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\begin{document}

\title
[Affable equivalence relations and orbits
of Cantor dynamical systems]
{Affable equivalence relations and orbit
structure of Cantor dynamical systems}

\author{Thierry Giordano}

\address{Thierry Giordano,
Department of Mathematics and Statistics, University of Ottawa,
585 King Edward, Ottawa, K1N 6N5, Canada}

\email{giordano@science.uottawa.ca}

\author{Ian Putnam}

\address{Ian Putnam,
Department of Mathematics and Statistics, University of Victoria,
Victoria, B.C. V8W 3P4, Canada}

\email{putnam@math.uvic.ca}

\author{Christian Skau}

\address{Christian Skau,
Department of Mathematical Sciences,
Norwegian University of Science and Technology (NTNU),
N-7491 Trondheim, Norway}

\email{csk@math.ntnu.no}

\begin{abstract}
We prove several new results about
$AF$-equivalence relations, and relate these to Cantor minimal systems (i.e.
to minimal $Z$-actions). The results we obtain turn out to be crucial for the
study of the topological orbit structure of more general countable group
actions (as homeomorphisms) on Cantor sets, which will be the topic of a
forthcoming paper. In all this, Bratteli diagrams and their dynamical
interpretation, are indispensable tools.
\end{abstract}

\maketitle

\section{Introduction}

In the present paper we will prove some new results about $AF$-equivalence
relations (cf. Definition 3.7) that, besides being of interest in their own
right, turn out to be powerful new tools for the study of the topological
orbit structure of countable group actions as homeomorphisms on Cantor sets.
In a forthcoming paper we will apply these new techniques to prove that
certain minimal and free $\mathbb{Z}^{2}$-actions on Cantor sets are
(topologically) orbit equivalent to Cantor minimal systems $\left(
X,T\right)  $, or, equivalently, to minimal $\mathbb{Z}$ actions. The strategy
is to prove that the equivalence relation $R_{\mathbb{Z}^{2}}$ associated to
the given $\mathbb{Z}^{2}$-action is \textit{affable }(''$AF$-able''), i.e.
may be given an $AF$-equivalence structure. To prove affability of
$R_{\mathbb{Z}^{2}}$ we need a delicate ''glueing'' procedure, the technical
part of which is stated in this paper as Lemma 4.15, the ''key lemma''. We
demonstrate the power of Lemma 4.15 by establishing results that are highly
non-trivial, concerning the intimate link that exists between minimal
$AF$-equivalence relations and Cantor minimal systems (Thm. 4.16, Thm. 4.17;
cf. also Thm. 4.8).

Our ultimate goal is to attack the following question (which is an analogue in
the topological dynamical setting to the celebrated Connes-Feldman-Weiss
result in the measure-theoretic setting [CFW]):

\textit{Let }$G$\textit{\ be a countable, amenable group acting minimally
(i.e. every }$G$\textit{-orbit is dense) and freely (i.e. }$gx=x$\textit{\ for
some }$x\in X$\textit{, implies }$g=$\textit{the identity element of }%
$G$\textit{) as homeomorphisms on the Cantor set }$X$\textit{. Then }$\left(
X,G\right)  $\textit{\ is topologically orbit equivalent to a Cantor minimal
system }$\left(  Y,T\right)  $\textit{, i.e. there exists a homeomorphism
}$F:X\rightarrow Y$\textit{\ mapping }$G$\textit{-orbits onto }$T$\textit{-orbits.}

[By Theorem 4.8 and Theorem 4.16 this is equivalent to show that the
equivalence relation $R_{G}$ associated to $G$ (cf. Example 2.7$(i)$) is
affable. Previously, this is known to be true for locally finite groups $G$
(in fact, in this case we do not need to require free action, only that
$fix\left(  g\right)  =\{x\in X|gx=x\}$ is a clopen set for each $g\in G$),
and for $\mathbb{Z}^{n}$-actions that split as Cartesian products, and also
for the case that $G=\mathbb{Z\times}H$, where $H$ is a finite cyclic group.
These facts can be deduced from results contained in [Kr], [GPS], [Jo]. It is
also noteworthy that in the (standard) Borel setting the analogous question
has an affirmative answer for $\mathbb{Z}^{n}$-actions [W].]\smallskip

We shall need the key concept of a Bratteli diagram, both ordered and
unordered, and we refer to [HPS] and [GPS] for details and proofs of basic
results. (Cf. also Example 2.7(ii).) We will state two results that we shall
need in the sequel, concerning the interplay between Bratteli diagrams and
Cantor minimal systems.

\begin{Proposition}
[HPS; Section 3]Let \ $\left(  V,E\right)  $ be a simple (standard) Bratteli
diagram, and let $X=X_{\left(  V,E\right)  }$ be the Cantor set consisting of
the (infinite) paths associated to $\left(  V,E\right)  .$ Let $x,y\in X$ be
two paths that are not cofinal. There exists a Cantor minimal system $\left(
X,T\right)  $ such that $T$ preserves cofinality, except that $Tx=y$.
\end{Proposition}

\begin{proof}
Without loss of generality we may assume (by appropriately telescoping the original
diagram), that for every $n=0,1,2,...$, there is at least one edge between every vertex
$v$ at level $n$ and every vertex $w$ at level $n+1$ of $(V,E)$. Furthermore, we may
assume that the $n$'th edge of $x$ is distinct from the $n$'th edge of $y$. It is an easy
observation that $(V,E)$ may be given a proper ordering (also called simple ordering)
such that $x$ becomes the unique max path, and $y$ becomes the unique min path.
The associated Bratteli-Vershik system $(X,T)$ has the desired properties.
\end{proof}

\begin{Theorem}
[GPS;\ Lemma 5.1]Let $\left(  X,T\right)  $ be\ Cantor minimal system, and let
$Y$ be a closed (non-empty) subset of $X$ that meets each $T$-orbit at most
once. There exists an ordered Bratteli diagram $B_{Y}=\left(  V,E,\geq\right)
$, where $\left(  V,E\right)  $ is a simple (standard) Bratteli diagram, such
that the associated Bratteli-Vershik system $\left(  X_{\left(  V,E\right)
},T_{B_{Y}}\right)  $ is conjugate to $\left(  X,T\right)  $. The conjugating
map $F:X\rightarrow X_{\left(  V,E\right)  }$ maps $Y$ onto the set of maximal
paths, and $T\left(  Y\right)  $ onto the set of minimal paths. Furthermore,
if $y\in Y$, the backward orbit of $y$, $\{T^{n}y|n\leq0\}$, is mapped onto
the set of paths cofinal with $F\left(  y\right)  $, while the forward orbit
$\{T^{n}y|n\geq1\}$ is mapped onto the set of paths cofinal with $F\left(
Ty\right)  $. Any $T$-orbit, $\{T^{n}x|n\in\mathbb{Z}\}$, that does not meet
$Y$ is mapped onto the set of paths cofinal with $F\left(  x\right)  .$
\end{Theorem}

\begin{Corollary}
[HPS;\ Theorem 4.7]If $Y=\{y\}$, then $B_{\{y\}}=\left(  X,V,\geq\right)  $ is
a properly ordered (also called simply ordered) Bratteli diagram, with
$F\left(  y\right)  $ equal to the unique max path and $F\left(  Ty\right)  $
equal to the unique min path.
\end{Corollary}

\section{\'{E}tale equivalence relations}

Let $X$ be a Hausdorff locally compact, second countable (hence metrizable)
space. For the most part we shall be considering the case when $X$ is
zero-dimensional, i.e., $X$ has a countable basis of closed and open (clopen)
sets. (This is equivalent to $X$ being totally disconnected.) Of particular
importance will be the case when $X$ is a \textit{Cantor set}, i.e., $X$ is a
totally disconnected compact metric space with no isolated points\unskip
\thinspace---\hskip
.16667em\ignorespaces it is a well known fact (going back to Cantor and
Hausdorff) that all such spaces are homeomorphic.

We shall be considering countable equivalence relations $R$ on $X$, i.e.
$R\subset X\times X$ is an equivalence relation so that each equivalence class
$[x]_{R}=\{y\in X\mid(x,y)\in R\}$ is countable (or finite) for each $x$ in
$X$. $R$ has a natural (principal) \textit{groupoid} structure, with
\textit{unit space} equal to the diagonal set $\Delta=\{(x,x)\mid x\in X\}$,
which we may identify with $X$. Specifically, if $(x,y)$, $(y,z)\in R$, then
the product of this \textit{composable} pair is defined as
\[
(x,y)\cdot(y,z)=(x,z),
\]
and the inverse of $(x,y)\in R$ is $(x,y)^{-1}=(y,x)$. The unit space of $R$
is by definition the set consisting of products of elements of $R$ with their
inverses, and so equals $\Delta$. Assume $R$ is given a Hausdorff locally
compact, second countable (hence metrizable) topology $\mathcal{T}$, so that
the product of composable pairs (with the topology inherited from the product
topology on $R\times R$) is continuous. Also, the inverse map on $R$ shall be
a homeomorphism. With this structure ($R,\mathcal{T}$) is a \textit{locally
compact} \textit{(principal)} \textit{groupoid}, cf.\ [Re1].

The range map $r:R\rightarrow X$ and the source map: $s:R\rightarrow X$ are
defined by $r((x,y))=x$ and $s((x,y))=y$, respectively, where $(x,y)\in
R$\unskip
\thinspace---\hskip.16667em\ignorespaces both maps being surjective.

\begin{Definition}
[\'{E}tale equivalence relation]The locally compact groupoid $\left(
R,\mathcal{T}\right)  $, where $R$ is a countable equivalence relation on the
locally compact metric space $X$, is \textit{\'{e}tale} if $r:R\rightarrow X$
is a local homeomorphism, i.e. for every $(x,y)\in R$ there exists an open
neighborhood $U^{(x,y)}\in\mathcal{T}$ of $(x,y)$ so that $r(U^{(x,y)})$ is
open in $X$ and $r:U^{(x,y)}\rightarrow r(U^{(x,y)})$ is a homeomorphism. In
particular, r is an open map. If $X$ is zero-dimensional, we may clearly
choose $U^{(x,y)}$ to be a clopen set.

We will call $\left(  R,\mathcal{T}\right)  $ an \textit{\'{e}tale equivalence
relation} on $X$, and we will occasionally refer to the local homeomorphism
condition as the \textit{\'{e}taleness condition}.
\end{Definition}

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\begin{Remark}
The definition we have given of \'{e}taleness\unskip\thinspace---\hskip
.16667em\ignorespaces which is the most convenient to use for our objects of
study\unskip\thinspace---\hskip.16667em\ignorespaces is equivalent to the
various definitions of an \'{e}tale (or $r$-discrete) locally compact groupoid
(applied to our setting) that can be found in the literature. Confer for
instance [Re1; Def.\ 2.6 and Prop.\ 2.8] and [Pat; Def.\ 2.2.1 and
Def.\ 2.2.3]\unskip\thinspace---\hskip.16667em\ignorespaces the existence of
an (essential) unique Haar system consisting of counting measures follows from
our definition, cf.\ [Pat; Prop.\ 2.2.5]. Furthermore, one can prove that the
diagonal $\Delta=\Delta_{X}=\{(x,x)\mid x\in X\}$ is a clopen subset of $R$
[Re1; Prop.\ 2.8]. Also, $\Delta$ is homeomorphic to $X$, and so we are
justified in identifying $\Delta$ with $X$.

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We observe that $s$ is a local homeomorphism, since $s((x,y))=r((x,y)^{-1})$.
It is easily deduced that $r^{-1}(x)=\{(x,y)\in R\}$, as well as
$s^{-1}(x)=\{(y,x)\in R\}$, are (countable) discrete topological spaces in the
relative topology for each $x\in X$. Clearly $R$ can be written as a union of
graphs of local homeomorphisms of the form $s\circ r^{-1}.$
\end{Remark}

Note that the topology $\mathcal{T}$ on $R$ ($\ \subset X\times X$) is rarely
the topology $\mathcal{T}_{\mathrm{rel}}$ inherited from the product topology
on $X\times X$. Necessarily $\mathcal{T}$ is finer than $\mathcal{T}%
_{\mathrm{rel}}$. For details on topological groupoids, in particular locally
compact and \'{e}tale ($r$-discrete) groupoids, and the associated $C^{\ast}%
$-algebras, we refer to [Re1], [Pat], [Put].
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The following proposition is the analogue in our setting of Theorem 1 of [FM],
where countable (standard) Borel equivalence relations were studied. Our proof
mimics the proof in [FM].

\begin{Proposition}
Let $\left(  R,\mathcal{T}\right)  $ be an \'{e}tale equivalence relation on
the zero-dimensional space $X$. There exists a countable group $G$ of
homeomorphisms of $X$ so that $R=R_{G}$, where $R_{G}=\{(x,gx)\mid x\in X,g\in
G\}$.
\end{Proposition}

\begin{proof}
Let $\{{C_{k}\}}_{k=1}^{\infty}$ be a clopen partition of $R\setminus
\Delta$\thinspace, where $\Delta$ is the diagonal $\{(x,x)\mid x\in X\}$ \thinspace, so that for each $k$, the maps $r$ and $s$ are homeomorphisms
from $C_{k}$ onto the clopen sets $r(C_{k})\subset X$ and $s(C_{k})\subset
X$ , respectively. We refine the partition $\{C_{k}\}$ so that $r(C_{k})\cap
s(C_{k})=\emptyset$ for each $k$. In fact, this may be achieved as follows.
Let $\{I_{i}\times J_{i}\}_{i=1}^{\infty}$ be a clopen covering of $\left(
X\times X\right) \setminus\Delta$ so that $I_{i}\cap J_{i}=\emptyset$ for
every $i$, and each $I_{i}$ and $J_{i}$ are clopen. Define $
D_{k}^{i}=C_{k}\cap(I_{i}\times J_{i})$. Then $\{D_{k}^{i}\}_{i,k=1}^{
  \infty}$ is a clopen partition of $R\setminus\Delta$, %page 6 
so that $r$ and $s$ are homeomorphisms of $D_{k}^{i}$ onto the clopen sets $ r(D_{k}^{i})\subset X$ and $s(D_{k}^{i})\subset X$, respectively, and $ r(D_{k}^{i})\cap s(D_{k}^{i})=\emptyset$ for all $i,k$. We relabel the
non-empty sets in $\{{D_{k}^{i}\}}$, and so get a sequence $ \{E_{i}\}_{i=1}^{\infty}$, which is a clopen refinement of $ \{C_{k}\}_{k=1}^{\infty}$, with the property that $r(E_{i})\cap
s(E_{i})=\emptyset$ for every $i$. For each $i$ we define the continuous
function
\begin{equation*}
g(x)= \cases{
y(=s_ir^{-1}_i(x))&if $(x,y)\in E_i$,\cr
y(=r_is^{-1}_i(x))&if $(y,x)\in E_i$,\cr
x&otherwise,}
\end{equation*}
where $r_{i}$ and $s_{i}$ denote the restriction to $E_{i}$ of $r$ and $s$,
respectively. Observe that $g_{i}^{2}=\id$, and so $g_{i}$ is a
homeomorphism. The graph $\Gamma(g_{i})$ of $g_{i}$ is easily seen to be $ E_{i}\cup\theta(E_{i})\cup(\Delta\cap(F_{i}\times F_{i}))$, where $ \theta:X\times X\rightarrow X\times X$ denotes the flip map $ (x,y)\rightarrow(y,x)$, and $F_{i}=X\setminus(r(E_{i})\cup s(E_{i}))$.
Hence $\Gamma(g_{i})\subset R$. Let $G$ be the (countable) group generated
by the $g_{i}$'s. Clearly $R_{G}\subset R$. On the other hand, $ \bigcup_{i=1}^{\infty}\Gamma(g_{i})\supset R\setminus\Delta$, since $ \{E_{i}\}$ is a covering of $R\setminus\Delta$. Since clearly $\Delta
\subset R_{G}$, we conclude that $R=R_{G}$.
\end{proof}

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\begin{op}
%Open problem
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Assume that for every $x\in X$, the equivalence class $[x]_{R}$ is dense in
$X$, where $(R,\mathcal{T})$ is as in Proposition 2.3. Is it possible to
choose $G$ so that $R=R_{G}$ and $G$ acts freely on $X$ (i.e., $gy=y$ for some
$y\in X$, $g\in G$, implies that $g={}$identity element)?

The analogous question has a negative answer in the setting of countable
(standard) Borel equivalence relations [Ad]. Furthermore, in the ergodic,
measure-preserving case there are examples of countable equivalence relations
that cannot be generated by an essentially free action of a countable group
[Fu; Theorem $D$].
\end{op}

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Let $(R_{1},\mathcal{T}_{1})$ and $(R_{2},\mathcal{T}_{2})$ be two \'{e}tale
equivalence relations on $X_{1}$ and $X_{2}$, respectively. There is an
obvious notion of isomorphism, namely a homeomorphism of $(R_{1}%
,\mathcal{T}_{1})$ onto $(R_{2},\mathcal{T}_{2})$ respecting the groupoid
operations. Since the unit spaces $\Delta_{i}=\{(x,x)\mid x\in X_{i}\}$%
\unskip\thinspace---\hskip.16667em\ignorespaces which we identify with $X_{i}%
$\unskip\thinspace---\hskip.16667em\ignorespaces
are equal to $\{aa^{-1}\mid a\in R_{i}\}$, $i=1,2$, the definition of
isomorphism may be given as follows.

\begin{Definition}
[Isomorphism and orbit equivalence]
%Definition 2.4 
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Let $(R_{1},\mathcal{T}_{1})$ and $(R_{2},\mathcal{T}_{2})$ be two \'{e}tale
equivalence relations on $X_{1}$ and $X_{2}$ respectively. $(R_{1}%
,\mathcal{T}_{1})$ is \textit{isomorphic} to $(R_{2},\mathcal{T}_{2})$%
\unskip\thinspace---\hskip.16667em\ignorespaces we use the notation
$(R_{1},\mathcal{T}_{1})\cong(R_{2},\mathcal{T}_{2})$\unskip\thinspace
---\hskip.16667em\ignorespaces if there exists a homeomorphism $F:X_{1}%
\rightarrow X_{2}$ so that

\begin{enumerate}
\item [(i)]$(x,y)\in R_{1}\iff(F(x),F(y))\in R_{2}$

\item[(ii)] $F\times F:(R_{1},\mathcal{T}_{1})\rightarrow(R_{2},\mathcal{T}%
_{2})$ is a homeomorphism, where $F\times F((x,y))=(F(x),F(y))$, $(x,y)\in
R_{1}$. We say $F$ \textit{implements }an isomorphism between $\left(
R_{1},\mathcal{T}_{1}\right)  $ and $\left(  R_{2},\mathcal{T}_{2}\right)  .$
\end{enumerate}

We say that $(R_{1},\mathcal{T}_{1})$, or $R_{1}$, is \textit{orbit
equivalent} to $(R_{2},\mathcal{T}_{2})$, or to $R_{2}$, if $(i)$ is
satisfied, and we call $F$ an \textit{orbit map} in this case.
\end{Definition}

\begin{Remark}
Observe that $(R_{1},\mathcal{T}_{1})$ is orbit equivalent to $(R_{2}%
,\mathcal{T}_{2})$ via the orbit map $F:X_{1}\rightarrow X_{2}$ if and only if
$F([x]_{R_{1}})=[F(x)]_{R_{2}}$ for each $x\in X_{1}$. So $F$ maps equivalence
classes onto equivalence classes.

Note that if $R_{i}=R_{G_{i}}$ for some countable group $G_{i}$, $i=1,2$, then
the equivalence classes coincide with $G_{i}$-orbits, and so the term orbit
equivalence is appropriate, cf.\ Proposition 2.3.

There is a notion of invariant probability measure associated to an \'{e}tale
groupoid $\left(  R,\mathcal{T}\right)  $. Suffice to say here that the
probability measure $\mu$ on $X$ is $\left(  R,\mathcal{T}\right)  $-invariant
iff $\mu$ is $G$-invariant, where $G$ is as in Proposition 2.3. It is
straightforward to show that if $\left(  R_{1},\mathcal{T}_{1}\right)  $ and
$\left(  R_{2},\mathcal{T}_{2}\right)  $ on $X_{1}$ and $X_{2}$, respectively,
are orbit equivalent via the orbit map $F:X_{1}\rightarrow X_{2}$, then $F$
maps the set of $\left(  R_{1},\mathcal{T}_{1}\right)  $-invariant probability
measures injectively onto the set of $\left(  R_{2},\mathcal{T}_{2}\right)
$-invariant probability measures.
\end{Remark}

\begin{Remark}
It is very important to be aware of the fact that a countable equivalence
relation $R$ on $X$ may be given distinct non-isomorphic topologies
$\mathcal{T}_{1}$ and $\mathcal{T}_{2}$, so that $(R,\mathcal{T}_{1})$ and
$(R,\mathcal{T}_{2})$ are \'{e}tale equivalence relations. In fact, one may
give examples of non-isomorphic $(R,\mathcal{T}_{1})$ and $(R,\mathcal{T}%
_{2})$ where every equivalence class is dense. Specifically, $(R,\mathcal{T}%
_{1})$ may be chosen to be the \'{e}tale equivalence relation associated to a
Cantor minimal system $(X,T)$, while $(R,\mathcal{T}_{2})$ is the cofinal
relation associated to a (simple) standard Bratteli diagram, appropriately
topologized\unskip\thinspace---\hskip.16667em\ignorespaces see the description
given in the two examples below. (Confer also Section 4 and [GPS, Thm.\ 2.3]).

The above fact contrasts with the situation in the countable (standard) Borel
equivalence relation setting, where the Borel structure is uniquely determined
by $R(\subset X\times X)$.
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\end{Remark}

\begin{Examples}
[Two \'{e}tale equivalence relations]
\end{Examples}

\textbf{(i)} Let $G$ be a countable discrete group acting freely as
homeomorphisms on the locally compact metric space $X$. Let
\[
R_{G}=\{(x,gx)\mid x\in X,g\in G\}\subset X\times X,
\]
i.e., the $R_{G}$-equivalence classes are simply the $G$-orbits. Topologize
$R_{G}$ by transferring the product topology on $X\times G$ to $R_{G}$ via the
bijection $(x,g)\rightarrow(x,gx)$. (This is a bijection since $G$ acts freely
on $X$.) Then it is easily verified that $R_{G}$ becomes an \'{e}tale
equivalence relation. (If $G$ do not act freely, we get a bijection between
$R_{G}$ and a closed subset of $X\times G\times X$ by the map $\left(
x,gx\right)  \rightarrow\left(  x,g,gx\right)  $, and we transfer the product
topology on $X\times G\times X$ to $R_{G}$.)

We shall be especially interested in the situation when $G$ acts
\textit{minimally} (and freely) on the Cantor set $X$, i.e., each orbit
$Gx=\{gx\mid g\in G\}$ is dense in $X$ for every $x$ in $X$. In particular,
when $G=\mathbb{Z}$, we let $(X,T)$, where $T$ is the (necessarily minimal)
homeomorphism corresponding to $1\in\mathbb{Z}$, denote the associated
\textit{Cantor} (dynamical) system. We will use the term ``$T$-orbit'' instead
of ``$\mathbb{Z}$-orbit''in this case.

Let $\left(  X,T\right)  $ and $\left(  Y,S\right)  $ be two Cantor minimal
systems, and denote the associated \'{e}tale equivalence relations by
$R\left(  X,T\right)  $ and $R\left(  Y,S\right)  ,$ respectively. We claim
that $R\left(  X,T\right)  \cong R\left(  Y,S\right)  $ if and only if
$\left(  X,T\right)  $ is flip conjugate to $\left(  Y,S\right)  $ (i.e.
$\left(  X,T\right)  $ is conjugate to either $\left(  Y,S\right)  $ or
$\left(  Y,S^{-1}\right)  $). In fact, one direction is obvious. Conversely,
assume $R\left(  X,T\right)  $ is isomorphic to $R\left(  Y,S\right)  $ via
the implementing map $F:X\rightarrow Y.$ Let $\left(  x_{i}\right)  $ be a
sequence in $X$\ so that $x_{i}\rightarrow x.$ Then $\left(  x_{i}%
,Tx_{i}\right)  \rightarrow\left(  x,Tx\right)  $ in $R\left(  X,T\right)  . $
Now $F\times F\left(  \left(  x,Tx\right)  \right)  =\left(  Fx,S^{m}\left(
Fx\right)  \right)  ,$ $F\times F\left(  \left(  x_{i},Tx_{i}\right)  \right)
=\left(  Fx_{i},S^{m_{i}}\left(  Fx_{i}\right)  \right)  ,$ for some
$m,m_{i}\in$ $\mathbb{Z}.$ Since $F\times F\left(  \left(  x_{i}%
,Tx_{i}\right)  \right)  \rightarrow F\times F\left(  \left(  x,Tx\right)
\right)  $ in $R\left(  Y,S\right)  ,$ this implies that $m_{i}=m$ for all but
finitely many $i$'s. By a theorem of M.Boyle (cf. Thm. 1.4. of [GPS]), this
implies that $\left(  X,T\right)  $ and $\left(  Y,S\right)  $ are flip conjugate.\smallskip

\textbf{(ii)} We begin with a special infinite directed graph $(V,E)$, called
a \textit{Bratteli diagram}, which consists of a vertex set $V$ and an edge
set $E$, where $V$ and $E$ can be written as a countable disjoint union of
non-empty finite sets:
\[
V=V_{0}\cup V_{1}\cup V_{2}\cup\cdots\quad\hbox{and}\quad E=E_{0}\cup
E_{1}\cup E_{2}\cup\cdots
\]
with the following property: An edge $e$ in $E_{n}$ goes from a vertex in
$V_{n-1}$ to one in $V_{n}$, which we denote by $i(e)$ and $f(e)$,
respectively. We call $i$ the \textit{source map} and $r$ the \textit{range
map}. We require that there are no \textit{sinks}, i.e. $i^{-1}(v)\neq
\emptyset$ for all $v\in V$.

It is convenient to give a diagrammatic presentation of a Bratteli diagram
with $V_{n}$ the vertices at level $n$ and $E_{n}$ the edges (downward
directed) between $V_{n-1}$ and $V_{n}$, see Figure 1.%

%TCIMACRO{\FRAME{ftbFU}{210.25pt}{210.75pt}{0pt}{\Qcb{ }}{}{Figure
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\begin{figure}
[tb]
\begin{center}
\includegraphics[
natheight=320.125000pt,
natwidth=319.375000pt,
height=210.75pt,
width=210.25pt
]%
{fig1.eps}%
\caption{ }%
\end{center}
\end{figure}
%EndExpansion

We let $X=X_{\left(  V,E\right)  }$ denote the space of infinite paths in the
diagram beginning at some \textit{source} $v\in V$, i.e., $f^{-1}%
(v)=\emptyset$. Say $v\in V_{n}$ is a source, let
\[
X_{v}=\{(e_{n+1},e_{n+2},\dots)\mid i(e_{n+1})=v,\,i(e_{k+1})=f(e_{k}),\,k>n\}
\]
which is given the relative topology of the product space $\prod_{k>n}E_{k}$,
and is therefore compact, metrizable and zero-dimensional. We let $X$ be the
disjoint union of the $X_{v}$'s with the topological sum topology. Then $X$ is
locally compact, metrizable and zero-dimensional, which has a basis consisting
of clopen \textit{cylinder} sets, i.e. sets of the form $U_{\left(
e_{n+1},\ldots,e_{m}\right)  }=\{\left(  f_{n+1},f_{n+2},\ldots\right)  \in X$
$\mid f_{n+1}=e_{n+1},\ldots,f_{m}=e_{m};i\left(  e_{n+1}\right)  \in V_{n}$
is a source$\}$. The equivalence $R$ on $X$ shall be \textit{cofinal} or
\textit{tail equivalence}: two paths are equivalent if they agree from some
level on. For $N=0,1,2,\dots$, let
\[
R_{N}=\{\left(  (e_{m+1},e_{m+2},...),\,(e_{n+1}^{\prime},e_{n+2}^{\prime
},...)\right)  \in X\times X\mid m,n\leq N\hbox{ and }e_{k}=e_{k}^{\prime
}\hbox{
for all }k>N\}.
\]

Give $R_{N}$ the relative topology $\mathcal{T}_{N}$ of $X\times X$. Then
$R_{N}$ is compact and is an open subset of $R_{N+1}$ for all $N$. Let
$R=\bigcup_{N=0}^{\infty}R_{N}$, and give $R$ the inductive limit topology
$\mathcal{T}$, so that a set $U$ is in $\mathcal{T}$ if and only if $U\cap
R_{N}$ is in $\mathcal{T}_{N}$ for each $N$. This means that a sequence
$\{{(x_{n},y_{n})\}}$ in $R$ converges to $(x,y)$ in $R$ if and only if
$\{{x_{n}\}}$ converges to $x$, $\{{y_{n}\}}$ converges to $y$ (in $X$) and,
for some $N$, $(x_{n},y_{n})$ is in $R_{N}$ for all but finitely many $n$.

It is now a simple task to verify that $(R,\mathcal{T})$ is an \'{e}tale
equivalence relation. We shall prove in Section 3 (Theorem 3.9) that this
Bratteli diagram example is the prototype of an AF-equivalence relation\unskip
\thinspace---\hskip.16667em\ignorespaces the latter will be defined in Section
3 (Definition 3.7). Therefore we will denote $\left(  R,\mathcal{T}\right)  $
by $AF\left(  V,E\right)  $.

There is an obvious (countable) locally finite group $G$ of homeomorphisms of
$X$ so that $R=R_{G}=\{\left(  x,gx\right)  \mid x\in X,g\in G\}$, where the
fixed point set, $\ fix\left(  g\right)  =\{x\in X\mid g\left(  x\right)
=x\}$, is a clopen subset of $X$ for every $g\in G$ (cf. Proposition 2.3). In
fact, $G=\bigcup\limits_{n=0}^{\infty}G_{n}$, where $\{id\}=G_{0}\subset
G_{1}\subset G_{2}\subset\cdots$ is an increasing sequence of finite groups
with $G_{n}=\bigoplus\limits_{v\in V_{n}}G_{v}$. Here $G_{v}$ is the group
consisting of those homeomorphisms $g$ of $X=X_{\left(  V,E\right)  }$, such
that $g\left(  x\right)  =x$ for those paths $x\in X$ that do not pass through
$v$, and otherwise $g\left(  x\right)  $ is obtained by permuting the initial
segments (above level $n$) of the various $x$'s passing through $v$, leaving
the tails unchanged. We omit the details. Conversely, let $G$ be a locally
finite group acting as homeomorphisms on a zero-dimensional space $X$, such
that the fixed point set of each $g\in G$ is clopen. Then one can construct a
Bratteli diagram $\left(  V,E\right)  $ so that $X$ may be identified with
$X_{\left(  V,E\right)  }$, and $R_{G}$ will coincide with the cofinal
equivalence relation, cf. [Kr] and [Jo].

%page 14
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If the Bratteli diagram $(V,E)$ has only one source $v_{0}\in V$%
\unskip\thinspace---\hskip.16667em\ignorespaces which necessarily entails that
$V_{0}=\{v_{0}\}$\unskip\thinspace---\hskip.16667em\ignorespaces we will call
$(V,E)$ a \textit{standard }Bratteli diagram. The path space $X_{\left(
V,E\right)  }$ associated to a standard Bratteli diagram $\left(  V,E\right)
$ is compact. We observe that if $\left(  V^{\prime},E^{\prime}\right)  $ is a
\textit{telescope} of $\left(  V,E\right)  ,$ i.e. $\left(  V^{\prime
},E^{\prime}\right)  $ is obtained from $\left(  V,E\right)  $ by telescoping
$\left(  V,E\right)  $ to certain levels $0<n_{1}<n_{2}<n_{3}<\cdots,$ then
$AF\left(  V,E\right)  \cong AF\left(  V^{\prime},E^{\prime}\right)  .$ In
fact, there is a natural homeomorphism $\alpha:X_{\left(  V,E\right)
}\rightarrow X_{\left(  V^{\prime},E^{\prime}\right)  },$ and $\alpha$ clearly
implements the isomorphism, according to the description we have given of
convergence in $AF\left(  V,E\right)  ,$ respectively $AF\left(  V^{\prime
},E^{\prime}\right)  .$

The standard Bratteli diagram $\left(  V,E\right)  $ is \textit{simple} if for
each $n$ there is an $m>n$ so that by telescoping the diagram between levels
$n$ and $m$, every vertex $v$ in $V_{n}$ is connected to every vertex $w$ in
$V_{m}$. It is a simple observation that $\left(  V,E\right)  $ is simple if
and only if every $AF\left(  V,E\right)  $-equivalence class is dense in
$X_{\left(  V,E\right)  }$.
%page 12
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\section{AF-equivalence relations}

We recall some terminology that we shall use. Let $R$ be an equivalence
relation on $X$ and let $A\subset X$. We will denote by $R|_{A}$ the
\textit{restriction} of $R$ to $A$, that is, $R|_{A}=R\cap(A\times A)$. We say
that $A$ is \textit{R-invariant} if $(x,y)\in R$ and $x\in A$, implies $y\in
A$. In other words, every $R$-equivalence class that meets $A$ lies entirely
inside $A$.

If $R^{\prime}$ is another equivalence relation on $X$, we say that
$R^{\prime}$ is a \textit{subequivalence relation} of $R$ if $R^{\prime
}\subset R$.

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\begin{Definition}
[Compact \'{e}tale equivalence relation (CEER)]Let $(R,\mathcal{T})$ be an
\'{e}tale equivalence relation on the locally compact space $X$, and let
$\Delta=\Delta_{X}\subset R$ be the diagonal in $X\times X$ (i.e., the unit
space of $R$). We say that $(R,\mathcal{T})$ is a \textit{compact \'{e}tale
equivalence relation} (CEER for short) if $R\setminus\Delta$ is a compact
subset of $R$. If $X$ itself is compact this is equivalent to say that $R$ is
compact, since $\Delta$ then is compact.
\end{Definition}

\begin{Proposition}
Let $(R,\mathcal{T})$ be a CEER on $X$. Then:

(i) $\mathcal{T}$ is the relative topology from $X\times X$.

(ii) $R$ is a closed subset of $X\times X$ (with the product topology), and
the quotient topology of the quotient space $X/R$ is Hausdorff.

(iii) $R$ is uniformly finite, that is, there is a natural number $N$ such
that the number $\#\left(  [x]_{R}\right)  $ of elements in each equivalence
class $[x]_{R}$ is at most $N$.
\end{Proposition}

\begin{proof}
(i) The (identity) map $r\times s:R\setminus\Delta\rightarrow X\times X$
%page 16  
is continuous and injective, and so is a homeomorphism onto its image, since
$R\setminus\Delta$ is compact. Now $\Delta$ is a clopen subset of $R$, and
its relative topology with respect to $\mathcal{T}$ and the product
topology of $X\times X$ coincide (Remark 2.2). Hence the assertion follows.
(ii) Clearly $R$ is a closed subset of $X\times X$ by (i). Now the quotient
map $q:X\to X/R$ is open since $r$ and $s$ are open maps. In fact, if $U$ is
an open subset of $X$, the set $q(U)=s(r^{-1}(U))$ is open in $X/R$. By [Ke;
Thm.\ 11], $X/R$ is Hausdorff.
(iii) Let $X_{1}=r(R\setminus\Delta)$. The subset $X_{1}\subset X$ is compact,
open (since $\Delta$ is clopen in $R$) and $R$-invariant, in the sense that
$x\in X_{1}$ implies $[x]_{R}\subset X_{1}$. Since $r^{-1}(x)$ is a closed,
discrete subset of $R$, it follows that $\#([x]_{R})<\infty$ for every $x\in
X_{1}$. Clearly
%page 17  
$\#([x]_{R})=1$ for $x\in X\setminus X_{1}$. To prove (iii) it is sufficient
to show that $\Phi:x\rightarrow\#([x]_{R})$ is a continuous map from $X$ into
$\N$. Now $\Phi$ is always lower semicontinuous. In fact, let $\Phi(x)=k$ for
some $x\in X$, and let $r^{-1}(x)=\{(x,y_{1}),\dots,(x,y_{k})\}$. By the
\'{e}taleness condition there exist in $\TT$ disjoint clopen neighbourhoods
$U^{(x,y_{1})}$,..., $U^{(x,y_{k})}$ of $(x,y_{1})$,..., $(x,y_{k})$,
respectively, such that the restriction of $r$ to each $U^{(x,y_{i})}$ is a
homeomorphism onto its open image. Hence there is an open neighbourhood
$U=\bigcap_{i=1}^{k}r(U^{(x,y_{i})})$ of $x$ such that $x^{\prime}\in U$
implies $\#([x^{\prime}]_{R})\geq k$. Consequently $\Phi$ is lower
semicontinuous at $x$.
To prove that $\Phi$ is upper semicontinuous at $x$, we need to use our
assumption that $R\setminus\Delta$ is compact. It is enough to establish upper
semicontinuity at a point $x$ in the open, compact subset $X_{1}$ of $X$. It
follows from our assumption that $R\cap(X_{1}\times X_{1})$ is a compact
subset of $R$. We retain the notation we used above, letting $\Phi(x)=k$, with
$r^{-1}(x)=\{(x,y_{1}),\,\dots,\,(x,y_{k})\}$, etc. Let $x_{n}\rightarrow x$,
where
%page 18 
$\{x_{n}\}$ is a sequence in $X_{1}$, and assume by contradiction that
$\Phi(x_{n})>k$ for every $n$. We choose $k+1$ distinct points $y_{n,1},\dots,y_{n,k+1}$ for each $n$ so that $r^{-1}(x_{n})=\{(x_{n},y_{n,1} ),\,\dots,\,(x_{n},y_{n,k+1})\}$, and we may assume that $(x_{n},y_{n,i})\in
U^{(x,y_{i})}$ for $i\in1,2,\dots,k$ and every $n$. Passing to a subsequence
we may assume by compactness that $(x_{n},y_{n,k_{+}1})\rightarrow(x^{\prime
},y^{\prime})$, where $(x^{\prime},y^{\prime})\in R\setminus\bigcup_{i=1} ^{k}U^{(x,y_{i})}$. Clearly $x^{\prime}=x$. But this yields a contradiction,
since
\[
(x,y^{\prime})\notin\{(x,y_{1}),\,\dots,\,(x,y_{k})\}=r^{-1}(x).
\]
This completes the proof.
\end{proof}

%page 19
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\begin{Remark}
It is not true that an \'{e}tale equivalence relation satisfying (ii) and
(iii) of Proposition 3.2 is CEER, even when $X$ is compact. In fact, let $X$
be the unit interval $[0,1]$, and let the graph of $R$ in $X\times X$ be the
union of the diagonal $\Delta$ and the graph of the function $f(t)=1-t$. The
equivalence classes have cardinality two, except the equivalence class of
$\frac{1}{2}$, which has cardinality one. Clearly $R$ is a closed (hence
compact) subset of $X\times X$. It is easy to see that $R$ may be given a
topology $\mathcal{T}$ so that $(R,\mathcal{T})$ is an \'{e}tale equivalence
relation on $X$. However, $\mathcal{T}$ is not the relative topology of
$X\times X$, and so is not CEER. [One can construct a similar example with $X$
equal to the Cantor set.]

One can show that an \'{e}tale equivalence relation $(R,\mathcal{T})$ on $X$
satisfying (i) of Proposition 3.2 is characterized by the property that the
map $r$ is a local homeomorphism of $R$, when the latter is given the relative
topology of $X\times X$.
\end{Remark}

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The \textit{disjoint union} of a finite set of \'{e}tale equivalence relations
is defined in the obvious way: Let $(R_{i},\mathcal{T}_{i})$ be an \'{e}tale
equivalence relation on $X_{i}$ for $i=1,2,\dots,k$, where the $X_{i}$'s are
disjoint. Let $X=\bigsqcup_{i=1}^{k}X_{i}$ be the disjoint union and let
$R=\bigsqcup_{i=1}^{k}R_{i}$ be the equivalence relation on $X$ defined in the
obvious way. Let $\mathcal{T}=\sqcup_{i=1}^{k}\mathcal{T}_{i} $ be the
disjoint union topology (also called the sum topology). Then $(R,\mathcal{T}%
)$\unskip\thinspace---\hskip.16667em\ignorespaces the \textit{disjoint union}
of $\{(R_{i},\mathcal{T}_{i})\}_{i=1}^{k}$\unskip\thinspace---\hskip
.16667em\ignorespaces is an \'{e}tale equivalence relation on $X$.

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The \textit{product} of two \'{e}tale equivalence relations is also defined in
the obvious way: Let $(R_{i},\mathcal{T}_{i})$ be an \'{e}tale equivalence
relation on $X_{i}$, for $i=1,2$. The \textit{product} $(R,\mathcal{T})$ of
$(R_{1},\mathcal{T}_{1})$ and $(R_{2},\mathcal{T}_{2})$ is an \'{e}tale
equivalence relation on $X=X_{1}\times X_{2}$ in an obvious way, where
$R=R_{1}\times R_{2}$ is given the product topology.\bigskip

The following lemma, besides giving the structure of CEERs on zero-dimensional
spaces, will be used below in connection with the construction of various
Bratteli diagrams that we shall associate to AF-equivalence relations. We also
point out the relevance of Remark 3.6 in this regard.

\begin{Lemma}
Let $(R,\mathcal{T})$ be a CEER \textnormal{(Definition 3.1)} on $X$, where $X$ is
a zero-dimensional space. Then $(R,\mathcal{T})$ is isomorphic to a finite
disjoint union of CEERs $\bigl\{(R_{i},\mathcal{T}_{i})\bigr\}_{i=1}^{k}$ of
type $m_{1},\dots,m_{k}$, respectively, where $X=\bigsqcup_{i=1}^{k}X_{i}$ and
$R_{i}$ is an equivalence relation on $X_{i}$ of type $m_{i}$ for
$i=1,\dots,k$. Specifically, $X_{i}$ is (homeomorphic to) $Y_{i}%
\times\{1,2,\dots,m_{i}\}$ for some natural number $m_{i}$, where $R_{i}$ is
the product of the \textit{trivial equivalence relation} on $\{1,2,\dots
,m_{i}\}$ (all points are equivalent) with the \textit{cotrivial equivalence
relation} (the identity relation) on $Y_{i}$. Furthermore, if $X$ is
non-compact, then $Y_{k}$ is the only non-compact set of the $Y_{i}$'s, and
$m_{k}=1$.

Conversely, an \'{e}tale equivalence relation of the type described is CEER.

The family of sets
\[
\mathcal{O}=\bigl\{Y_{i}\times\{j\}\mid1\leq j\leq m_{i}\hbox{ for }%
i=1,2,\dots,k\bigr\}%
\]
is a finite clopen partition of $X$. If $\mathcal{P}$ is an initially given
finite clopen partition of $X$, we may choose the $X_{i}$'s so that
$\mathcal{O}$ is finer than $\mathcal{P}$. Furthermore, $\mathcal{O}$ gives
rise to a clopen partition $\mathcal{O}^{\prime}$ of $R$ in a natural way,
namely $\mathcal{O}^{\prime}$ consists of the graphs of the local
homeomorphisms $\gamma_{lm}^{(i)}:Y_{i}\times\{l\}\rightarrow Y_{i}%
\times\{m\}$ where
\[
\gamma_{lm}^{(i)}\left(  \left(  y,l\right)  \right)  =\left(  y,m\right)
\text{ };\text{ }1\leq l,m\leq m_{i}\text{ and }i=1,...,k.
\]
(So the maps $\gamma_{lm}^{(i)}$ are of the form $s\circ r^{-1}$,
appropriately restricted.) If $\mathcal{P}^{\prime}$ is an initially given
finite clopen partition of $R$, we may choose the $X_{i}$'s so that
$\mathcal{O}^{\prime}$ is finer than $\mathcal{P}^{\prime}$.
\end{Lemma}

\begin{proof}
Let $X_{1}=r(R\setminus\Delta)$. Then $X_{1}$ is a compact, clopen subset
of $X$ that is $R$-invariant. Now the restriction $R|_{X\setminus X_{1}}$ of
$R$ to the invariant set $X\setminus X_{1}$ coincides with $\Delta
|_{X\setminus X_{1}}$. Hence $\left( R,\mathcal{T}\right) $ is the disjoint
union of $R|_{X_{1}}$ and $\Delta|_{X\setminus X_{1}}$ (with the relative
topologies), and so we may assume at the outset that $X$ is compact, and
hence a fortiori $\left( R,\mathcal{T}\right) $ is compact. For $x\in X$,
let $r^{-1}\left( x\right)
=\bigl\{(x,y_{1}),\dots,(x,y_{m})\bigr\}$\emdash
in other words, $[x]_{R}=\{y_{1},\dots,y_{m}\}$. (We may %page 23 assume that $y_{1}=x$.) We claim that we can find a clopen neighbourhood $ U^{(y_{i},y_{j})}$ of ${(y_{i},y_{j})}\in R$ for every $i,j$, so that the
restrictions of $r$ and $s$, respectively, to $U^{(y_{i},y_{j})}$ are
homeomorphisms onto their clopen images. Furthermore, $r\bigl( U^{(y_{i},y_{j})}\bigr)=U^{y_{i}}$ is independent of $j$ and is a clopen
neighbourhood of $y_{i}$, and the sets $U^{y_{1}},\dots,U^{y_{m}}$ are
disjoint. Likewise, $s\bigl(U^{(y_{i},y_{j})}\bigr)$ equals $U^{y_{j}}$ and
so is independent of $i$. In fact, by the \'{e}taleness condition it follows
easily that there exist disjoint clopen neighborhoods $U^{(y_{1},y_{j})}$,
for $j=1,\dots,m$, such that both $r$ and $s$, restricted to $U^{\left(
y_{1},y_{j}\right) }$, are homeomorphisms onto their respective clopen
images, with $r\left( U^{\left( y_{1},y_{j}\right) }\right) =$ $U^{y_{1}}$
for every $j$. Set $U^{y_{j}}=s\bigl(U^{(y_{1},y_{j})}\bigr)$. By
appropriately restricting the $r$ and $s$ maps we construct from this
homeomorphisms $\gamma_{ij}:U^{y_{i}}\rightarrow U^{y_{j}}$ for each $i,j$.
The graph of $\gamma_{ij}$ is $U^{(y_{i},y_{j})}$. We omit the details. Let
$\tilde{U}^{x}=\bigcup_{i=1}^{m}U^{y_{i}}$, and recall that $U^{y_{1}}%
=U^{x}$. It is tempting to restrict $R$ to $\tilde{U}^{x}$, that is, $R\cap( \tilde{U}^{x}\times\tilde{U}^{x})$, which is easily seen to be isomorphic
to the product of the cotrivial and the trivial equivalence relations on $U^{x}\times\{1,\dots,m\}$. However, $\tilde{U}^{x}$ need not be $R$-invariant, so we have to proceed more carefully. Therefore, let $ W^{x}=\bigcup_{i,j=1}^{m}U^{(y_{i},y_{j})}$. Then $\{W^{x}\mid x\in X\}$ is
a clopen covering of $R$, and so by compactness of $R$ there exists a finite
set $\{x_{1},\dots,x_{n}\}$ in $X$ such that $R=\bigcup_{i=1}^{n}W^{x_{i}}$. Observe that $r(W^{x_{i}})=\tilde{U}^{x_{i}}$ for $i=1,\dots,n$, where we
retain the notation introduced above. Assume we have ordered the $x_{i}$'s
so that
\begin{equation*}
\#\bigl([x_{1}]_{R}\bigr)\geq\#\bigl([x_{2}]_{R}\bigr)\geq\cdots\geq\# \bigl([x_{n}]_{R}\bigr).
\end{equation*}
Now
\begin{equation*}
V^{x_{1}}=\tilde{U}^{x_{1}},V^{x_{2}}=\tilde{U}^{x_{2}}\setminus\tilde{U} ^{x_{1}},\dots,V^{x_{n}}=\tilde{U}^{x_{n}}\setminus\bigcup_{i=1}^{n-1} \tilde{U}^{x_{i}}
\end{equation*}
is a clopen partition of $X$, and one verifies that these sets are $R$ -invariant. Hence $(R,\mathcal{T})$ is isomorphic to the disjoint union of $R
$ restricted to those sets that are non-empty (with the relative
topologies). Each of these restrictions is of the desired form. We omit the
details. Clearly an \'{e}tale equivalence relation of the described type is
CEER. To ensure that the associated partition $\mathcal{O}$ of $X$ is finer
than the given $\mathcal{P}$, simply choose for each $x\in X$ the $U^{x}$ so
small that each $U^{y_{i}}$ that occurs in $\tilde{U}^{x}= \bigcup_{i=1}^{m}U^{y_{i}}$ is contained in some element of $\mathcal{P}$,
where we again use the notation above. Similarly we may choose $U^{x}$ so
small that the graph $U^{\left( y_{i},y_{j}\right) }$ of the local
homeomorphism $\gamma_{ij}:U^{y_{i}}\rightarrow U^{j}$ is contained in some
element of $\mathcal{P}^{\prime}$ for each $1\leq i,j\leq m$. This will
ensure that $\mathcal{O}^{\prime}$ is finer than $\mathcal{P}^{\prime}$.
\end{proof}

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\begin{Corollary}
Let $\left(  R,\mathcal{T}\right)  $ be a CEER on the zero-dimensional space
$X $. If $\mathcal{P}$ and $\mathcal{P}^{\prime}$ are finite clopen partitions
of $X$ and $R$, respectively, there exist clopen partitions $\mathcal{O}$ and
$\mathcal{O}^{\prime}$ of $X$ and $R$, respectively, as described in Lemma
3.4, so that $\mathcal{O}$ is finer than $\mathcal{P}$ and $\mathcal{O}%
^{\prime}$ is finer than $\mathcal{P}^{\prime}$. In fact, $\mathcal{O}%
=\Delta\cap\mathcal{O}^{\prime},$ where $\Delta=\Delta_{X}$ is the diagonal in
$X\times X.$ (We make the obvious identification between $X$ and $\Delta$.)
\end{Corollary}

\begin{Remark}
It is instructive to draw a picture to illustrate the content of the above
lemma: $X$ can be composed into $n$ (disjoint) compact, clopen towers\unskip
\thinspace---\hskip.16667em\ignorespaces the $k$-th tower being $V^{x_{k}}$ of
height $\#([x_{k}]_{R})$\unskip\thinspace---\hskip.16667em\ignorespaces with
clopen floors, and possibly one non-compact tower of height 1. See Figure 2
(we assume that the various towers are non-empty). The equivalence classes of
$R$ are formed by the sets of points lying vertically above or below one
another in each tower. The clopen partition $\mathcal{O}$ of $X$ is the set
consisting of the floors of the various towers. The partition $\mathcal{O}%
^{\prime}$ of $R$ can also be easily described in terms of the towers in
Figure 2. In fact, for each tower there is between every pair of floors a
local homeomorphism of the form $s\circ r^{-1}$(appropriately restricted). The
graphs of these maps make up $\mathcal{O}^{\prime}$. With this picture it is
obvious how we may identify $\mathcal{O}$ with $\mathcal{O}^{\prime}\cap
\Delta_{X},$ where $\Delta_{X}$ is the diagonal in $X\times X$. In Figure 2 we
have also shown the ''contribution'' of one of the towers, say $V^{x_{i}}$ of
height three (i.e. $\#\left(  [x_{i}]_{R}\right)  =3$), to the partition
$\mathcal{O}^{\prime}$ of $R$ --- we have drawn the graphs of the local
homeomorphism associated to $V^{x_{i}}$ in boldface.

Notice that $\mathcal{O}^{\prime}$ is a very special partition of $R$. In
fact, $\mathcal{O}^{\prime}$ has a natural (abstract) principal groupoid
structure, with unit space identifiable with $\mathcal{O}$, that we shall now
describe. If we define $U\cdot V$ for two subsets $U,V$ of $R$ to be
\[
U\cdot V=\{\left(  x,z\right)  |\left(  x,y\right)  \in U,\left(  y,z\right)
\in V\text{ for some }y\in X\}
\]
then we can list the properties of $\mathcal{O}^{\prime}$ as follows:

(i) $\mathcal{O}^{\prime}$ is a finite clopen partition of $R$ finer than
$\{\Delta,R\setminus\Delta\}$.

(ii) For all $U\in\mathcal{O}^{\prime}$, the maps $r,s:U\rightarrow X$ are
local homeomorphisms, and if $U\in\mathcal{O}^{\prime}\cap\left(
R\setminus\Delta\right)  $, then $r\left(  U\right)  \cap s\left(  U\right)
=\emptyset$.

(iii) For all $U,V\in\mathcal{O}^{\prime}$, we have $U\cdot V=\emptyset$ or
$U\cdot V\in\mathcal{O}^{\prime}$. Also, $U^{-1}\left(  =\{\left(  y,x\right)
|\left(  x,y\right)  \in U\}\right)  $ is in $\mathcal{O}^{\prime}$ for every
$U $ in $\mathcal{O}^{\prime}$.

(iv) With $\mathcal{O}^{\prime\left(  2\right)  }=\{\left(  U,V\right)
|U,V\in\mathcal{O}^{\prime},U\cdot V\neq\emptyset\}$, define $\left(
U,V\right)  \in\mathcal{O}^{\prime\left(  2\right)  }\longrightarrow U\cdot
V\in\mathcal{O}^{\prime}$. Then $\mathcal{O}^{\prime}$ becomes a principal
groupoid with unit space equal to $\{U\in\mathcal{O}^{\prime}|U\subset
\Delta\}$, which clearly may be identified with $\mathcal{O}$.

We will call $\mathcal{O}^{\prime}$ a \textit{groupoid partition} of $\left(
R,\mathcal{T}\right)  ,$ or simply $R$. Notice that if we define the
equivalence relation $\sim_{\mathcal{O}^{\prime}}$ on $\mathcal{O}$ by
$A\sim_{\mathcal{O}^{\prime}}B$ if there exists $U\in\mathcal{O}^{\prime}$ so
that $U^{-1}\cdot U=A,$ $U\cdot U^{-1}=B$, then the equivalence classes,
denoted $[$ $]_{\mathcal{O}^{\prime}},$ are exactly the towers in Figure 2.%

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\end{Remark}

The compact \'{e}tale equivalence relations (CEER) are the building blocks
with which we will define an \textit{AF-equivalence relations}. (AF stands for
``approximately finite-dimensional'', and we refer to [Re1] and [GPS] for
further explanation of the terminology.)

\begin{Definition}
[AF-equivalence relation]Let $\{(R_{n},\mathcal{T}_{n})\}_{n=1}^{\infty}$ be a
sequence of CEERs on a zero-dimensional (second countable, locally compact
Hausdorff) space $X$, so that $R_{n}$ is an open subequivalence relation of
$R_{n+1}$, i.e. $R_{n}\subset R_{n+1}$ and $R_{n}\in\mathcal{T}_{n+1}$ for
every $n$. (Note that this implies that $R_{n}$ is a clopen subset of
$R_{n+1}$, since $R_{n}\setminus\Delta$ is compact.) Let $(R,\mathcal{T})$ be
the \textit{inductive limit} of $\{\left(  R_{n},\mathcal{T}_{n}\right)  \}$
with the \textit{inductive limit topology} $\mathcal{T}$, i.e., $R=\bigcup
_{n=1}^{\infty}R_{n}$ and $U\in\mathcal{T}$ if and only if $U\cap R_{n}%
\in\mathcal{T}_{n}$ for every $n$. We say that $(R,\mathcal{T})$ is an
\textit{AF-equivalence relation} on $X$, and we use the notation
$(R,\mathcal{T})=\lim\limits_{\longrightarrow}(R_{n},\mathcal{T}_{n}%
).$\newline We say that $\left(  R,\mathcal{T}\right)  $ is \textit{minimal}
if each equivalence class $[x]_{R},$ $x\in X,$ is dense in $X$.
\end{Definition}

\begin{Theorem}
Let $G$ be a countable group acting minimally and freely on the Cantor set $X
$, and let $\left(  R_{G},\mathcal{T}\right)  $ be the associated \'{e}tale
equivalence relation (cf. Example 2.7(i)). Then $\left(  R_{G},\mathcal{T}%
\right)  $ is $AF$ if and only if $G$ is locally finite.
\end{Theorem}

\begin{proof}
Assume $G$ is locally finite, and let $G_{1}\subset G_{2}\subset\cdots
\subset G=\bigcup\limits_{n}G_{n},$ where $G_{n}$ is a finite group for
every $n$. It is easy to see that $\left( R_{G},\mathcal{T}\right) $ $ =\lim\limits_{\longrightarrow}(R_{G_{n}},\mathcal{T}_{n}),$ where $\left(
R_{G_{n}},\mathcal{T}_{n}\right) $ is obviously CEER. In fact, $R_{G_{n}}$
may be identified with $X\times G_{n},$ and $\mathcal{T}_{n}$ is the product
topology. Hence $\left( R_{G},\mathcal{T}\right) $ is an $AF$-equivalence
relation.
Conversely, if $\left( R_{G},\mathcal{T}\right)
=\lim\limits_{\longrightarrow}(R_{n},\mathcal{T}_{n})$ is an $AF$-equivalence relation, then --- identifying $R_{G}$ with $X\times G$ --- let
$F$ be a finite subset of $G.$ To show that $G$ is locally finite it
suffices to show that the subgroup generated by $F$ is finite. Since $X\times F$ is compact, it is contained in some $R_{n}.$ Define $H=\{g\in
G$ $|$ $X\times\{g\}\subset R_{n}\}.$ Clearly $F\subset H,$ and since $R_{n}$
is an equivalence relation it follows that $H$ is a subgroup of $G.$ As $R_{n}$ is compact, $H$ is finite, and so the subgroup generated by $F$ is
finite.
\end{proof}

It is straightforward to verify that an AF-equivalence relation is an
\'{e}tale equivalence relation. Furthermore, one verifies that the \'{e}tale
equivalence relation of Example 2.7(ii) is an $AF$-equivalence relation. The
following theorem is the converse result, as alluded to in Example 2.7(ii).

\begin{Theorem}
Let $(R,\mathcal{T})=\lim\limits_{\longrightarrow}(R_{n},\mathcal{T}_{n})$ be
an AF-equivalence relation on $X$. There exists a Bratteli diagram $(V,E)$
such that $(R,\mathcal{T})$ is isomorphic to the AF-equivalence relation
$AF\left(  V,E\right)  $ associated to $(V,E)$.

If $X$ is compact, the Bratteli diagram $(V,E)$ may be chosen to be standard.
Furthermore, $\left(  V,E\right)  $ is simple if and only if $\left(
R,\mathcal{T}\right)  $ is minimal.
\end{Theorem}

\begin{proof}
Choose an increasing sequence $\{K_{m}\}_{m=1}^{\infty}$ of compact, clopen
subsets of $X$, such that $X=\bigcup_{m=1}^{\infty}K_{m}$ (if $X$ is itself
compact, we let $K_{1}=K_{2}=\cdots=X$), and such that $r(R_{m}\setminus
\Delta)\subset K_{m}$ for each $m$. We will choose and increasing sequence $\mathcal{P}_{1}\prec\mathcal{P}_{2}\prec\cdots$ of finite clopen
partitions of $X$ whose union generates the topology of $X$, and which is
related to the sequence $\{K_{m}\}$ in a way we shall describe. In fact, if $\mathcal{P}_{m}=\{A_{1},\dots,A_{n_{m}}\}$, we require that $K_{m}=\bigcup_{i=1}^{n_{m}-1}A_{i}$, and hence $A_{n_{m}}=X\setminus K_{m}$.
(In particular, if $X$ is compact, $A_{n_{m}}=\emptyset$. Henceforth we
will assume that $X$ is not compact\emdash and so $A_{n_{m}}\neq\emptyset$
for every $m$\emdash the compact case being of course a simplified version.
We will also assume that $K_{m}\subset K_{m+1}$ for every $m$, so that $A_{n_{m}}\setminus A_{n_{m+1}}\neq\emptyset$ for every $m$.) It is easily
seen that $K_{m}$ and $X\setminus K_{m}$ are $R_{m}$-invariant sets, so $(R_{m},\mathcal{T}_{m})$ is the disjoint union of the restrictions of $R_{m}$
to $K_{m}$ and $X\setminus K_{m}$, respectively. (It is to be understood
that when we restrict we take the relative topology.) Now $R_{m}|_{X\setminus K_{m}}$ is equal to $\Delta|_{X\setminus K_{m}}$. By
Lemma 3.4 we can decompose $K_{m}$ into a finite number of disjoint towers,
with $R_{m}|_{K_{m}}$ being as described in Lemma 3.4 and in Remark 3.6. We
will use the terminology suggested by Remark 3.6, including the terms
``tower'', ``floor'', ``height'' and ''groupoid partition''. 
%page 29 
Keeping the notation above, as well as then one used in Lemma 3.4 and Remark
3.6, we now describe how to construct the Bratteli diagram $(V,E)$. First,
let $V_{0}=\{v_{0}\}$ be a one-point set. We will do the next two steps in
the construction, which should make it clear how one proceeds. \newline
\ \textbf{Step 1. }Applying Lemma 3.4 to $\left( R_{1},\mathcal{T}_{1}\right) $, we get a groupoid partition $\mathcal{O}_{1}^{\prime}$ of $R_{1}$, so that $R_{1}|_{K_{1}}$ is represented (as explained in Remark 3.6)
by $l_{1}$ (compact) towers of heights $h_{1},...,h_{l_{1}}$ and $R_{1}|_{X\setminus K_{1}}=\Delta|_{X\setminus K_{1}}$ is a single
non-compact tower of height one, so that the associated clopen partition $\mathcal{O}_{1}$ of $X$ is finer than $\mathcal{P}_{1}$ and contains $A_{n_{1}}=X\setminus K_{1}.$ We let $V_{1}=\{v_{1},...,v_{l_{1}},v_{l_{1}+1}\}$, where $v_{i}$ corresponds to the i'th tower of height $h_{i}$ when $1\leq i\leq l_{1}$, and $v_{l_{1}+1}$ corresponds to the non-compact tower $A_{n_{1}}$ of height one. The number of edges between $v_{0}$ and $v_{i}$ is
the height $h_{i}$ of the tower $v_{i}$ for $1\leq i\leq l_{1}$. There are
no edges between $v_{0}$ and $v_{l_{1}+1}$ (so $v_{l_{1}+1}$ will be a
source in $V$). This defines $E_{1}$. \newline
\ \textbf{Step 2.} Let $\mathcal{\tilde{P}}_{2}=\mathcal{P}_{2}\vee\mathcal{O}_{1}$ be the join of $\mathcal{P}_{2}$ and $\mathcal{O}_{1}$, and so $\mathcal{\tilde{P}}_{2}$ is a finite clopen partition of $X$. Applying
Corollary 3.5 to $\left( R_{2},\mathcal{T}_{2}\right) $ we get a groupoid
partition $\mathcal{O}_{2}^{\prime}$ of $R_{2}$ that is finer than $\{\mathcal{O}_{1}^{\prime},R_{2}\setminus R_{1}\}$, and so that the
associated clopen partition $\mathcal{O}_{2}$ of $X$ is finer than $\mathcal
{\tilde{P}}_{2}$ and contains $A_{n_{2}}=X\setminus K_{2}$. Hence $R|_{K_{2}}$
will be represented by $l_{2}$ (compact) towers of heights $\tilde{h}_{1},...,\tilde{h}_{l_{2}}$, and $R|_{X\setminus K_{2}}=\Delta|_{X\setminus
K_{2}}$ a single non-compact tower of height one. We let $V_{2}=\{\tilde{v}_{1},...,\tilde{v}_{l_{2}},\tilde{v}_{l_{2}+1}\}$, where $\tilde{v}_{j}$
corresponds to the j'th tower of height $\tilde{h}_{j}$ when $1\leq j\leq
l_{2}$, and $\tilde{v}_{l_{2}+1}$ corresponds to the non-compact tower $A_{n_{2}}$ of height one. Let $\mathcal{O}_{2}^{\prime\prime}=\{U\in
\mathcal{O}_{2}^{\prime}|U\subset R_{1}\}$. It is a simple observation that
$\mathcal{O}_{2}^{\prime\prime}$ is a groupoid partition of $R_{1}$ that
is finer than $\mathcal{O}_{1}^{\prime}$, and with unit space equal to $\mathcal{O}_{2}$. The set $E_{2}$ of edges between $V_{1}$ and $V_{2}$ is
labelled by $\mathcal{O}_{2}\setminus\{A_{n_{2}}\}$ modulo $\mathcal{O}_{2}^{\prime\prime}$, i.e. $E_{2}$ consists of $\sim_{\mathcal{O}_{2}^{\prime\prime}}$ equivalence classes (denoted by $[$ $]_{\mathcal{O}_{2}^{\prime\prime}}$) of $\mathcal{O}_{2}\setminus\{A_{n_{2}}\}$.
Specifically, if $A,B\in\mathcal{O}_{2}\setminus\{A_{n_{2}}\},$ we have $A$
$\sim_{\mathcal{O}_{2}^{\prime\prime}}B$ if there exists $U\in\mathcal{O}_{2}^{\prime\prime}$ so that $U^{-1}\cdot U=A$, $U\cdot U^{-1}=B$. As we
explained in Remark 3.6, the vertex set $V_{j}$ ( $j=1,2$) may be identified
with the $\sim_{\mathcal{O}_{j}^{\prime}}$ equivalence classes $[$ $]_{\mathcal{O}_{j}^{\prime}}$ of $\mathcal{O}_{j}$. Doing this we may write
down the source and range maps $i:E_{2}\rightarrow V_{1},f:E_{2}\rightarrow
V_{2},$ associated to the Bratteli diagram. In fact, if $[A]_{\mathcal{O}_{2}^{\prime\prime}}\in E_{2}$, where $A\in\mathcal{O}_{2}\setminus
\{A_{n_{2}}\}$, then $f\left( [A]_{\mathcal{O}_{2}^{\prime\prime}}\right)
=[A]_{\mathcal{O}_{2}^{\prime}}$, $i\left( [A]_{\mathcal{O}_{2}^{\prime
\prime}}\right) =[B]_{\mathcal{O}_{1}^{\prime}}$, where $B$ is the unique
element of $\mathcal{O}_{1}$ such that $A\subset B$. The vertex $v_{l_{2}+1}\in V_{2}$ (corresponding to the tower $A_{n_{2}}$) will be a
source in $V$. \textbf{[}Using the pictorial presentation of Figure 2, we
can give a more intuitive explanation of our construction of the edge set $E_{2}$ between $V_{1}$ and $V_{2}$. In fact, for $1\leq j\leq l_{2}$, let $x$
be any point in the tower $\tilde{v}_{j}\in V_{2}$. The $R_{2}$-equivalence
class $[x]_{R_{2}}$ of $x$ consists of the $\tilde{h}_{j}$ vertically lying
points of $\tilde{v}_{j}$ including $x$. Now $[x]_{R_{2}}$ is a \ disjoint
union of distinct $R_{1}$-equivalence classes. If $[x]_{R_{2}}$ contains \ $h_{ij}$ distinct $R_{1}$-equivalence classes ''belonging'' to the tower $v_{i}\in V_{1}$, we connect $v_{i}$ to $\tilde{v}_{j}$ by $h_{ij}$ edges.
There are no edges between $\tilde{v}_{l_{2}+1}$ and any vertex in $V_{1}$.\textbf{]} Continuing in the same manner we construct the Bratteli diagram $\left( V,E\right) $. There is an obvious map $F$ from $X$ to the path space
associated to $\left( V,E\right) $. In fact, if $x\in X$ let \ $v_{i}\in
V_{n-1}$ be the tower at level $n-1$ that contains $x$. Likewise let $\tilde
{v}_{j}\in V_{n}$ be the tower at level $n$ that contains $x$, and assume
that $x$ lies in floor $A$($\in\mathcal{O}_{n}$) of $\tilde{v}_{j}$. The
n'th edge $e_{n}\in E_{n}$ of $F\left( x\right) =\left(
e_{1},e_{2},...\right) $ is then $[A]_{\mathcal{O}_{n}^{\prime\prime}}$.
(We use similar notation at level $n$ as we used above at level two.) One
verifies that $F$ is an homeomorphism between $X$ and the path space
associated to $\left( V,E\right) $. Furthermore, it is straightforward to
show that the map $F$ establishes an isomorphism between $\left( R,\mathcal
{T}\right) $ and the AF-equivalence relation associated to $\left( V,E\right
) $. In fact, $\left( x,y\right) \in R$ and $\left( x,y\right) \notin R_{n-1}$,
$\left( x,y\right) \in R_{n}$ if and only if $F\left( x\right) $ and $F\left( y\right) $ become cofinal from level $n$ on. We omit the details.
\\
The two last assertions of the theorem are immediate consequences of our
construction and the comments we made in Example 2.7 (ii).
\end{proof}

\begin{Remark}
Even though the Bratteli diagram model for $(R,\mathcal{T})$ is not unique, it
is true that two such models give rise to isomorphic dimension groups, and so
the diagrams themselves are related by a telescoping procedure, cf.\ Example
2.7(ii) and Lemma 4.13.\newline (A relevant reference for Theorem 3.9 is [Re2;
Theorem 3.1].)
\end{Remark}

We now consider the situation that $\left(  V,E\right)  $ is a \ standard
Bratteli diagram and $\left(  W,F\right)  $ is a (standard) subdiagram, i.e.
$W\subset V,$ $F\subset E$ and $W_{0}=V_{0}=\{v_{0}\}.$ It is easy to see that
$X_{\left(  W,F\right)  }$ is a closed subset of $X_{\left(  V,E\right)  }.$
It is also clear that $AF\left(  W,F\right)  $ is the intersection of
\newline $AF$ $\left(  V,E\right)  $ with $X_{(W,F)}\times X_{(W,F)}$.
Moreover, it is easy to check that the relative topology on $AF(W,F)$ coming
from $AF(V,E)$ agrees with the usual topology on $AF(W,F)$. We have the
following realization theorem for this situation.

\begin{Theorem}
Let $\left(  R,\mathcal{T}\right)  $ be an $AF$-equivalence relation on the
compact (zero-dimensional) space $X$. Suppose that $Z$ is a closed subset of
$X$ such that $R\cap(Z\times Z),$ with the relative topology from $\left(
R,\mathcal{T}\right)  ,$ is an \'{e}tale equivalence relation on $Z$. Then
there exists a Bratteli diagram $(V,E)$, a subdiagram $(W,F)$ and a
homeomorphism $h:X_{(V,E)}\rightarrow X$ such that

\begin{enumerate}
\item [(i)]$h$ implements an isomorphism between $AF(V,E)$ and $\left(
R,\mathcal{T}\right)  $.

\item[(ii)] $h(X_{(W,F)})=Z$ and the restriction of $h$ to $X_{(W,F)}$
implements on isomorphism between $AF(W,F)$ and $R\cap(Z\times Z)$.
\end{enumerate}
\end{Theorem}

\begin{proof}
Let us take $\left( R^{\prime},\mathcal{T}^{\prime}\right) $, a compact
open subequivalence relation of $\left( R,\mathcal{T}\right) ,$ where $\mathcal{T}^{\prime}$ is the relative topology. As noted in Proposition
3.2, the topology $\mathcal{T}^{\prime}$ on $R^{\prime}$ coincides with
the relative topology from $X\times X$. We note that $R^{\prime}\cap\left(
Z\times Z\right) $ is a clopen, compact subequivalence relation of $R\cap
\left( Z\times Z\right) $ and therefore, in particular, is \'{e}tale.
For each pair $(x,y)$ in $R^{\prime}$, we choose a clopen neighbourhood $U$
in $\mathcal{T}^{\prime}$ (hence in $\mathcal{T}$) as follows. First, if $(x,y)$ is not in $Z\times Z$, we use the fact that $Z$, and hence $Z\times
Z$, is closed to find a clopen neighbourhood $U$ which is disjoint from $Z\times Z$. If $(x,y)$ is in $Z\times Z$, then we use the fact that $R^{\prime}\cap\left( Z\times Z\right) $ is \'{e}tale to find a clopen
subset $U\subset R^{\prime}$ such that $r:U\cap\left( Z\times Z\right)
\rightarrow r(U)\cap Z$ and $s:U\cap\left( Z\times Z\right) \rightarrow
s(U)\cap Z$ are both homeomorphisms. [To achieve this, let $V\in\mathcal{T}^{\prime}$ be a clopen neighbourhood of $\left( x,y\right) $ such that $r:V\cap\left( Z\times Z\right) \rightarrow A\cap Z$ and $s:V\cap\left(
Z\times Z\right) \rightarrow B\cap Z$ are homeomorphisms, where $A$ and $B$
are clopen subsets of $X.$ Choose $U$ to be $V\cap r^{-1}\left( A\right)
\cap s^{-1}\left( B\right) .$] Moreover, we can choose $U$ sufficiently
small so that $r,s$ are also local homeomorphisms from $U$ to $r(U)$ and $s(U)$, respectively.
In this way, we cover $R^{\prime}$ with clopen sets. We then extract a
finite subcover and then choose a groupoid partition of $R^{\prime}$ which
is finer than this subcover, such that each member of the partition
satisfies the properties just listed. (Cf. the proof of Lemma 3.4 and Remark
3.6.) In the end, we obtain a groupoid partition $\mathcal{O}^{\prime}$ of $R^{\prime}$ such that
\begin{equation*}
\mathcal{O}^{\prime}\mid Z=\{U\cap(Z\times Z)\mid U\in\mathcal{O}^{\prime
},U\cap(Z\times Z)\neq\emptyset\}
\end{equation*}
is a groupoid partition of $R^{\prime}\cap(Z\times Z)$. Moreover, it is
clear from the construction that these may both be made finer than any pair
of pre-assigned clopen partitions of $R^{\prime}$ and $R^{\prime}\cap
(Z\times Z)$. We also note that it is easy to verify from the construction,
any element $U$ of $\mathcal{O}^{\prime}$ meets $Z\times Z$ if and only if $r(U)$ and $s(U)$ both meet $Z$. (Referring to Figure 2 as representing $\mathcal{O}^{\prime}$, this means that if $Z$ intersects any two floors in
the same tower, these intersections project onto each other.)
Now we consider a sequence
\begin{equation*}
R_{1}\subset R_{2}\subset R_{3}\subset\cdots
\end{equation*}
of CEERs in $R,$ with union $R,$ each open in the next, so that $\left( R,\mathcal{T}\right) $ is the inductive limit. For each $n,$ we construct a
groupoid partition $\mathcal{O}_{n}^{\prime}$ of $R_{n}$ with the
properties as above. We do this so that the restriction of $\mathcal{O}_{n+1}^{\prime}$ to $R_{n}$ is finer than $\mathcal{O}_{n}^{\prime},$ and
so that the restriction of $\mathcal{O}_{n+1}^{\prime}\mid Z$ to $R_{n}\cap
\left( Z\times Z\right) $ is finer than $\mathcal{O}_{n}^{\prime}\mid Z.$
This sequence is also chosen so as to generate the respective topologies of $R$ and $R\cap\left( Z\times Z\right) .$ In the proof of Theorem 3.9, it was
shown how $\mathcal{O}_{n}^{\prime}$ defines a Bratteli diagram, $\left(
V,E\right) .$ We now describe the subdiagram $\left( W,F\right) .$ Let $\Delta_{X}$ and $\Delta_{Z}$ denote the diagonals in $X\times X$ and $Z\times Z,$ respectively (which we identify with $X$ and $Z,$ respectively).
The vertices of $V_{n}$ correspond to towers in $\mathcal{O}_{n}^{\prime};$
that is, equivalence classes of sets in $\Delta_{X}\cap\mathcal{O}_{n}^{\prime}$, modulo $\mathcal{O}_{n}^{\prime}.$ The vertices of $W_{n}$
are those classes having a representative which meets $\Delta_{Z}.$ (Again
referring to Figure 2, the vertices of $V_{n}$ correspond to the towers,
while the vertices of $W_{n}$ correspond to those towers that meet $Z$.) The
edges of $E_{n}$ are the equivalence classes of sets in $\Delta_{X}\cap
\mathcal{O}_{n}^{\prime}$, modulo $\mathcal{O}_{n}^{\prime\prime}=\{U\in
\mathcal{O}_{n}^{\prime}|U\subset R_{n-1}\}.$ (For an interpretation of
this in terms of Figure 2, we refer to the remarks made in the proof of
Theorem 3.9.) We let $F_{n}$ be those equivalence classes having a
representative which meets $\Delta_{Z}.$ It follows from the properties of
the partitions $\mathcal{O}_{n}^{\prime}$ described above that $\left(
V,E\right) $ and $\left( W,F\right) $ satisfy the desired conclusion. We
leave the details to the reader.
\end{proof}

The inductive limit $(R,\mathcal{T})$ of a sequence of \'{e}tale equivalence
relations $\{(R_{n},\mathcal{T}_{n})\}_{n=1}^{\infty}$ on $X$ (notation:
$(R,\mathcal{T})=\lim\limits_{\longrightarrow}(R_{n},\mathcal{T}_{n})$) is
defined as in Definition 3.7, with the obvious modifications. It is an easy
exercise to show that $(R,\mathcal{T})$ is an \'{e}tale equivalence relation.
The following proposition is a stabilization result with respect to
AF-equivalence relations.

\begin{Proposition}
(i) Let $(R,\mathcal{T})=\lim\limits_{\longrightarrow}(R_{n},\mathcal{T}_{n})
$ be an inductive limit of a sequence $\{(R_{n},\mathcal{T}_{n})\}$ of
AF-equivalence relations on $X$. Then $(R,\mathcal{T})$ is an AF-equivalence
relation on $X$.

(ii) Let $(R,\mathcal{T})$ be an AF-equivalence relation on $X$, and let
$R^{\prime}\subset R$ be a subequivalence relation which is open, i.e.,
$R^{\prime}\in\mathcal{T}$. Then $(R^{\prime},\mathcal{T}^{\prime})$ is an
AF-equivalence relation, where $\mathcal{T}^{\prime}$ is the relative topology
of $R$.
\end{Proposition}

\begin{proof}
(i). For each $n$, let $(R_{n},\mathcal{T}_{n})=\lim\limits_{\longrightarrow
}(R_{n,k},\mathcal{T}_{n,k})$, where $(R_{n,k},\mathcal{T}_{n,k})$ is a CEER
(Definition 3.1) on $X$ for every $k$. So we have
\begin{align*}
R_{1}& \subset R_{2}\subset R_{3}\subset...\subset
R=\bigcup\limits_{n=1}^{\infty}R_{n}\text{,} \\
R_{n,1}& \subset R_{n,2}\subset R_{n,3}\subset...\subset
R_{n}=\bigcup\limits_{k=1}^{\infty}R_{n,k}\text{ };n=1,2,...,
\end{align*}
where each set is open in the next one containing it with respect to the
relevant topology. Define $R_{1}^{\prime}=R_{{1,1}}$. Now $R_{1,2}\subset
R_{1}\subset R_{2}=\bigcup_{k=1}^{\infty}R_{2,k}$, and so we may choose $k_{2}\geq2$ so large that $R_{1,2}\subset R_{2,k_{2}}$. Define $R_{2}^{\prime}=R_{2,k_{2}}$. Continuing in this manner we get an ascending
sequence $\{R_{n}^{\prime}\}_{n=1}^{\infty}$ of equivalence relations on $X
$ so that $R_{n}^{\prime}$ contains all $R_{l,m}$'s, provided $l$ and $m$
are at most $n$. Clearly $R=\bigcup_{n=1}^{\infty}R_{n}^{\prime}$, and we
claim that $(R,\mathcal{T})=\lim\limits_{\longrightarrow}(R_{n}^{\prime},\mathcal{T}_{n}^{\prime})$, where $(R_{n}^{\prime},\mathcal{T}_{n}^{\prime
})=(R_{n,k_{n}},\mathcal{T}_{n,k_{n}})$ is a CEER for each $n$. This will
finish the proof of (i). (Note that $R_{n}^{\prime}$ is open in $R_{n+1}^{\prime}$, i.e. $R_{n}^{\prime}\in\mathcal{T}_{n+1}^{\prime}$
for every $n$.) Now, if $U\in\mathcal{T}$ then $U\cap R_{n}\in\mathcal{T}_{n}$ for every $n$. Hence $U\cap R_{n}^{\prime}=U\cap R_{n,k_{n}}=(U\cap
R_{n})\cap R_{n,k_{n}}\in\mathcal{T}_{n,k_{n}}=\mathcal{T}_{n}^{\prime}$
for every $n$, and so $U\in\tilde{\mathcal{T}}$, where by definition $(R,\tilde{\mathcal{T}})$ equals $\lim\limits_{\longrightarrow}R_{n}^{\prime},\mathcal{T}_{n}^{\prime})$. Hence $\mathcal{T}\subset\tilde{\mathcal{T}}$.
Conversely, assume $U\in\tilde{\mathcal{T}}$. Then
\begin{equation*}
U\cap R_{n}^{\prime}=U\cap R_{n,k_{n}}\in\mathcal{T}_{n}^{\prime}=\mathcal{T}_{n,k_{n}}=\mathcal{T}_{n}\cap R_{n,k_{n}}\subset\mathcal{T}_{n}
\end{equation*}
for every $n$. To show that $U\in\mathcal{T}$ we must show that $U\cap
R_{m}\in\mathcal{T}_{m}$ for any given $m$. This is again equivalent to
show that
\begin{equation*}
U\cap R_{m,l}=(U\cap R_{m})\cap R_{m,l}\in\mathcal{T}_{m,l}
\end{equation*}
for any $l$. Now choose $n\geq\max\{m,l\}$. Then $R_{m,l}\subset
R_{n}^{\prime}$ and, since clearly $R_{m,l}\subset R_{m}$, we get $U\cap
R_{m,l}=((U\cap R_{n}^{\prime})\cap R_{m})\cap R_{m,l}\in\mathcal{T}_{m,l}$
from the fact that $U\cap R_{n}^{\prime}\in\mathcal{T}_{n}$ and $R_{m,l}$
is open in $R_{m}$, which again is open in $R_{n}$. This proves that $U\in
\mathcal{T}$, and so $\mathcal{\tilde{T}}\subset\mathcal{T}$, which
finishes the proof of (i).
(ii). $R^{\prime}$ being an open subequivalence relation of $R$ implies that
$(R^{\prime},\mathcal{T}^{\prime})$ is an \'{e}tale equivalence relation,
where $\mathcal{T}^{\prime}$ is the relative topology of $R$\emdash a fact
that is easily shown. Let $(R,\mathcal{T})=\lim\limits_{\longrightarrow
}(R_{n},\mathcal{T}_{n})$, where $(R_{n},\mathcal{T}_{n})$ is a CEER on $X$
for every $n$. It is easily verified that $(R^{\prime},\mathcal{T}^{\prime
})=\lim\limits_{\longrightarrow}(R_{n}^{\prime},\mathcal{T}_{n}^{\prime})$, where $R_{n}^{\prime}=R^{\prime}\cap R_{n}$ is an open subequivalence
relation of $R_{n}$, and $\mathcal{T}_{n}^{\prime}$ is the relative
topology of $R_{n}$. By (i) it is sufficient to show that $(R_{n}^{\prime},\mathcal{T}_{n}^{\prime})$ is an AF-equivalence relation for every $n$. So
the proof boils down to showing that an open subequivalence relation of a
CEER is an AF-equivalence relation. This again may be reduced further by
Lemma 3.4 to showing that an open subequivalence relation of a CEER of type $m$, with $m\geq2$, on a compact space is an AF-equivalence relation.
(Indeed, it is straightforward to show that a finite disjoint union of
AF-equivalence relations is again an AF-equivalence relation).
So we may assume that $(R,\mathcal{T})$ is equal to the product of the
cotrivial and trivial equivalence relations on $X=Y\times\{1,\dots,m\}$,
where $Y$ is compact, and $R^{\prime}\subset R$ is an open subequivalence
relation. The proof will be completed by showing that $(R^{\prime},\mathcal{T}^{\prime})$ is an AF-equivalence relation, where $\mathcal{T}^{\prime}$
is the relative topology of $R$. By Proposition 3.2(i) we know that both $\mathcal{T}$ and $\mathcal{T}^{\prime}$ are the relative topology from $X\times X$. For $y\in Y$, let
\begin{equation*}
R^{\prime}(y)=\bigl\{(i,j)\mid((y,i),\,(y,j))\in R^{\prime}\bigr\}
\end{equation*}
and observe that $R^{\prime}(y)$ is an equivalence relation on $\{1,\dots
,m\}$. For $(i,j)\in R^{\prime}(y)$, there exists a clopen neighbourhood $U_{i,j}$ of $y$ such that
\begin{equation*}
\bigl\{((y^{\prime},i),\,(y^{\prime},j))\mid y^{\prime}\in U_{i,j}\bigr
\}\subset R^{\prime},
\end{equation*}
since $R^{\prime}$ is open in $R$. Let $U_{y}=\bigcup_{(i,j)\in R^{\prime
}(y)}U_{i,j}$. Then $U_{y}$ is a clopen neighbourhood of $y$ such that $R^{\prime}(y)\subset R^{\prime}(y^{\prime})$ for all $y^{\prime}\in U_{y}
$.
Fix $\epsilon>0$. For each $y\in Y$ select $U_{y}$ as above so that $U_{y}\subset B(y,\epsilon)$, where $B(y,\epsilon)$ is the open ball around
$y$ of radius $\epsilon$. Select a finite subcover $U_{y_{1}},\dots
,U_{y_{k}}$ of the clopen cover $\{U_{y}\mid y\in Y\}$ of $Y$. For $y\in Y$,
let $R^{\prime\prime}(y)=\bigcap\limits_{\{i|y\in U_{i}\}}R^{\prime
}(y_{i})$. Let $\mathcal{P}$ be the clopen partition of $Y$ generated by $\{U_{y_{1}},\dots,U_{y_{k_{n}}}\}$. Then $R^{\prime\prime}|_{E}$ is
constant for every $E\in\mathcal{P}$. In an obvious way $R^{\prime\prime}$
defines an equivalence relation on $X=Y\times\{1,\dots,m\}$. Furthermore,
we have shown that $R^{\prime\prime}$ is a subequivalence relation of $R^{\prime}$ which is a CEER in the relative topology. Clearly $R^{\prime
\prime}$ is an open subset of $R^{\prime}$.
Now we let $R_{1}^{\prime}$ be some $R^{\prime\prime}$ corresponding to $\epsilon=1$. Assume $R_{n}^{\prime}$ has been defined to be some $R^{\prime\prime}$ corresponding to $\epsilon=1/n$, and let $\{U_{y_{1}},\dots,U_{y_{k_{n}}}\}$ be the associated clopen cover of $Y$ as
explained above. For $\epsilon=1/(n+1)$, choose a finite clopen cover $\{U_{z_{1}},\dots,U_{z_{k_{n+1}}}\}$ of $Y$ as before so that every $U_{z_{i}}$ is contained in some $U_{y_{j}}$, and define $R_{n+1}^{\prime}$
to be the associated $R^{\prime\prime}$. One observes that $R_{n}^{\prime
}\subset R_{n+1}^{\prime}$, and that $R_{n}^{\prime}$ is open in $R_{n+1}^{\prime}$, where each equivalence relation is given the relative
topology of $X\times X$. Now $\bigcup_{n=1}^{\infty}R_{n}^{\prime}\subset
R^{\prime}$, since each $R_{n}^{\prime}$ is contained in $R^{\prime}$. On
the other hand, let $y\in Y$ and let $U_{y}$ be, as before, a clopen
neighbourhood of $y$ such that $y^{\prime}\in U_{y}\Rightarrow R(y)\subset
R(y^{\prime})$. Choose $n$ so large that $B(y,1/n)\subset U_{y}$. With $\epsilon=1/n$ let $\{U_{y_{1}},\dots,U_{y_{k_{n}}}\}$ be the clopen cover
of $Y$ associated to $R_{n}^{\prime}$. In particular, $U_{y_{i}}\subset
B(y_{i},1/n)$ for each $i$. If $y\in U_{y_{i}}$, then $y\in B(y_{i},1/n)$
and so $y_{i}\in B(y,1/n)\subset U_{y}$. Hence $R^{\prime}(y)\subset
R^{\prime}(y_{i})$, and so we get $R^{\prime}(y)\subset R_{n}^{\prime}(y)$. Consequently, $R^{\prime}=\bigcup_{n=1}^{\infty}R_{n}^{\prime}$ and one
verifies easily that $(R^{\prime},\mathcal{T}^{\prime
})=\lim\limits_{\longrightarrow}(R_{n}^{\prime},\mathcal{T}_{n}^{\prime})$, where $\mathcal{T}_{n}^{\prime}$ is the relative topology of $X\times X$.
This finishes the proof of (ii).
\end{proof}

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\section{Affable equivalence relations}

\begin{Definition}
[Affable equivalence relation]Let $R$ be a countable equivalence relation on
the zero-dimensional (Hausdorff locally compact, second countable) space $X$.
We say that $R$ is \textit{affable} if $R$ may be given a topology
$\mathcal{T}$ so that $(R,\mathcal{T})$ is an AF-equivalence relation.
\end{Definition}

\begin{Remark}
To say that the countable equivalence relation $R$ on $X$ is affable is the
same as to say that $R$ is orbit equivalent (cf. Definition 2.4) to some
$(R^{\prime},\mathcal{T}^{\prime})$, where $(R^{\prime},\mathcal{T}^{\prime})$
is an AF-equivalence relation on $X^{\prime}$, i.e. there exists a
homeomorphism $F:X\rightarrow X^{\prime}$ such that $(x,y)\in R\iff\bigl
(F(x),F(y)\bigr)\in R^{\prime}$. In fact, using $F$ to pull back the topology
$\mathcal{T}^{\prime}$ on $R^{\prime}$ to get the topology $\mathcal{T}$ on
$R$, we get that $(R,\mathcal{T})$ is an AF-equivalence relation on $X$.
\end{Remark}

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The next result is an immediate consequence of Corollary 1.3. Because of its
importance we state it is as a theorem.

\begin{Theorem}
Let $(X,T)$ be a Cantor minimal system, and let $(R,\mathcal{T})$ be the
associated \'{e}tale equivalence relation on $X$ \textnormal{(cf. Example
2.7(i))}. Let $x$ be an arbitrary point of $X$. The subequivalence relation
$R_{\{x\}}$ of $R$ whose equivalence classes are the full $T$-orbits, except
that the $T$-orbit of $x$ is split into two at $x$\unskip\thinspace---\hskip
.16667em\ignorespaces the forward orbit $\{T^{n}x\mid n\geq1\}$ and the
backward orbit $\{T^{n}x\mid n\leq0\}$\unskip\thinspace---\hskip
.16667em\ignorespaces is open in $R$. Furthermore, $(R_{\{x\}},\mathcal{T}%
_{\{x\}})$ is an AF-equivalence relation on $X$, where $\mathcal{T}_{\{x\}}$
is the relative topology. In particular, $R_{\{x\}}$ is affable.
\end{Theorem}

\begin{proof}
By Corollary 1.3 we may assume that $\left(  X,T\right)  $ is the
Bratteli-Vershik system associated to the properly ordered Bratteli diagram
$\left(  V,E,\geq\right)  $, and where $x$, respectively $Tx$, is the unique
max path, respectively the unique min path. The set of paths that are cofinal
with the unique max path equals the backward orbit of $x$, and the set of
paths that are cofinal with the unique min path equals the forward orbit of
$x$. As for the other $T$-orbits, they agree with the cofinal equivalence
relation of $(V,E)$. Thus $R_{\{x\}}$ coincides with the cofinal equivalence
relation associated to $(V,E)$. Let $(y,T^{k}y)\in R_{\{x\}}$, where
$y=(e_{1},e_{2},\dots)\in X$ and $k$ is an integer. So $y$ and $T^{k}y$ are
paths that agree from a certain level on, say $N$. Let $U$ be the open (and
closed) neighbourhood of $y$ defined by
\[
U=U_{\left(  e_{1},e_{2},\ldots,e_{N}\right)  }=\bigl\{(f_{1},f_{2},\dots)\in
X_{B}\mid(f_{1},\dots,f_{N})=(e_{1},\dots,e_{N})\bigr\}.
\]
Then $W=\bigl\{(z,T^{k}z)\mid z\in U\bigr\}$ is an open neighbourhood of
$(y,T^{k}y)$ in $(R,\TT)$. Furthermore, $W\subset R_{\{x\}}$ since $z$ and
$T^{k}z$ agree from level $N$ on for every $z\in U$. Hence $R_{\{x\}}$ is open
in $R$. The argument we have just given also shows that $\TT_{\{x\}}$
coincides with the topology associated to $(V,E)$ as described in Example
2.7(ii), and so $\left(  R_{\{x\}},\mathcal{T}_{\{x\}}\right)  =AF\left(
V,E\right)  $. Hence $(R_{\{x\}},\TT_{\{x\}})$ is an AF-equivalence relation
on $X$.
\end{proof}

\begin{Definition}
Let $(X,T)$ be a Cantor minimal system, and let $Y$ be a \ non-empty closed
subset of $X$. We say that $Y$ is \textit{regular} (with respect to $(X,T)$)
if the positive and negative return time maps $\lambda^{+}$ and $\lambda^{-}$
for $T$ on $Y$ are continuous, where $\lambda^{+},\lambda^{-}:Y\rightarrow
\mathbb{N}\cup\{+\infty\}$ are given by
\begin{align*}
\lambda^{+}\left(  y\right)   &  =\inf\{k\geq1,+\infty|T^{k}y\in Y\}\text{,}\\
\lambda^{-}\left(  y\right)   &  =\inf\{k\geq1,+\infty|T^{-k}y\in Y\}\text{,}%
\end{align*}
and $\mathbb{N}\cup\{+\infty\}$ is given the ``one-point compactification topology''.
\end{Definition}

\begin{Remark}
Let $(X,T)$ be a Cantor minimal system. If $Y$ is a closed subset of $X$ that
meets each $T$-orbit at most once, then $Y$ is regular. In fact, in this case
$\lambda^{+}(y)=\lambda^{-}(y)=+\infty$ for each $y\in Y$.
\end{Remark}

The following theorem, besides being a considerable generalization of Theorem
4.3, turns out to be a useful tool in the study of affability of equivalence
relations associated to certain group actions on the Cantor set. (See also the
remarks prior to Theorem 4.8.)

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\begin{Theorem}
Let $(X,T)$ be a Cantor minimal system, and let $(R,\mathcal{T})$ be the
associated \'{e}tale equivalence relation on $X$. Let $Y\subset X$ be a
regular set that contains a point $y\in Y$ so that $\lambda^{+}(y)=\lambda
^{-}(y)=+\infty$, i.e. $Y$ meets the $T$-orbit of $y$ at $y$ only. Let $R_{Y}
$ be the subequivalence relation of $R$ defined by
\[
R_{Y}=\{(x,T^{k}x),(T^{k}x,x)\mid x\in X,k\geq0,\ \#(\{0\leq i<k\mid T^{i}x\in
Y\})\text{ is an even number}\}.
\]

In particular, if $Y$ meets each $T$-orbit at most once, $R_{Y}$ is obtained
from $R$ by splitting the $T$-orbits meeting $Y$ in the forward and backward
orbits at $Y$.

Then $(R_{Y},\mathcal{T}_{Y})$ is an AF-equivalence relation on $X$, where
$\mathcal{T}_{Y}$ is the relative topology from $R$. In particular, $R_{Y}$ is affable.
\end{Theorem}

\begin{proof}
Clearly $R_{Y}$ is a subequivalence relation of $R_{\{y\}}$. By Theorem 4.3 we
have that $R_{\{y\}}$ is an open subrelation of $R$ and $\left(
R_{\{y\}},\mathcal{T}_{\{y\}}\right)  $ is an AF-equivalence relation,
$\mathcal{T}_{\{y\}}$ being the relative topology from $R$. So by Proposition
3.12 (ii) the proof will be completed if we can show that $R_{Y}$ is an open
subset of $R$, i.e. $R_{Y}\in\mathcal{T}$.
Let $x\in X$ and $k\geq1$. It is obviously sufficient to show that
$(x,T^{k}x)\in R_{Y}$ has an open neighbourhood $U\in\mathcal{T}$ so that
$U\subset R_{Y}$. Let $T^{i_{1}}x,\dots,T^{i_{l}}x$, where $0\leq i_{1}<\dots<i_{l}\leq k-1$, be the points on the $T$-orbit of $x$ lying between $x$
and $T^{k-1}x$ that meet $Y$. By assumption $l$ is an even number.
Obviously\bigskip\[\begin{array}
[c]{c}\lambda^{+}\left(  T^{i_{1}}x\right)  =i_{2}-i_{1}\text{, \ \ \ }\\
\lambda^{+}\left(  T^{i_{j}}x\right)  =i_{j+1}-i_{j\text{, \ \ \ }}\\
\lambda^{+}\left(  T^{i_{l}}x\right)  >k-1-i_{l\text{, }}\end{array}\begin{array}
[c]{c}\lambda^{-}\left(  T^{i_{1}}x\right)  >i_{1}\text{,
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\
\lambda^{-}\left(  T^{i_{j}}x\right)  =i_{j}-i_{j-1}\text{ if }1<j<l\text{;}\\
\lambda^{-}\left(  T^{i_{l}}x\right)  =i_{l}-i_{l-1}\text{.
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\end{array}
\]
By continuity of $\lambda^{+}$ and $\lambda^{-}$ on $Y$, we may find an open
neighbourhood $V$ of $x$ so that if $x^{\prime}\in V$, the number $l^{\prime}$
of points in $Y$ lying on the $T$-orbit of $x^{\prime}$ between $x^{\prime}$
and $T^{k-1}x^{\prime}$ is even. In fact, we choose $V$ so small that for
$1\leq i\leq k-1$, $T^{i}V$ does not meet $Y$ if $i\notin\{i_{1},\dots
,i_{l}\}$, and if $T^{i_{j}}x^{\prime}\in Y,1\leq j\leq l$, then $\lambda
^{+}(T^{i_{j}}x^{\prime})=\lambda^{+}(T^{i_{j}}x)$ and $\lambda^{-}(T^{i_{j}}x^{\prime})=\lambda^{-}(T^{i_{j}}x)$ if these values are finite (with obvious
modification if some of these values are $+\infty$). By a simple argument it
follows that if $T^{i_{j}}x^{\prime}\in Y$ for some $i_{j}\in\{i_{1},...,i_{l}\}$, then $l^{\prime}=l$. Thus either $l^{\prime}=l$ or $l^{\prime
}=0$. In either case $l^{\prime}$ is even. This shows that $U=\{(x^{\prime
},T^{k}x^{\prime})\mid x^{\prime}\in V\}$ is contained in $R_{Y}$. By
definition of the topology $\mathcal{T}$ we have $U\in\mathcal{T}$.
\end{proof}

\begin{Remark}
We illustrate by a figure the equivalence classes of $R_{Y}$. Let us draw a
$T$-orbit (ordered from left to right) as dots, and circle those dots that are
in $Y$, see Figure 3. The $T$-orbit splits into two $R_{Y}$-equivalence
classes (assuming the orbit in question meets $Y$). One class is everything
inside the boxes, while the other class is everything else. Note that if $Y$
meets the $T$-orbit at exactly one point, say $y$, we get the splitting at $y
$ into the forward and backward $T$-orbits.
\end{Remark}%

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\underline{\emph{Henceforth all Bratteli diagrams }$\left(  V,E\right)
$\emph{\ that we shall consider will be standard}}.\textit{\ }This entails in
particular that the associated path space $X_{\left(  V,E\right)  }$ is compact.\medskip

One of our aims is to prove that the \'{e}tale equivalence relation $\left(
R,\mathcal{T}\right)  $ associated to a Cantor minimal system $\left(
X,T\right)  $ is affable. But more than that, we want to prove that $\left(
X,T\right)  $\thinspace---or $\left(  R,\mathcal{T}\right)  $\thinspace--- is
orbit equivalent (cf. Remark 4.2) to the subequivalence relation $R_{Y}$ of
$R$, obtained by splitting the $T$-orbit in the forward and backward orbits at
$Y$, where $Y$ is any (non-empty) closed subset $Y$ of $X$ that meets each
$T$-orbit at most once. By Theorem 4.6 we know that $R_{Y}$ with the relative
topology from $\left(  R,\mathcal{T}\right)  $ is an $AF$-equivalence
relation. We shall obtain our result without involving the full power of the
main theorem, Theorem 2.2 in [GPS], and hence we will avoid using homological
algebra as well as $C^{\ast}$-algebra ingredients in our proof. More
importantly, the key lemma, Lemma 4.15, that will give us our desired result
as a corollary, is of independent interest. It turns out to be a powerful tool
in handling more general group actions, something that will be treated in a
forthcoming paper.

First, however, we want to prove a converse of the above \thinspace--- a
result which is more easily accessible. This converse result is actually an
immediate corollary of Lemma 6.1 in [GPS], but we give here a simplified and
more direct proof avoiding any mentioning of $K$-Theory and $C^{\ast}$-algebras.

\begin{Theorem}
Let $\left(  R,\mathcal{T}\right)  $ be a minimal $AF$-equivalence relation on
the Cantor set $X$. Then $\left(  R,\mathcal{T}\right)  $ is orbit equivalent
to a Cantor minimal system $\left(  Y,S\right)  ,$ i.e. there exists a
homeomorphism $F:X\rightarrow Y$ so that $F\left(  [x]_{R}\right)
=orbit_{S}\left(  F\left(  x\right)  \right)  $ for every $x\in X$.
\end{Theorem}

\begin{proof}
By Theorem 3.9 we may assume at the outset that $\left( R,\mathcal{T}\right)
$ is the $AF$-equivalence relation $AF\left( V,E\right) $ associated to a
simple Bratteli diagram $\left( V,E\right) $, where $X$ is the path space $X_{\left( V,E\right) }$ associated to $\left( V,E\right) $. Choose a proper
ordering on $\left( V,E\right) $, and denote the associated Bratteli-Vershik
system by $\left( X,T\right) $. Let $x_{\max}$, respectively $x_{\min}$,
be the unique maximal, respectively minimal, path in $X$. Choose a point $x_{0}\in X$ which is not cofinal with $x_{\max}$ or $x_{\min}$\thinspace
---in other words, $x_{0}\notin orbit_{T}\left( x_{\max}\right) $. Let $Z=\{x_{\max},x_{0}\}$. By Theorem 1.2 there exists an ordered Bratteli
diagram $B_{Z}=(\tilde{V},\tilde{E},\geq)$, where $(\tilde{V},\tilde{E})$
is simple, such that $\left( X,T\right) $ is conjugate to the
Bratteli-Vershik system, denoted $(\tilde{X},\tilde{T})$, associated to $B_{Z}$. Here $\tilde{X}$ is the path space $X_{\left( \tilde{V},\tilde{E}\right) }$ associated to $(\tilde{V},\tilde{E})$. Let $F:X\rightarrow\tilde{X}$ be the conjugating map, and let $(\tilde{R},\mathcal{\tilde{T})}=AF(\tilde{V},\tilde{E})$ be the $AF$-equivalence relation associated to $(\tilde{V},\tilde{E}).$ By Theorem 1.2 we get that $F$ implements an orbit
equivalence between $AF\left( V,E\right) $ and the equivalence relation $\tilde{R}\subset\tilde{X}\times\tilde{X}$ generated by $AF(\tilde{V},\tilde{E})$($\subset\tilde{X}\times\tilde{X}$) and $\left( F\left(
x_{0}\right) ,F\left( Tx_{0}\right) \right) \in\tilde{X}\times\tilde{X}$.
Since $F\left( x_{0}\right) $, respectively $F\left( Tx_{0}\right) $, is a
max path, respectively a min path, in $\tilde{X}$ with respect to the
ordering $\geq$ on $(\tilde{V},\tilde{E})$, we may give $(\tilde{V},\tilde{E})$ a new ordering $\geq^{\prime}$, which is proper, such that $F\left(
x_{0}\right) $, respectively $F\left( Tx_{0}\right) $, is the unique max
path, respectively unique min path. Let $\left( Y,S\right) $ denote the
associated Bratteli-Vershik system, where $Y=\tilde{X}$. Then the map $F:X\rightarrow Y$ above implements an orbit equivalence between $\left( R,\mathcal{T}\right) =AF\left( V,E\right) $ and $\left( Y,S\right) $.
\end{proof}

Before we prove Lemma 4.15, alluded to above, we need some definitions and
preliminary results.

\begin{Definition}
Let $\left(  R.\mathcal{T}\right)  $ be an \'{e}tale relation on the space
$X.$ Let $\mu$ be a probability measure on $X$. Define the two $\sigma$-finite
(regular) measures $\nu_{r}$ and $\nu_{s}$ on $\left(  R,\mathcal{T}\right)  $
by
\[
\nu_{r}\left(  C\right)  =\int\limits_{X}\#\left(  r^{-1}\left(  x\right)
\cap C\right)  d\mu\left(  x\right)  ,\text{ }\nu_{s}\left(  C\right)
=\int\limits_{X}\#\left(  s^{-1}\left(  x\right)  \cap C\right)  d\mu\left(
x\right)
\]
where $C$ is a Borel subset of $R$, and $\#\left(  S\right)  $ denotes the
cardinality of a set $S$. We say that $\mu$ is $R$\textit{-invariant} if
$\nu_{r}=\nu_{s}$.

By [FM;Section 2] this is equivalent to say that $\mu$ is $G$-invariant, where
$G$ is any countable group such that $R=R_{G}$.
\end{Definition}

\begin{Remark}
Let $\left(  R,\mathcal{T}\right)  $ be a minimal $AF$-equivalence relation on
the Cantor set $X.$ By Theorem 3.9 we may assume that $\left(  R,\mathcal{T}%
\right)  =AF\left(  V,E\right)  ,$ where $\left(  V,E\right)  $ is a simple
Bratteli diagram with $X=X_{\left(  V,E\right)  }.$ Then $\mu$ is a
$R$-invariant probability measure iff $\mu$ is $T$-invariant, where $\left(
X,T\right)  $ is the Bratteli-Vershik system associated to any properly
ordered diagram $\left(  V,E,\geq\right)  ,$ cf. [HPS; Theorem 5.5].
\end{Remark}

\begin{Definition}
Let $\left(  R,\mathcal{T}\right)  $ be an \'{e}tale equivalence relation on
the space $X.$ We say that\textit{\ }a (non-empty) closed subset $Z$ of $X$ is
\textit{thin} (with respect to $\left(  R,\mathcal{T}\right)  $) if
$\mu\left(  Z\right)  =0,$ for all $R$-invariant probability measures $\mu$ on
$X. $

Let $\left(  V,E\right)  $ be a Bratteli diagram and $\left(  W,F\right)  $ be
a subdiagram. If $X_{\left(  W,F\right)  }$ is a thin subset of $X_{\left(
V,E\right)  }$ (with respect to $AF\left(  V,E\right)  $), then we say that
$\left(  W,F\right)  $ is a \textit{thin} subdiagram of $\left(  W,F\right)  .$
\end{Definition}

\begin{Lemma}
Let $\left(  W,F\right)  $ be a thin subdiagram of $\left(  V,E\right)  $,
where $\left(  V,E\right)  $ is a simple Bratteli diagram. Given $m\geq
1,K\geq1$, there exists $n\geq m$ so that for $w\in W_{m}$, $w^{\prime}\in
W_{n}$,
\[
K\cdot\#\left(  \{\text{paths in }\left(  W,F\right)  \text{ from }w\text{ to
}w^{\prime}\}\right)  \leq\#\left(  \{\text{paths in }\left(  V,E\right)
\text{ from }w\text{ to }w^{\prime}\}\right)
\]
\end{Lemma}

\begin{proof}
For $n\geq m$, let $U_{n}=\bigcup\{U_{\left(  f_{1},,\ldots,f_{n}\right)
}\mid\left(  f_{1},\ldots,f_{n}\right)  \in P_{n}\left(  W,F\right)  \}$,
where $P_{n}\left(  W,F\right)  $ denotes the paths in $\left(  W,F\right)  $
from the top vertex $v_{0}\in V_{0}=W_{0}$ to a vertex $w\in W_{n}$. Here
$U_{\left(  f_{1},\ldots,f_{n}\right)  }$ is the clopen cylinder set in
$X_{\left(  V,E\right)  }$ defined by $U_{\left(  f_{1},\ldots,f_{n}\right)
}=\{\left(  e_{1},e_{2},\ldots\right)  \in X_{\left(  V,E\right)  }\mid\left(
e_{1},e_{2},\ldots,e_{n}\right)  =\left(  f_{1},f_{2},\ldots,f_{n}\right)
\}$. Clearly $\bigcap\limits_{n=1}^{\infty}U_{n}=X_{\left(  W,F\right)  }$.
Let
\begin{equation}
0<\delta<\frac{1}{K}\cdot\inf\{\mu\left(  U_{\left(  f_{1,}\ldots
,,f_{m}\right)  }\right)  \mid\left(  f_{1},\ldots,f_{m}\right)  \in
P_{m}\left(  W,F\right)  ;\mu\in M\left(  V,E\right)  \}\tag{$\ast$}\end{equation}
where $M\left(  V,E\right)  $ denotes the $\left(  V,E\right)  $-invariant
probability measures. (Note that $\mu\left(  A\right)  >0$ for any non-empty
clopen set $A\subset X$ and any $\mu\in M\left(  V,E\right)  $, since $\left(
V,E\right)  $ \ is simple. In fact, $M\left(  V,E\right)  $ may be identified
with the set of states on the simple dimension group $K_{0}\left(  V,E\right)
,$ and $A$ (being a finite union of cylinder sets) may naturally be identified
with a non-zero element $[A]$ in $K_{0}\left(  V,E\right)  ^{+}$.)
Let $M_{n}=\{\mu\in M\left(  V,E\right)  \mid\mu\left(  U_{n}\right)
\geq\delta\}$. Then $M_{n}$ is $w^{\ast}$-compact, since the characteristic
function $\chi_{U_{n}}$ is continuous. Since obviously $M_{m}\supseteq
M_{m+1}\supseteq\cdots$, we get by thinness of $X_{\left(  W,F\right)  }$ that
$\bigcap\limits_{m=n}^{\infty}M_{n}=\emptyset$. By compactness there exists an
$n_{1}\geq m$ so that $M_{n_{1}}=\emptyset$, and so $\mu\left(  U_{n_{1}}\right)  <\delta$ for all $\mu\in M\left(  V,E\right)  $. By ($\ast$), we
then get $\mu\left(  U_{\left(  f_{1},\ldots,f_{m}\right)  }\right)
>K\mu\left(  U_{n_{1}}\right)  $ for all $\mu\in M\left(  V,E\right)  $ and
all $\left(  f_{1},\ldots,f_{m}\right)  \in P_{m}\left(  W,F\right)  $. Hence
$[U_{\left(  f_{1},\ldots,f_{m}\right)  }]>K[U_{n_{1}}]$ in $K_{0}\left(
V,E\right)  $ for all $\left(  f_{1},\ldots,f_{m}\right)  \in P_{m}\left(
W,F\right)  $, cf. [Ef; Ch.4]. This means that there exists $n\geq
n_{1}$ so that
$\chi_{U_{\left(  f_{1},\ldots,f_{m}\right)  }}\left(  v\right)
>K\chi_{U_{n_{1}}}\left(  v\right)  $ for all $v\in V_{n}$ and all $\left(
f_{1},\ldots,f_{m}\right)  \in P_{m}\left(  W,F\right)  $, where we have made
the obvious identification of the characteristic function of a clopen set
(being a finite union of cylinder sets) with a group element in $K_{0}\left(
V,E\right)  $. Now let $w\in W_{m},w^{\prime}\in W_{n}$, and choose $\left(
f_{1},\ldots,f_{m}\right)  \in P_{m}\left(  W,F\right)  $ so that $f\left(
f_{m}\right)  =w$. Then\medskip\
\begin{align*}
& \#\left(  \{\text{paths from }w\text{ to }w^{\prime}\text{ in }\left(
V,E\right)  \}\right)  =\chi_{U_{\left(  f_{1},\ldots,f_{m}\right)  }}\left(
w^{\prime}\right)  >K\cdot\chi_{U_{n_{1}}}\left(  w^{\prime}\right)  \\
& =K\cdot\#\left(  \{\text{paths }\left(  e_{1},\ldots,e_{n}\right)  \text{
from }v_{0}\text{ to }w^{\prime}\text{ in }\left(  V,E\right)  \text{ such
that }\left(  e_{1},\ldots,e_{n_{1}}\right)  \in P_{n_{1}}\left(  W,F\right)
\}\text{ }\right)  \\
& \geq K\cdot\#\left(  \{\text{paths from }v_{0}\text{ to }w^{\prime}\text{ in
}\left(  W,F\right)  \}\right)  \geq K\cdot\#\left(  \{\text{paths from
}w\text{ to }w^{\prime}\text{ in }\left(  W,F\right)  \}\right)
\end{align*}
This completes the proof.
\end{proof}

\begin{Lemma}
Let $\left(  V,E\right)  $ and $\left(  V^{\prime},E^{\prime}\right)  $ be two
Bratteli diagrams. The following are equivalent:

\begin{enumerate}
\item [(i)]$AF\left(  V,E\right)  \cong AF\left(  V^{\prime},E^{\prime
}\right)  $

\item[(ii)] $K_{0}\left(  V,E\right)  \cong K_{0}\left(  V^{\prime},E^{\prime
}\right)  $, i.e. $K_{0}\left(  V,E\right)  $ is order-isomorphic to
$K_{0}\left(  V^{\prime},E^{\prime}\right)  $ by a map preserving the
canonical order units.

\item[(iii)] There exists a so-called ''aggregate'' Bratteli diagram
$(\tilde{V},\tilde{E}),$ so that telescoping $(\tilde{V},\tilde{E})$ to odd
levels $0<1<3<5<\cdots$ yields a telescope of $\left(  V,E\right)  ,$ while
telescoping $(\tilde{V},\tilde{E})$ to even levels $0<2<4<6<\cdots$ yields a
telescope of $\left(  V^{\prime},E^{\prime}\right)  .$

\item[(iv)] $\left(  V,E\right)  \sim\left(  V^{\prime},E^{\prime}\right)  ,$
where $\sim$ denotes the equivalence relation on Bratteli diagrams generated
by telescoping.
\end{enumerate}
\end{Lemma}

\begin{proof}
The equivalence of (ii), (iii) and (iv) is well known, cf. [GPS; Section 3].
The implication (iii)$\Rightarrow$(i) is immediate from the observation we
made in Example 2.7 (ii) concerning telescoping of Bratteli diagrams. In
fact, $AF(\tilde{V},\tilde{E})$ is isomorphic to both $AF(\tilde{V}_{o},\tilde{E}_{o})$ and $AF(\tilde{V}_{e},\tilde{E}_{e}),$ where $(\tilde{V}_{o},\tilde{E}_{o}),$ respectively $(\tilde{V}_{e},\tilde{E}_{e}),$ is the
telescope of $(\tilde{V},\tilde{E})$ to odd levels, respectively even levels.
\\
We prove (i)$\Rightarrow$(iii). Let $AF\left( V,E\right)
=\lim\limits_{\longrightarrow}\left( R_{N},\mathcal{T}_{N}\right) ,$ $AF\left( V^{\prime},E^{\prime}\right) =\lim\limits_{\longrightarrow
}\left( R_{N}^{\prime},\mathcal{T}_{N}^{\prime}\right) ,$ where $R_{N},\mathcal{T}_{N},$ respectively $R_{N}^{\prime},\mathcal{T}_{N}^{\prime},$
have the same meaning as in Example 2.7(ii). There is an obvious groupoid
partition associated to $R_{N},$ respectively $R_{N}^{\prime},$ which
corresponds to the vertex set $V_{N}\in V,$ respectively $V_{N}^{\prime}\in
V^{\prime}.$ Let $\alpha:X_{\left( V,E\right) }\rightarrow X_{\left(
V^{\prime},E^{\prime}\right) }$ implement the isomorphism between $AF\left( V,E\right) $ and $AF\left( V^{\prime},E^{\prime}\right) .$
Because of compactness of $R_{1,}$ $\alpha\times\alpha\left( R_{1}\right)
$ is contained in $R_{n_{1}}$ for some $n_{1}\geq1.$ We may choose $n_{1}$
so large that the groupoid partition associated to $R_{n_{1}}^{\prime}$ is finer than the one associated to $\alpha\times
\alpha\left( R_{1}\right) .$ By the same procedure as in the proof of
Theorem 3.9 we associate edges between the vertices in $V_{1}$ and $V_{n_{1}}^{\prime},$ keeping in mind that the groupoid partitions
associated to $R_{1}$ and $\alpha\times\alpha\left( R_{1}\right) ,$
respectively, are isomorphic in an obvious way. Next we consider $\alpha
^{-1}\times\alpha^{-1}\left( R_{n_{1}}^{\prime}\right) ,$ which by
compactness of $R_{n_{1}}^{\prime}$ is contained in some $R_{n_{2}}.$ We
choose $n_{2}>1$ so large that the groupoid partition associated to $R_{n_{2}}$ is finer than the one associated to $\alpha^{-1}\times\alpha
^{-1}\left( R_{n_{1}}^{\prime}\right) .$ Similarly as above we associate
edges between $V_{n_{1}}^{\prime}$ and $V_{n_{2}}.$ Continuing in this way
we construct a Bratteli diagram $(\tilde{V},\tilde{E}).$ It is now a simple
matter to show that $(\tilde{V},\tilde{E})$ is an aggregate diagram with
respect to $\left( V,E\right) $ and $\left( V^{\prime},E^{\prime}\right) .$
We omit the details.
\end{proof}

\begin{Remark}
Let $\left(  X,T\right)  $ and $\left(  Y,S\right)  $ be two Cantor minimal
systems with associated \'{e}tale equivalence relations $R$ and $R^{\prime}, $
respectively, cf. Example 2.7(i). Let $x\in X,$ $y\in Y,$ and let $R_{x},$
respectively $R_{y}^{\prime},$ denote the $AF$-subequivalence relation of $R,$
respectively $R^{\prime}$ (cf. Theorem 4.3). Then $R_{x}\cong R_{y}^{\prime}$
iff $\left(  X,T\right)  $ and $\left(  Y,S\right)  $ are strong orbit
equivalent. This is an immediate consequence of the lemma and Corollary 1.3,
in combination with Theorem 2.1 of [GPS].
\end{Remark}

\begin{Lemma}
[Key lemma]Let $\left(  R,\mathcal{T}\right)  $ be isomorphic to $\left(
R^{\prime},\mathcal{T}^{\prime}\right)  ,$ where $\left(  R,\mathcal{T}%
\right)  $ and $\left(  R^{\prime},\mathcal{T}^{\prime}\right)  $ are minimal
$AF$-equivalence relations on the Cantor sets $X$ and $X^{\prime},$
respectively. Let $Z$ and $Z^{\prime}$ be closed, thin subsets of $X$ and
$X^{\prime}$ (with respect to $\left(  R,\mathcal{T}\right)  $ and $\left(
R^{\prime},\mathcal{T}^{\prime}\right)  $), respectively. Assume that

\begin{enumerate}
\item [(i)]the (not necessarily minimal) restrictions $R|_{Z}=R\cap\left(
Z\times Z\right)  $ (respectively $R^{\prime}|_{Z^{\prime}}=R^{\prime}%
\cap\left(  Z^{\prime}\times Z^{\prime}\right)  $) with the relative
topologies are \'{e}tale equivalence relations on $Z$ (respectively
$Z^{\prime}$).

\item[(ii)] there exists a homeomorphism $\alpha:Z\rightarrow Z^{\prime}$
which implements an isomorphism between $R|_{Z}$ and $R^{\prime}|_{Z^{\prime}}.$
\end{enumerate}

There exists an extension $\tilde{\alpha}:X\rightarrow X^{\prime}$ of $\alpha$
such that $\tilde{\alpha}$ implements an isomorphism between $\left(
R,\mathcal{T}\right)  $ and $\left(  R^{\prime},\mathcal{T}^{\prime}\right)  .$
\end{Lemma}

\begin{proof}
By Theorem 3.11 we may assume that $\left( R,\mathcal{T}\right) =AF\left(
V,E\right) ,$ $\left( R^{\prime},\mathcal{T}^{\prime}\right) =AF\left
( V^{\prime
},E^{\prime}\right) ,$ and $R|_{Z}=AF\left( W,F\right) ,$ $R^{\prime
}|_{Z^{\prime}}=AF\left( W^{\prime},F^{\prime}\right) ,$ where $\left(
W,F\right) $ and $\left( W^{\prime},F^{\prime}\right) $ are thin
subdiagrams of the (simple) Bratteli diagrams $\left( V,E\right) $ and $\left( V^{\prime},E^{\prime}\right) ,$ respectively. So $X=X_{\left(
V,E\right) },$ $X^{\prime}=X_{\left( V^{\prime},E^{\prime}\right) },$ $Z=X_{\left( W,F\right) },$ $Z^{\prime}=X_{\left( W^{\prime},F^{\prime
}\right) }.$
The idea of the proof is to construct a Bratteli diagram $(\tilde{V}%
,\tilde{E}),$ together with a subdiagram $(\tilde{W},\tilde{F}),$ so that $(\tilde{V},\tilde{E})$ is an aggregate diagram with respect to $\left(
V,E\right) $ and $\left( V^{\prime},E^{\prime}\right) ,$ while $(\tilde{W},\tilde{F})$ is an aggregate diagram with respect to $\left( W,F\right) $ and
$\left( W^{\prime},F^{\prime}\right) .$ Furthermore, we will do this in
such way that we can ''read off'' the map $\alpha:X_{\left( W,F\right)
}\rightarrow X_{\left( W^{\prime},F^{\prime}\right) }$ from $(\tilde{W},\tilde{F})$ as an ''intertwining map'' (to be explained below). \newline
We will then use $(\tilde{V},\tilde{E})$ to extend $\alpha$ to $\tilde{\alpha}:X_{\left( V,E\right) }\rightarrow X_{\left( V^{\prime},E^{\prime
}\right) },$ $\alpha$ being again an intertwining map. We begin the proof
by first noticing that by telescoping $\left( V,E\right) ,$ say, we
automatically get a corresponding telescoping of $\left( W,F\right) ,$ which
again will be a thin subdiagram. Now there is a natural
homeomorphism between the path spaces associated to a Bratteli diagram and a
telescope of it (cf. Example 2.7(ii)). Thus we may by condition (i) ---
invoking Lemma 4.13 --- assume at the start that there is a Bratteli diagram
$(\overline{V},\overline{E}),$ so that telescoping $(\overline{V}%
,\overline{E})$ to odd levels $0<1<3<\cdots$ we get $\left( V,E\right) ,$ while
telescoping $(\overline{V},\overline{E})$ to even levels $0<2<4<\cdots$ we
get $\left( V^{\prime},E^{\prime}\right) .$ Using the notation introduced
in Example 2.7(ii), let $V=V_{0}\cup V_{1}\cup V_{3}\cup\cdots,$ $E=E_{1}\cup E_{2}\cup\cdots,$ $W=W_{0}\cup W_{1}\cup W_{2}\cup\cdots,$ $F=F_{1}\cup F_{2}\cup\cdots,$ and similarly for $V^{\prime},$ $E^{\prime
},$ $W^{\prime},$ $F^{\prime}.$ Also, let $AF\left( W,E\right)
=\lim\limits_{\longrightarrow}\left( R_{N}^{\prime},\mathcal{T}_{N}^{\prime}\right) ,$ $AF\left( W^{\prime},F^{\prime}\right)
=\lim\limits_{\longrightarrow}\left( R_{N}^{\prime},\mathcal{T}_{N}^{\prime}\right) $. Since $R_{1}$ is compact, we get by assumption
(iii) that there exists $n_{1}^{\prime}\geq1$ so that $\alpha\times
\alpha\left( R_{1}\right) \subset R_{n_{1}^{\prime}}^{\prime}.$
Furthermore, we may assume $n_{1}^{\prime}$ is chosen so large that the
groupoid partition associated to $R_{n_{1}^{\prime}}^{\prime}$ is finer
than the one associated to $\alpha\times\alpha\left( R_{1}\right) .$ As
in the proof of (i)$\Rightarrow$(iii) in Lemma 4.13 we associate edges
between vertices in $W_{1}$ and $W_{n_{1}^{\prime}}^{\prime}.$ Denote
these edges by $L.$ Now $V_{1}$ and $V_{n_{1}^{\prime}}^{\prime}$
correspond to level $n_{1}=1$ and level $2n_{1}^{\prime},$ respectively, of
$(\overline{V},\overline{E}),$ and so by telescoping between these levels we
get edges connecting vertices in $V_{1}=V_{n_{1}}$ with vertices in $V_{n_{1}^{\prime}}^{\prime}.$ Denote these edges by $M.$ Since $\left(
W^{\prime},F^{\prime}\right) $ is thin in $\left( V^{\prime},E^{\prime
}\right) $ by condition (ii), we may apply Lemma 4.12 and choose $n_{1}^{\prime}$ so large that for any $v\in W_{1}\subset V_{1}$ and $v^{\prime}\in W_{n_{1}^{\prime}}^{\prime}\subset V_{n_{1}^{\prime
}}^{\prime},$ the number of edges in $M$ between $v$ and $v^{\prime}$ is
larger than the number of edges in $L$ between $v$ and $v^{\prime}.$
\newline
Next we consider $\alpha^{-1}\times\alpha^{-1}\left( R_{n_{1}^{\prime
}}^{\prime}\right) .$ Arguing the same way as above we may find $n_{2}>n_{1}^{\prime}$ with edge set $L,$ respectively $M,$ between vertices
in $W_{n_{1}^{\prime}}^{\prime}$ and $W_{n_{2}},$ respectively $V_{n_{1}^{\prime}}^{\prime}$ and $V_{n_{2}},$ with the same properties as
above. Continuing in this way we construct a Bratteli diagram $(\tilde{V},\tilde{E}),$ which we will show has the desired properties. In fact, by its
very construction there is a Bratteli subdiagram $(\tilde{W},\tilde
{F})$ of $(\tilde{V},\tilde{E}),$ so that $(\tilde{W},\tilde{F})$ is an aggregate
diagram with respect to $\left( W,F\right) $ and $\left( W^{\prime
},F^{\prime}\right) ,$ while $(\tilde{V},\tilde{E})$ itself is an aggregate
diagram with respect to $\left( V,E\right) $ and $\left( V^{\prime
},E^{\prime}\right) .$ In fact, by our previous deliberations we may assume
that telescoping $(\tilde{V},\tilde{E}),$ respectively $(\tilde{W},\tilde{F}),$ to odd levels $0<1<3<\cdots,$ we get $\left( V,E\right) ,$ respectively
$\left( W,F\right) $ --- not just a telescope of these. Likewise,
telescoping $(\tilde{V},\tilde{E}),$ respectively $(\tilde{W},\tilde{F}),$
to even levels $0<2<4<\cdots,$ we get $\left( V^{\prime},E^{\prime
}\right) ,$ respectively $\left( W^{\prime},F^{\prime}\right) .$ Let $\tilde{E}=\tilde{E}_{1}\cup\tilde{E}_{2}\cup\cdots$ and $\tilde{F}=\tilde{F}_{1}\cup\tilde{F}_{2}\cup\cdots.$ We define the composition of edges
between levels $k-1$ and $k+1$, where $k\geq1,$ by
\begin{eqnarray*}
\tilde{E}_{k}\circ\tilde{E}_{k+1} &=&\{\left( \tilde{e}_{k},\tilde{e}_{k+1}\right) |\tilde{e}_{k}\in\tilde{E}_{k},\tilde{e}_{k+1}\in\tilde{E}_{k+1},f\left( \tilde{e}_{k}\right) =i\left( \tilde{e}_{k+1}\right) \} \\
\tilde{F}_{k}\circ\tilde{F}_{k+1} &=&\{(\tilde{f}_{k},\tilde{f}_{k+1})|\tilde{f}_{k}\in\tilde{F}_{k},\tilde{f}_{k+1}\in\tilde{F}_{k+1},f(\tilde{f}_{k})=i(\tilde{f}_{k+1})\}.
\end{eqnarray*}
We will establish bijections
\begin{equation*}
\left( a\right) \left\{
\begin{array}{l}
\tilde{E}_{k-1}\circ\tilde{E}_{k}\longleftrightarrow E_{\frac{k}{2}}^{\prime} \\
\tilde{E}_{k}\circ\tilde{E}_{k+1}\longleftrightarrow E_{\frac{k}{2}+1}
\end{array}
\right. \text{ \ \ \bigskip\ \ \ }(b)\left\{
\begin{array}{l}
\tilde{F}_{k-1}\circ\tilde{F}_{k}\longleftrightarrow F_{\frac{k}{2}}^{\prime} \\
\tilde{F}_{k}\circ\tilde{F}_{k+1}\longleftrightarrow F_{\frac{k}{2}+1}
\end{array}
\right.
\end{equation*}
for every even number $k=2,4,\ldots,$ which will respect the range and
source maps --- recalling that $\tilde{V}_{m}=V_{\frac{m+1}{2},}$ $\tilde{W}_{m}=W_{\frac{m+1}{2}}$ for $m$ odd, and $\tilde{V}_{m}=V_{\frac{m}{2}}^{\prime},$ $\tilde{W}_{m}=W_{\frac{m}{2}}^{\prime}$ for $m$ even. (In
addition we have $\tilde{E}_{1}=E_{1},$ $\tilde{F}_{1}=F,$ and $\tilde{V}_{0}=V_{0}=V_{0}^{\prime}=\tilde{W}_{0}=W_{0}=W_{0}^{\prime},$ all being
equal to the top vertex of $(\tilde{V},\tilde{E}).$) The bijections will be
chosen successively, and in such a way that we will be able to read off the
given map $\alpha:X_{\left( W,F\right) }\rightarrow X_{\left( W^{\prime
},F^{\prime}\right) }$ by this intertwining process. Furthermore, the
bijections in $(a)$ shall extend the ones in $\left( b\right) $, keeping in
mind that the various edge sets occurring in $\left( b\right) $ are
contained in the corresponding ones occurring in $\left( a\right) $. \newline
We first consider the bijections $\left( b\right) ,$ starting with $\tilde
{F}_{1}=F_{1}.$ By the embedding scheme that we outlined above, the inclusion $\alpha\times\alpha\left( R_{1}\right) \subset R_{1}^{\prime}$ determines
uniquely the edges $\tilde{F}_{2}$ between $\tilde{W}_{1}=W_{1}$ and $\tilde
{W}_{2}=W_{1}^{\prime}.$ This in turn sets up a bijection $\tilde{F}_{1}\circ\tilde{F}_{2}\longleftrightarrow F_{1}^{\prime}$ in an obvious
way, respecting the range and source maps. Similarly, the inclusion $\alpha
^{-1}\times\alpha^{-1}\left( R_{1}^{\prime}\right) \subset R_{2}$
determines uniquely the edges $\tilde{F}_{3}$ between $\tilde{W}_{2}=W_{1}^{\prime}$ and $\tilde{W}_{3}=W_{2},$ and this in turn sets up a
bijection $\tilde{F}_{2}\circ\tilde{F}_{3}\longleftrightarrow F_{2}$ in an
obvious way, respecting the range and source maps. Continuing in this way we
get all the bijections in $\left( b\right) .$ Now these bijections induce a
map between $X_{\left( W,F\right) }$ and $X_{\left( W^{\prime},F^{\prime
}\right) },$ which will be equal to $\alpha.$ In fact, if $x=\left(
f_{1},f_{2},\ldots\right) \in X_{\left( W,F\right) },$ where $f_{i}\in
F_{i},$ then the successive bijections $\tilde{F}_{k}\circ\tilde{F}_{k+1}\longleftrightarrow F_{\frac{k}{2}+1}$ of $\left( b\right) ,$ starting
with $\tilde{F}_{1}=F_{1},$ will determine a unique path in $X_{(\tilde{W},\tilde{F})}.$ Then, using the bijections $\tilde{F}_{k-1}\circ\tilde{F}_{k}\longleftrightarrow F_{\frac{k}{2}}^{\prime}$ of $\left( b\right) ,$ we
conclude that $x$ determines a unique path $y=(\tilde{f}_{1},\tilde{f}_{2},\ldots)\in X_{(W^{\prime},F^{\prime})},$ where $f_{i}^{\prime}\in
F_{i}^{\prime}.$ One shows easily that $y=\alpha\left( x\right) $ --- we
omit the details. We say that $\alpha:X_{\left( W,F\right) }\rightarrow
X_{\left( W^{\prime},F^{\prime}\right) }$ is defined as an \textit{intertwining map }(via the aggregate diagram $(\tilde{W},\tilde{F})$). (Cf.
also [GPS;Theorem 2.1, proof of (ii)$\Rightarrow$(i)].) \newline
To conclude the proof of the lemma we extend the bijection in $\left(
b\right) $ to $\left( a\right) $. We do this successively, starting with $\tilde{E}_{1}=E_{1}.$ We may choose\ the various bijections in $\left(
a\right) ,$ which extend the ones in $\left( b\right) ,$ in an arbitrary
way, the only proviso being that the source and range maps are respected.
The associated intertwining map $\tilde{\alpha}:X_{\left( V,E\right)
}\rightarrow X_{\left( V^{\prime},E^{\prime}\right) }$ will clearly be a
homeomorphism that extends $\alpha.$ By its very construction, $\tilde{\alpha}$ clearly preserves cofinality. Furthermore, the $n$ first edges $\{f_{1}^{\prime},f_{2}^{\prime},\ldots,f_{n}^{\prime}\}$ of $y=\tilde{\alpha}\left( x\right) =\left( f_{1}^{\prime},f_{2}^{\prime},\ldots
\right) \in X_{\left( V^{\prime},E^{\prime}\right) },$ $f_{i}^{\prime}\in
E_{i}^{\prime},$ is determined by the $n$ first edges of $x=\left(
e_{1},e_{2},\ldots\right) \in X_{\left( V,E\right) },$ $e_{i}\in E_{i}.$ (A
similar statement is true for $\tilde{\alpha}^{-1}.)$ This implies that $\tilde{\alpha}$ implements an isomorphism between $AF\left( V,E\right) $ and
$AF\left( V^{\prime},E^{\prime}\right) $ (cf. the description we gave of
convergence in Example 2.7(ii)).
\end{proof}

We are now in a position to prove the following non-trivial result, which is a
converse to Theorem 4.8.

\begin{Theorem}
Let $\left(  X,T\right)  $ be a Cantor minimal system, and let $R$ denote the
equivalence relation associated to $\left(  X,T\right)  ,$ i.e. the
$R$-equivalence classes are the $T$-orbits. Then $R$ is affable.

In fact, more is true: $R$ is orbit equivalent to $R_{Y},$ where $Y$ is any
non-empty closed subset of $X$ that meets each $T$-orbit at most once. (Cf.
Theorem 4.6 for the definition of $R_{Y},$ and the proof that $R_{Y}$ is affable.)
\end{Theorem}

\begin{proof}
The main ingredient of the proof will be Lemma 4.15, in combination with
Theorem 1.2. However, we need a preliminary result in order to set the
stage, so to say, before we can apply Lemma 4.15. So let $Y$ be any
non-empty closed subset of $X$ that meets each $T$-orbit at most once. We
shall need the following technical result, which we state as a sublemma, and
whose proof we postpone to the end in order not to interrupt the flow of the
main argument. \medskip\newline
\textbf{Sublemma. }\textit{There exists a point }$y\in X$ \textit{and a
sequence of pairwise disjoint, non-empty clopen sets }$\{U_{n}\}_{n=1}^{\infty},$\textit{\ each of which are disjoint from }$Y,$\textit{\ together
with a sequence of non-empty closed sets }$\{Y_{n}\}_{n=1}^{\infty},$\textit{\ where }$Y_{n}\subset$\textit{\ }$U_{n}$\textit{\ for each }$n$\textit{, so that}
\begin{enumerate}
\item[(i)]  $U_{n}\rightarrow y$\textit{\ (i.e. for each neighbourhood }$U$\textit{\ of }$y,$\textit{\ there exists }$N$\textit{\ so that }$n\geq N$\textit{\ implies }$U_{n}\subset U$\textit{).}
\item[(ii)]  \textit{The set }$Z=Y\cup\{y\}\cup
\bigcup\limits_{n=1}^{\infty}Y_{n}$\textit{\ is a closed set such that }$Z$\textit{\ meets each }$T$\textit{-orbit at most once.}
\item[(iii)]  \textit{There exists a sequence }$\{h_{n}\}_{n=0}^{\infty}$\textit{\ of homeomorphisms }$h_{0}:Y\rightarrow
Y_{1},h_{1}:Y_{1}\rightarrow Y_{2},h_{2}:Y_{2}\rightarrow Y_{3},\ldots
\medskip$
\end{enumerate}
Applying the sublemma we may define a homeomorphism $h:Z\rightarrow
Z^{\prime},$ where $Z^{\prime}=\{y\}\cup\bigcup\limits_{n=1}^{\infty
}Y_{n}.$ In fact, define $h$ by $h|_{Y_{n}}=h_{n},$ $n=0,1,2,\ldots,$ and
set $h\left( y\right) =y.$ Using Theorem 1.2 we construct a Bratteli-Vershik
model for $\left( X,T\right) $ --- which we still will denote by $\left(
X,T\right) $ --- based on the closed set $Z,$ and we let $B=\left( V,E,\geq
\right) $ denote the associated ordered Bratteli diagram. Hence $X=X_{\left(
V,E\right) },$ and $T$ is the Vershik map. Furthermore, since $Z$ are the
maximal paths and $T\left( Z\right) $ are the minimal paths in $X$, $Z\cup
T\left( Z\right) $ will be the path space associated to a thin subdiagram $\left( W,F\right) $ of $\left( V,E\right) ,$ i.e. $Z\cup T\left( Z\right)
=X_{\left( W,F\right) }.$ In fact, $\left( W,F\right) $ is a \textit{tree }and it is obviously thin because of condition (ii) of the sublemma. By
Theorem 1.2, $R_{Z}$ is equal to the cofinal relation associated to $\left(
V,E\right) .$ Furthermore, it is clear that
\begin{equation}
R=R_{Z}\vee\{\left( z,Tz\right) |z\in Z\}\medskip\text{ \ \ },\medskip
\text{ \ \ }R_{Y}=R_{Z}\vee\{\left( z,Tz\right) |z\in Z^{\prime}\},
\tag{$\ast$}
\end{equation}
\newline
where we here let $A\vee B$ denote the equivalence relation on $X$ generated
by the two subsets $A,B$ of $X\times X.$ \newline
Set $\left( V^{\prime},E^{\prime}\right) =\left( V,E\right) ,$ and let $\left( W^{\prime},E^{\prime}\right) $ be the thin subdiagram of $\left(
V^{\prime},E^{\prime}\right) $ associated to $Z^{\prime}\cup T\left(
Z^{\prime}\right) ,$ i.e. $Z^{\prime}\cup T\left( Z^{\prime}\right)
=X_{\left( W^{\prime},F^{\prime}\right) }.$ (Note that $\left( W^{\prime
},F^{\prime}\right) $ is a subdiagram of $\left( W,F\right) .$) There is a
homeomorphism $\alpha:X_{\left( W,F\right) }\rightarrow X_{\left( W^{\prime
},F^{\prime}\right) },$ namely $\alpha|_{Z}=h$ and $\alpha\left(
Tz\right) =T\left( h\left( z\right) \right) ,z\in Z.$ Because of condition
(ii) of the sublemma, $\alpha$ is well-defined and is a homeomorphism.
Also, $\alpha$ clearly implements an isomorphism between $AF\left(
W,F\right) $ and $AF\left( W^{\prime},F^{\prime}\right) ,$ which is an
immediate consequence of the fact that both the Bratteli diagrams $\left(
W,F\right) $ and $\left( W^{\prime},F^{\prime}\right) $ are trees. Hence
all the conditions of Lemma 4.15 are satisfied. Let $\tilde{\alpha}:X_{\left( V,E\right) }\rightarrow X_{\left( V^{\prime},E^{\prime}\right) }
$ be the extension of $\alpha,$ with the properties stated in Lemma 4.15.
Clearly $\tilde{\alpha}\times\tilde{\alpha}\left( R_{Z}\right) =R_{Z},$
since $\tilde{\alpha}$ preserves the cofinal relation associated to $\left(
V,E\right) =\left( V^{\prime},E^{\prime}\right) .$ Also, $\tilde{\alpha}\times\tilde{\alpha}\left( \{\left( z,Tz\right) |z\in Z\}\right) =\alpha
\times\alpha\left( \{\left( z,Tz\right) |z\in Z\}\right) =\{\left(
z,Tz\right) |z\in Z^{\prime}\}.$ Hence, by $\left( \ast\right) ,$ $\tilde{\alpha}\times\tilde{\alpha}\left( R\right) =R_{Y}$ $.\smallskip$
\textit{Proof of sublemma. }We first show that if $V$ is a non-empty clopen
subset of $X$ that is disjoint from $Y,$ then we may find a closed subset $Y^{\prime}$ of $V,$ such that $Y\cup Y^{\prime}$ meets each $T$-orbit at
most once, and, furthermore, there exists a homeomorphism $h:Y\rightarrow
Y^{\prime}$. To obtain this we pick a finite partition $\{A_{1},\ldots
,A_{n_{1}}\}$ of $Y$ consisting of non-empty clopen sets ($Y$ is given the
relative topology from $X$) such that the diameters of each $A_{i}$ is less
than $\frac{1}{2}$. Also, pick $B_{1},\ldots B_{n_{1}}$ to be non-empty
clopen and pairwise disjoint subsets of $V$, each of diameter less than $\frac{1}{2}$, such that
\begin{equation*}
\begin{array}{lll}
\left( i\right)  & T^{i}\left( B_{k}\right) \cap T^{j}\left( B_{l}\right)
=\emptyset& \text{for }-2\leq i,j\leq2,\text{ \ }1\leq k,l\leq n_{1}. \\
&  &  \\
\left( ii\right)  & B_{k}\cap T^{i}\left( Y\right) =\emptyset& \text{for }-2\leq i\leq2,\text{ \ }1\leq k\leq n_{1}.
\end{array}
\end{equation*}
This can clearly be achieved --- as for $\left( ii\right) $ we note that $\bigcup\limits_{i=-2}^{2}T^{i}\left( Y\right) $ is a closed subset of $X$
with empty interior. Next we partition each of the $A_{i}$'s into non-empty
clopen sets of diameter less than $\frac{1}{3}$, with corresponding picking
of clopen subsets of the $B_{i}$'s. For example, say the partition of $A_{1}$
is $\{A_{1}^{1},\ldots,A_{1}^{m_{1}}\}$. We pick $m_{1}%
$ non-empty and pairwise disjoint clopen
subsets $B_{1}^{1},\ldots,B_{1}^{m_{1}}$ of $B_{1}$, each with diameter
less than $\frac{1}{3}$, such that they satisfy properties $\left( i\right) $
and $\left( ii\right) $, where $i$ and $j$ now range between $-3$ and $3$,
and $k$ and $l$ range from $1$ to $m_{1}$. Continuing like this we get
nested sequences of $A$'s converging to every point in $Y$, together with
nested sequences of $B$'s converging to points in $V$, and we denote this
set of points by $Y^{\prime}$. There is a map $h:Y\rightarrow
Y^{\prime}$ induced by the obvious $1-1$ correspondence between nested $A$-
and $B$-sequences, and it is a routine matter to verify that $Y^{\prime}$
and $h$ satisfy the desired properties.
To finish the proof of the sublemma, choose $y\in X\setminus Y$ so that $Y_{0}^{\prime}=Y\cup\{y\}$ meets each $T$-orbit at most once, and let $\{U_{n}\}_{n=1}^{\infty}%
$ be a sequence of pairwise disjoint, non-empty clopen sets (each disjoint from  $Y$)
converging to $y$. Applying what we just have proved, choose successively
non-empty closed sets $Y_{1}^{\prime},Y_{2}^{\prime},Y_{3}^{\prime
},\ldots$ such that $Y_{1}^{\prime}\subset U_{1},Y_{2}^{\prime}\subset
U_{2},Y_{3}^{\prime}\subset U_{3},\ldots$ and homeomorphisms $h_{0}^{\prime}:Y_{0}^{\prime}\rightarrow Y_{1}^{\prime},h_{1}^{\prime
}:Y_{0}^{\prime}\cup Y_{1}^{\prime}\rightarrow Y_{2}^{\prime
},h_{2}^{\prime}:Y_{0}^{\prime}\cup Y_{1}^{\prime}\cup Y_{2}^{\prime
}\rightarrow Y_{3}^{\prime},\ldots$ so that for each $k$, $Y_{0}^{\prime
}\cup Y_{1}^{\prime}\cup Y_{2}^{\prime}\cup\ldots\cup Y_{k}^{\prime}$
meets each $T$-orbit at most once. Define successively $Y_{0}=Y,Y_{1}=h_{1}^{\prime}\left( Y_{0}\right) ,Y_{2}=h_{1}^{\prime
}\left( Y_{1}\right) ,Y_{3}=h_{2}^{\prime}\left( Y_{2}\right) ,\ldots$ and
$h_{k}=h_{k}^{\prime}|_{Y_{k}}$ for $k\geq0$. Then $Z=Y\cup\{y\}\cup
\bigcup\limits_{n=1}^{\infty}Y_{n}$ is a closed set that meets each $T$-orbit at most once. This finishes the proof of the sublemma, and
consequently the proof of the theorem.
\end{proof}

The following theorem, which essentially is corollary of Theorem 4.16, can
succinctly be stated to say that a finite extension of a minimal
$AF$-equivalence relation is affable.

\begin{Theorem}
Let $\left(  R,\mathcal{T}\right)  $ be a minimal $AF$-equivalence relation on
the Cantor set $X$. Let $\left(  x_{1},y_{1}\right)  ,\ldots,\left(
x_{n},y_{n}\right)  $ be $n$ pairs of points in $X\times X$. Let $R^{\prime}$
be the equivalence relation on $X$ generated by $R$ and $\left(  x_{1}%
,y_{1}\right)  ,\ldots,\left(  x_{n},y_{n}\right)  $. The $R^{\prime}$ is affable.
\end{Theorem}

\begin{proof}
Clearly it is enough to prove the statement for the special case $n=1$,
since the general case follows by induction. By Theorem 3.9 we may assume
that $\left( R,\mathcal{T}\right) =AF\left( V,E\right) $, where $\left(
V,E\right) $ is a simple, standard Bratteli diagram, and $X=X_{\left(
V,E\right) }$. If $x_{1}$ and $y_{1}$ are cofinal paths, then $R^{\prime}=R$, and there is nothing to prove. So assume $x_{1}$ and $y_{1}$ are not
cofinal. According to Proposition 1.1, there exists a Cantor minimal system $\left( X,T\right) $ such that $T$ respects cofinality, except that $Tx_{1}=y_{1}$. By Theorem 4.16 the equivalence relation $R^{\prime}$
associated to $\left( X,T\right) $ is affable. Now $R^{\prime}$ is the
equivalence relation generated by $R$ and $\left( x_{1},y_{1}\right) $.
\end{proof}

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\end{document}
