbolic dynamics has shown to
provide a very useful source of approximating systems, see [22, 31, 48]. An
early example occurs with the existence of a Markov partition on several
classes of dynamical systems, which is proved to imply that the system under
study is a factor of a corresponding shift associated to the Markov partition,
see [48, Proposition 6.5.8].

It has recently been shown in [39] that any dynamical system $(X,f)$, where
$X$ is a compact Hausdorff space, can be expressed as an inverse limit of one-
sided shifts of finite type, and that the important property of shadowing
holds for $(X,f)$ if and only if the inverse system satisfies the Mittag-
Leffler property. See also [53] and [54] for results concerning the
approximation of zero-dimensional systems in terms of various combinatorial
structures.

In a different setting, Bratteli diagrams were used in [18] to classify
approximately finite-dimensional $C^{*}$-algebras. Herman, Putnam and Skau
[40] showed that there is a bijective correspondence (modulo suitable
equivalence relations) between essentially minimal homeomorphisms $f$ on the
Cantor set $X$ and a class of ordered Bratteli diagrams. These results were
later used to obtain classification theorems for the corresponding crossed
product $C^{*}$-algebras $C^{*}(X)\rtimes{\mathbb{Z}}$ associated to the
action of ${\mathbb{Z}}$ on $X$ induced by the homeomorphism $f$, see for
instance [35]. This kind of combinatorial structure has been studied in more
generality, for some dynamical systems not necessarily essentially minimal,
see for example [17]. In addition, the above correspondences have been
recently upgraded to suitable category equivalences by Amini, Elliott and
Golestani, see [2] and [3].

In this paper we develop a new approximation method for any dynamical system
$(X,\sigma)$, where $\sigma$ is a surjective local homeomorphism and $X$ is a
totally disconnected compact metrizable space. This is a very large class of
dynamical systems, including all homeomorphisms on totally disconnected
compact metrizable spaces, in particular all minimal Cantor systems [51], all
one-sided shifts of finite type, all two-sided shifts, covering maps of
compact totally disconnected metrizable spaces (see [29], [33] and Remark
3.17), the one-sided shifts associated to topological graphs
$E=(E^{0},E^{1},r,s)$, where $E^{0}$ and $E^{1}$ are compact totally
disconnected metrizable spaces and $s$ is surjective (see [42], [43], [44],
[45]), one-sided shifts on infinite path spaces of finite ultragraphs with no
sinks [37], and the covers of one-sided subshifts [20].

We show that our approximation is well-behaved with respect to the associated
$C^{*}$-algebras. Specifically, we prove that the groupoid $C^{*}$-algebra
$C^{*}({\pazocal G}(X,\sigma))$ of the Deaconu-Renault groupoid ${\pazocal
G}(X,\sigma)$ associated to $(X,\sigma)$ is naturally isomorphic to the
colimit of the groupoid $C^{*}$-algebras of the approximating systems, see
Theorem 1.4 below. This groupoid $C^{*}$-algebra generalizes the crossed
product construction mentioned above. A new basic building block emerges from
our approach, generalizing both one-sided and two-sided shifts of finite type,
which we have termed a generalized shift of finite type. The main
combinatorial tool employed is the theory of separated graphs. A separated
graph $(E,C)$ consists of a graph $E=(E^{0},E^{1},r,s)$ together with a
prescribed partition $C_{v}$ of the set of edges arriving at $v$, for each
vertex $v\in E^{0}$. In our work, we use separated Bratteli diagrams, which
are Bratteli diagrams which are also separated graphs, so that the edges on
every layer of the diagram are given a specific separation.

It is worth to mention that for a large class of dynamical systems, including
the ones considered in the present paper, it has been shown in [16] that one
can characterize conjugacy of the dynamical systems in terms of the associated
$C^{*}$-algebras and also in terms of the associated Deaconu-Renault
groupoids. See also [19], [21], [24] and [38] for other studies on the
interconnections between local homeomorphisms, groupoids and $C^{*}$-algebras.

Let LHomeo be the family of dynamical systems $(X,\sigma)$ consisting of a
totally disconnected compact metrizable space $X$, together with a surjective
local homeomorphism $\sigma:X\rightarrow X$. In the spirit of the work done in
[40], we provide a bijective correspondence (modulo suitable equivalence
relations) between LHomeo and a class of separated Bratteli diagrams, which we
call $l$-diagrams, see Definition 3.2.

Let $E$ be a finite graph having neither sinks nor sources. It is well-known
that out of it one can construct a dynamical system
$(X_{E},\sigma_{E})\in\textsf{LHomeo}$ as follows. The space $X_{E}$ is taken
to be the set of one-sided infinite paths in the graph, endowed with the
topology generated by the cylinder sets, and then $\sigma_{E}$ is defined as
the one-sided shift. These are the well-known one-sided shifts of finite type.
If we consider the space of bi-infinite paths instead of $X_{E}$, we obtain
the definition of a two-sided shift of finite type, which is a homeomorphism.
These are basic objects in symbolic dynamics.

From an arbitrary directed graph $E$ one can construct two important
${\mathbb{C}}$-algebras in a universal way. The first one is a purely
algebraic construction, called the Leavitt path algebra of the directed graph
$E$, denoted by $L_{{\mathbb{C}}}(E)$. Leavitt path algebras were
independently introduced by Abrams and Aranda Pino [1] and by the first-named
author, Moreno and Pardo [15]. The second construction, which chronologically
precedes the first, also contains analytic information and is called the graph
$C^{*}$-algebra of the directed graph $E$, denoted by $C^{*}(E)$. Graph
$C^{*}$-algebras were first studied in detail in the papers [47] and [46]. It
turns out that $C^{*}(E)$ is the enveloping $C^{*}$-algebra of
$L_{{\mathbb{C}}}(E)$ for each directed graph $E$, in the sense of Definition
2.4.

The connection with the theory of topological groupoids was already explored
in [47], where it is shown that the graph $C^{*}$-algebra of a locally finite
graph is isomorphic to the $C^{*}$-algebra of a suitably defined graph
groupoid ${\pazocal G}_{E}$ associated to $E$. This was later generalized to
arbitrary graphs. When the graph is finite and has neither sinks nor sources,
the graph groupoid ${\pazocal G}_{E}$ turns out to be the Deaconu-Renault
groupoid of the one-sided shift $(X_{E},\sigma_{E})$, and thus we obtain the
isomorphism $C^{*}(E)\cong C^{*}({\pazocal G}(X_{E},\sigma_{E}))$. This
connection was extended to the purely algebraic situation in [56] and [26]
through the consideration of the Steinberg algebra $A_{{\mathbb{C}}}({\pazocal
G})$ of an ample étale groupoid ${\pazocal G}$. Concretely, it is shown in
[26, Proposition 4.3] that there is a natural $*$-algebra isomorphism
$L_{{\mathbb{C}}}(E)\cong A_{{\mathbb{C}}}({\pazocal G}_{E})$ for each
directed graph $E$. There are multiple interesting connections between these
groupoid algebras and symbolic dynamics, see for instance the recent papers
[16] and [23].

The first-named author and Goodearl introduced in [12] and [13] the Leavitt
path algebra $L_{{\mathbb{C}}}(E,C)$ and the graph $C^{*}$-algebras
$C^{*}(E,C)$ associated to a separated graph $(E,C)$. These are defined in a
similar way to the Leavitt path algebras and graph $C^{*}$-algebras of
ordinary directed graphs, but taking into account the partitions $\\{C_{v}\mid
v\in E^{0}\\}$ in the defining relations of such algebras (see Definitions 2.1
and 2.2 for the precise definitions). Thenceforth there has been a
considerable amount of work towards the theory of separated graph algebras [4,
5, 9, 11, 14, 36, 49, 50], not only because they provide a whole rich class of
examples of algebras not being realizable as graph algebras, but because they
proved to be useful in solving, either in the positive or in the negative,
important conjectures (see e.g. [6, 7, 8, 10]).

The relation between dynamical systems and separated graphs was first
demonstrated in a paper by the first-named author and Exel [10]. The authors
attach to each finite bipartite separated graph $(E,C)$ a partial dynamical
system $(\Omega(E,C),{\mathbb{F}},\theta)$, where
${\mathbb{F}}={\mathbb{F}}(E^{1})$ is the free group on the set of edges, in
such a way that the algebraic crossed product
$C_{{\mathbb{C}}}(\Omega(E,C))\rtimes_{\theta^{*}}^{\text{alg}}{\mathbb{F}}$
is canonically isomorphic to a well-determined quotient
$L_{{\mathbb{C}}}(E,C)/J$ of $L_{{\mathbb{C}}}(E,C)$. They also found similar
results in the $C^{*}$-algebraic setting. As an application of these results,
the authors were able to construct non-paradoxical topological actions of
discrete groups on Cantor spaces which do not admit an invariant finitely
additive measure, thus providing counterexamples to the topological version of
Tarski’s alternative, see [10, Section 7] for details. In connection with this
work, it was shown in [14] that the study of the above mentioned dynamical
systems associated to separated graphs is equivalent to the study of convex
subshifts, a notion coined by the first-named author and Lolk in [14],
introduced as a far-reaching unification proposal for the well-known one-sided
and two-sided subshifts.

The first main result of the paper is the following:

######  Theorem 1.1 (Theorem 3.14).

Out of a dynamical system $(X,\sigma)\in\emph{{LHomeo}}$, one can construct an
$l$-diagram $(F,D)$. Conversely, out of an $l$-diagram $(F,D)$, one can
construct a dynamical system $(X,\sigma)\in\emph{{LHomeo}}$.

Moreover, these correspondences are inverse of each other, so we obtain,
modulo appropriate equivalence relations, a bijective correspondence between
_LHomeo_ and _LDiag_ , the class of $l$-diagrams.

In particular, if we restrict our attention to homeomorphisms, the
corresponding diagrams have a simpler form. We call such separated Bratteli
diagrams $h$-diagrams (see Definition 4.1) and we denote by Homeo the subclass
of LHomeo consisting of those dynamical systems $(X,\sigma)$ such that
$\sigma$ is a homeomorphism.

######  Theorem 1.2 (Theorem 4.8).

The restriction of the above correspondences gives a bijective correspondence
between the subclass of homeomorphisms Homeo and the subclass _HDiag_ of
$h$-diagrams.

It follows from Theorem 1.2 and [40, Theorem 4.7] that there is a bijective
map between equivalence classes of essentially simple ordered Bratteli
diagrams and a subset of equivalence classes of $h$-diagrams. It would be
interesting to determine explicitly the range of this bijective map. Moreover,
one can try to build a category of $h$-diagrams, in the spirit of [3], and
then extend the above bijection to a suitable equivalence of categories.
Another line of future research is extending the techniques developed in the
present paper to cover the case of partial (local) homeomorphisms, and relate
this extension with the work done by Giordano, Gonçalves and Starling in [34].

Let $(E,C)$ be a finite bipartite separated graph satisfying the conditions
needed to be the first layer of an $l$-diagram. These conditions are
explicitly stated in Definition 5.6, where these separated graphs are named
generalized finite shift graphs. By applying a process called the canonical
resolution of the bipartite separated graph we obtain an $l$-diagram $(F,D)$
(Proposition 5.11) which, by an application of Theorem 1.1, gives rise to a
dynamical system $(X,\sigma)\in\textsf{LHomeo}$. Those dynamical systems
arising by applying this construction are named generalized finite shifts,
since its class includes both one-sided and two-sided finite shifts. Our next
main result states that any dynamical system $(X,\sigma)\in\textsf{LHomeo}$
can be approximated, in a sense made precise below, by generalized finite
shifts. This is done by working with an explicit realization of the
generalized finite shifts, which is based on the work done in [14].

######  Theorem 1.3 (Theorem 5.26).

Given $(X,\sigma)\in\emph{{LHomeo}}$, there exists an inverse system
$\\{(X_{n},\sigma_{n}),\psi_{n,m}\\}$ in _LHomeo_ consisting of generalized
finite shift graphs such that

$(X,\sigma)\cong\varprojlim(X_{n},\sigma_{n}).$

Let $(X,\sigma)\in\textsf{LHomeo}$ and $(F,D)$ be the corresponding
$l$-diagram. In the last section of the paper we apply the results obtained in
order to study the relationship between the Steinberg algebra and the
$C^{*}$-algebra of the Deaconu-Renault groupoid associated with $(X,\sigma)$
and a certain corner of an inductive limit of Leavitt path algebras and graph
$C^{*}$-algebras corresponding to the finite levels of $(F,D)$. This study
culminates in Theorems 6.16 and 6.17, where it is shown that the
aforementioned algebras are isomorphic.

We close the paper by further using Theorem 1.3 and various other tools to
obtain a similar approximation result at the level of the Steinberg algebras
and the groupoid $C^{*}$-algebras. Here $A_{K}({\pazocal G}(X,\sigma))$
denotes the Steinberg algebra of ${\pazocal G}(X,\sigma)$, the Deaconu-Renault
groupoid associated with $(X,\sigma)$, with coefficients in an arbitrary field
with involution $(K,*)$.

######  Theorem 1.4 (Theorem 6.24).

Let $(X,\sigma)\in\emph{{LHomeo}}$ and write it as an inverse limit of a
sequence of generalized finite shifts $\\{(X_{n},\sigma_{n})\\}_{n}$ as in
Theorem 1.3. Then

$A_{K}({\pazocal G}(X,\sigma))\cong\varinjlim A_{K}({\pazocal
G}(X_{n},\sigma_{n})).$

At the level of $C^{*}$-algebras, we also get

$C^{*}({\pazocal G}(X,\sigma))\cong\varinjlim C^{*}({\pazocal
G}(X_{n},\sigma_{n})).$

The results of this paper, in conjunction with the results in [10] and [11],
will be used in forthcoming work by the authors, to study both the stable and
the non-stable $K$-theory of the algebras attached to the dynamical systems
$(X,\sigma)\in\textsf{LHomeo}$, as well as the type semigroup of $(X,\sigma)$.

The paper is organized as follows. In Section 2 we provide the reader some
basic concepts such as the definition of a separated graph, and its associated
Leavitt path algebra and $C^{*}$-algebra. We also recall the construction of
the Deaconu-Renault groupoid associated with a dynamical system $(X,\sigma)$
formed by a surjective local homeomorphism $\sigma$ on a totally disconnected
compact metrizable space $X$, and the construction of its groupoid
$C^{*}$-algebra, and its Steinberg algebra over an arbitrary field with
involution. We present in Theorem 3.14 the bijective correspondence between
surjective local homeomorphisms on totally disconnected compact metrizable
spaces and a certain subfamily of the family of separated graphs, called
$l$-diagrams, in Section 3. Section 4 is then devoted to characterize the
class of $l$-diagrams which arises from the class of dynamical systems given
by homeomorphisms on totally disconnected compact metrizable spaces, called
$h$-diagrams. In Section 5 we show how shifts of finite type fit in our
theory, and we present a generalization of them, which we call generalized
finite shifts. We provide yet another different point of view of such
dynamical systems using the work from [14], culminating in Theorem 5.26 which
shows that any dynamical system of our kind is an inverse limit of a sequence
of generalized finite shits. We use this ‘approximation theorem’ in order to
give similar characterizations, in terms of colimits, of the Steinberg algebra
(respectively, $C^{*}$-algebra) of the Deaconu-Renault groupoid associated
with the dynamical system in terms of the sequence of Steinberg algebras
(respectively, $C^{*}$-algebras) of the Deaconu-Renault groupoids associated
with the generalized finite shifts (Theorem 6.24). This is done in Section 6.

##  2\. Preliminaries

In this section we provide background definitions and concepts needed
throughout the paper.

###  2.1. Separated graphs

We follow [10]. A separated graph is a pair $(E,C)$ where
$E=(E^{0},E^{1},r,s)$ is a directed graph ($E^{0}$ is the set of vertices,
$E^{1}$ the set of edges, and $r,s\colon E^{1}\to E^{0}$ are the range and
source maps, respectively), and $C=\bigsqcup_{v\in E^{0}}C_{v}$, where each
$C_{v}$ is a partition of the set of edges $r^{-1}(v)$ into pairwise disjoint,
non-empty subsets, for each vertex $v\in E^{0}$. We set $C_{v}=\emptyset$ in
case $r^{-1}(v)=\emptyset$. We say that $(E,C)$ is finitely separated if all
the sets in $C$ are finite, that is, if $|X|<\infty$ for all $X\in C$.

A bipartite separated graph is a separated graph $(E,C)$ where we can write
$E^{0}=E^{0,0}\sqcup E^{0,1}$ in such a way that $s(E^{1})=E^{0,1}$ and
$r(E^{1})=E^{0,0}$.

We will follow the conventions from [10] and [14] concerning sources and
ranges of edges of a graph, and the relations (E), (SCK1) and (SCK2) below. In
particular, a non-trivial path in the graph $E$ will be a sequence
$e_{1}e_{2}\cdots e_{n}$ of edges in $E$ such that $s(e_{i})=r(e_{i+1})$ for
$i=1,\dots,n-1$. (A trivial path consists of just one vertex $v$ in $E$.) We
warn the reader that different conventions are used in various other sources.

We can now define our algebras of interest, namely the Leavitt path algebra
and the $C^{*}$-algebra of a separated graph $(E,C)$.

######  Definition 2.1.

The Leavitt path algebra of the separated graph $(E,C)$ with coefficients in a
field $K$ with involution $*$, denoted by $L_{K}(E,C)$, is the $*$-algebra
with generators $E^{0}\sqcup E^{1}\sqcup(E^{1})^{*}$ together with the
relations

  1. (V)

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

  2. (E)

$r(e)e=es(e)=e$ for $e\in E^{1}$,

  3. (SCK1)

$e^{*}f=\delta_{e,f}s(e)$ for $e,f\in Y$, for all $Y\in C$,

  4. (SCK2)

$v=\sum_{e\in Y}ee^{*}$ for every $Y\in C_{v}$ finite, $v\in E^{0}$.

######  Definition 2.2.

The $C^{*}$-algebra of the separated graph $(E,C)$, denoted by $C^{*}(E,C)$,
is the universal $C^{*}$-algebra with generators $E^{0}\sqcup
E^{1}\sqcup(E^{1})^{*}$ together with the relations (V),(E),(SCK1) and (SCK2)
from Definition 2.1. In other words, $C^{*}(E,C)$ is the enveloping
$C^{*}$-algebra of $L_{{\mathbb{C}}}(E,C)$.

###  2.2. Groupoids and associated algebras

Given a groupoid ${\pazocal G}$, we will write ${\pazocal G}^{(2)}$ for the
set of pairs of composable elements, and by ${\pazocal G}^{(0)}$ for its set
of units. The range and source maps will be denoted by $r:{\pazocal
G}\rightarrow{\pazocal G}^{(0)}$ and $s:{\pazocal G}\rightarrow{\pazocal
G}^{(0)}$, respectively.

In this paper, a topological groupoid will be a groupoid ${\pazocal G}$
possessing a locally compact Hausdorff topology, in which the range and source
maps $r,s$, the inverse map and the multiplication map are all continuous
maps. We refer to [52, 55, 56, 26] for general background on topological
groupoids and their associated $C^{*}$-algebras.

######  Definition 2.3.

Let $X$ and $Y$ be topological spaces. A local homeomorphism from $X$ to $Y$
is a continuous map $h\colon X\to Y$ such that every $x\in X$ has an open
neighborhood $U$ such that $h(U)$ is open in $Y$ and $h|_{U}$ is a
homeomorphism from $U$ onto its image $h(U)$. Whenever $X=Y$ we say that $h$
is a local homeomorphism on $X$.

A topological groupoid ${\pazocal G}$ is étale if the range map
$r\colon{\pazocal G}\to{\pazocal G}^{(0)}$ is a local homeomorphism. Then the
source map is also a local homeomorphism and ${\pazocal G}^{(0)}$ is an open
subset of ${\pazocal G}$ (see [55, Lemma 8.4.2]).

An open subset ${\pazocal W}\subseteq{\pazocal G}$ is called an open bisection
if both $r({\pazocal W}),s({\pazocal W})$ are open subsets of ${\pazocal
G}^{(0)}$, and $r|_{{\pazocal W}},s|_{{\pazocal W}}$ are homeomorphisms onto
their respective images. Note that a topological groupoid is étale if and only
if its topology has a basis of open bisections.

A topological groupoid ${\pazocal G}$ is called ample if its topology admits a
basis consisting of compact open bisections. Clearly all ample groupoids are
étale.

From now on ${\pazocal G}$ will denote a second countable, Hausdorff and ample
groupoid, thus admitting a countable basis for the topology consisting of
compact open bisections.

######  Definition 2.4.

Let $(K,*)$ be a field endowed with an involution $*$. The Steinberg algebra
$A_{K}({\pazocal G})$ of ${\pazocal G}$ is the $*$-algebra of all the locally
constant functions $f\colon{\pazocal G}\to K$ with compact support. The
multiplication is just the convolution product

$(f\ast g)(\gamma)=\sum_{s(\alpha)=s(\gamma)}f(\gamma\alpha^{-1})g(\alpha)$

for $f,g\in A_{K}({\pazocal G}),\gamma\in{\pazocal G}$, and the involution is
given by the rule

$f^{*}(\gamma)=f(\gamma^{-1})^{*}$

for $f\in A_{K}({\pazocal G}),\gamma\in{\pazocal G}$.

The full groupoid $C^{*}$-algebra of ${\pazocal G}$, denoted by
$C^{*}({\pazocal G})$, is the enveloping $C^{*}$-algebra of
$A_{{\mathbb{C}}}({\pazocal G})$, in the sense that there is an injective
$*$-homomorphism $\iota\colon A_{{\mathbb{C}}}({\pazocal G})\to
C^{*}({\pazocal G})$ such that any $*$-homomorphism from
$A_{{\mathbb{C}}}({\pazocal G})$ to a $C^{*}$-algebra $B$ uniquely extends to
a $*$-homomorphism from $C^{*}({\pazocal G})$ to $B$. For second countable,
ample Hausdorff groupoids, this definition agrees with the usual definition of
the full groupoid $C^{*}$-algebra given by Renault in [52], by [55, Lemma
9.2.4] and [27, Theorem 7.1(1)].

Every element of $A_{K}({\pazocal G})$ can be written as a finite linear
combination of characteristic functions $\mathds{1}_{U_{i}}$, where
$\\{U_{i}\\}$ are mutually disjoint compact open bisections. If $U$ and $V$
are two compact open bisections in ${\pazocal G}$, then we have

$\mathds{1}_{U}\cdot\mathds{1}_{V}=\mathds{1}_{UV}$

in $A_{K}({\pazocal G})$, where

$UV:=\\{uv\in{\pazocal G}\mid u\in U,v\in V,s(u)=r(v)\\}$

is again a compact open bisection.

###  2.3. Dynamical systems

We will concentrate on dynamical systems $(X,\sigma)$ formed by a totally
disconnected compact metrizable space $X$ together with a surjective local
homeomorphism $\sigma\colon X\to X$ (see Definition 2.3).

######  Definition 2.5.

By a partition of $X$ we mean a finite family ${\pazocal P}$ consisting on
non-empty, pairwise disjoint clopen subsets of $X$, such that
$X=\bigsqcup_{Z\in{\pazocal P}}Z$.

Given two partitions ${\pazocal P}_{1},{\pazocal P}_{2}$ of $X$, we say that
${\pazocal P}_{2}$ is finer than ${\pazocal P}_{1}$, written as ${\pazocal
P}_{1}\precsim{\pazocal P}_{2}$, if every element $Z\in{\pazocal P}_{2}$ is
contained in a (necessarily unique) element $Z^{\prime}\in{\pazocal P}_{1}$,
that is $Z\subseteq Z^{\prime}$. Given two partitions ${\pazocal
P}_{1},{\pazocal P}_{2}$ of $X$, there is a canonical way of constructing a
new partition which refines both ${\pazocal P}_{1},{\pazocal P}_{2}$; this new
partition is given by the wedge of ${\pazocal P}_{1}$ and ${\pazocal P}_{2}$,
and is defined by

${\pazocal P}_{1}\vee{\pazocal P}_{2}:=\\{Z_{1}\cap Z_{2}\mid
Z_{1}\in{\pazocal P}_{1},Z_{2}\in{\pazocal P}_{2}\text{ with }Z_{1}\cap
Z_{2}\neq\emptyset\\}.$

The diameter of an element $Z\in{\pazocal P}$, being ${\pazocal P}$ a
partition of $X$, is defined by

$\text{diam}(Z):=\sup\\{d(x,y)\mid x,y\in Z\\},$

where $d$ is the distance witnessing the metrizability of $X$. The diameter of
the partition ${\pazocal P}$ is thus

$\text{diam}({\pazocal P}):=\max\\{\text{diam}(Z)\mid Z\in{\pazocal P}\\}.$

We now recall the definition of the Deaconu-Renault groupoid associated with
the dynamical system $(X,\sigma)$. Define

${\pazocal G}(X,\sigma):=\bigcup_{m,n\in{\mathbb{N}}_{0}}\\{(x,m-n,y)\mid
x,y\in X,\sigma^{m}(x)=\sigma^{n}(y)\\}\subseteq X\times{\mathbb{Z}}\times X.$

This set is given the structure of a groupoid via the following. The set of
composable elements will be

${\pazocal G}(X,\sigma)^{(2)}=\\{((x,p,y),(z,q,t))\mid
y=z\\}\subseteq{\pazocal G}(X,\sigma)^{2}.$

The multiplication is then given by $(x,p,y)\cdot(y,q,z)=(x,p+q,z)$, and the
inverse of an element is simply $(x,p,y)^{-1}=(y,-p,x)$. Thus the unit space
of this groupoid is

${\pazocal G}(X,\sigma)^{(0)}=\\{(x,0,x)\mid x\in X\\}\simeq X.$

The range and source maps $r,s:{\pazocal G}(X,\sigma)\rightarrow X$ are given,
respectively, by $r(x,p,y)=x$ and $s(x,p,y)=y$.

This groupoid becomes a topological Hausdorff groupoid in the topology
generated by the basis of cylinder sets ${\pazocal B}=\\{Z(U,m,n,V)\mid
U,V\subseteq X\text{ open, and }m,n\in{\mathbb{N}}_{0}\\}$, where each
cylinder $Z(U,m,n,V)$ is defined by

$Z(U,m,n,V):=\\{(x,m-n,y)\mid x\in U,y\in V,\sigma^{m}(x)=\sigma^{n}(y)\\}.$

In fact, ${\pazocal G}(X,\sigma)$ becomes an étale groupoid. More
specifically, by fixing $m,n\in{\mathbb{N}}_{0}$ and letting $U,V\subseteq X$
be open subsets such that both $\sigma^{m}|_{U},\sigma^{n}|_{V}$ are
homeomorphisms onto their images, then $Z=Z(U,m,n,V)$ is such that
$r|_{Z},s|_{Z}$ are homeomorphisms onto open subsets of ${\pazocal
G}(X,\sigma)$. In particular, they are also local homeomorphisms onto open
subsets of ${\pazocal G}(X,\sigma)^{(0)}=X$, and so $Z$ is in fact an open
bisection.

Moreover, since we can take $U,V$ to be clopen subsets of $X$ (as $X$ is
totally disconnected), we can take $Z(U,m,n,V)$ to be also compact. As a
consequence, ${\pazocal G}(X,\sigma)$ is an ample, Hausdorff groupoid, with a
basis of open compact bisections consisting of the cylinder sets $Z(U,m,n,V)$
such that $m,n\in{\mathbb{N}}_{0}$ and $U,V\subseteq X$ are clopen with the
property that $\sigma^{m}|_{U},\sigma^{n}|_{V}$ are homeomorphisms onto their
images.

##  3\. The correspondence between local homeomorphisms and $l$-diagrams

In this section we establish a bijective correspondence between a specific
family of separated graphs and surjective local homeomorphisms of a second
countable totally disconnected compact space $X$.

We first describe the type of separated graphs we consider. It is a subclass
of the class of all separated Bratteli diagrams considered in [14]. We recall
the definition of these graphs.

######  Definition 3.1.

A separated (or colored) Bratteli diagram is an infinite separated graph
$(F,D)$ with the following properties:

  1. (a)

The vertex set $F^{0}$ is the union of finite, non-empty, pairwise disjoint
sets $F^{0,j}$, $j\geq 0$.

  2. (b)

The edge set $F^{1}$ is the union of finite, non-empty, pairwise disjoint sets
$F^{1,j}$, $j\geq 0$.

  3. (c)

The range and source maps satisfy $r(F^{1,j})=F^{0,j}$ and
$s(F^{1,j})=F^{0,j+1}$ for all $j\geq 0$, respectively.

Note that a Bratteli diagram is just a separated Bratteli diagram with the
trivial separation.

Now we can define our class of graphs.

######  Definition 3.2.

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

  1. (a)

(Separations for the even layers) For each $j\geq 0$ and $v\in F^{0,2j}$, the
separation $D_{v}$ is given by

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

for some (non-empty) subsets $B_{v},R_{v}\subseteq r^{-1}(v)$. The elements of
$B_{v}$ will be called blue edges, and the elements of $R_{v}$ red edges.

  2. (b)

(Separations for the odd layers) For each $j\geq 1$ and $v\in F^{0,2j-1}$, the
separation $D_{v}$ is given by

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

for some (non-empty) subsets $B_{v},R(f)\subseteq r^{-1}(v)$. The elements of
$B_{v}$ will be called blue edges. We write $R_{v}=\bigcup_{f\in s^{-1}(v)\cap
R_{r(f)}}R(f)$, and call the elements of $R_{v}$ generically red edges.

  3. (c)

(Vertices at even layers) For $j\geq 1$, we have

$F^{0,2j}=\bigsqcup_{v\in F^{0,2j-1}}s(B_{v})=\bigsqcup_{v\in
F^{0,2j-1}}s(R_{v}).$

Moreover, for any $v\in F^{0,2j-1}$, we have $s(e)\neq s(e^{\prime})$ for all
distinct edges $e,e^{\prime}\in B_{v}$, and similarly $s(f)\neq s(f^{\prime})$
for all distinct edges $f,f^{\prime}\in R_{v}$.

  4. (d)

(Vertices at the first odd layer) We have

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

Moreover, $s(e)\neq s(e^{\prime})$ for all distinct edges $e,e^{\prime}$ with
$e\in B_{v_{1}}$, $e^{\prime}\in B_{v_{2}}$, for any $v_{1},v_{2}\in F^{0,2j}$
with $j\geq 0$.

  5. (e)

(Compatibility at even layers) For each $j\geq 0$ and $v\in F^{0,2j}$, we have

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

  6. (f)

(Rombs at odd layers, red edges) For each $j\geq 1$, each $v\in F^{0,2j-1}$
and each pair $(f_{0},f_{1})$ of red edges such that $r(f_{0})=v$ and
$s(f_{0})=r(f_{1})$, there exists a (unique) pair $(e_{0},e_{1})$ of blue
edges such that $r(e_{0})=r(f_{0})=v$, $s(e_{0})=r(e_{1})$ and
$s(e_{1})=s(f_{1})$.

  7. (g)

(Rombs at odd layers, blue edges) For each $j\geq 1$, each $v\in F^{0,2j-1}$,
each red edge $g\in R_{r(g)}$ such that $s(g)=v$, and each pair
$(e_{0},e_{1})$ of blue edges such that $r(e_{0})=v$ and $s(e_{0})=r(e_{1})$,
there exists a unique pair $(f_{0},f_{1})$ of red edges such that $f_{0}\in
R(g)$, $r(f_{0})=r(e_{0})=v$, $s(f_{0})=r(f_{1})$ and $s(f_{1})=s(e_{1})$.

The family of $l$-diagrams will be denoted by LDiag.

From the very definition of an $l$-diagram we can infer two properties: the
first one is that the decompositions given in condition (d) above can be
extended to any odd layer; the second one is that one can automatically
construct rombs at even layers, using condition (e) above together with the
first property already mentioned. These are proven in the next proposition.

######  Proposition 3.3.

Let $(F,D)$ be an $l$-diagram.

  1. (i)

_(Vertices at odd layers)_ For $j\geq 1$, we have

$F^{0,2j-1}=\bigsqcup_{v\in F^{0,2j-2}}s(B_{v})=\bigcup_{v\in
F^{0,2j-2}}s(R_{v}).$

  2. (ii)

_(Rombs at even layers)_ Let $j\geq 0$ and $v\in F^{0,2j}$. Given a pair
$(e_{0},e_{1})$ of blue edges 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 $r(f_{0})=v$,
$r(f_{1})=s(f_{0})$, $s(f_{1})=s(e_{1})$ and $f_{1}\in R(f_{0})$.

Conversely, given a pair $(f_{0},f_{1})$ of red edges such that $r(f_{0})=v$
and $f_{1}\in R(f_{0})$, 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 $s(e_{1})=s(f_{1})$.

$e_{1}$$e_{0}$$f_{1}$$f_{0}$$v$$F^{0,2j}$$F^{0,2j+1}$$F^{0,2j+2}$

###### Proof.

  1. (i)

We can prove this by induction. The case $j=1$ holds by definition, so assume
the statement is true for all indices up to some $j-1\geq 1$ and let us prove
it for $j$. Given a vertex $v\in F^{0,2j-1}$, being $(F,D)$ a separated
Bratteli diagram implies that there exists some edge $e\in F^{1,2j-2}$ with
source $v$, which can be either blue or red.

In case $e=e_{1}\in B_{r(e_{1})}$ is a blue edge, then condition (c) above
applied to $F^{0,2j-2}$ tells us that there is another blue edge $e_{0}\in
B_{r(e_{0})}$ such that $s(e_{0})=r(e_{1})$. Moreover $r(e_{0})\in
F^{0,2j-3}$, thus by the induction hypothesis there is some red edge $g\in
R_{r(g)}$ such that $s(g)=r(e_{0})$. Now we can apply condition (g) to get a
unique pair $(f_{0},f_{1})$ of red edges such that $f_{0}\in R(g)$,
$r(f_{0})=r(e_{0})$, $s(f_{0})=r(f_{1})$ and $s(f_{1})=s(e_{1})=v$. This says
precisely that $v$ is the source of some red edge from $F^{1,2j-2}$.

In case $e=f_{1}\in R_{r(f_{1})}$ is a red edge, then condition (c) again
tells us that there is another red edge $f_{0}\in R_{r(f_{0})}$ such that
$s(f_{0})=r(f_{1})$. We apply condition (f) to get a unique pair
$(e_{0},e_{1})$ of blue edges such that $r(e_{0})=r(f_{0})$,
$s(e_{0})=r(e_{1})$ and $s(e_{1})=s(f_{1})=v$. This says precisely that $v$ is
the source of some blue edge from $F^{1,2j-2}$.

This proves that

$F^{0,2j-1}=\bigcup_{v\in F^{0,2j-2}}s(B_{v})=\bigcup_{v\in
F^{0,2j-2}}s(R_{v}).$

The disjointness of the first union follows from the last part of property
(d).

  2. (ii)

Let $(e_{0},e_{1})$ be a pair of blue edges such that $r(e_{0})=v$ and
$r(e_{1})=s(e_{0})$. By condition (e) we have

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

so there exists a pair of red edges $(f_{0},f_{1})$ such that
$s(e_{1})=s(f_{1})$, with $f_{1}\in R(f_{0})$ and $f_{0}\in R_{v}$ (so that
$r(f_{1})=s(f_{0})$ and $r(f_{0})=v$). 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
$r(f^{\prime}_{0})=v$, $s(f^{\prime}_{0})=r(f^{\prime}_{1})$,
$s(f^{\prime}_{1})=s(e_{1})=s(f_{1})$ and $f_{1}^{\prime}\in
R(f_{0}^{\prime})$. Condition (c) ensures that $f^{\prime}_{1}=f_{1}$; in
particular $s(f^{\prime}_{0})=r(f^{\prime}_{1})=r(f_{1})=s(f_{0})$. Since
$f_{1}^{\prime}=f_{1}\in R(f_{0})\cap R(f_{0}^{\prime})$, we conclude that
$f_{0}=f_{0}^{\prime}$. That is, the pair $(f_{0},f_{1})$ is unique.

The proof of the converse is analogous, although the last part of condition
(d) is needed.

∎

######  Notation 3.4.

We will sometimes write $(e_{0},e_{1},f_{0},f_{1})$ when referring to the romb
associated to either the blue pair of edges $(e_{0},e_{1})$ or to the red pair
of edges $(f_{0},f_{1})$.

######  Remarks 3.5.

  1. (1)

By condition (c) in Definition 3.2, we deduce that for each $j\geq 1$ and
$v\in F^{0,2j}$ there are exactly two different edges $e,f$ such that
$s(e)=s(f)=v$, one of them, say $e$, belongs to a blue set $e\in B_{r(e)}$ and
the other, $f$, belongs to $R(g)$ for a unique red edge $g$, that is, there is
a unique $g\in R_{r(g)}$ such that $s(g)=r(f)$ and $f\in R(g)$.

On the other hand, by part (i) of Proposition 3.3 and the last part of
condition (d), we deduce that for each $j\geq 1$ and $v\in F^{0,2j-1}$ there
is exactly one blue edge $e\in B_{r(e)}$ such that $s(e)=v$. There also exists
a red edge $f\in R_{r(f)}$ such that $s(f)=v$, but such red edge may not be
the unique red edge departing from $v$.

  2. (2)

We have exactly 2 “colors” at each vertex $v$ from an even layer $F^{0,2j}$,
namely the blue edges $B_{v}$ and the red edges $R_{v}$.

On the other hand, by remark (1) above, we have $|s^{-1}(v)|$ different
“colors” at each vertex $v\in F^{2j-1}$, $j\geq 1$, each one corresponding to
a different edge departing from $v$.

  3. (3)

The disjointness of both unions in condition (e) of Definition 3.2 is
automatic, due to condition (c) of the same definition and remark (1) above.

  4. (4)

The uniqueness of the pair $(e_{0},e_{1})$ in condition (f) of Definition 3.2
is also automatic, due to remark (1) above.

  5. (5)

An $l$-diagram cannot have double red edges, except possibly at the first
layer. More concretely, for $n\geq 2$, given vertices $v\in F^{0,n-1}$ and
$w\in F^{0,n}$, there is at most one red edge $f$ such that $s(f)=w$ and
$r(f)=v$. This is clear if $n$ is even by remark (1). Assume then that
$n=2j+1$ for some $j\geq 1$, and suppose that there are two red edges
$f_{1},f^{\prime}_{1}$ such that $s(f_{1})=s(f^{\prime}_{1})=w$ and
$r(f_{1})=r(f^{\prime}_{1})=v$. Let $f_{0}$ be the unique red edge such that
$s(f_{0})=v$, so there is a unique red edge $g\in R_{r(g)}$ such that
$f_{0}\in R(g)$ (see again remark (1)). By condition (f) in Definition 3.2,
there is a unique pair $(e_{0},e_{1})$ of blue edges such that
$r(e_{0})=r(f_{0}),s(e_{0})=r(e_{1})$ and $s(e_{1})=s(f_{1})=w$. But now
$(f_{0},f_{1})$ and $(f_{0},f^{\prime}_{1})$ are two pairs of red edges such
that $f_{0}\in R(g)$, $r(f_{0})=r(e_{0}),s(f_{0})=r(f_{1})=r(f^{\prime}_{1})$
and $s(f_{1})=s(f^{\prime}_{1})=s(e_{1})$, and this contradicts the uniqueness
statement in condition (g) of Definition 3.2. This shows that there is at most
one red edge from $w$ to $v$.

######  Example 3.6.

Figure 1 presents the first three layers of an $l$-diagram. Here we have, for
example, that the separation $D_{u}$ associated with the vertex $u\in F^{0,1}$
is written as

$D_{u}=\\{B_{u},R(f_{1}),R(f_{2})\\},$

where the set $B_{u}$ consists of the two blue edges having range $u$,
$R(f_{1})$ consists of the unique straight red edge having range $u$ (which we
call $g_{1}$), and $R(f_{2})$ consists of the unique dotted red edge having
range $u$. Similarly,

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

where here $B_{v}$ consists of the unique blue edge having range $v$, and
$R_{v}$ consists of the two dotted red edges having range $v$. It is
straightforward to check that conditions (a)-(g) from Definition 3.2 are
fulfilled, at least for these first three levels.

$f_{1}$$f_{2}$$g_{1}$$e_{1}$$g_{2}$$e_{2}$$u$$v$$w$$F^{0,0}$$F^{1,0}$$F^{0,1}$$F^{1,1}$$F^{0,2}$$F^{1,2}$$F^{0,3}$$F^{1,3}$
Figure 1. The first three levels of an $l$-diagram.

In the same Figure 1 we show two examples of rombs. For instance, the first
romb, based on the vertex $w\in F^{0,3}$ and ending at the vertex $u\in
F^{0,1}$, corresponds to condition (f) for the pair of red edges
$(g_{1},g_{2})$ determining a unique pair of blue edges $(e_{1},e_{2})$
satisfying the romb condition, and also corresponds to condition (g) for
$g=f_{1}$ and the pair of blue edges $(e_{1},e_{2})$ determining a unique pair
of red edges $(g_{1},g_{2})$ satisfying the romb condition and such that
$g_{1}\in R(f_{1})$. The second romb, based on the vertex $v$, is a
consequence of Proposition 3.3.

Our correspondence will associate, to every $l$-diagram $(F,D)$, a dynamical
system $(X,\sigma)$ where $X$ is a totally disconnected metrizable compact
space, and $\sigma:X\rightarrow X$ is a surjective local homeomorphism; and,
conversely, an $l$-diagram $(F,D)$ will be associated to every such dynamical
system $(X,\sigma)$.

######  Notation 3.7.

For ease of notation, we will denote by LHomeo the family of dynamical systems
$(X,\sigma)$ where $X$ is a totally disconnected metrizable compact space, and
$\sigma:X\rightarrow X$ is a surjective local homeomorphism.

Of course, the aforementioned constructions will hold ‘up to equivalences’ in
appropriate senses.

######  Definition 3.8.

Let $\sigma_{1}:X_{1}\rightarrow X_{1}$, $\sigma_{2}:X_{2}\rightarrow X_{2}$
be two surjective local homeomorphisms defined on totally disconnected
metrizable compact spaces $X_{1},X_{2}$. We say that $(X_{1},\sigma_{1})$ and
$(X_{2},\sigma_{2})$ are topologically conjugate if there exists a
homeomorphism $h:X_{1}\rightarrow X_{2}$ intertwining $\sigma_{1}$ and
$\sigma_{2}$, that is $h\circ\sigma_{1}=\sigma_{2}\circ h$.

We write $(X_{1},\sigma_{1})\sim_{t.c.}(X_{2},\sigma_{2})$ in case the two
dynamical systems are topologically conjugate. It is clear that $\sim_{t.c.}$
is an equivalence relation.

######  Remark 3.9.

The definition of conjugacy from [16, Definition 2.1] requires the additional
condition that $\sigma_{1}\circ h^{-1}=h^{-1}\circ\sigma_{2}$. However this
condition is automatic in our situation, as can be seen by pre- and post-
composing the identity $h\circ\sigma_{1}=\sigma_{2}\circ h$ with $h^{-1}$ (see
also [16, Lemma 2.6]).

For $l$-diagrams, the notion of equivalence that fits in our context (apart
from the natural notion of ‘up to isomorphism’ which we will define later) is
the notion of telescoping, or contraction. For an $l$-diagram $(F,D)$ and a
given pair $j<l$ of non-negative integers, we denote

$P_{j,l}=\\{(e_{j},e_{j+1},\dots,e_{l-1})\mid e_{i}\in
F^{1,i},s(e_{i})=r(e_{i+1})\\}\subseteq\prod_{i=j}^{l-1}F^{1,i}.$

The source of $\overline{e}=(e_{j},e_{j+1},\dots,e_{l-1})\in P_{j,l}$ is
$s(\overline{e}):=s(e_{l-1})$, and its range is $r(\overline{e}):=r(e_{j})$.

######  Definition 3.10.

Given an $l$-diagram $(F,D)$ and a sequence $(m_{n})_{n\geq 0}$, $0\leq
m_{0}<m_{1}<m_{2}<\cdots$ in ${\mathbb{N}}\cup\\{0\\}$, with $m_{0}$ even and
satisfying the property

(CR) $m_{n+1}-m_{n}\equiv 1\text{ }(\text{mod }2),$

we define the telescoping (or contraction) of $(F,D)$ with respect to
$(m_{n})_{n\geq 0}$ as the separated graph $(F^{\prime},D^{\prime})$ where:

  1. (a)

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

  2. (b)

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

$P^{B}_{j,l}:=\\{(e_{j},\dots,e_{l-1})\in P_{j,l}\mid e_{i}\in
B_{r(e_{i})},j\leq i\leq l-1\\},$ $P^{R}_{j,l}:=\\{(f_{j},\dots,f_{l-1})\in
P_{j,l}\mid f_{i}\in R_{r(f_{i})},\text{ }j\leq i\leq l-1\text{ and
}f_{i+1}\in R(f_{i}),\text{ $i$ even}\\}.$

  3. (c)

For $v\in(F^{\prime})^{0,2j}$ with $j\geq 0$,
$D^{\prime}_{v}:=\\{B^{\prime}_{v},R^{\prime}_{v}\\}$, where

$B^{\prime}_{v}=\\{\overline{e}\in P_{m_{2j},m_{2j+1}}^{B}\mid
r(\overline{e})=v\\},\quad R^{\prime}_{v}=\\{\overline{f}\in
P_{m_{2j},m_{2j+1}}^{R}\mid r(\overline{f})=v\\}.$

  4. (d)

For $v\in(F^{\prime})^{0,2j+1}$ with $j\geq 0$,
$D^{\prime}_{v}:=\\{B^{\prime}_{v}\\}\cup\\{R^{\prime}(\hat{f})\mid\hat{f}\in(F^{\prime})^{1,2j},s(\hat{f})=v,\hat{f}\in
R^{\prime}_{r(\hat{f})}\\}$, where

$B^{\prime}_{v}=\\{\overline{e}\in P_{m_{2j+1},m_{2j+2}}^{B}\mid
r(\overline{e})=v\\},$
$R^{\prime}(\hat{f})=\\{\overline{f}=(f_{m_{2j+1}},\dots,f_{m_{2j+2}-1})\in
P_{m_{2j+1},m_{2j+2}}^{R}\mid f_{m_{2j+1}}\in R(f^{\prime}_{m_{2j+1}-1})\\},$

where $\hat{f}=(f^{\prime}_{m_{2j}},\dots,f^{\prime}_{m_{2j+1}-1})$.

We note that both $m_{n}$ and $n$ have the same parity due to condition (CR)
and the fact that $m_{0}$ is even.

######  Lemma 3.11.

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

###### Proof.

First observe that $(F^{\prime},D^{\prime})$ is a separated Bratteli diagram
since $(F,D)$ is. To prove that it is indeed an $l$-diagram, we need to check
conditions (a)-(g) from Definition 3.2. Conditions (a) and (b) follow by
definition.

  1. (c)

We first show that, for $j\geq 1$,

$\bigsqcup_{v\in
F^{0,m_{2j-1}}}s(B^{\prime}_{v})\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny(c.1)}}}{=}}F^{0,m_{2j}}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny(c.2)}}}{=}}\bigsqcup_{v\in
F^{0,m_{2j-1}}}s(R^{\prime}_{v}).$

The inclusion $\subseteq$ in (c.1) is obvious. For the other one we use
condition (c) of Definition 3.2 and part (i) of Proposition 3.3. Given $v\in
F^{0,m_{2j}}$, there exists $v_{1}\in F^{0,m_{2j}-1}$ and $e_{m_{2j}-1}\in
B_{v_{1}}$ such that $v=s(e_{m_{2j}-1})$. Now since $v_{1}\in F^{0,m_{2j}-1}$,
by part (i) of Proposition 3.3 there exists $v_{2}\in F^{0,m_{2j}-2}$ and
$e_{m_{2j}-2}\in B_{v_{2}}$ such that $v_{1}=s(e_{m_{2j}-2})$. Proceeding by
induction, we construct a sequence $(e_{m_{2j}-i})_{i=1}^{m_{2j}-m_{2j-1}}$ of
blue edges satisfying $s(e_{m_{2j}-i-1})=r(e_{m_{2j}-i})$ and
$v=s(e_{m_{2j}-1})$. Then $e:=(e_{m_{2j-1}},\dots,e_{m_{2j}-1})\in
P^{B}_{m_{2j-1},m_{2j}}$ is such that $s(e)=v$, so the inclusion is now
proved. Now in order to prove that the union is indeed disjoint, take $v\in
s(B^{\prime}_{v_{1}})\cap s(B^{\prime}_{v_{2}})$ with $v_{1}\neq v_{2}\in
F^{0,m_{2j-1}}$. There exist then $e_{1}\in B^{\prime}_{v_{1}}$ and $e_{2}\in
B^{\prime}_{v_{2}}$ with $s(e_{1})=v=s(e_{2})$. Let $N$ be the integer such
that $(e_{1})_{N}\neq(e_{2})_{N}$ and $(e_{1})_{j}=(e_{2})_{j}$ for $j>N$. It
exists since $v_{1}\neq v_{2}$, so necessarily $e_{1}\neq e_{2}$. But then
$s((e_{1})_{N})=s((e_{2})_{N})$ and $(e_{1})_{N}\neq(e_{2})_{N}$, and this
contradicts either condition (c) of Definition 3.2 or part (i) of Proposition
3.3 at the $N^{\text{th}}$ layer, depending on the parity of $N$.

The same arguments hold exactly for (c.2), but special care needs to be taken
when proving disjointness of the union. This is because we need to make a
distinction in terms of the parity of $N$ which is of different nature to the
above distinction. Given $v\in s(R^{\prime}_{v_{1}})\cap
s(R^{\prime}_{v_{2}})$ with $v_{1}\neq v_{2}\in F^{0,m_{2j-1}}$, there exist
$f_{1}\in R^{\prime}_{v_{1}}$ and $f_{2}\in R^{\prime}_{v_{2}}$ with
$s(f_{1})=v=s(f_{2})$. We can again consider the integer $N$ such that
$(f_{1})_{N}\neq(f_{2})_{N}$ and $(f_{1})_{j}=(f_{2})_{j}$ for $j>N$. Thus
$s((f_{1})_{N})=s((f_{2})_{N})$ and $(f_{1})_{N}\neq(f_{2})_{N}$. If $N$ is
odd, the contradiction arises due to condition (c) of Definition 3.2. If $N$
is even, then $s((f_{1})_{N})=s((f_{2})_{N})$ belongs to an odd layer, thus
$(f_{1})_{N+1}=(f_{2})_{N+1}\in R((f_{1})_{N})\cap R((f_{2})_{N})$ by
definition of $P_{m_{2j-1},m_{2j}}^{B}$. This is a contradiction because
$R((f_{1})_{N})\cap R((f_{2})_{N})=\emptyset$ since
$(f_{1})_{N}\neq(f_{2})_{N}$.

Now for the second part. Take any $v\in(F^{\prime})^{0,2j-1}=F^{0,m_{2j-1}}$
and $e_{1},e_{2}\in B^{\prime}_{v}$. If it were the case that
$s(e_{1})=s(e_{2})$, then by applying properties (c) and (d) of Definition 3.2
at the different layers in between $F^{0,m_{2j-1}}$ and $F^{0,m_{2j}}$ we
would deduce that $e_{1}=e_{2}$. An argument similar to the one given in the
above paragraph shows the same property for $f_{1},f_{2}\in R^{\prime}_{v}$,
analyzing the parity of the different layers.

  2. (d)

The first two parts (1) and (2) follow mutatis mutandis the same arguments as
in (c) above, using conditions (c) and (d) of Definition 3.2 and part (i) of
Proposition 3.3 at the different layers.

The second property follows from the same results again, applied to the
different layers.

  3. (e)

We need to show that, for $v\in F^{0,m_{2j}}$,

$\bigsqcup_{e\in B^{\prime}_{v}}s(B^{\prime}_{s(e)})=\bigsqcup_{f\in
R^{\prime}_{v}}s(R^{\prime}(f)).$

Note that we do not need to prove disjointness of the unions, since these will
follow by Remark 3.5 (3). Let $w=s(e^{(1)})$, with
$e^{(1)}=(e_{m_{2j+1}}^{(1)},\dots,e_{m_{2j+2}-1}^{(1)})\in
B^{\prime}_{s(e^{(0)})}$ and
$e^{(0)}=(e_{m_{2j}}^{(0)},\dots,e_{m_{2j+1}-1}^{(0)})\in B^{\prime}_{v}$.
Note that $(e^{(0)},e^{(1)})\in P^{B}_{m_{2j},m_{2j+2}}$, and

$m_{2j+2}-m_{2j}\equiv(2j+2)-2j\equiv 0\text{ (mod }2),$

so we have an even number of edges from $w$ to $v$ through the blue path
$(e^{(0)},e^{(1)})$. We can then write the path as
$(e^{(0)},e^{(1)})=(e_{1,0},e_{1,1},\dots,e_{n_{j},0},e_{n_{j},1})$, being
$n_{j}$ the natural number satisfying $2n_{j}=m_{2j+2}-m_{2j}$. For each layer
$1\leq i\leq n_{j}$, we can make use of condition (e) of Definition 3.2 to
conclude that there exist pairs of red edges $(f_{i,0},f_{i,1})$ such that
$f_{i,0}\in R_{r(e_{i,0})},f_{i,1}\in R(f_{i,0})$, and
$s(f_{i,1})=s(e_{i,1})$. We then construct two red paths

$(f^{(0)},f^{(1)})=(f_{1,0},f_{1,1},\dots,f_{n_{j},0},f_{n_{j},1})$

of lengths $m_{2j+2}-m_{2j+1}$ and $m_{2j+1}-m_{2j}$ respectively, satisfying
$f^{(0)}\in R^{\prime}_{v}$ and $f^{(1)}\in R^{\prime}(f^{(0)})$. Since
$w=s(f^{(1)})$, we see that $w\in s(R^{\prime}(f^{(0)}))$ with $f^{(0)}\in
R^{\prime}_{v}$, thus completing the proof of the first inclusion. The other
inclusion is completely analogous, and so we omit it.

  4. (f)

Let $v\in(F^{\prime})^{0,2j-1}=F^{0,m_{2j-1}}$ and $(f^{(0)},f^{(1)})$ a pair
of red edges such that $r(f^{(0)})=v$ and $s(f^{(0)})=r(f^{(1)})\in
F^{0,m_{2j}}$. Note that $w:=s(f^{(1)})\in F^{0,m_{2j+1}}$ and

$m_{2j+1}-m_{2j-1}\equiv(2j+1)-(2j-1)\equiv 0\text{ (mod }2),$

so we have an even number of edges from $w$ to $v$ through the red path
$(f^{(0)},f^{(1)})$. We can then write the path as
$(f^{(0)},f^{(1)})=(f_{1,0},f_{1,1},\dots,f_{n_{j},0},f_{n_{j},1})$, being
$n_{j}$ the natural number satisfying $2n_{j}=m_{2j+1}-m_{2j-1}$. For each
layer $1\leq i\leq n_{j}$, we can make use of condition (f) of Definition 3.2
to conclude that there exist pairs of blue edges $(e_{i,0},e_{i,1})$ such that
$r(e_{i,0})=r(f_{i,0})$, $s(e_{i,0})=r(e_{i,1})$ and
$s(e_{i,1})=s(f_{i,1})=r(f_{i+1,0})=r(e_{i+1,0})$. We then construct two blue
paths

$(e^{(0)},e^{(1)})=(e_{1,0},e_{1,1},\dots,e_{n_{j},0},e_{n_{j},1})$

of lengths $m_{2j+1}-m_{2j}$ and $m_{2j}-m_{2j-1}$ respectively, satisfying
$r(e^{(0)})=r(e_{1,0})=r(f_{1,0})=v$, $s(e^{(0)})=r(e^{(1)})$ and
$s(e^{(1)})=s(e_{n_{j},1})=s(f_{n_{j},1})=w$. As noticed in Remark 3.5 (4), it
is not necessary to prove uniqueness of this blue pair of edges
$(e^{(0)},e^{(1)})$.

  5. (g)

Let $v\in(F^{\prime})^{0,2j-1}=F^{0,m_{2j-1}}$, a red edge $g\in
R^{\prime}_{r(g)}$ such that $s(g)=v$, and $(e^{(0)},e^{(1)})$ a pair of blue
edges such that $r(e^{(0)})=v$ and $s(e^{(0)})=r(e^{(1)})\in F^{0,m_{2j}}$.
Note that $w:=s(e^{(1)})\in F^{0,m_{2j+1}}$, and again

$m_{2j+1}-m_{2j-1}\equiv(2j+1)-(2j-1)\equiv 0\text{ (mod }2),$

so we have an even number of edges from $w$ to $v$ through the blue path
$(e^{(0)},e^{(1)})$. We can then write the path as
$(e^{(0)},e^{(1)})=(e_{1,0},e_{1,1},\dots,e_{n_{j},0},e_{n_{j},1})$, being
$n_{j}$ the natural number satisfying $2n_{j}=m_{2j+1}-m_{2j-1}$. For each
layer $1\leq i\leq n_{j}$, we can make use of condition (g) of Definition 3.2
to conclude that there exist unique pairs of red edges $(f_{i,0},f_{i,1})$
such that $f_{1,0}\in R(g_{0}),f_{i,0}\in R(f_{i-1,1})$ for $2\leq i\leq
n_{j}$, and $r(f_{i,0})=r(e_{i,0}),s(f_{i,0})=r(f_{i,1})$ and
$s(f_{i,1})=s(e_{i,1})$. Here $g_{0}$ is the first red edge of the path $g$,
i.e. the one having source $v$. We then construct two red paths

$(f^{(0)},f^{(1)})=(f_{1,0},f_{1,1},\dots,f_{n_{j},0},f_{n_{j},1})$

with $f^{(0)}\in P^{R}_{m_{2j-1},m_{2j}}$ and $f^{(1)}\in
P^{R}_{m_{2j},m_{2j+1}}$, satisfying $r(f^{(0)})=r(f_{1,0})=r(e_{1,0})=v$,
$s(f^{(0)})=r(f^{(1)})$ and $s(f^{(1)})=s(f_{n_{j},1})=s(e_{n_{j},1})=w$.
Moreover, $f^{(0)}\in R^{\prime}(g)$ by the definition of $R^{\prime}(g)$
(Definition 3.10 (d)). This is the desired red pair.

To prove uniqueness, assume that $(\widetilde{f}^{(0)},\widetilde{f}^{(1)})$
is another pair of red edges satisfying the same conditions as the pair
$(f^{(0)},f^{(1)})$, namely that $\widetilde{f}^{(0)}\in R^{\prime}(g)$,
$r(\widetilde{f}^{(0)})=r(e^{(0)})=v$,
$s(\widetilde{f}^{(0)})=r(\widetilde{f}^{(1)})$ and
$s(\widetilde{f}^{(1)})=s(e^{(1)})$. Write it as

$(\widetilde{f}^{(0)},\widetilde{f}^{(1)})=(\widetilde{f}_{1,0},\widetilde{f}_{1,1},\dots,\widetilde{f}_{n_{j},0},\widetilde{f}_{n_{j},1}).$

We apply condition (f) of Definition 3.2 to each pair of red edges
$(\widetilde{f}_{i,0},\widetilde{f}_{i,1})$ to conclude that there exist pairs
$(\widetilde{e}_{i,0},\widetilde{e}_{i,1})$ of blue edges such that
$r(\widetilde{e}_{i,0})=r(\widetilde{f}_{i,0})$,
$s(\widetilde{e}_{i,0})=r(\widetilde{e}_{i,1})$ and
$s(\widetilde{e}_{i,1})=s(\widetilde{f}_{i,1})$. Now,

$s(\widetilde{e}_{n_{j},1})=s(\widetilde{f}^{(1)})=s(e^{(1)})=s(e_{n_{j},1})\in
F^{0,m_{2j+1}}.$

By condition (d) of Definition 3.2, we necessarily have
$\widetilde{e}_{n_{j},1}=e_{n_{j},1}$. Thus in particular
$s(\widetilde{e}_{n_{j},0})=r(\widetilde{e}_{n_{j},1})=r(e_{n_{j},1})=s(e_{n_{j},0})\in
F^{0,m_{2j+1}-1}$. Now condition (c) of Definition 3.2 ensures that
$\widetilde{e}_{n_{j},0}=e_{n_{j},0}$ necessarily. Therefore
$r(\widetilde{f}_{n_{j},0})=r(\widetilde{e}_{n_{j},0})=r(e_{n_{j},0})=r(f_{n_{j},0})$.
We continue in this way to get that $\widetilde{e}_{i,j}=e_{i,j}$ for all
$i,j$, and thus

$r(f_{i,0})=r(e_{i,0})=r(\widetilde{e}_{i,0})=r(\widetilde{f}_{i,0})\quad\text{
for all }1\leq i\leq n_{j}.$

Now observe that $\widetilde{f}_{i,0}\in R(g_{0})$, and thus by the uniqueness
part of condition (g) of Definition 3.2, we get that the pairs
$(\widetilde{f}_{1,0},\widetilde{f}_{1,1})$ and $(f_{1,0},f_{1,1})$ are the
same. By applying this argument inductively (using that
$f^{(0)},f^{(1)},\widetilde{f}^{(0)},\widetilde{f}^{(1)}$ belong to
$P^{R}_{j,l}$ for suitable $j,l$), we end up concluding that
$\widetilde{f}^{(0)}=f^{(0)}$ and $\widetilde{f}^{(1)}=f^{(1)}$. This proves
the uniqueness property we wanted to show.

The lemma is proved. ∎

######  Definition 3.12.

We let $\sim_{l}$ denote the equivalence relation on $l$-diagrams generated by
‘up to isomorphism’ equivalence and ‘up to contraction’ equivalence, defined
as follows.

  1. (a)

(Isomorphism of $l$-diagrams) The $l$-diagrams $(F_{1},D_{1})$ and
$(F_{2},D_{2})$ are said to be isomorphic, written
$(F_{1},D_{1})\cong(F_{2},D_{2})$, if there exists
$\phi:(F_{1},D_{1})\rightarrow(F_{2},D_{2})$ an isomorphism of separated
diagrams (see [13]) such that, for any $j\geq 0$, we have

$\phi_{1}((B_{1})_{v})=(B_{2})_{\phi_{0}(v)}\quad\text{ and
}\quad\phi_{1}((R_{1})_{v})=(R_{2})_{\phi_{0}(v)}$

in case $v\in(F_{1})^{0,2j}$, and

$\phi_{1}((B_{1})_{v})=(B_{2})_{\phi_{0}(v)}\quad\text{ and
}\quad\phi_{1}(R_{1}(f))=R_{2}(\phi_{1}(f))$

in case $v\in(F_{1})^{0,2j+1}$, with $f\in R_{r(f)}$ and $s(f)=v$. Here
$R_{1}(f)\subseteq(R_{1})_{v}$ and
$R_{2}(\phi_{1}(f))\subseteq(R_{2})_{\phi_{0}(v)}$.

  2. (b)

(Contraction of $l$-diagrams) The $l$-diagrams $(F_{1},D_{1})$ and
$(F_{2},D_{2})$ are said to be equivalent by contraction, written
$(F_{1},D_{1})\sim_{\text{ctr}}(F_{2},D_{2})$, if there exists a sequence of
$l$-diagrams $\\{(E_{i},C_{i})\\}_{i=1}^{n}$ with
$(E_{1},C_{1})=(F_{1},D_{1})$ and $(E_{n},C_{n})=(F_{2},D_{2})$ so that
$(E_{i+1},C_{i+1})$ is obtained from $(E_{i},C_{i})$ by contraction (as
defined in Definition 3.10), or vice versa. Symbolically, we will write

${(E_{1},C_{1})}$${(E_{2},C_{2})}$${\cdots}$${(E_{n},C_{n})}$

in case $(E_{1},C_{1})\sim_{\text{ctr}}(E_{n},C_{n})$.

It is clear that both relations $\cong$ and $\sim_{\text{ctr}}$ define
equivalence relations, so that $\sim_{l}$ is well-defined.

######  Lemma 3.13.

$(E_{1},C_{1})\sim_{\emph{ctr}}(E_{n},C_{n})$ if and only if there exists a
sequence of $l$-diagrams $\\{(E_{i},C_{i})\\}_{i=2}^{n-1}$ such that

${(E_{1},C_{1})}$${(E_{2},C_{2})}$${(E_{3},C_{3})}$${\cdots}$${(E_{n},C_{n})}$

That is, each $(E_{2i},C_{2i})$ can be obtained by contracting
$(E_{2i-1},C_{2i-1})$ and also by contracting $(E_{2i+1},C_{2i+1})$.

###### Proof.

It suffices to prove that the composition of two contractions is again a
contraction. To this aim, suppose that $(F_{3},D_{3})$ is a contraction of
$(F_{2},D_{2})$ and $(F_{2},D_{2})$ is a contraction of $(F_{1},D_{1})$. Thus
there exist strictly increasing sequences $(m_{n})_{n\geq 0}$ and
$(m^{\prime}_{n})_{n\geq 0}$ in ${\mathbb{N}}\cup\\{0\\}$, with $m_{0}$ and
$m^{\prime}_{0}$ even and satisfying (CR), and such that
$(F_{2},D_{2})=(F^{\prime}_{1},D^{\prime}_{1})$ and
$(F_{3},D_{3})=(F^{\prime}_{2},D^{\prime}_{2})$. Then the sequence
$M_{n}:=m^{\prime}_{m_{n}}$ also satisfies (CR):

$M_{n+1}-M_{n}=m^{\prime}_{m_{n+1}}-m^{\prime}_{m_{n}}\equiv
m_{n+1}-m_{n}\equiv 1\text{ (mod }2).$

Moreover, $M_{0}=m^{\prime}_{m_{0}}\equiv m_{0}\equiv 0\text{ (mod }2)$, and
thus $M_{0}$ is also even. The rest is now clear. ∎

Our first aim is to prove the following theorem.

######  Theorem 3.14.

There exists a bijective correspondence between equivalence classes of
$l$-diagrams under $\sim_{l}$ and topological conjugacy classes of surjective
local homeomorphisms on totally disconnected, compact metrizable spaces.

More concretely, there are well-defined maps

$\Phi:\emph{{LHomeo}}/\sim_{t.c.}\longrightarrow\emph{{LDiag}}/\sim_{l},\quad\Psi:\emph{{LDiag}}/\sim_{l}\longrightarrow\emph{{LHomeo}}/\sim_{t.c.}$

which are inverse of each other.

The rest of the section is devoted to the proof of Theorem 3.14.

###  3.1. From LHomeo to LDiag

We start by associating an $l$-diagram to a surjective local homeomorphism.
Let $(X,\sigma)\in\textsf{LHomeo}$. First, some general remarks about
$\sigma$.

######  Lemma 3.15.

There always exists a partition ${\pazocal P}$ of $X$ such that the
restriction $\sigma|_{Z}$ is a homeomorphism for any $Z\in{\pazocal P}$.
Moreover, each $\sigma(Z)$ is also clopen.

As a consequence, the set $\sigma^{-1}(\\{x\\})$ is finite for any $x\in X$,
and in fact uniformly bounded by $|{\pazocal P}|$ above.

###### Proof.

Since $\sigma$ is a local homeomorphism, for any $x\in X$ we can find an open
set $U_{x}$ such that $\sigma(U_{x})$ is open in $X$ and $\sigma|_{U_{x}}$ is
a homeomorphism. In fact, since the space $X$ is totally disconnected, we can
assume that each $U_{x}$ is clopen. Then $X=\bigcup_{x\in X}U_{x}$, so by
compactness we can find a finite subfamily $\\{U_{1},\dots,U_{n}\\}$ of
$\\{U_{x}\\}_{x\in X}$ such that $X=\bigcup_{i=1}^{n}U_{i}$, each $U_{i}$ is
clopen and $\sigma|_{U_{i}}$ is a homeomorphism. By disjontifying
$\\{U_{1},\dots,U_{n}\\}$, we can construct another finite family
$\\{V_{1},\dots,V_{m}\\}$ such that its elements are clopen and pairwise
disjoint, $X=\bigcup_{i=1}^{m}V_{i}$, and each $\sigma|_{V_{i}}$ is a
homeomorphism. This family is our desired partition ${\pazocal P}$, so
$|{\pazocal P}|=m$.

Note that $\sigma(V_{i})$ is open in $X$ for $1\leq i\leq m$. Moreover, since
each $V_{i}$ is closed in $X$, it is in particular compact, thus
$\sigma(V_{i})$ is also compact. Since $X$ is a Hausdorff space, this implies
that $\sigma(V_{i})$ is closed. That is, each $\sigma(V_{i})$ is clopen.

To conclude, assume that there is some $x\in X$ with at least $m+1$ different
pre-images $y_{1},\dots,y_{m+1}\in X$. Since $X=\bigcup_{i=1}^{m}V_{i}$, there
are at least two pre-images of $x$ that lie inside the same $V_{i}$, say
$y_{1},y_{2}\in V_{i}$. But this contradicts the fact that $\sigma$ is
injective when restricted to $V_{i}$. The lemma now follows. ∎

Let $N(\sigma):=\text{sup}\\{|\sigma^{-1}(\\{x\\})|\mid x\in X\\}$, which is a
finite quantity by the lemma above. For each $1\leq i\leq N(\sigma)$, define

${\pazocal U}_{i}:=\\{x\in X\mid|\sigma^{-1}(\\{x\\})|=i\\}.$

Since $\sigma$ is surjective, the family $\\{{\pazocal U}_{1},\dots,{\pazocal
U}_{N(\sigma)}\\}$ covers $X$. In fact, they form a partition of $X$ into
clopen sets, as we prove in the following lemma.

######  Lemma 3.16.

The family $\\{{\pazocal U}_{1},\dots,{\pazocal U}_{N(\sigma)}\\}$ forms a
partition of $X$ into clopen sets.

###### Proof.

It is clear that the sets ${\pazocal U}_{1},\dots,{\pazocal U}_{N(\sigma)}$
are disjoint and cover $X$. Thus it is enough to show that each ${\pazocal
U}_{i}$ is clopen. We will in fact show that each set

${\pazocal U}_{\geq i}:=\\{x\in X\mid|\sigma^{-1}(\\{x\\})|\geq i\\},$

for $1\leq i\leq N(\sigma)$, is clopen, from which the result will follow
immediately since ${\pazocal U}_{i}={\pazocal U}_{\geq i}\backslash{\pazocal
U}_{\geq i+1}$ for $1\leq i<N(\sigma)$, and ${\pazocal
U}_{N(\sigma)}={\pazocal U}_{\geq N(\sigma)}$.

By letting ${\pazocal P}$ be any partition satisfying the properties of Lemma
3.15, we see that

${\pazocal U}_{\geq
i}=\bigcup_{\begin{subarray}{c}(Z_{1},\dots,Z_{i})\in{\pazocal P}^{i}\\\
\text{all different}\end{subarray}}\sigma(Z_{1})\cap\cdots\cap\sigma(Z_{i}).$

Indeed, given $x\in{\pazocal U}_{\geq i}$, there exist at least $i$ different
pre-images of $x$, namely $y_{1},\dots,y_{i}$. Since $\sigma$ restricts to a
homeomorphism on each $Z\in{\pazocal P}$, it is clear that $y_{j}\in Z_{j}$
for different $Z_{j}$’s. Thus $x\in\sigma(Z_{1})\cap\cdots\cap\sigma(Z_{i})$
for $(Z_{1},\dots,Z_{i})\in{\pazocal P}^{i}$ all different. The converse
inclusion is trivial, since any element in
$\sigma(Z_{1})\cap\cdots\cap\sigma(Z_{i})$ for
$(Z_{1},\dots,Z_{i})\in{\pazocal P}^{i}$ all different has, by definition, at
least $i$ different pre-images.

The result follows since each $\sigma(Z)$ is clopen. ∎

######  Remark 3.17.

Any covering map of topological spaces (in the sense of [25]) is a surjective
local homeomorphism, and any surjective local homeomorphism on a compact space
is a covering map. Hence the results from [29] and [33] apply to the family of
dynamical systems studied in the present paper. In particular it follows from
[33, Theorem 9.1] that, if $(X,\sigma)\in\textsf{LHomeo}$, then the groupoid
$C^{*}$-algebra $C^{*}({\pazocal G}(X,\sigma))$ is isomorphic to the crossed
product $C(X)\rtimes_{\alpha,\mathscr{L}}\mathbb{N}$, where $\alpha\colon
C(X)\to C(X)$ is the $*$-homomorphism given by $\alpha(f)=f\circ\alpha$ and
$\mathscr{L}\colon C(X)\to C(X)$ is the transfer operator

$\mathscr{L}(f)(x)=\frac{1}{|\sigma^{-1}(x)|}\sum_{y\in\sigma^{-1}(x)}f(y).$

Note that in [25, Example 4.5] it is constructed an example of a surjective
local homeomorphism $f\colon X\to Y$ which is not a covering map, where $X$
and $Y$ are totally disconnected, locally compact, second-countable spaces.
Take $Z:=\bigsqcup_{n=0}^{\infty}Y_{n}\sqcup\bigsqcup_{m=0}^{\infty}X_{m}$,
where $Y_{n}=Y$ for all $n\geq 0$ and $X_{m}=X$ for all $m\geq 0$. Denoting by
$y_{n}$ the copy of $y\in Y$ in $Y_{n}$, and by $x_{m}$ the copy of $x\in X$
in $X_{m}$, we define $\sigma\colon Z\to Z$ by $\sigma(y_{m})=y_{m+1}$ for
$m\geq 0$, $\sigma(x_{n})=x_{n-1}$ for all $n\geq 1$, and
$\sigma(x_{0})=f(x)_{0}$. Then $Z$ is a totally disconnected, locally compact,
second countable space, and $\sigma$ is a surjective local homeomorphism on
$Z$ which is not a covering map. Another nice example of a surjective local
homeomorphism which is not a covering map is the map
$f\colon{\mathbb{C}}\setminus\\{0\\}\to{\mathbb{C}}\setminus\\{0\\}$ defined
by $f(z)=e^{z}$.

######  Definition 3.18.

For a partition ${\pazocal P}$ of $X$ such that $\sigma$ restricts to a
homeomorphism on each $Z\in{\pazocal P}$, we define new partitions

  1. (a)

${\pazocal
P}^{\sigma}:=\bigcup_{k=1}^{N(\sigma)}\\{\sigma(Z_{1})\cap\cdots\cap\sigma(Z_{k})\backslash{\pazocal
U}_{\geq k+1}\neq\emptyset\mid Z_{i}\in{\pazocal P}\text{ all different}\\}$;

  2. (b)

$\sigma^{-1}({\pazocal P}):=\\{\sigma^{-1}(Z)\mid Z\in{\pazocal P}\\}$.

The next lemma shows that ${\pazocal P}^{\sigma}$ and $\sigma^{-1}({\pazocal
P})$ are indeed well-defined.

######  Lemma 3.19.

The sets ${\pazocal P}^{\sigma}$ and $\sigma^{-1}({\pazocal P})$ constitute
partitions of $X$. Also, if ${\pazocal P}_{1}$ and ${\pazocal P}_{2}$ are
partitions of $X$, then

$({\pazocal P}_{1}\vee{\pazocal P}_{2})^{\sigma}={\pazocal
P}_{1}^{\sigma}\vee{\pazocal P}_{2}^{\sigma}\quad\text{ and
}\quad\sigma^{-1}({\pazocal P}_{1}\vee{\pazocal P}_{2})=\sigma^{-1}({\pazocal
P}_{1})\vee\sigma^{-1}({\pazocal P}_{2}).$

###### Proof.

The statement for $\sigma^{-1}({\pazocal P})$ is clear. Let us prove it for
${\pazocal P}^{\sigma}$.

First, note that each non-empty
$\overline{Z}=\sigma(Z_{1})\cap\cdots\cap\sigma(Z_{k})\backslash{\pazocal
U}_{\geq k+1}\in{\pazocal P}^{\sigma}$ is clopen in $X$, since ${\pazocal
U}_{\geq k+1}$ is clopen and each $\sigma(Z_{i})$ is clopen too.

Second, take
$\overline{Z}_{1}=\sigma(Z_{1})\cap\cdots\cap\sigma(Z_{k})\backslash{\pazocal
U}_{\geq k+1}$ and
$\overline{Z}_{2}=\sigma(Z^{\prime}_{1})\cap\cdots\cap\sigma(Z^{\prime}_{k^{\prime}})\backslash{\pazocal
U}_{\geq k^{\prime}+1}$ elements from ${\pazocal P}^{\sigma}$, so that
$(Z_{1},\dots,Z_{k})$ are all different, and so are
$(Z^{\prime}_{1},\dots,Z^{\prime}_{k^{\prime}})$. If say $k>k^{\prime}$, then
$(Z_{1},\dots,Z_{k})$ constitute $k>k^{\prime}$ different pairwise disjoint
clopen sets, thus $\sigma(Z_{1})\cap\cdots\cap\sigma(Z_{k})\subseteq{\pazocal
U}_{\geq k^{\prime}+1}$. But then

$\overline{Z}_{1}\cap\overline{Z}_{2}=\sigma(Z_{1})\cap\cdots\cap\sigma(Z_{k})\cap\sigma(Z^{\prime}_{1})\cap\cdots\cap\sigma(Z^{\prime}_{k^{\prime}})\backslash({\pazocal
U}_{\geq k+1}\cup{\pazocal U}_{\geq k^{\prime}+1})\subseteq{\pazocal U}_{\geq
k^{\prime}+1}\backslash{\pazocal U}_{\geq k^{\prime}+1}=\emptyset.$

The argument for $k^{\prime}>k$ is similar, so we may assume that
$k^{\prime}=k$. But then

$\overline{Z}_{1}\cap\overline{Z}_{2}=\sigma(Z_{1})\cap\cdots\cap\sigma(Z_{k})\cap\sigma(Z^{\prime}_{1})\cap\cdots\cap\sigma(Z^{\prime}_{k})\backslash{\pazocal
U}_{\geq k+1}.$

If some $Z_{i}$ is different from all $Z^{\prime}_{j}$’s, then
$(Z_{i},Z^{\prime}_{1},\dots,Z^{\prime}_{k})$ are all different, thus
$\sigma(Z_{i})\cap\sigma(Z^{\prime}_{1})\cap\cdots\cap\sigma(Z^{\prime}_{k})\subseteq{\pazocal
U}_{\geq k+1}$. This gives, as before,
$\overline{Z}_{1}\cap\overline{Z}_{2}=\emptyset$. This proves that the
elements of ${\pazocal P}^{\sigma}$ are pairwise disjoint.

To conclude, simply note that
$\\{\sigma(Z_{1})\cap\cdots\cap\sigma(Z_{k})\backslash{\pazocal U}_{\geq
k+1}\neq\emptyset\mid Z_{i}\in{\pazocal P}\text{ all different}\\}$ gives a
partition of ${\pazocal U}_{k}$, thus $\bigcup_{\overline{Z}\in{\pazocal
P}^{\sigma}}\overline{Z}={\pazocal U}_{1}\sqcup\cdots\sqcup{\pazocal
U}_{N(\sigma)}=X$.

The rest of the lemma is straightforward. ∎

We now define the key concept of a $\sigma$-refined sequence of partitions.

######  Definition 3.20.

We say that a sequence of partitions $\\{{\pazocal P}_{n}\\}_{n\geq 0}$ of $X$
is a $\sigma$-refined sequence of partitions of $X$ if the following
properties hold:

  1. (a)

$\sigma$ restricts to a homeomorphism $\sigma|_{Z}$ for any $Z\in{\pazocal
P}_{0}$ (the existence of such a partition is guaranteed by Lemma 3.15).

  2. (b)

${\pazocal P}_{n}\precsim{\pazocal P}_{n+1}$, that is ${\pazocal P}_{n+1}$ is
finer than ${\pazocal P}_{n}$, for all $n\geq 0$.

  3. (c)

The union $\bigcup_{n\geq 0}{\pazocal P}_{n}$ generates the topology of $X$.

  4. (d)

$\text{diam}({\pazocal P}_{n})$ tends to $0$ as $n\rightarrow\infty$.

  5. (e)

We have

${\pazocal P}_{2j}\vee{\pazocal P}_{2j}^{\sigma}\precsim{\pazocal
P}_{2j+1},\quad\text{and}\quad{\pazocal P}_{2j+1}\vee\sigma^{-1}({\pazocal
P}_{2j+1})\precsim{\pazocal P}_{2j+2}$

for all $j\geq 0$.

Note that conditions (a) and (b) together imply that, for any $n\geq 0$,
$\sigma$ restricts to a homeomorphism $\sigma|_{Z}$ for any $Z\in{\pazocal
P}_{n}$.

######  Remark 3.21.

Since $X$ is compact and all our partitions are finite, it is straightforward
to check that any $\sigma$-refined sequence of partitions of $X$ in the sense
of Definition 3.20 is a complete, tame defining sequence in the sense of [28,
Definitions 3.1.1 and 3.1.8].

######  Lemma 3.22.

Let $(X,\sigma)\in\emph{{LHomeo}}$. Then $X$ admits a $\sigma$-refined
sequence of partitions.

###### Proof.

Take a sequence of partitions ${\pazocal P}^{\prime}_{n}$ satisfying
conditions (a)-(d) in Definition 3.20. It always exists by our assumptions on
the space $X$. Now define new partitions inductively by

$\displaystyle{\pazocal P}_{0}:={\pazocal P}^{\prime}_{0};$
$\displaystyle{\pazocal P}_{2j+1}:={\pazocal P}^{\prime}_{2j+1}\vee{\pazocal
P}_{2j}\vee{\pazocal P}_{2j}^{\sigma}$ $\displaystyle\text{ for all }j\geq 0;$
$\displaystyle{\pazocal P}_{2j+2}:={\pazocal P}^{\prime}_{2j+2}\vee{\pazocal
P}_{2j+1}\vee\sigma^{-1}({\pazocal P}_{2j+1})$ $\displaystyle\text{ for all
}j\geq 0.$

It is clear that the sequence $\\{{\pazocal P}_{n}\\}_{n\geq 0}$ satisfies the
required properties. ∎

Thus from now on we will assume that a $\sigma$-refined sequence of partitions
$\\{{\pazocal P}_{n}\\}_{n\geq 0}$ is given.

######  Examples 3.23.

  1. 1.

Take $X=\\{0,1\\}^{{\mathbb{Z}}}$ as a model for the Cantor set, and define
the two-sided shift

$\varphi\colon X\to X,\quad\varphi(x)_{i}=x_{i+1}.$

A basis for the topology of $X$ is again given by the cylinder sets: for
integers $r,l\geq 0$ and $\varepsilon_{-l},...,\varepsilon_{r}\in\\{0,1\\}$,
these are given by

$[\varepsilon_{-l}\cdots\underline{\varepsilon_{0}}\cdots\varepsilon_{r}]:=\\{x\in
X\mid x_{i}=\varepsilon_{i}\text{ for }-l\leq i\leq r\\}.$

With this topology, $(X,\varphi)\in\textsf{LHomeo}$, and in fact $\varphi$ is
even a homeomorphism. The sequence of partitions $\\{{\pazocal
P}_{n}\\}_{n\geq 0}$, given, for $n\geq 0$, by

${\pazocal
P}_{2n}=\\{[\varepsilon_{-n}\cdots\underline{\varepsilon_{0}}\cdots\varepsilon_{n}]\mid\varepsilon_{i}\in\\{0,1\\}\text{
for }-n\leq i\leq n\\},$ ${\pazocal
P}_{2n+1}=\\{[\varepsilon_{-n-1}\varepsilon_{-n}\cdots\underline{\varepsilon_{0}}\cdots\varepsilon_{n}]\mid\varepsilon_{i}\in\\{0,1\\}\text{
for }-n-1\leq i\leq n\\},$

is a $\varphi$-refined sequence of partitions of $X$.

  2. 2.

Take $X^{+}=\\{0,1\\}^{{\mathbb{N}}_{0}}$ as another model for the Cantor set,
but consider now the one-sided shift

$\sigma\colon X^{+}\to X^{+},\quad\sigma(x)_{i}=x_{i+1}.$

A basis for the topology of $X^{+}$ is again given by the cylinder sets, where
now $l=0$. Even though $\sigma$ is not a homeomorphism, it is certainly a
surjective local homeomorphism, so $(X^{+},\sigma)\in\textsf{LHomeo}$. The
sequence of partitions $\\{{\pazocal P}_{n}\\}_{n\geq 0}$, given, for $n\geq
0$, by

${\pazocal
P}_{n}=\\{[\underline{\varepsilon_{0}}\cdots\varepsilon_{n}]\mid\varepsilon_{i}\in\\{0,1\\}\text{
for }0\leq i\leq n\\},$

is a $\sigma$-refined sequence of partitions of $X^{+}$.

The main idea to construct the $l$-diagram $(F,D)$ from $(X,\sigma)$ is the
following: the vertices will correspond to the sets $Z\in{\pazocal P}_{n}$,
with ${\pazocal P}_{n}$ a $\sigma$-refined sequence of partitions; the blue
edges will correspond to inclusion maps $Z\subseteq Z^{\prime}$, and the red
edges will correspond to the action of $\sigma$ or $\sigma^{-1}$, which in the
latter case will give rise to various red edges departing from the same
vertex.

######  Construction 3.24.

Take $\\{{\pazocal P}_{n}\\}_{n\geq 0}$ to be any $\sigma$-refined sequence of
partitions of $X$. We construct $(F,D)$ inductively, as follows.

  1. (a)

(Vertices) For $n\geq 0$ we set $F^{0,n}={\pazocal P}_{n}$, so that the
vertices at the $n^{\text{th}}$ layer are the clopen sets in the partition
${\pazocal P}_{n}$.

  2. (b)

(Edges) For $n\geq 0$, we let $F^{1,n}=B^{(n)}\sqcup R^{(n)}$, where the set
$B^{(n)}$ will consist on the blue edges, and the set $R^{(n)}$ will be the
set of red edges, which we now define. We also set, for $Z\in F^{0,n}$,

$B_{Z}=\\{e\in B^{(n)}\mid r(e)=Z\\},\quad R_{Z}=\\{f\in R^{(n)}\mid
r(f)=Z\\}.$

  3. (b.1)

(Blue edges) The blue edges are defined as follows. For any pair
$Z\in{\pazocal P}_{n},Z^{\prime}\in{\pazocal P}_{n+1}$ such that
$Z^{\prime}\subseteq Z$, we write down a blue edge $e(Z^{\prime},Z)$ with
source $Z^{\prime}$ and range $Z$. Thus

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

  4. (b.2)

(Red edges) For the definition of red edges, we need to distinguish between
even and odd layers.

We start with an even layer $n=2j,j\geq 0$. For each pair $Z\in{\pazocal
P}_{2j}$ and $Z^{\prime}\in{\pazocal P}_{2j+1}$ such that
$Z^{\prime}\subseteq\sigma(Z)$, we write down a red edge $f(Z^{\prime},Z)$
with source $Z^{\prime}$ and range $Z$. Thus

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

Note that it may be the case that there are different red edges departing from
the same vertex $Z^{\prime}$, simply because $Z^{\prime}$ is in principle
contained in an intersection $\sigma(Z_{1})\cap\cdots\cap\sigma(Z_{l})$ for
some $Z_{i}\in{\pazocal P}_{2j}$.

Now we pay our attention to an odd layer $n=2j+1,j\geq 0$. For
$Z^{\prime}\in{\pazocal P}_{2j+1}$ and $Z^{\prime\prime}\in{\pazocal
P}_{2j+2}$ such that $\sigma(Z^{\prime\prime})\subseteq Z^{\prime}$ (or
equivalently $Z^{\prime\prime}\subseteq\sigma^{-1}(Z^{\prime})$), we write
down a red edge $f(Z^{\prime\prime},Z^{\prime})$ with source
$Z^{\prime\prime}$ and range $Z^{\prime}$. Thus

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

Note that here, in contrast to what happens for even layers, there is only one
red edge departing from each vertex $Z^{\prime\prime}\in{\pazocal P}_{2j+2}$.

  5. (c)

(Separations) We again distinguish between even and odd layers. For an even
layer $n=2j$, $j\geq 0$, the separation for $Z\in F^{0,2j}$ is defined to be

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

For an odd layer $n=2j+1$, $j\geq 0$, the separation for $Z^{\prime}\in
F^{0,2j+1}$ is defined to be

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

where here $R(f)$ denotes the set of all those red edges $g\in R_{Z^{\prime}}$
such that $s(g)\subseteq r(f)$. Thus $R_{Z^{\prime}}=\bigcup_{f\in
s^{-1}(Z^{\prime})\cap R_{r(f)}}R(f)$.

$Z\in F^{0,2j}$$Z^{\prime}\in F^{0,2j+1}$$Z^{\prime\prime}\in
F^{0,2j+2}$$Z^{\prime\prime}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\rightarrow
Z^{\prime}\text{ if }Z^{\prime\prime}\subseteq
Z^{\prime}$$Z^{\prime\prime}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\rightarrow
Z^{\prime}\text{ if }\sigma(Z^{\prime\prime})\subseteq
Z^{\prime}$$Z^{\prime}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\rightarrow
Z\text{ if }Z^{\prime}\subseteq
Z$$Z^{\prime}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\rightarrow
Z\text{ if }Z^{\prime}\subseteq\sigma(Z)$ Figure 2. Construction of the edges
of $(F,D)$.

In this way we build a separated Bratteli diagram $(F,D)$. We now show that
$(F,D)$ is an $l$-diagram. Indeed, it is an $l$-diagram of a special type,
since by definition it does not have double red edges in any of its layers.
This motivates the following definition.

######  Definition 3.25.

Let $(F,D)$ be an $l$-diagram. We say that $(F,D)$ is a refined $l$-diagram if
for any two vertices $w\in F^{0,n}$ and $v\in F^{0,n-1}$, $n\geq 1$, there is
at most one red edge $f$ such that $s(f)=w$ and $r(f)=v$. By Remark 3.5 (5),
this is equivalent to asking the property only at the first layer, that is,
for $n=1$.

The following lemma will be useful in the sequel.

######  Lemma 3.26.

Let $\\{{\pazocal P}_{n}\\}_{n\geq 0}$ be a $\sigma$-refined sequence of
partitions of $X$.

  1. i)

For $j\geq 0$, let $Z_{1}\in{\pazocal P}_{2j}$, and let $g\in R_{Z_{1}}$, with
$s(g)=Z$. Then $Z\subseteq\sigma(Z_{1})$, and we have

$(\sigma|_{Z_{1}})^{-1}(Z)=\sigma^{-1}(Z)\cap Z_{1}=\bigsqcup_{f\in
R(g)}s(f).$

  2. ii)

For $j\geq 1$, let $Z\in{\pazocal P}_{2j-1}$, $Z^{\prime\prime}\in{\pazocal
P}_{2j+1}$ and $Z_{1}\in{\pazocal P}_{2j-2}$ be such that

$Z^{\prime\prime}\subseteq Z\subseteq\sigma(Z_{1}).$

Then there is a unique $Z^{\prime}\in{\pazocal P}_{2j}$ such that

$Z^{\prime}\subseteq Z_{1}\qquad\text{and}\qquad
Z^{\prime\prime}\subseteq\sigma(Z^{\prime})\subseteq Z\subseteq\sigma(Z_{1}).$

###### Proof.

  1. i)

First note that $Z\subseteq\sigma(Z_{1})$ by definition of the red edges at
even layers.

Now note that since ${\pazocal P}_{2j}\vee\sigma^{-1}({\pazocal
P}_{2j+1})\precsim{\pazocal P}_{2j+1}\vee\sigma^{-1}({\pazocal
P}_{2j+1})\precsim{\pazocal P}_{2j+2}$, we can write the set
$\sigma^{-1}(Z)\cap Z_{1}$ as a disjoint union of sets
$Z^{\prime\prime}\in{\pazocal P}_{2j+2}$. If $Z^{\prime\prime}\in{\pazocal
P}_{2j+2}$ and $Z^{\prime\prime}\subseteq\sigma^{-1}(Z)\cap Z_{1}$, then there
is a unique red edge $f$ such that $s(f)=Z^{\prime\prime}$ and $r(f)=Z$, and
moreover $s(f)=Z^{\prime\prime}\subseteq Z_{1}=r(g)$, so that $f\in R(g)$.
Conversely any $f\in R(g)$ gives rise to a set
$Z^{\prime\prime}:=s(f)\in{\pazocal P}_{2j+2}$ with
$Z^{\prime\prime}\subseteq\sigma^{-1}(Z)\cap Z_{1}$. This shows the result.

  2. ii)

Take $j\geq 1$, $Z\in{\pazocal P}_{2j-1}$, $Z^{\prime\prime}\in{\pazocal
P}_{2j+1}$, and $Z_{1}\in{\pazocal P}_{2j-2}$ such that
$Z^{\prime\prime}\subseteq Z\subseteq\sigma(Z_{1})$. Since ${\pazocal
P}_{2j}^{\sigma}$ is coarser than ${\pazocal P}_{2j+1}$, we can find sets
$Z^{\prime}_{1},Z^{\prime}_{2},\dots,Z^{\prime}_{k}\in{\pazocal P}_{2j}$ such
that

$Z^{\prime\prime}\subseteq\sigma(Z^{\prime}_{1})\cap\cdots\cap\sigma(Z^{\prime}_{k})\setminus{\pazocal
U}_{\geq k+1}.$

We have that $Z^{\prime\prime}\subseteq\sigma(Z^{\prime}_{i})$ for $1\leq
i\leq k$ and $Z^{\prime\prime}\cap\sigma(\overline{Z})=\emptyset$ for all
$\overline{Z}\in{\pazocal P}_{2j}$ distinct from
$Z^{\prime}_{1},\dots,Z^{\prime}_{k}$. On the other hand note that, since
$Z\subseteq\sigma(Z_{1})$, there is a unique red edge $g$ such that $s(g)=Z$
and $r(g)=Z_{1}$. By i), we have

$(\sigma|_{Z_{1}})^{-1}(Z)=\sigma^{-1}(Z)\cap Z_{1}=\bigsqcup_{f\in
R(g)}s(f).$

Since $\sigma$ is injective on all sets of the partitions ${\pazocal P}_{n}$,
we obtain that

$Z=\bigsqcup_{f\in R(g)}\sigma(s(f)).$

Since $Z^{\prime\prime}\subseteq Z$, it follows that
$Z^{\prime\prime}\cap\sigma(s(f))\neq\emptyset$ for some $f\in R(g)$. But by
the property mentioned above, this means that $s(f)$ is one of the sets
$Z_{1}^{\prime},\dots,Z_{k}^{\prime}$ and thus
$Z^{\prime\prime}\subseteq\sigma(s(f))$. Since the sets $\sigma(s(f))$ are
mutually disjoint for $f\in R(g)$, it follows that there is exactly one $f\in
R(g)$ such that $Z^{\prime\prime}\subseteq\sigma(s(f))$. Let $f_{0}$ be this
unique edge in $R(g)$, and set $Z^{\prime}:=s(f_{0})$. Since $f_{0}\in R(g)$
we have $Z^{\prime}=s(f_{0})\subseteq r(g)=Z_{1}$ and moreover

$Z^{\prime\prime}\subseteq\sigma(s(f_{0}))=\sigma(Z^{\prime})\subseteq
Z\subseteq\sigma(Z_{1}),$

as desired.

∎

######  Lemma 3.27.

The separated graph $(F,D)$ associated to a $\sigma$-refined sequence of
partitions $\\{{\pazocal P}_{n}\\}_{n\geq 0}$ is a refined $l$-diagram.

###### Proof.

The condition of not having double red edges follows directly from the
definition. We need to check conditions (a)-(g) from Definition 3.2.
Conditions (a) and (b) follow by definition.

  1. (c)

We first show that, for $j\geq 1$,

$\bigsqcup_{Z\in{\pazocal
P}_{2j-1}}s(B_{Z})\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny(c.1)}}}{=}}{\pazocal
P}_{2j}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny(c.2)}}}{=}}\bigsqcup_{Z\in{\pazocal
P}_{2j-1}}s(R_{Z}).$

We prove the non-trivial inclusion in (c.1). Given $Z^{\prime}\in{\pazocal
P}_{2j}$, we can find a (unique) $Z\in{\pazocal P}_{2j-1}$ such that
$Z^{\prime}\subseteq Z$ since ${\pazocal P}_{2j-1}$ is coarser than ${\pazocal
P}_{2j}$. Therefore there exists, by definition, a blue edge $e(Z^{\prime},Z)$
with source $Z^{\prime}$ and range $Z$. This proves (c.1). Disjointness of the
left union follows from the uniqueness of the set $Z$. Indeed, given
$Z^{\prime}\in s(B_{Z_{1}})\cap s(B_{Z_{2}})$ with $Z_{1}\neq
Z_{2}\in{\pazocal P}_{2j-1}$, we have $Z^{\prime}\subseteq Z_{1}\cap
Z_{2}=\emptyset$, a contradiction.

Let us now prove the non-trivial inclusion in (c.2). Since
$\sigma^{-1}({\pazocal P}_{2j-1})$ is a partition coarser than ${\pazocal
P}_{2j}$, given $Z^{\prime}\in{\pazocal P}_{2j}$ we can find $Z\in{\pazocal
P}_{2j-1}$ such that $Z^{\prime}\subseteq\sigma^{-1}(Z)$, equivalently
$\sigma(Z^{\prime})\subseteq Z$. This says that there is a red edge
$f(Z^{\prime},Z)$ with source $Z^{\prime}$ and range $Z$. This proves the
desired inclusion. The disjointness of the right union follows by analogous
arguments as in the above paragraph.

The second part of condition (c) follows by the definition of the edges in
$(F,D)$.

  2. (d)

The fact that

$\bigsqcup_{Z\in{\pazocal P}_{0}}s(B_{Z})={\pazocal
P}_{1}=\bigcup_{Z\in{\pazocal P}_{0}}s(R_{Z})$

follows from identical arguments as in (c), using now the fact that the
partition ${\pazocal P}_{0}$ is coarser than the partition ${\pazocal P}_{1}$.
Disjointness of the left union is proved in a similar way as in (c).

The second part of condition (d) follows by the definition of the blue edges
in $(F,D)$.

  3. (e)

We need to show that, for $j\geq 0$ and $Z\in{\pazocal P}_{2j}$,

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

Disjointness of the unions will follow from condition (c) (see Remark 3.5
(3)).

Let $Z^{\prime\prime}=s(e_{1})$ for some $e_{1}\in B_{s(e_{0})}$ and some
$e_{0}\in B_{Z}$, so that $Z^{\prime\prime}\subseteq s(e_{0})\subseteq Z$.
Since $\sigma^{-1}({\pazocal P}_{2j+1})$ is a coarser partition than
${\pazocal P}_{2j+2}$ we can find $Z^{\prime}\in{\pazocal P}_{2j+1}$ such that
$\sigma(Z^{\prime\prime})\subseteq Z^{\prime}$. Thus there exists a red edge
$f_{1}=f(Z^{\prime\prime},Z^{\prime})\in R_{Z^{\prime}}$ such that
$Z^{\prime\prime}=s(f_{1})$. Note now that
$Z^{\prime}\cap\sigma(Z)\neq\emptyset$, so by the construction of the
partition ${\pazocal P}_{2j}^{\sigma}$ and the fact that ${\pazocal
P}_{2j}^{\sigma}$ is coarser than ${\pazocal P}_{2j+1}$, we necessarily have
$Z^{\prime}\subseteq\sigma(Z)$. Therefore there exists another red edge
$f_{0}=f(Z^{\prime},Z)\in R_{Z}$ with $Z^{\prime}=s(f_{0})$. Since
$s(f_{1})=Z^{\prime\prime}\subseteq Z=r(f_{0})$, we have $f_{1}\in R(f_{0})$.
This proves the inclusion $\subseteq$.

For the other inclusion, let $Z^{\prime\prime}=s(f_{1})$ for some $f_{1}\in
R(f_{0})$ and some $f_{0}\in R_{Z}$, so that
$Z^{\prime\prime}=s(f_{1})\subseteq r(f_{0})=Z$. Since ${\pazocal P}_{2j+1}$
is coarser than ${\pazocal P}_{2j+2}$, we can find $Z^{\prime}\in{\pazocal
P}_{2j+1}$ with $Z^{\prime\prime}\subseteq Z^{\prime}$. But then
$Z^{\prime}\cap Z\neq\emptyset$, so we necessarily have that
$Z^{\prime}\subseteq Z$ by the fact that ${\pazocal P}_{2j}$ is coarser than
${\pazocal P}_{2j+1}$. This means precisely that there exist blue edges
$e_{0}\in B_{Z}$ with $Z^{\prime}=s(e_{0})$ and $e_{1}\in B_{s(e_{0})}$ with
$Z^{\prime\prime}=s(e_{1})$. This concludes the proof of the desired
inclusion.

  4. (f)

Take $j\geq 1$, $Z\in{\pazocal P}_{2j-1}$ and a pair $(f_{0},f_{1})$ of red
edges such that $f_{0}\in R_{Z},f_{1}\in R_{s(f_{0})}$. By letting
$Z^{\prime\prime}:=s(f_{1})$, we have by definition
$Z^{\prime\prime}\subseteq\sigma(\overline{Z}^{\prime})\subseteq Z$ where
$\overline{Z}^{\prime}:=s(f_{0})=r(f_{1})$. Since ${\pazocal P}_{2j}$ is
coarser than ${\pazocal P}_{2j+1}$, we can find $Z^{\prime}\in{\pazocal
P}_{2j}$ with $Z^{\prime\prime}\subseteq Z^{\prime}$. But then $Z^{\prime}\cap
Z\neq\emptyset$, so necessarily $Z^{\prime}\subseteq Z$ since ${\pazocal
P}_{2j-1}$ is coarser than ${\pazocal P}_{2j}$. This means precisely that
there exist blue edges $e_{0}\in B_{Z}$ with $Z^{\prime}=s(e_{0})$ and
$e_{1}\in B_{s(e_{0})}$ with $Z^{\prime\prime}=s(e_{1})$. The pair of blue
edges $(e_{0},e_{1})$ is unique by Remark 3.5 (4). This shows condition (f).

  5. (g)

Take $j\geq 1$, $Z\in{\pazocal P}_{2j-1}$, a red edge $g\in R_{r(g)}$ with
$Z=s(g)$ and $Z_{1}:=r(g)$, and a pair $(e_{0},e_{1})$ of blue edges such that
$e_{0}\in B_{Z}$ and $s(e_{0})=r(e_{1})$. By letting
$Z^{\prime\prime}:=s(e_{1})$, we have by definition
$Z^{\prime\prime}\subseteq\overline{Z}^{\prime}\subseteq Z$ where
$\overline{Z}^{\prime}:=s(e_{0})=r(e_{1})$. We obtain the chain

$Z^{\prime\prime}\subseteq\overline{Z}^{\prime}\subseteq
Z\subseteq\sigma(Z_{1}).$

By Lemma 3.26 (ii), there is a unique $Z^{\prime}\in{\pazocal P}_{2j}$ such
that $Z^{\prime}\subseteq Z_{1}$ and

$Z^{\prime\prime}\subseteq\sigma(Z^{\prime})\subseteq
Z\subseteq\sigma(Z_{1}).$

Let $f_{0}$ be the red edge such that $Z^{\prime}=s(f_{0})$ and $Z=r(f_{0})$,
and observe that $f_{0}\in R(g)$ because $s(f_{0})\subseteq r(g)$. We then
have $Z^{\prime\prime}\subseteq\sigma(s(f_{0}))$, and this means that there is
a unique red edge $f_{1}$ such that $Z^{\prime\prime}=s(f_{1})$ and
$Z^{\prime}=r(f_{1})=s(f_{0})$. We have that $r(f_{0})=Z=r(e_{0})$ and
$s(f_{1})=Z^{\prime\prime}=s(e_{1})$. Therefore we have proved that there is a
unique pair $(f_{0},f_{1})$ of red edges such that $f_{0}\in R(g)$,
$r(f_{0})=r(e_{0})$, $s(f_{0})=r(f_{1})$ and $s(f_{1})=s(e_{1})$, as required.

This proves the lemma. ∎

######  Examples 3.28.

This is a continuation of Examples 3.23.

  1. 1.

Take $X=\\{0,1\\}^{{\mathbb{Z}}}$ and $\varphi\colon X\to X$ to be the two-
sided shift. Using the $\varphi$-refined sequence of partitions $\\{{\pazocal
P}_{n}\\}_{n\geq 0}$ given in Examples 3.23 1), we obtain the $l$-diagram
shown in Figure 3.

$[\underline{0}]$$[\underline{1}]$$[0\underline{0}]$$[1\underline{0}]$$[0\underline{1}]$$[1\underline{1}]$$[0\underline{0}0]$$[0\underline{0}1]$$[1\underline{0}0]$$[1\underline{0}1]$$[0\underline{1}0]$$[0\underline{1}1]$$[1\underline{1}0]$$[1\underline{1}1]$${\pazocal
P}_{0}$${\pazocal P}_{1}$${\pazocal P}_{2}$ Figure 3. First two levels of the
$l$-diagram associated with $(X,\varphi)$.

  2. 2.

Take $X^{+}=\\{0,1\\}^{{\mathbb{N}}_{0}}$ and $\sigma\colon X^{+}\to X^{+}$ to
be the one-sided shift. Using the $\sigma$-refined sequence of partitions
$\\{{\pazocal P}_{n}\\}_{n\geq 0}$ given in Examples 3.23 2), we obtain the
$l$-diagram shown in Figure 4.

$[\underline{0}]$$[\underline{1}]$$[\underline{0}0]$$[\underline{0}1]$$[\underline{1}0]$$[\underline{1}1]$$[\underline{0}00]$$[\underline{0}01]$$[\underline{0}10]$$[\underline{0}11]$$[\underline{1}00]$$[\underline{1}01]$$[\underline{1}10]$$[\underline{1}11]$${\pazocal
P}_{0}$${\pazocal P}_{1}$${\pazocal P}_{2}$ Figure 4. First two levels of the
$l$-diagram associated with $(X^{+},\sigma)$.

We now proceed by showing that the equivalence class, under $\sim_{l}$, of the
$l$-diagram constructed from $(X,\sigma)$ does not depend on the choice of the
$\sigma$-refined sequence of partitions $\\{{\pazocal P}_{n}\\}_{n\geq 0}$.
First, a proposition.

######  Proposition 3.29.

Let $\\{{\pazocal P}_{n}\\}_{n\geq 0}$ be a $\sigma$-refined sequence of
partitions of $X$. Let $(F,D)$ be the $l$-diagram obtained by applying the
construction 3.24. Let also $(m_{n})_{n\geq 0}$ be a sequence in
${\mathbb{N}}\cup\\{0\\}$ with $0\leq m_{0}<m_{1}<\cdots$, $m_{0}$ even and
satisfying _(CR)_ , and let

  1. $(I)$

$(F^{\prime},D^{\prime})$ be the $l$-diagram obtained by telescoping $(F,D)$
with respect to $(m_{n})_{n\geq 0}$;

  2. $(II)$

$(F^{\prime\prime},D^{\prime\prime})$ be the $l$-diagram obtained from the
$\sigma$-refined sequence of partitions $\\{{\pazocal P}_{m_{n}}\\}_{n\geq 0}$
by applying the construction 3.24.

Then $(F^{\prime},D^{\prime})\cong(F^{\prime\prime},D^{\prime\prime})$ as
$l$-diagrams.

###### Proof.

First of all, note that $(F,D)$ and $(F^{\prime},D^{\prime})$ make sense by
Lemmas 3.27 and 3.11 respectively. Also $(F^{\prime\prime},D^{\prime\prime})$
makes sense by Lemma 3.27 since the sequence $(m_{n})_{n\geq 0}$ is strictly
increasing, and thus $\\{{\pazocal P}_{m_{n}}\\}_{n\geq 0}$ is indeed a
$\sigma$-refined sequence of partitions.

Clearly the vertex sets for $(F^{\prime},D^{\prime})$ and
$(F^{\prime\prime},D^{\prime\prime})$ are the same, since
$(F^{\prime})^{0,n}=F^{0,m_{n}}={\pazocal
P}_{m_{n}}=(F^{\prime\prime})^{0,n}$. We now define a bijective correspondence
between $(F^{\prime})^{1,n}=P^{B}_{m_{n},m_{n+1}}\sqcup P^{R}_{m_{n},m_{n+1}}$
and $(F^{\prime\prime})^{1,n}=B^{\prime\prime(n)}\sqcup R^{\prime\prime(n)}$.

  1. (i)

Take $e=(e_{m_{n}},\dots,e_{m_{n+1}-1})\in P^{B}_{m_{n},m_{n+1}}$. Put
$Z=r(e)=r(e_{m_{n}})\in{\pazocal P}_{m_{n}}$ and
$Z^{\prime}=s(e)=s(e_{m_{n+1}-1})\in{\pazocal P}_{m_{n+1}}$. By construction,
there exists a sequence $(Z_{0},Z_{1},\dots,Z_{m_{n+1}-m_{n}})$ of clopen
sets, with $Z_{i}\in{\pazocal P}_{m_{n+1}-i}$, satisfying

$Z^{\prime}=Z_{0}\subseteq Z_{1}\subseteq\cdots\subseteq
Z_{m_{n+1}-m_{n}-1}\subseteq Z_{m_{n+1}-m_{n}}=Z,$

and also $s(e_{m_{n+1}-i})=Z_{i-1}$, $r(e_{m_{n+1}-i})=Z_{i}$ for each $1\leq
i\leq m_{n+1}-m_{n}$. Therefore $Z^{\prime}\subseteq Z$, so $e$ corresponds to
a unique blue edge $\overline{e}\in(F^{\prime\prime})^{1,n}$ with source
$Z^{\prime}$ and range $Z$. This defines an injective map

$P_{m_{n},m_{n+1}}^{B}\rightarrow B^{\prime\prime(n)},\quad
e\mapsto\overline{e}.$

To show bijectivity of this map, take $\overline{e}\in B^{\prime\prime(n)}$
with source $Z^{\prime}\in{\pazocal P}_{m_{n+1}}$ and range $Z\in{\pazocal
P}_{m_{n}}$. Since ${\pazocal P}_{m}$ is coarser than ${\pazocal P}_{m+1}$ for
all $m\geq 0$, there exists a unique sequence
$(Z_{0},\dots,Z_{m_{n+1}-m_{n}})$ of clopen sets, with $Z_{i}\in{\pazocal
P}_{m_{n+1}-i}$, such that

$Z^{\prime}=Z_{0}\subseteq Z_{1}\subseteq\cdots\subseteq
Z_{m_{n+1}-m_{n}-1}\subseteq Z_{m_{n+1}-m_{n}}=Z.$

This corresponds to a unique blue path $e=(e_{m_{n}},\dots,e_{m_{n+1}-1})$
with $r(e_{m_{n+1}-i})=Z_{i}=s(e_{m_{n+1}-i-1})$ for all $1\leq i\leq
m_{n+1}-m_{n}-1$, and $Z=r(e),Z^{\prime}=s(e)$. Therefore $e\in
P^{B}_{m_{n},m_{n+1}}$. It is clear that $e$ maps to $\overline{e}$ under the
above map $P^{B}_{m_{n},m_{n+1}}\rightarrow B^{\prime\prime(n)}$. This proves
the desired bijectivity.

  2. (ii)

We now deal with the red edges for the even layers, so $n=2j$ for $j\geq 0$.
Note that in this case $m_{n}$ is also even. Take
$f=(f_{m_{2j}},\dots,f_{m_{2j+1}-1})\in P^{R}_{m_{2j},m_{2j+1}}$. Put
$Z=r(f)=r(f_{m_{2j}})\in{\pazocal P}_{m_{2j}}$ and
$Z^{\prime}=s(f)=s(f_{m_{2j+1}-1})\in{\pazocal P}_{m_{2j+1}}$. By
construction, there exists a sequence
$(Z_{0},Z_{1},\dots,Z_{m_{2j+1}-m_{2j}})$ of clopen sets, with
$Z_{i}\in{\pazocal P}_{m_{2j+1}-i}$, satisfying

$Z^{\prime}=Z_{0}\subseteq\sigma(Z_{1})\subseteq Z_{2}\subseteq\cdots\subseteq
Z_{m_{2j+1}-m_{2j}-1}\subseteq\sigma(Z_{m_{2j+1}-m_{2j}})=\sigma(Z),$

$s(f_{m_{2j+1}-i})=Z_{i-1}$ and $r(f_{m_{2j+1}-i})=Z_{i}$ for each $1\leq
i\leq m_{2j+1}-m_{2j}$, and also

$Z_{1}\subseteq Z_{3}\subseteq\cdots\subseteq Z_{m_{2j+1}-m_{2j}-2}\subseteq
Z_{m_{2j+1}-m_{2j}}=Z.$

Therefore $Z^{\prime}\subseteq\sigma(Z)$, so $f$ corresponds to a unique red
edge $\overline{f}\in(F^{\prime\prime})^{1,2j}$ with source $Z^{\prime}$ and
range $Z$. This defines an injective map

$P_{m_{2j},m_{2j+1}}^{R}\rightarrow R^{\prime\prime(2j)},\quad
f\mapsto\overline{f}.$

We now proceed to show its bijectivity. Take $\overline{f}\in
R^{\prime\prime(2j)}$ with source $Z^{\prime}\in{\pazocal P}_{m_{2j+1}}$ and
range $Z\in{\pazocal P}_{m_{2j}}$.

######  Claim.

There exists a unique sequence $(Z_{0},\dots,Z_{m_{2j+1}-m_{2j}})$ of clopen
sets $Z_{i}\in{\pazocal P}_{m_{2j+1}-i}$ such that

$Z^{\prime}=Z_{0}\subseteq\sigma(Z_{1})\subseteq Z_{2}\subseteq\cdots\subseteq
Z_{m_{2j+1}-m_{2j}-1}\subseteq\sigma(Z_{m_{2j+1}-m_{2j}})=\sigma(Z)$

and

$Z_{1}\subseteq Z_{3}\subseteq\cdots\subseteq Z_{m_{2j+1}-m_{2j}-2}\subseteq
Z_{m_{2j+1}-m_{2j}}=Z.$

###### Proof.

First we set $Z_{0}:=Z^{\prime}$ and $Z_{m_{2j+1}-m_{2j}}:=Z$. Now since the
partition ${\pazocal P}_{m}$ is coarser than the partition ${\pazocal
P}_{m+1}$ for all $m\geq 0$, we see that there are unique $Z_{2}\in{\pazocal
P}_{m_{2j+1}-2}$, $Z_{4}\in{\pazocal
P}_{m_{2j+1}-4},\dots,Z_{m_{2j+1}-m_{2j}-1}\in{\pazocal P}_{m_{2j}+1}$ with

$Z^{\prime}=Z_{0}\subseteq Z_{2}\subseteq Z_{4}\subseteq\cdots\subseteq
Z_{m_{2j+1}-m_{2j}-1}.$

Since $Z^{\prime}\subseteq\sigma(Z)$ and ${\pazocal P}_{m_{2j}}^{\sigma}$ is
coarser than ${\pazocal P}_{m_{2j}+1}$, it follows that
$Z_{m_{2j+1}-m_{2j}-1}\subseteq\sigma(Z)$. Therefore we have the inclusions

$Z_{m_{2j+1}-m_{2j}-3}\subseteq Z_{m_{2j+1}-m_{2j}-1}\subseteq\sigma(Z).$

By Lemma 3.26 (ii), there is a unique $Z_{m_{2j+1}-m_{2j}-2}\in{\pazocal
P}_{m_{2j}+2}$ such that $Z_{m_{2j+1}-m_{2j}-2}\subseteq Z$ and

$Z_{m_{2j+1}-m_{2j}-3}\subseteq\sigma(Z_{m_{2j+1}-m_{2j}-2})\subseteq
Z_{m_{2j+1}-m_{2j}-1}\subseteq\sigma(Z_{m_{2j+1}-m_{2j}})=\sigma(Z).$

Iteration of this argument produces the desired sequence
$(Z_{0},\dots,Z_{m_{2j+1}-m_{2j}})$. This proves the claim. ∎

The sequence just constructed corresponds to a unique red path
$f=(f_{m_{2j}},\dots,f_{m_{2j+1}-1})$ with
$r(f_{m_{2j+1}-i})=Z_{i}=s(f_{m_{2j+1}-i-1})$ for all $1\leq i\leq
m_{2j+1}-m_{2j}-1$, and $Z=r(f),Z^{\prime}=s(f)$. Therefore $f\in
P^{R}_{m_{2j},m_{2j+1}}$. It is clear that $f$ maps to $\overline{f}$ under
the above map $P_{m_{2j},m_{2j+1}}^{R}\rightarrow R^{\prime\prime(2j)}$.
Bijectivity now follows.

  3. (iii)

Now for the red edges at the odd layers, so $n=2j-1$ for $j\geq 1$. Note that
in this case $m_{n}$ is also odd. Take $f=(f_{m_{2j-1}},\dots,f_{m_{2j}-1})\in
P^{R}_{m_{2j-1},m_{2j}}$. Put $Z=r(f)=r(f_{m_{2j-1}})\in{\pazocal
P}_{m_{2j-1}}$ and $Z^{\prime}=s(f)=s(f_{m_{2j}-1})\in{\pazocal P}_{m_{2j}}$.
Take also $g=(g_{m_{2j-2}},\dots,g_{m_{2j-1}-1})\in P^{R}_{m_{2j-2},m_{2j-1}}$
to be such that $f\in R^{\prime}(g)$. By construction, there exists a sequence
$(Z_{0},Z_{1},\dots,Z_{m_{2j}-m_{2j-1}})$ of clopen sets, with
$Z_{i}\in{\pazocal P}_{m_{2j}-i}$, satisfying

$\sigma(Z^{\prime})=\sigma(Z_{0})\subseteq
Z_{1}\subseteq\sigma(Z_{2})\subseteq\cdots\subseteq\sigma(Z_{m_{2j}-m_{2j-1}-1})\subseteq
Z_{m_{2j}-m_{2j-1}}=Z,$

$s(f_{m_{2j}-i})=Z_{i-1}$ and $r(f_{m_{2j}-i})=Z_{i}$ for each $1\leq i\leq
m_{2j}-m_{2j-1}$, and also

$Z^{\prime}=Z_{0}\subseteq Z_{2}\subseteq Z_{4}\subseteq\cdots\subseteq
Z_{m_{2j}-m_{2j-1}-1}.$

Therefore $\sigma(Z^{\prime})\subseteq Z$, so $f$ corresponds to a unique red
edge $\overline{f}\in(F^{\prime\prime})^{1,2j-1}$ with source $Z^{\prime}$ and
range $Z$. This defines an injective map

$P_{m_{2j-1},m_{2j}}^{R}\rightarrow R^{\prime\prime(2j-1)},\quad
f\mapsto\overline{f}.$

Moreover, since $g\in P^{R}_{m_{2j-2},m_{2j-1}}$, there exists a sequence
$(\widetilde{Z}_{0},\widetilde{Z}_{1},\dots,\widetilde{Z}_{m_{2j-1}-m_{2j-2}})$
of clopen sets $\widetilde{Z}_{i}\in{\pazocal P}_{m_{2j-1}-i}$ such that

$Z=\widetilde{Z}_{0}\subseteq\sigma(\widetilde{Z}_{1})\subseteq\widetilde{Z}_{2}\subseteq\cdots\subseteq\widetilde{Z}_{m_{2j-1}-m_{2j-2}-1}\subseteq\sigma(\widetilde{Z}_{m_{2j-1}-m_{2j-2}})=\sigma(\widetilde{Z})$

and

$\widetilde{Z}_{1}\subseteq\widetilde{Z}_{3}\subseteq\cdots\subseteq\widetilde{Z}_{m_{2j-1}-m_{2j-2}}=\widetilde{Z},$

where $Z=s(g)$, $\widetilde{Z}=r(g)$, and
$s(g_{m_{2j-1}-i})=\widetilde{Z}_{i-1}$, $r(g_{m_{2j-1}-i})=\widetilde{Z}_{i}$
for each $1\leq i\leq m_{2j-1}-m_{2j-2}$. Since $f\in R^{\prime}(g)$, it
follows by the definition of $R^{\prime}(g)$ (Definition 3.10 (d)) that
$f_{m_{2j-1}}\in R(g_{m_{2j-1}-1})$, and so

$Z_{m_{2j}-m_{2j-1}-1}=s(f_{m_{2j-1}})\subseteq
r(g_{m_{2j-1}-1})=\widetilde{Z}_{1}.$

Combining this with the previous inclusions we obtain

$s(\overline{f})=Z^{\prime}=Z_{0}\subseteq Z_{2}\subseteq\cdots\subseteq
Z_{m_{2j}-m_{2j-1}-1}\subseteq\widetilde{Z}_{1}\subseteq\cdots\subseteq\widetilde{Z}=r(\overline{g}),$

where $\overline{g}$ is the unique red edge in $(F^{\prime\prime})^{1,2j-2}$
constructed from $g$ (see (ii) above). This shows that $\overline{f}\in
R^{\prime\prime}(\overline{g})$ in the $l$-diagram
$(F^{\prime\prime},D^{\prime\prime})$.

We now show bijectivity of the map $P_{m_{2j-1},m_{2j}}^{R}\rightarrow
R^{\prime\prime(2j-1)}$. Take $\overline{f}\in R^{\prime\prime(2j-1)}$ with
source $Z^{\prime}\in{\pazocal P}_{m_{2j}}$ and range $Z\in{\pazocal
P}_{m_{2j-1}}$. Take also $\overline{g}\in R^{\prime\prime(2j-2)}$ such that
$\overline{f}\in R^{\prime\prime}(\overline{g})$. Since $m_{2j-2}$ is even, we
know from (ii) above that there is a unique sequence
$(\widetilde{Z}_{0},\widetilde{Z}_{1},\dots,\widetilde{Z}_{m_{2j-1}-m_{2j-2}})$
of clopen sets $\widetilde{Z}_{i}\in{\pazocal P}_{m_{2j-1}-i}$ such that

$Z=\widetilde{Z}_{0}\subseteq\sigma(\widetilde{Z}_{1})\subseteq\widetilde{Z}_{2}\subseteq\cdots\subseteq\widetilde{Z}_{m_{2j-1}-m_{2j-2}-1}\subseteq\sigma(\widetilde{Z}_{m_{2j-1}-m_{2j-2}})=\sigma(\widetilde{Z})$

and

$\widetilde{Z}_{1}\subseteq\widetilde{Z}_{3}\subseteq\cdots\subseteq\widetilde{Z}_{m_{2j-1}-m_{2j-2}}=\widetilde{Z},$

where $Z=s(\overline{g})$, $\widetilde{Z}=r(\overline{g})$. Let $g\in
P^{R}_{m_{2j-2},m_{2j-1}}$ be the corresponding red edge.

######  Claim.

There exists a unique sequence $(Z_{0},\dots,Z_{m_{2j}-m_{2j-1}})$ of clopen
sets $Z_{i}\in{\pazocal P}_{m_{2j}-i}$ such that

$\sigma(Z^{\prime})=\sigma(Z_{0})\subseteq
Z_{1}\subseteq\sigma(Z_{2})\subseteq\cdots\subseteq\sigma(Z_{m_{2j}-m_{2j-1}-1})\subseteq
Z_{m_{2j}-m_{2j-1}}=Z$

and

$Z^{\prime}=Z_{0}\subseteq Z_{2}\subseteq Z_{4}\subseteq\cdots\subseteq
Z_{m_{2j}-m_{2j-1}-1}\subseteq\widetilde{Z}_{1}.$

###### Proof.

Note that the hypothesis that $\overline{f}\in R^{\prime\prime}(\overline{g})$
gives us the condition

$Z^{\prime}=s(\overline{f})\subseteq r(\overline{g})=\widetilde{Z}.$

Since $\sigma^{-1}({\pazocal P}_{m_{2j}-1})$ is coarser than ${\pazocal
P}_{m_{2j}}$, there is a unique $Z_{1}\in{\pazocal P}_{m_{2j}-1}$ such that
$\sigma(Z^{\prime})\subseteq Z_{1}$. We can now find
$Z_{3},Z_{5},\dots,Z_{m_{2j}-m_{2j-1}}$ such that $Z_{2l+1}\in{\pazocal
P}_{m_{2j}-(2l+1)}$ and

$Z_{1}\subseteq Z_{3}\subseteq Z_{5}\subseteq\cdots\subseteq
Z_{m_{2j}-m_{2j-1}}.$

Observe that $\sigma(Z^{\prime})\subseteq Z_{1}\subseteq Z_{m_{2j}-m_{2j-1}}$
and also $\sigma(Z^{\prime})\subseteq Z$ by hypothesis. Since ${\pazocal
P}_{m_{2j-1}}$ is a partition, we must have $Z_{m_{2j}-m_{2j-1}}=Z$.

Now note that we have inclusions

$Z_{m_{2j}-m_{2j-1}-2}\subseteq Z\subseteq\sigma(\widetilde{Z}_{1}).$

By Lemma 3.26 (ii), we get a unique $Z_{m_{2j}-m_{2j-1}-1}\in{\pazocal
P}_{m_{2j-1}+1}$ such that $Z_{m_{2j}-m_{2j-1}-1}\subseteq\widetilde{Z}_{1}$
and

$Z_{m_{2j}-m_{2j-1}-2}\subseteq\sigma(Z_{{m_{2}j}-m_{2j-1}-1})\subseteq
Z\subseteq\sigma(\widetilde{Z}_{1}).$

Continuing in this way, we find unique clopen sets
$Z_{2},Z_{4},\dots,Z_{m_{2j}-m_{2j-1}-3}$ such that $Z_{2l}\in{\pazocal
P}_{m_{2j}-2l}$ for $1\leq l\leq\frac{m_{2j}-m_{2j-1}-3}{2}$ and

$Z_{2}\subseteq Z_{4}\subseteq\cdots\subseteq Z_{m_{2j}-m_{2j-1}-3}\subseteq
Z_{m_{2j}-m_{2j-1}-1},$ $Z_{1}\subseteq\sigma(Z_{2})\subseteq
Z_{3}\subseteq\cdots\subseteq\sigma(Z_{m_{2j}-m_{2j-1}-1})\subseteq
Z\subseteq\sigma(\widetilde{Z}_{1}).$

We show that $Z^{\prime}\subseteq Z_{2}$. Indeed, we have
$\sigma(Z^{\prime})\subseteq Z_{1}\subseteq\sigma(Z_{2})$. Moreover we have
that $Z^{\prime}\subseteq\widetilde{Z}$ by our hypothesis that
$\overline{f}\in R^{\prime\prime}(\overline{g})$, and also we have the
inclusions

$Z_{2}\subseteq Z_{4}\subseteq\cdots\subseteq
Z_{m_{2j}-m_{2j-1}-1}\subseteq\widetilde{Z}_{1}\subseteq\widetilde{Z}_{3}\subseteq\cdots\subseteq\widetilde{Z}_{m_{2j-1}-m_{2j-2}}=\widetilde{Z}.$

Hence both $Z^{\prime}$ and $Z_{2}$ are subsets of $\widetilde{Z}$. Since the
restriction of $\sigma$ to $\widetilde{Z}$ is injective and
$\sigma(Z^{\prime})\subseteq\sigma(Z_{2})$, we get that $Z^{\prime}\subseteq
Z_{2}$. Hence taking $Z_{0}:=Z^{\prime}$ we complete our proof of the
existence and uniqueness of the sequence $(Z_{0},\dots,Z_{m_{2j}-m_{2j-1}})$,
thus proving the claim. ∎

The sequence just constructed corresponds to a unique red path
$f=(f_{m_{2j-1}},\dots,f_{m_{2j}-1})$ with
$r(f_{m_{2j}-i})=Z_{i}=s(f_{m_{2j}-i-1})$ for all $1\leq i\leq
m_{2j}-m_{2j-1}-1$, and $Z=r(f),Z^{\prime}=s(f)$. Therefore $f\in
P^{R}_{m_{2j-1},m_{2j}}$, and moreover $f\in R^{\prime}(g)$ because
$Z_{m_{2j}-m_{2j-1}-1}\subseteq\widetilde{Z}_{1}$. It is clear that $f$ maps
to $\overline{f}$ under the map $P_{m_{2j-1},m_{2j}}^{R}\rightarrow
R^{\prime\prime(2j-1)}$.

One can easily check that all the defined bijective maps preserve the
separations as in Definition 3.12 (a). The result follows. ∎

######  Theorem 3.30.

The equivalence class, under $\sim_{l}$, of the $l$-diagram constructed from
$(X,\sigma)$ by using the construction 3.24 does not depend on the choice of
the $\sigma$-refined sequence of partitions $\\{{\pazocal P}_{n}\\}_{n\geq
0}$.

###### Proof.

Suppose we have two $\sigma$-refined sequences of partitions $\\{{\pazocal
P}_{n}\\}_{n\geq 0}$, $\\{\widetilde{{\pazocal P}}_{n}\\}_{n\geq 0}$. Passing
to subsequences (satisfying (CR) and with the first element of the sequence
being an even number) does not affect the equivalence class of the resulting
$l$-diagram by Proposition 3.29, so we may assume that

${\pazocal P}_{n}\precsim\widetilde{{\pazocal P}}_{n+1}\precsim{\pazocal
P}_{n+2}\quad\text{for all }n\geq 0.$

We can create a new sequence of partitions $\\{\overline{{\pazocal
P}}_{n}\\}_{n\geq 0}$ by

$\overline{{\pazocal P}}_{n}:=\begin{cases}{\pazocal P}_{3n}&\text{for
}n\equiv 0\text{ (mod }3);\\\ \widetilde{{\pazocal
P}}_{3n}&\text{otherwise}.\end{cases}$

This sequence of partitions also satisfies the required properties:

  1. (a)

Since $\overline{{\pazocal P}}_{0}={\pazocal P}_{0}$, $\sigma$ restricts to a
homeomorphism $\sigma|_{Z}$ for each $Z\in\overline{{\pazocal P}}_{0}$.

  2. (b)

For $n\geq 0$, we have

$\begin{cases}\overline{{\pazocal P}}_{3n}={\pazocal
P}_{9n}\precsim\widetilde{{\pazocal P}}_{9n+1}\precsim\widetilde{{\pazocal
P}}_{9n+3}=\overline{{\pazocal P}}_{3n+1},\\\ \overline{{\pazocal
P}}_{3n+1}=\widetilde{{\pazocal P}}_{9n+3}\precsim\widetilde{{\pazocal
P}}_{9n+6}=\overline{{\pazocal P}}_{3n+2},\\\ \overline{{\pazocal
P}}_{3n+2}=\widetilde{{\pazocal P}}_{9n+6}\precsim{\pazocal
P}_{9n+7}\precsim{\pazocal P}_{9n+9}=\overline{{\pazocal
P}}_{3n+3}.\end{cases}$

  3. (c)

Clearly the union $\bigcup_{n\geq 0}\overline{{\pazocal P}}_{n}$ generates the
topology of $X$, since both $\bigcup_{n\geq 0}{\pazocal P}_{n}$ and
$\bigcup_{n\geq 0}\widetilde{{\pazocal P}}_{n}$ generate the topology of $X$.

  4. (d)

We have $\text{diam}(\overline{{\pazocal P}}_{n})\leq\text{diam}({\pazocal
P}_{3n+1})$, which tends to zero as $n\rightarrow\infty$.

  5. (e)

We compute

$\overline{{\pazocal P}}_{2n}\vee\overline{{\pazocal
P}}_{2n}^{\sigma}=\begin{cases}{\pazocal P}_{6n}\vee{\pazocal
P}_{6n}^{\sigma}\precsim{\pazocal P}_{6n+1}\precsim\widetilde{{\pazocal
P}}_{6n+3}=\overline{{\pazocal P}}_{2n+1}&\text{if }2n\equiv 0\text{ (mod
}3),\\\ \widetilde{{\pazocal P}}_{6n}\vee\widetilde{{\pazocal
P}}_{6n}^{\sigma}\precsim\widetilde{{\pazocal
P}}_{6n+1}\precsim\widetilde{{\pazocal P}}_{6n+3}=\overline{{\pazocal
P}}_{2n+1}&\text{if }2n\equiv 1\text{ (mod }3),\\\ \widetilde{{\pazocal
P}}_{6n}\vee\widetilde{{\pazocal P}}_{6n}^{\sigma}\precsim\widetilde{{\pazocal
P}}_{6n+1}\precsim{\pazocal P}_{6n+3}=\overline{{\pazocal P}}_{2n+1}&\text{if
}2n\equiv 2\text{ (mod }3).\end{cases}$

In either case

$\overline{{\pazocal P}}_{2n}\vee\overline{{\pazocal
P}}_{2n}^{\sigma}\precsim\overline{{\pazocal P}}_{2n+1}.$

  6. (f)

Also,

$\overline{{\pazocal P}}_{2n+1}\vee\sigma^{-1}(\overline{{\pazocal
P}}_{2n+1})=\begin{cases}\widetilde{{\pazocal
P}}_{6n+3}\vee\sigma^{-1}(\widetilde{{\pazocal
P}}_{6n+3})\precsim\widetilde{{\pazocal P}}_{6n+4}\precsim\widetilde{{\pazocal
P}}_{6n+6}=\overline{{\pazocal P}}_{2n+2}&\text{if }2n\equiv 0\text{ (mod
}3),\\\ \widetilde{{\pazocal P}}_{6n+3}\vee\sigma^{-1}(\widetilde{{\pazocal
P}}_{6n+3})\precsim\widetilde{{\pazocal P}}_{6n+4}\precsim{\pazocal
P}_{6n+6}=\overline{{\pazocal P}}_{2n+2}&\text{if }2n\equiv 1\text{ (mod
}3),\\\ {\pazocal P}_{6n+3}\vee\sigma^{-1}({\pazocal
P}_{6n+3})\precsim{\pazocal P}_{6n+4}\precsim\widetilde{{\pazocal
P}}_{6n+6}=\overline{{\pazocal P}}_{2n+2}&\text{if }2n\equiv 2\text{ (mod
}3).\end{cases}$

In either case

$\overline{{\pazocal P}}_{2n+1}\vee\sigma^{-1}(\overline{{\pazocal
P}}_{2n+1})\precsim\overline{{\pazocal P}}_{2n+2}.$

We let $(F_{1},D_{1}),(F_{2},D_{2})$ and $(F,D)$ be the $l$-diagrams
constructed from the $\sigma$-refined sequences of partitions $\\{{\pazocal
P}_{n}\\}_{n\geq 0},\\{\widetilde{{\pazocal P}}_{n}\\}_{n\geq 0}$ and
$\\{\overline{{\pazocal P}}_{n}\\}_{n\geq 0}$, respectively, using the
construction given in 3.24. Consider the two sequences

$(m_{n})_{n\geq 0}:=(0,3,6,9,\dots,3n,\dots),\qquad(\widetilde{m}_{n})_{n\geq
0}:=(2,5,\dots,3n+2,\dots),$

which clearly satisfy (CR), and $m_{0}\equiv\widetilde{m}_{0}\equiv 0\text{
(mod }2)$. The contraction of $(F,D)$ with respect to $(m_{n})_{n\geq 0}$
gives an $l$-diagram with associated partitions

${\pazocal P}_{0},{\pazocal P}_{9},{\pazocal P}_{18},{\pazocal
P}_{27},\dots,{\pazocal P}_{9n},\dots$

so it is itself a contraction of $(F_{1},D_{1})$ by Proposition 3.29. Hence
$(F_{1},D_{1})\sim_{l}(F,D)$. Also, the contraction of $(F,D)$ with respect to
$(\widetilde{m}_{n})_{n\geq 0}$ gives an $l$-diagram with associated
partitions

$\widetilde{{\pazocal P}}_{6},\widetilde{{\pazocal
P}}_{15},\widetilde{{\pazocal P}}_{24},\dots,\widetilde{{\pazocal
P}}_{9n+6},\dots$

so it is itself a contraction of $(F_{2},D_{2})$ by Proposition 3.29 again.
Hence $(F_{2},D_{2})\sim_{l}(F,D)$.

Altogether we deduce that

$(F_{1},D_{1})\sim_{l}(F,D)\sim_{l}(F_{2},D_{2}),$

as required. ∎

Using the results obtained so far, we obtain a well-defined map

$\widetilde{\Phi}:\textsf{LHomeo}\longrightarrow\textsf{LDiag}/\sim_{l},\quad(X,\sigma)\mapsto[(F,D)]$

where $(F,D)$ is the $l$-diagram obtained from $(X,\sigma)$ by applying the
construction 3.24. Indeed, $\widetilde{\Phi}(X,\sigma)$ does not depend on the
choice of the $\sigma$-refined sequence of partitions used to define it by
Theorem 3.30.

In the next theorem we prove the last thing left in order to properly define
$\Phi$, namely that the equivalence class of $(F,D)$ does not depend on the
representative chosen for the class of $(X,\sigma)$ under the equivalence
relation $\sim_{t.c.}$.

######  Theorem 3.31.

Suppose we have $(X_{1},\sigma_{1}),(X_{2},\sigma_{2})\in\emph{{LHomeo}}$
which are topologically conjugate through the homeomorphism
$h:X_{1}\rightarrow X_{2}$. Then
$\widetilde{\Phi}(X_{1},\sigma_{1})=\widetilde{\Phi}(X_{2},\sigma_{2})$.

###### Proof.

First, notice that $N(\sigma_{1})=N(\sigma_{2})=:N$ since $h$ is a
homeomorphism. Also, $h({\pazocal U}^{(1)}_{k})={\pazocal U}^{(2)}_{k}$ for
each $1\leq k\leq N$, being

${\pazocal U}^{(i)}_{k}=\\{x\in X_{i}\mid|\sigma^{-1}_{i}(\\{x\\})|=k\\}.$

Take $\\{{\pazocal P}_{n}\\}_{n\geq 0}$ a $\sigma_{1}$-refined sequence of
partitions of $X_{1}$, and $\\{{\pazocal P}^{\prime}_{n}\\}_{n\geq 0}$ a
$\sigma_{2}$-refined sequence of partitions of $X_{2}$. Let $(F_{i},D_{i})$
denote the $l$-diagram constructed from $(X_{i},\sigma_{i})$ using the
construction 3.24, for $i=1,2$. We define, for $n\geq 0$, new partitions

${\pazocal P}^{(1)}_{n}:={\pazocal P}_{n}\vee h^{-1}({\pazocal
P}^{\prime}_{n})\text{ for $X_{1}$, and }\text{ }{\pazocal
P}^{(2)}_{n}:=h({\pazocal P}_{n})\vee{\pazocal P}^{\prime}_{n}\text{ for
$X_{2}$.}$

Then the sequence $\\{{\pazocal P}^{(i)}_{n}\\}_{n\geq 0}$ is a
$\sigma_{i}$-refined sequence of partitions of $X_{i}$, for $i=1,2$:

  1. (a)

Since $\sigma_{1}$ (resp. $\sigma_{2}$) restricts to a homeomorphism over each
$Z\in{\pazocal P}_{0}$ (resp. $Z\in{\pazocal P}^{\prime}_{0}$), it is clear
that each $\sigma_{i}$ restricts to a homeomorphism $\sigma_{i}|_{Z}$, for
$Z\in{\pazocal P}^{(i)}_{0}$.

  2. (b)

${\pazocal P}^{(1)}_{n}\precsim{\pazocal P}^{(1)}_{n+1}$ and ${\pazocal
P}^{(2)}_{n}\precsim{\pazocal P}^{(2)}_{n+1}$, since ${\pazocal
P}_{n}\precsim{\pazocal P}_{n+1}$ and ${\pazocal
P}^{\prime}_{n}\precsim{\pazocal P}^{\prime}_{n+1}$.

  3. (c)

Their unions generate the topology of $X_{1}$ and $X_{2}$, respectively, since
this property holds true for the partitions $\\{{\pazocal P}_{n}\\}_{n\geq 0}$
of $X_{1}$ and $\\{{\pazocal P}_{n}^{\prime}\\}_{n\geq 0}$ of $X_{2}$.

  4. (d)

$\text{diam}({\pazocal P}_{n}^{(1)})\leq\text{diam}({\pazocal P}_{n})$ and
$\text{diam}({\pazocal P}_{n}^{(2)})\leq\text{diam}({\pazocal
P}_{n}^{\prime})$, so they both tend to zero as $n\rightarrow\infty$.

  5. (e)

First, one may notice that $h({\pazocal P}^{\sigma_{1}})=h({\pazocal
P})^{\sigma_{2}}$ for any $\sigma_{1}$-refined partition ${\pazocal P}$ of
$X_{1}$. This follows from the fact that $h({\pazocal U}^{(1)}_{k})={\pazocal
U}^{(2)}_{k}$. Now, for $X_{1}$ and $n\geq 0$, we compute

$\displaystyle{\pazocal P}^{(1)}_{2n}\vee({\pazocal
P}^{(1)}_{2n})^{\sigma_{1}}$ $\displaystyle={\pazocal P}_{2n}\vee
h^{-1}({\pazocal P}^{\prime}_{2n})\vee{\pazocal
P}_{2n}^{\sigma_{1}}\vee(h^{-1}({\pazocal P}^{\prime}_{2n}))^{\sigma_{1}}$
$\displaystyle={\pazocal P}_{2n}\vee{\pazocal P}_{2n}^{\sigma_{1}}\vee
h^{-1}({\pazocal P}^{\prime}_{2n})\vee h^{-1}\big{(}({\pazocal
P}^{\prime}_{2n})^{\sigma_{2}}\big{)}$ $\displaystyle=\big{(}{\pazocal
P}_{2n}\vee{\pazocal P}_{2n}^{\sigma_{1}}\big{)}\vee h^{-1}\big{(}{\pazocal
P}^{\prime}_{2n}\vee({\pazocal P}^{\prime}_{2n})^{\sigma_{2}}\big{)}$
$\displaystyle\precsim{\pazocal P}_{2n+1}\vee h^{-1}({\pazocal
P}^{\prime}_{2n+1})={\pazocal P}^{(1)}_{2n+1}$

and

$\displaystyle{\pazocal P}^{(1)}_{2n+1}\vee\sigma_{1}^{-1}({\pazocal
P}^{(1)}_{2n+1})$ $\displaystyle={\pazocal P}_{2n+1}\vee h^{-1}({\pazocal
P}^{\prime}_{2n+1})\vee\sigma_{1}^{-1}({\pazocal
P}_{2n+1})\vee\sigma_{1}^{-1}(h^{-1}({\pazocal P}^{\prime}_{2n+1}))$
$\displaystyle=\big{(}{\pazocal P}_{2n+1}\vee\sigma_{1}^{-1}({\pazocal
P}_{2n+1})\big{)}\vee h^{-1}\big{(}{\pazocal
P}^{\prime}_{2n+1}\vee\sigma_{2}^{-1}({\pazocal P}^{\prime}_{2n+1})\big{)}$
$\displaystyle\precsim{\pazocal P}_{2n+2}\vee h^{-1}({\pazocal
P}^{\prime}_{2n+2})={\pazocal P}^{(1)}_{2n+2}.$

Similar computations hold for $X_{2}$, that is, for $n\geq 0$, we have

${\pazocal P}^{(2)}_{2n}\vee({\pazocal
P}^{(2)}_{2n})^{\sigma_{2}}\precsim{\pazocal P}^{(2)}_{2n+1}\quad\text{ and
}\quad{\pazocal P}^{(2)}_{2n+1}\vee\sigma_{2}^{-1}({\pazocal
P}^{(2)}_{2n+1})={\pazocal P}^{(2)}_{2n+2}.$

Note that $h({\pazocal P}^{(1)}_{n})=h({\pazocal P}_{n}\vee h^{-1}({\pazocal
P}^{\prime}_{n}))=h({\pazocal P}_{n})\vee{\pazocal P}^{\prime}_{n}={\pazocal
P}^{(2)}_{n}$. Therefore the vertices of $(F_{1},D_{1})$ are in bijective
correspondence with the vertices of $(F_{2},D_{2})$ at each layer $n\geq 0$.
To conclude the theorem, we need to show that the edges $F_{1}^{1,n}$ are in
bijective correspondence with $F_{2}^{1,n}$ for all $n\geq 0$, also preserving
the separations as in Definition 3.12 (a). These follow directly from the
following considerations:

  1. (i)

For $Z\in{\pazocal P}^{(1)}_{2n}$ and $Z^{\prime}\in{\pazocal
P}^{(1)}_{2n+1}$, $Z^{\prime}\subseteq Z$ if and only if
$h(Z^{\prime})\subseteq h(Z)$. Also $Z^{\prime}\subseteq\sigma_{1}(Z)$ if and
only if $h(Z^{\prime})\subseteq h(\sigma_{1}(Z))=\sigma_{2}(h(Z))$.

  2. (ii)

For $Z^{\prime}\in{\pazocal P}^{(1)}_{2n+1}$ and $Z^{\prime\prime}\in{\pazocal
P}^{(1)}_{2n+2}$, $Z^{\prime\prime}\subseteq Z^{\prime}$ if and only if
$h(Z^{\prime\prime})\subseteq h(Z^{\prime})$. Also
$\sigma_{1}(Z^{\prime\prime})\subseteq Z^{\prime}$ if and only if
$\sigma_{2}(h(Z^{\prime\prime}))=h(\sigma_{1}(Z^{\prime\prime}))\subseteq
h(Z^{\prime})$.

This concludes the proof of the theorem. ∎

We finally obtain the desired map $\Phi$.

######  Theorem 3.32.

There exists a well-defined map

$\Phi:\emph{{LHomeo}}/\sim_{t.c.}\longrightarrow\emph{{LDiag}}/\sim_{l},\quad[(X,\sigma)]\mapsto[(F,D)],$

where $(F,D)$ is the $l$-diagram obtained from $(X,\sigma)$ by applying the
construction 3.24.

###### Proof.

This follows directly from the definition of $\widetilde{\Phi}$ and Theorem
3.31. ∎

###  3.2. From LDiag to LHomeo

We now describe how to construct a surjective local homeomorphism $(X,\sigma)$
from an $l$-diagram $(F,D)$.

Let $(F,D)\in\textsf{LDiag}$. We first construct the space $X$. It will be
given by the set of infinite paths

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

Note that all edges are blue. The topology on $X$ is the one with basis
consisting of the cylinder sets

$Z(e_{0},e_{1},\dots,e_{n})=\\{(x_{i})_{i\geq 0}\in X\mid
x_{0}=e_{0},\dots,x_{n}=e_{n}\\},$

for $(e_{0},\dots,e_{n})\in P_{0,n}^{B}$. This is indeed a basis for a
topology because the intersection of two cylinder sets is either empty or
again a cylinder set.

######  Lemma 3.33.

The space $X$ is a totally disconnected, compact metrizable space.

###### Proof.

The decompositions

$X=\bigsqcup_{(e_{0},\dots,e_{n})\in P_{0,n}^{B}}Z(e_{0},\dots,e_{n}),$

for any $n\geq 0$, imply that each cylinder set is a clopen set. Therefore $X$
is totally disconnected. It is clearly Hausdorff, second countable and
regular, hence by Urysohn’s Metrization Theorem, $X$ becomes metrizable.

We can regard $X$ as a topological subspace of $\prod_{n\geq 0}F^{1,n}$
endowed with the product topology, where each set $F^{1,n}$ is endowed with
the discrete topology. This space is compact by Tychonoff’s Theorem, and $X$
is a closed subspace of it. Therefore $X$ is compact. ∎

We now aim to define a natural surjective local homeomorphism
$\sigma:X\rightarrow X$, as follows.

######  Construction 3.34.

Let $(e_{0},e_{1},e_{2},\dots)\in X$. For each $j\geq 0$ we may complete the
pair of blue edges $(e_{2j},e_{2j+1})$ to a romb
$(e_{2j},e_{2j+1},f_{2j},f_{2j+1})$ using part (ii) of Proposition 3.3, so
that $r(f_{2j})=r(e_{2j})$, $s(f_{2j})=r(f_{2j+1})$, $s(f_{2j+1})=s(e_{2j+1})$
and $f_{2j+1}\in R(f_{2j})$. Note that $s(f_{2j-1})=r(f_{2j})$ for each $j\geq
1$. So, for $j\geq 1$, we may consider the red pair $(f_{2j-1},f_{2j})$ and
complete it to another romb
$(e_{2j-1}^{\prime},e_{2j}^{\prime},f_{2j-1},f_{2j})$ by using condition (f)
in Definition 3.2. We obtain a well-defined blue path

$(e_{1}^{\prime},e_{2}^{\prime},\dots)\in X$

starting at $v:=r(e_{1}^{\prime})\in F^{0,1}$. Now, part (1) of condition (d)
in Definition 3.2 entails that there is $w\in F^{0,0}$ and $e_{0}^{\prime}\in
B_{w}$ such that $s(e_{0}^{\prime})=v=r(e_{1}^{\prime})$. Also, by the last
part of this same condition (d) the edge $e_{0}^{\prime}$ is unique with this
property. Thus

$(e_{0}^{\prime},e_{1}^{\prime},e_{2}^{\prime},\dots)\in X.$

We define $\sigma\colon X\to X$ by

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

$e_{0}$$e_{1}$$e_{2}$$e_{3}$$e_{4}$$e_{5}$$f_{0}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$f_{5}$$e_{0}^{\prime}$$e_{1}^{\prime}$$e_{2}^{\prime}$$e_{3}^{\prime}$$e_{4}^{\prime}$$e_{5}^{\prime}$$v$$w$
Figure 5. Construction of the map $\sigma$.

######  Lemma 3.35.

The map $\sigma$ is a surjective local homeomorphism.

###### Proof.

We first show that $\sigma$ is continuous. Let $x=(e_{0},e_{1},e_{2},\dots)\in
X$, with $e_{i}\in B_{r(e_{i})}$ for all $i\geq 0$. Let
$y=\sigma(x)=(e^{\prime}_{0},e^{\prime}_{1},e^{\prime}_{2},\dots)$ and
consider an open neighborhood
$Z(e_{0}^{\prime},e^{\prime}_{1},e^{\prime}_{2},\dots,e^{\prime}_{2n})$ of
$y$. Then we have

$\sigma(Z(e_{0},e_{1},\dots,e_{2n+1}))\subseteq
Z(e^{\prime}_{0},e^{\prime}_{1},e^{\prime}_{2},\dots,e^{\prime}_{2n}),$

and $Z(e_{0},e_{1},\dots,e_{2n+1})$ is an open neighborhood of $x$. This shows
continuity of $\sigma$.

We now show that $\sigma$ is surjective. Indeed we will show that there are a
finite number of pre-images of each element $y\in X$.

Let $y=(e_{0}^{\prime},e_{1}^{\prime},e_{2}^{\prime},\dots)\in X$, with
$e_{i}^{\prime}\in B_{r(e_{i}^{\prime})}$ for all $i\geq 0$. By part (2) of
condition (d) in Definition 3.2, we can choose a red edge $f^{\prime}_{0}$
such that $s(f^{\prime}_{0})=s(e_{0}^{\prime})$. Note that existence of such
$f^{\prime}_{0}$ is ensured, but certainly not uniqueness, although we do have
a finite number of possibilities for $f^{\prime}_{0}$. As we show now, this is
the only choice we need.

By recursively applying property (g) in Definition 3.2, we construct a
sequence of red edges $f^{\prime}_{1},f^{\prime}_{2},\dots$ such that
$f^{\prime}_{2j+1}\in R(f^{\prime}_{2j})$ and
$(e^{\prime}_{2j+1},e^{\prime}_{2j+2},f^{\prime}_{2j+1},f^{\prime}_{2j+2})$ is
a romb for all $j\geq 0$. This can be done since each $s(f^{\prime}_{2j})\in
F^{0,2j+1}$. Note that the sequence $f^{\prime}_{1},f^{\prime}_{2},\dots$ is
uniquely determined by $y$ and by $f^{\prime}_{0}$. Finally we use part (ii)
of Proposition 3.3 to build pairs of blue edges $(e_{2j},e_{2j+1})$ such that
$(e_{2j},e_{2j+1},f^{\prime}_{2j},f^{\prime}_{2j+1})$ is a romb, for all
$j\geq 0$. Then $x=(e_{0},e_{1},e_{2},\dots)\in X$ is an element such that
$\sigma(x)=y$, as required. This shows in particular that $\sigma$ is
surjective.

Conversely, if $\sigma(x)=y$, with $x=(e_{0},e_{1},e_{2},\dots)$, and if
$(f_{0},f_{1})$ is the unique pair of red edges such that $f_{1}\in R(f_{0})$
and $(e_{0},e_{1},f_{0},f_{1})$ is a romb, then $x$ is the unique pre-image of
$y$ determined by the choice of $f_{0}$ as the red edge $f$ such that
$s(f)=s(e_{0}^{\prime})$. This means that there exist exactly
$|s^{-1}(s(e_{0}^{\prime}))|-1$ pre-images of $y$. Observe also that this
implies that $\sigma$ is injective when restricted to $Z(e_{0},e_{1})$,
because $f_{0}$ is determined by $(e_{0},e_{1})$.

To show that $\sigma$ is a local homeomorphism, it suffices to show that
$\sigma$ is an open map, since we already know that $\sigma$ is continuous and
locally injective. To show that $\sigma$ is an open map, it suffices to prove
that for any blue path $(e_{0},e_{1},\dots,e_{2n-1})\in P^{B}_{0,2n}$, with
$n\geq 1$, we have that $\sigma(Z(e_{0},e_{1},\dots,e_{2n-1}))$ is an open
subset of $X$. Now, if
$y=(e_{0}^{\prime},e_{1}^{\prime},\dots,e^{\prime}_{2n},e^{\prime}_{2n+1},\dots)=\sigma(x)$
for some $x\in Z(e_{0},e_{1},\dots e_{2n-1})$, then following the proof of
surjectivity given above we can easily see that

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

This shows that $\sigma$ is an open map, hence it is a local homeomorphism. ∎

Using the above construction, we obtain a well-defined map

$\widetilde{\Psi}:\textsf{LDiag}\longrightarrow\textsf{LHomeo}/\sim_{t.c.},\quad(F,D)\mapsto[(X,\sigma)]$

where $(X,\sigma)$ is the dynamical system obtained from $(F,D)$ by applying
the construction 3.34. In the next theorem we show that the equivalence class
of $(X,\sigma)$ does not depend on the representative chosen for the class
$(F,D)$ under the equivalence relation $\sim_{l}$.

######  Theorem 3.36.

Let $(F_{1},D_{1}),(F_{2},D_{2})\in\emph{{LDiag}}$ be two equivalent
$l$-diagrams under $\sim_{l}$. Write $(X_{i},\sigma_{i})$ for the dynamical
system obtained from $(F_{i},D_{i})$ by applying the construction 3.34, for
$i=1,2$. Then there exists a homeomorphism $h:X_{1}\rightarrow X_{2}$ such
that $h\circ\sigma_{1}=\sigma_{2}\circ h$.

###### Proof.

It suffices to consider the case where $(F_{2},D_{2})$ is a contraction of
$(F_{1},D_{1})$ with respect to a sequence $(m_{n})_{n\geq 0}$ in
${\mathbb{N}}\cup\\{0\\}$ with $0\leq m_{0}<m_{1}<\cdots$, satisfying (CR) and
with $m_{0}$ even. We write
$(F,D):=(F_{1},D_{1}),(F^{\prime},D^{\prime}):=(F_{2},D_{2})$, and
$(X,\sigma):=(X_{1},\sigma_{1}),(X^{\prime},\sigma^{\prime}):=(X_{2},\sigma_{2})$.

We have

$\displaystyle X=\\{(e_{0},e_{1},\dots)\mid e_{i}\in F^{1,i}\text{ blue edges,
}r(e_{i+1})=s(e_{i})\text{ for all }i\geq 0\\},$ $\displaystyle
X^{\prime}=\\{((e_{m_{0}},\dots,e_{m_{1}-1}),(e_{m_{1}},\dots,e_{m_{2}-1}),\dots)\mid(e_{m_{0}},...,e_{m_{1}},...,e_{m_{2}},...)\in
X\\}.$

There is an obvious identification $X\cong X^{\prime}$, which is in fact a
homeomorphism (it sends cylinder sets to cylinder sets). It clearly gives our
desired conjugacy $h$. ∎

We summarize our construction in the following theorem.

######  Theorem 3.37.

There exists a well-defined map

$\Psi:\emph{{LDiag}}/\sim_{l}\longrightarrow\emph{{LHomeo}}/\sim_{t.c.},\quad[(F,D)]\mapsto[(X,\sigma)],$

where $(X,\sigma)\in\emph{{LHomeo}}$ is the dynamical system obtained from
$(F,D)$ by applying the construction 3.34.

###### Proof.

This follows directly from the definition of $\widetilde{\Psi}$ and Theorem
3.36. ∎

###  3.3. Equivalence between both constructions

In this section we prove that both constructions 3.24 and 3.34 are inverses of
each other, modulo the stated equivalences in LHomeo and LDiag. First, a
technical lemma. Recall Definition 3.25 for the notion of a refined
$l$-diagram.

######  Lemma 3.38.

Let $(F,D)$ be an $l$-diagram. Then $(F,D)$ is a refined $l$-diagram if and
only if $\sigma|_{Z(v)}$ is injective for all $v\in F^{0,0}$. Here $Z(v)$ is
defined naturally as the set of those elements $(e_{0},e_{1},e_{2},\dots)\in
X$ such that $r(e_{0})=v$.

###### Proof.

By Remark 3.5 (5) it is enough to show that $\sigma|_{Z(v)}$ is injective for
all $v\in F^{0,0}$ if and only if there are no double red edges from $F^{0,1}$
to $F^{0,0}$, that is, given $w\in F^{0,1}$ and $v\in F^{0,0}$, there is at
most one red edge $f$ such that $s(f)=w$ and $r(f)=v$.

Suppose first that there are no double red edges from a vertex in $F^{0,1}$ to
a vertex in $F^{0,0}$. Let
$y=(e_{0}^{\prime},e_{1}^{\prime},e_{2}^{\prime},\dots)\in X$, where
$e_{i}^{\prime}\in B_{r(e_{i}^{\prime})}$ for all $i\geq 0$. By the proof of
Lemma 3.35, the set of pre-images of $y$ by $\sigma$ is in bijective
correspondence with the set of red edges $f_{0}$ such that
$s(f_{0})=s(e_{0}^{\prime})$. Hence there is at most one element $x\in Z(v)$
such that $\sigma(x)=y$, for each $v\in F^{0,0}$.

Conversely, if there are two red edges $f_{0},f_{0}^{\prime}$ such that
$s(f_{0})=s(f_{0}^{\prime})=w\in F^{0,1}$ and $r(f_{0})=r(f_{0}^{\prime})=v\in
F^{0,0}$, then any point
$y=(e_{0}^{\prime},e_{1}^{\prime},e_{2}^{\prime},\dots)\in X$ such that
$s(e_{0}^{\prime})=w$ has at least two pre-images in $Z(v)$, and thus
$\sigma|_{Z(v)}$ is not injective.

This concludes the proof of the lemma. ∎

With the help of the above lemma, we can now show our main result.

######  Theorem 3.39 (Theorem 3.14).

The maps $\Phi:\emph{{LHomeo}}/\sim_{t.c.}\rightarrow\emph{{LDiag}}/\sim_{l}$
and $\Psi:\emph{{LDiag}}/\sim_{l}\rightarrow\emph{{LHomeo}}/\sim_{t.c.}$
defined in Theorems 3.32 and 3.37, respectively, are mutually inverse maps.

Therefore the above procedures establish a bijective correspondence between
equivalence classes of $l$-diagrams (via isomorphism and contraction) and
topological conjugacy classes of surjective local homeomorphisms on totally
disconnected, compact metrizable spaces.

###### Proof.

We first show that $\Phi\circ\Psi=\text{id}_{\textsf{LDiag}/\sim_{l}}$. Let
$(F,D)$ be an $l$-diagram. By Remark 3.5 (5) the $l$-diagram obtained from
$(F,D)$ by using the sequence $(2,3,4,5,\dots)$ is a refined $l$-diagram,
which of course is $\sim_{l}$-equivalent to $(F,D)$. Hence we may assume that
$(F,D)$ is a refined $l$-diagram.

Let $(X,\sigma)$ be the dynamical system in LHomeo constructed from $(F,D)$ by
applying the construction 3.34. Consider the sequence of partitions given by
the cylinder sets

${\pazocal P}_{0}:=\\{Z(v)\mid v\in F^{0,0}\\},\quad{\pazocal
P}_{n}:=\\{Z(e_{0},\dots,e_{n-1})\mid(e_{0},\dots,e_{n-1})\in P_{0,n}^{B}\\}$

for $n\geq 1$. Note that there is a natural identification between ${\pazocal
P}_{n}$ and $F^{0,n}$. Recall that, by definition of $\sigma$, we have

$\sigma(Z(e_{0},\dots,e_{2n+1}))\subseteq
Z(e^{\prime}_{0},\dots,e^{\prime}_{2n})$

for some blue path $(e_{0}^{\prime},\dots,e_{2n}^{\prime})$ of length $2n+1$.

First of all, we show that $\\{{\pazocal P}_{n}\\}_{n\geq 0}$ is a
$\sigma$-refined sequence of partitions of $X$ (see Definition 3.20).

  1. (a)

By Lemma 3.38, $\sigma|_{Z(v)}$ is injective for all $v\in F^{0,0}$, since we
are assuming that $(F,D)$ is a refined $l$-diagram.

  2. (b)

${\pazocal P}_{n}\precsim{\pazocal P}_{n+1}$ because
$Z(e_{0},\dots,e_{n},e_{n+1})\subseteq Z(e_{0},\dots,e_{n})$.

  3. (c)

The union $\bigcup_{n\geq 0}{\pazocal P}_{n}$ generates the topology of $X$ by
definition.

  4. (d)

We may take the metric determined by the distance function $d$ given by

$d((e_{i}),(e_{i}^{\prime}))=\frac{1}{2^{N}},$

where $N$ is the smallest non-negative integer such that $e_{N}\neq
e_{N}^{\prime}$ for two different elements $(e_{i}),(e_{i}^{\prime})\in X$. It
is then clear that
$\text{diam}(Z(e_{0},\dots,e_{n}))\xrightarrow{n\rightarrow\infty}0$.

  5. (e)

For each $n\geq 0$ we first show that $\sigma^{-1}({\pazocal
P}_{2n+1})\precsim{\pazocal P}_{2n+2}$. Given a blue path
$(e_{0},e_{1},\dots,e_{2n+1})$ of length $2n+2$, we know that there exists a
blue path $(e_{0}^{\prime},\dots,e_{2n}^{\prime})$ of length $2n+1$ such that

$\sigma(Z(e_{0},\dots,e_{2n+1}))\subseteq
Z(e^{\prime}_{0},\dots,e^{\prime}_{2n}).$

Therefore
$Z(e_{0},\dots,e_{2n+1})\subseteq\sigma^{-1}(Z(e^{\prime}_{0},\dots,e^{\prime}_{2n}))$,
and so $\sigma^{-1}({\pazocal P}_{2n+1})\precsim{\pazocal P}_{2n+2}$.

We now show that ${\pazocal P}_{2n}^{\sigma}\precsim{\pazocal P}_{2n+1}$. Take
a blue path $(e_{0},e_{1},\dots,e_{2n})$ of length $2n+1$. As seen in the
proof of Lemma 3.35, each element of $Z(e_{0},\dots,e_{2n})$ has exactly $k$
pre-images by $\sigma$, where $k$ is the number of red edges $f_{0}$ in
$F^{1,0}$ such that $s(f_{0})=s(e_{0})$. Suppose first that $n\geq 1$. Given
such an edge $f_{j,0}$, for $1\leq j\leq k$, there are unique red edges
$f_{j,1},f_{j,2},\dots,f_{j,2n-1},f_{j,2n}$ such that $f_{j,1}\in R(f_{j,0})$,
$(e_{2i-1},e_{2i},f_{j,2i-1},f_{j,2i})$ is a romb for $1\leq i\leq n$, and
$f_{j,2i+1}\in R(f_{j,2i})$ for $1\leq i\leq n-1$. For $0\leq i\leq n-1$, let
$(e_{j,2i}^{\prime},e_{j,2i+1}^{\prime},f_{j,2i},f_{j,2i+1})$ be the romb
determined by $(f_{j,2i,},f_{j,2i+1})$. We then have

$Z(e_{0},\dots,e_{2n})\subseteq\Big{(}\bigcap_{j=1}^{k}\sigma(Z(e^{\prime}_{j,0},\dots,e^{\prime}_{j,2n-1}))\Big{)}\setminus{\pazocal
U}_{\geq k+1}.$

Observing that the set in the right-hand side belongs to ${\pazocal
P}_{2n}^{\sigma}$, we obtain that ${\pazocal P}_{2n}^{\sigma}\precsim{\pazocal
P}_{2n+1}$, as desired.

If $n=0$, then using that $(F,D)$ is a refined $l$-diagram we get

$Z(e_{0})\subseteq\Big{(}\bigcap_{j=1}^{k}\sigma(Z(v_{j}))\Big{)}\setminus{\pazocal
U}_{\geq k+1},$