# Wieler solenoids: non-Hausdorff expansiveness, Cuntz-Pimsner models, and
functorial properties

Robin J. Deeley  Robin J. Deeley, Department of Mathematics, University of
Colorado Boulder Campus Box 395, Boulder, CO 80309-0395, USA
<EMAIL_ADDRESS>,  Menevşe Eryüzlü  Menevşe Eryüzlü, Department of
Mathematics, University of Colorado Boulder Campus Box 395, Boulder, CO
80309-0395, USA<EMAIL_ADDRESS>,  Magnus Goffeng  Magnus
Goffeng, Centre for Mathematical Sciences, Lund University, Box 118, 221 00
LUND, Sweden<EMAIL_ADDRESS>and  Allan Yashinski  Allan
Yashinski, Department of Mathematics, University of Maryland, College Park, MD
20742-4015, USA<EMAIL_ADDRESS>
###### Abstract.

Building on work of Williams, Wieler proved that every irreducible Smale space
with totally disconnected stable sets can be realized via a stationary inverse
limit. Using this result, the first and fourth listed authors of the present
paper showed that the stable $C^{*}$-algebra associated to such a Smale space
can be obtained from a stationary inductive limit of a Fell algebra. Its
spectrum is typically non-Hausdorff and admits a self-map related to the
stationary inverse limit. With the goal of understanding the fine structure of
the stable algebra and the stable Ruelle algebra, we study said self-map on
the spectrum of the Fell algebra as a dynamical system in its own right. Our
results can be summarized into the statement that this dynamical system is an
expansive, surjective, local homeomorphism of a compact, locally Hausdorff
space and from its $K$-theory we can compute $K$-theoretical invariants of the
stable and unstable Ruelle algebra of a Smale space with totally disconnected
stable sets.

RJD was partially supported by NSF Grants DMS 2000057 and DMS 2247424. MG was
supported by the Swedish Research Council Grant VR 2018-0350.

## Introduction

In [41] Williams proved that every expanding attractor can be realized as a
solenoid (i.e., a stationary inverse limit). The space in Williams’
construction is a branched manifold and the map satisfies natural axioms. The
expanding attractors studied by Williams are examples of Smale spaces that
have totally disconnected stable sets. Building on Williams’ axioms, Wieler
[39] proved that every Smale space with totally disconnected stable sets can
be realized as a solenoid. Roughly speaking, Wieler replaces the differential
geometric axioms of Williams with axioms that only rely on topological/metric
space properties. This is necessary to include examples such as shifts of
finite type where the solenoid is constructed from a Cantor set. In terms of
notation, we let $(Y,g)$ denote the Wieler pre-solenoid and $(X,\varphi)$
denote the associated Smale space; as mentioned $(X,\varphi)$ is the
stationary inverse limit of $(Y,g)$.

The stable $C^{*}$-algebra associated to a Smale space is introduced in [31]
and [32]. Building on work of Gonçalves [17, 18] (also see [26] and [19]) the
first and fourth listed authors [11] showed that the stable algebra is
isomorphic to a stationary inductive limit when the Smale space is obtained
from a Wieler pre-solenoid. The relevant $C^{*}$-algebra in the inductive
limit is particularly nice; it is the Fell algebra associated to a local
homeomorphism and furthermore the spectrum of this algebra is related to but
typically not equal to the space in Wieler’s construction. This paper aims at
clarifying this relation.

More to the point, the spectrum of the Fell algebra is typically non-
Hausdorff. However, it is compact and locally Hausdorff. In addition, the
original map on the pre-solenoid lifts to a map on this compact, locally
Hausdorff space. We denote this space and map as a pair $(X^{u}({\bf
P})/{\sim_{0}},\tilde{g})$ where the notation is introduced in more detail in
Section 2. Importantly for us, the map $\tilde{g}$ is a local homeomorphism
and (in a non-Hausdorff sense that is introduced in the present paper)
expansive. In this sense, the dynamical system $(X^{u}({\bf
P})/{\sim_{0}},\tilde{g})$ is a “non-Hausdorff resolution” of the failure of
$(Y,g)$ to be an expansive local homeomorphism.

Thus, at the purely dynamical level, there is a trade-off between the
Hausdorffness of the space of the pre-solenoid and the map being a local
homeomorphism and expansive. For $C^{*}$-algebraic applications, the map being
a local homeomorphism and expansive is often more important than the space
being Hausdorff. Using $(X^{u}({\bf P})/{\sim_{0}},\tilde{g})$ as our primary
example, we study the following three areas:

_The notion of forward orbit expansiveness in the context of non-Hausdorff
spaces._ This builds on the work of Achigar, Artigue, and Monteverde [1] who
study expansiveness for homeomorphisms on non-Hausdorff spaces. We obtain
natural conditions on the pre-solenoid (which is non-Hausdoff) that ensures
that the associated solenoid is Hausdorff. In our prototypical example, we
show that the solenoid associated with $(X^{u}({\bf P})/{\sim_{0}},\tilde{g})$
is the original Smale space. Thus any Smale space with totally disconnected
stable sets can be realized as solenoid where the map of the pre-solenoid is
an expansive, local homeomorphism; of course, the space cannot always be taken
to be Hausdorff, see [11, Example 10.3]. This area is discussed in Section 3.

_The inductive limit and Cuntz-Pimsner models for the stable and stable Ruelle
algebra respectively._ Building on previous work in [8, 11], we relate
stationary inductive limits to Cuntz-Pimsner algebras in Section 5. Our main
result is a very general construction of a Cuntz-Pimsner model of a stationary
inductive limit. Furthermore, we discuss the construction of unital Cuntz-
Pimsner models under the assumption of the existence of a full projection. In
our prototypical example $(X^{u}({\bf P})/{\sim_{0}},\tilde{g})$, the relevant
Fell algebra always has a full projection as we show in Section 4 and the
stable algebra of the original Smale space is a stationary inductive limit of
this Fell algebra and hence is the core of our Cuntz-Pimsner model of the
stable Ruelle algebra.

_The functorial properties of the $K$-theory of a compact, locally Hausdorff
space._ The $K$-theory of a compact, locally Hausdorff space is defined using
Fell algebras so functorial properties are subtle. In particular, a continuous
map between compact, locally Hausdorff spaces need not induce a
$*$-homomorphism at the $C^{*}$-algebra level. Nevertheless, we show that
there are well-behaved right way and wrong way functors despite the failure of
functoriality at the Fell algebra level. In our prototypical example, the map
induced by $\tilde{g}$ plays a key role in computing the $K$-theory of the
stable algebra and the stable Ruelle algebra of the original Smale space.
These computations use the stationary inductive limit in the case of the
stable algebra and the Cuntz-Pimsner models in the case of the stable Ruelle
algebra; an explicit description of the wrong way map associated to
$\tilde{g}$ is the key to such computations. This area is discussed in Section
6 where a number of illustrative examples are discussed at the end of the
section.

For each of the three areas, we work at a quite general level. However,
everything we consider is rooted and motivated by the dynamical system
$(X^{u}({\bf P})/{\sim_{0}},\tilde{g})$ associated with a Wieler pre-solenoid.
Before entering the above mentioned areas, we present some preliminary
material in Section 1, on Smale spaces, Fell algebras, groupoids and
correspondences, as well as in Section 2, on $C^{*}$-algebras associated with
Wieler solenoids. The succeeding four sections of the paper treat the areas
listed above and can be read independently of one another. Even if these last
four sections of the paper are logically independent of each other, we believe
they form a coherent picture of Wieler-Smale spaces. Such a description is
missing in the literature except in the case of a Wieler-Smale space defined
from a local homeomorphism [8], see also Subsection 1.2 below.

To avoid confusion about Hausdorffness, we indicate spaces that are possibly
non-Hausdorff with a tilde. For instance, $X$ is Hausdorff while $\tilde{X}$
is possibly non-Hausdorff. The exception to this notation is $X^{u}({\bf
P})/{\sim_{0}}$ which in general can be non-Hausdorff.

Acknowledgements The authors wish to thank Jamie Gabe for the examples leading
up to Remark 4.8. The first listed author thanks the University of Hawaii and
the Fields Institute for visits during which time the paper was completed.

##  1\. Preliminaries

In this section, we will present some preliminary material from the
literature. As the paper aims for a rather broad view on Wieler-Smale spaces,
we carefully review the known results.

###  1.1. Smale spaces

Although we are only interested in Wieler solenoids some definitions and basic
properties of general Smale spaces are required. The reader can find more on
Smale spaces in [22, 31, 32, 34].

######  Definition 1.1.

A Smale space is a metric space $(X,d)$ along with a homeomorphism
$\varphi:X\rightarrow X$ with the following additional structure: there exists
global constants $\epsilon_{X}>0$ and $0<\lambda<1$ and a continuous map,
called the bracket map,

$[\ \cdot\ ,\ \cdot\ ]:\\{(x,y)\in X\times X:d(x,y)\leq\epsilon_{X}\\}\to X$

such that the following axioms hold

  * B1

$\left[x,x\right]=x$;

  * B2

$\left[x,[y,z]\right]=[x,z]$ assuming both sides are defined;

  * B3

$\left[[x,y],z\right]=[x,z]$ assuming both sides are defined;

  * B4

$\varphi[x,y]=[\varphi(x),\varphi(y)]$ assuming both sides are defined;

  * C1

For $x,y\in X$ such that $[x,y]=y$, $d(\varphi(x),\varphi(y))\leq\lambda
d(x,y)$;

  * C2

For $x,y\in X$ such that $[x,y]=x$,
$d(\varphi^{-1}(x),\varphi^{-1}(y))\leq\lambda d(x,y)$.

A Smale space is denoted simply by $(X,\varphi)$ and to avoid certain
trivialities, throughout the paper we assume that $X$ is infinite.

######  Definition 1.2.

Suppose $(X,\varphi)$ is a Smale space and $x$, $y$ are in $X$. We write

$x\sim_{s}y\hbox{ if
}\lim_{n\rightarrow\infty}d(\varphi^{n}(x),\varphi^{n}(y))=0$

and we write

$x\sim_{u}y\hbox{ if
}\lim_{n\rightarrow\infty}d(\varphi^{-n}(x),\varphi^{-n}(y))=0.$

The $s$ and $u$ stand for stable and unstable respectively.

The global stable and unstable set of a point $x\in X$ are defined as follows:

$X^{s}(x)=\\{y\in X\>|\>y\sim_{s}x\\}\hbox{ and }X^{u}(x)=\\{y\in
X\>|\>y\sim_{u}x\\}$

Furthermore, the local stable and unstable sets of $x$ are defined as follows:
Given, $0<\epsilon\leq\epsilon_{X}$, we have

$\displaystyle X^{s}(x,\epsilon)$ $\displaystyle=\\{y\in X\>|\>[x,y]=y\hbox{
and }d(x,y)<\epsilon\\}\hbox{ and }$ (1.1) $\displaystyle X^{u}(x,\epsilon)$
$\displaystyle=\\{y\in X\>|\>[y,x]=y\hbox{ and }d(x,y)<\epsilon\\}.$ (1.2)

The following result is a standard, see for example [31, 34].

######  Theorem 1.3.

Suppose $(X,\varphi)$ is a Smale space and $x$, $y$ are in $X$ with
$d(x,y)<\epsilon_{X}$. Then the following hold: for any
$0<\epsilon\leq\epsilon_{X}$

  1. (1)

$X^{s}(x,\epsilon)\cap X^{u}(y,\epsilon)=\\{[x,y]\\}$ or is empty;

  2. (2)

$\displaystyle
X^{s}(x)=\bigcup_{n\in\mathbb{N}}\varphi^{-n}(X^{s}(\varphi^{n}(x),\epsilon))$;

  3. (3)

$\displaystyle
X^{u}(x)=\bigcup_{n\in\mathbb{N}}\varphi^{n}(X^{u}(\varphi^{-n}(x),\epsilon))$.

A Smale space is mixing if for each pair of non-empty open sets $U$, $V$,
there exists $N$ such that $\varphi^{n}(U)\cap V\neq\emptyset$ for all $n\geq
N$. When $(X,\varphi)$ is mixing, $X^{u}(x)$ and $X^{s}(x)$ are each dense as
subsets of $X$. However, one can use the previous theorem to give $X^{u}(x)$
and $X^{s}(x)$ locally compact, Hausdorff topologies, see for example [22,
Theorem 2.10].

###  1.2. Background on Wieler solenoids

Inspired by work of Williams [41], Wieler [39] proved that every Smale space
with totally disconnected stable sets $X^{s}(x)$ can be realized as a
solenoid. Such Smale spaces will be referred to as Wieler solenoids or Wieler-
Smale spaces. More precisely, she characterized Wieler-Smale spaces in terms
of the following axioms on the pre-solenoid.

######  Definition 1.4 (Wieler’s Axioms).

Let $(Y,\mathrm{d}_{Y})$ be a compact metric space, and $g:Y\rightarrow Y$ be
a continuous surjective map. Then, the triple $(Y,\mathrm{d}_{Y},g)$ satisfies
Wieler’s axioms if there exists global constants $\beta>0$,
$K\in\mathbb{N}^{+}$, and $0<\gamma<1$ such that the following hold:

Axiom 1:

If $x,y\in Y$ satisfy $\mathrm{d}_{Y}(x,y)\leq\beta$, then

$\mathrm{d}_{Y}(g^{K}(x),g^{K}(y))\leq\gamma^{K}\mathrm{d}_{Y}(g^{2K}(x),g^{2K}(y)).$

Axiom 2:

For all $x\in V$ and $0<\epsilon\leq\beta$

$g^{K}(B(g^{K}(x),\epsilon))\subseteq g^{2K}(B(x,\gamma\epsilon)).$

######  Definition 1.5.

Suppose $(Y,\mathrm{d}_{Y},g)$ satisfies Wieler’s axioms and form the inverse
limit space

$X:=\varprojlim(Y,g)=\\{(y_{n})_{n\in\mathbb{N}}=(y_{0},y_{1},y_{2},\ldots)\>|\>g(y_{i+1})=y_{i}\hbox{
for each }i\geq 0\\}.$

Consider the map $\varphi:X\rightarrow X$ defined via

$\varphi(x_{0},x_{1},x_{2},\ldots)=(g(x_{0}),g(x_{1}),g(x_{2}),\ldots)=(g(x_{0}),x_{0},x_{1},\ldots).$

Following Wieler, we take a metric on $X$, $\mathrm{d}_{X}$, given by

$\mathrm{d}_{X}((x_{n})_{n\in\mathbb{N}},(y_{n})_{n\in\mathbb{N}})=\sum_{i=0}^{K}\gamma^{i}\mathrm{d}^{\prime}_{X}(\varphi^{i}(x_{n})_{n\in\mathbb{N}},\varphi^{i}(y_{n})_{n\in\mathbb{N}}),$

where
$\mathrm{d}^{\prime}_{X}((x_{n})_{n\in\mathbb{N}},(y_{n})_{n\in\mathbb{N}})=\operatorname{sup}_{n\in\mathbb{N}}(\gamma^{n}\mathrm{d}_{Y}(x_{n},y_{n}))$.
We note that the topology induced by $\mathrm{d}_{X}$ is the product topology.
The triple $(X,\mathrm{d}_{X},\varphi)$ is called a Wieler solenoid.

######  Remark 1.6.

We will assume that $Y$ is infinite. In particular, this ensures that $X$ is
infinite and that $g$ is not a homeomorphism, see [11]. The pair $(Y,g)$ will
be called a presolenoid and $(X,\varphi)$ the associated solenoid or Smale
space.

######  Theorem 1.7.

[39, Theorems A and B on page 4] Suppose that $(Y,\mathrm{d}_{Y},g)$ satisfies
Wieler’s axioms. Then the associated Wieler solenoid
$(X,\mathrm{d}_{X},\varphi)$ is a Smale space with totally disconnected stable
sets. The constants in Wieler’s definition give Smale space constants:
$\epsilon_{X}=\frac{\beta}{2}$ and $\lambda=\gamma$. Moreover, if ${\bf
x}=(x_{n})_{n\in\mathbb{N}}\in X$ and $0<\epsilon\leq\frac{\beta}{2}$, the
locally stable and unstable sets of $(X,\mathrm{d}_{X},\varphi)$ are given by

$X^{s}({\bf x},\epsilon)=\\{{\bf
y}=(y_{n})_{n\in\mathbb{N}}\>|\>y_{m}=x_{m}\hbox{ for }0\leq m\leq K-1\hbox{
and }\mathrm{d}_{X}({\bf x},{\bf y})\leq\epsilon\\}$

and

$X^{u}({\bf x},\epsilon)=\\{{\bf
y}=(y_{n})_{n\in\mathbb{N}}\>|\>\mathrm{d}_{Y}(x_{n},y_{n})<\epsilon\>\forall
n\hbox{ and }\mathrm{d}_{X}({\bf x},{\bf y})\leq\epsilon\\}$

respectively.

Conversely, if $(X,\varphi)$ is an irreducible Smale space with totally
disconnected stable sets, then there exists a triple $(Y,\mathrm{d}_{Y},g)$
satisfying Wieler’s axioms such that $(X,\varphi)$ is conjugate to the Wieler
solenoid associated to $(Y,\mathrm{d}_{Y},g)$.

######  Remark 1.8.

Wieler’s axioms and the previous theorem should be compared with work of
Williams [41]. As mentioned in the introduction, an important difference
between the two is that Wieler’s are purely metric space theoretic. If a
triple $(Y,\mathrm{d}_{Y},g)$ satisfies Williams’ axioms the inverse limit
space $X$ with $\varphi$ as in definition 1.5 is also Smale space and we will
refer to such Smale spaces as Williams solenoids. However, we are most
concerned with the more general case of a Wieler solenoid so we will not
review Williams’ axioms but rather direct the reader to [41] for more details.

An important special case occurs when $g$ is a local homeomorphism that
satisfies Wieler’s axioms. This special case was studied in detail in [8]
(also see [37, Section 4.5]). We recall a salient characterization of how
refinements of Wieler’s axioms (Definition 1.4) are equivalent to $g$ being a
local homeomorphism, more detailed statements can be found in [8, Section 3].

######  Theorem 1.9 (Lemma 3.7 and 3.8 of [8]).

Let $(Y,\mathrm{d}_{Y})$ be a compact metric space, and $g:Y\rightarrow Y$ be
a continuous surjective map. The following are equivalent:

  * •

$(Y,\mathrm{d}_{Y},g)$ satisfies Wieler’s axioms and $g$ is a local
homeomorphism.

  * •

$(Y,\mathrm{d}_{Y},g)$ satisfies Wieler’s axiom 1 and $g$ is open.

  * •

$(Y,\mathrm{d}_{Y},g)$ satisfies Wieler’s axiom $2$ and $g^{K}$ is locally
expanding (for the $K$ in Wieler’s axiom 2).

######  Remark 1.10.

The reader is encouraged to compare Theorem 1.9 to the more satisfying
situation arising from going to a non-Hausdorff setting in Theorem 3.10.

A list of examples of Wieler solenoids can be found in [11]. Three explicit
examples that are relevant and illustrative of the results in this paper are
the following:

######  Example 1.11 ($n$-solenoid).

Let $S^{1}\subseteq\mathbb{C}$ be the unit circle. Take $n>1$ and define
$g:S^{1}\rightarrow S^{1}$ via $z\mapsto z^{n}$. Since $g$ is open and
expansive (notice that $|g^{\prime}(z)|=n$), one readily verifies Wieler’s
axioms for $(S^{1},g)$ (see Theorem 1.9). Hence the associated inverse limit
is a Smale space. It is worth emphasizing that in this case $g$ is a local
homeomorphism.

######  Example 1.12 ($ab/ab$-solenoid).

Let $Y=S^{1}\vee S^{1}$ be the wedge sum of two circles as in Figure 1.

$p$$a$$b$$a$$b$ Figure 1. $ab/ab$ pre-solenoid

The map $g:Y\rightarrow Y$ is defined using Figure 1. In Figure 1, we consider
the outer circle to be the $a$-circle and the inner circle to be the
$b$-circle. Each line segment labelled with $a$ in Figure 1 is mapped onto the
$a$-circle (i.e., the outer circle); while, each line segment labelled with
$b$ in Figure 1 is mapped onto the $b$-circle (i.e., the inner circle). The
mapping is done in an orientation-preserving way, provided we have oriented
both circles the same way, say clockwise. Note that $g$ is not a local
homeomorphism in this example. For more details on this specific example and
one-solenoids in general, see [38, 40, 42]. The next example is also of this
form.

######  Example 1.13 ($aab/ab$-solenoid).

Again, we take $Y=S^{1}\vee S^{1}$ to be the wedge sum of two circles but with
labels as in Figure 2. This example has been studied in [10, 11].

$p$$a$$a$$b$$a$$b$ Figure 2. $aab/ab$ pre-solenoid

The map $g:Y\rightarrow Y$ is defined from Figure 2 via the same process as in
Example 1.12. The map $g$ is of course different than the one in Example 1.12
because the labels are different. Again, the resulting map $g$ is not a local
homeomorphism and is an example of a one-solenoid, again see [38, 40, 42] for
more on this class of examples.

###  1.3. Fell algebras and their spectrum

Let us recall the basic facts about Fell algebras that we make use of. The
spectrum of a separable $C^{*}$-algebra $A$ is defined as the set $\hat{A}$ of
equivalence classes of irreducible representations (see [12, Chapter 3]). The
spectrum $\hat{A}$ can be topologized in several different ways, for instance
in the Fell topology or the Jacobson topology. For a Fell algebra, the
topologies coincide. The spectrum $\hat{A}$ is locally quasi-compact by [12,
Corollary 3.3.8].

A $C^{*}$-algebra $A$ is a Fell algebra if every $[\pi_{0}]\in\hat{A}$ admits
a neighborhood $U$ and an element $b\in A$ such that $\pi(b)$ is a rank one
projection for all $[\pi]\in U$. An equivalent definition of a Fell algebra is
that $A$ is generated by its abelian elements. For details, see [20, Chapter
3]. A Fell algebra has locally Hausdorff spectrum (i.e. any $[\pi]\in\hat{A}$
has a Hausdorff neighborhood) by [2, Corollary 3.4]. The spectrum of a
$C^{*}$-algebra is always locally quasi-compact. The properties of the
spectrum of a Fell algebra can be summarized as being locally Hausdorff and
locally locally compact (see [6, Chapter 3]).

######  Definition 1.14.

Let $\tilde{Y}$ be a topological space. A Hausdorff resolution of $\tilde{Y}$
is a surjective local homeomorphism $\psi:X\to\tilde{Y}$ from a locally
compact, Hausdorff space $X$.

######  Example 1.15.

The main example of a Fell algebra that we will concern ourselves with arises
in a rather explicit way from a Hausdorff resolution. The construction can be
found in [6, Corollary 5.4]. Suppose that $\psi:X\to\tilde{Y}$ is a Hausdorff
resolution of a topological space $\tilde{Y}$. It follows that $\tilde{Y}$ is
locally Hausdorff and locally locally compact, and second countable if $X$ is.
We define the equivalence groupoid

$R(\psi):=X\times_{\psi}X:=\\{(y_{1},y_{2})\in X\times
X:\psi(y_{1})=\psi(y_{2})\\}.$

By declaring the domain and range mappings $d(y_{1},y_{2}):=y_{2}$ and
$r(y_{1},y_{2}):=y_{1}$ to be local homeomorphisms, $R(\psi)$ becomes an etale
groupoid over $X$. By [6, Corollary 5.4], $C^{*}(R(\psi))$ is a Fell algebra
with vanishing Dixmier-Douady invariant and spectrum $\tilde{Y}$. We also note
that $R(\psi)$ is amenable so $C^{*}(R(\psi))=C^{*}_{r}(R(\psi))$.

The theory of Dixmier-Douady invariants of Fell algebras was introduced and
developed in [20], also see [6]. We only need to consider Fell algebras with
vanishing Dixmier-Douady class in which case the following theorem (see [20,
6]) reduces the problem to a more manageable situation.

######  Theorem 1.16.

We have the following relationship between Fell algebras and non-Hausdorff
spaces.

  1. a.

Let $A$ be a separable Fell algebra with vanishing Dixmier-Douady invariant.
Then the locally Hausdorff and locally locally compact space $\hat{A}$
determines $A$ up to stable isomorphism in the sense that whenever
$A^{\prime}$ is a separable Fell algebra with vanishing Dixmier-Douady
invariant then a homeomorphism $h:\hat{A}\to\widehat{A^{\prime}}$ can be
lifted to a stable isomorphism $A\otimes\mathbb{K}\cong
A^{\prime}\otimes\mathbb{K}$.

  2. b.

A topological space $\tilde{Y}$ is locally Hausdorff and locally locally
compact if and only if it admits a Hausdorff resolution $\psi:X\to\tilde{Y}$.
If $\tilde{Y}$ is second countable then $X$ can also be choosen second
countable. In particular, any second countable, locally Hausdorff, locally
locally compact topological space is the spectrum of a separable Fell algebra
with vanishing Dixmier-Douady invariant.

######  Remark 1.17.

One can view the previous theorem as follows. Taking the spectra defines an
equivalence between the category of stable isomorphism classes of separable
Fell algebras with vanishing Dixmier-Douady invariants and that of second
countable, locally Hausdorff and locally locally compact spaces. However, it
should be emphasized that the morphisms in the category in the previous
sentence are homeomorphisms; one cannot generalize to the case of continuous
map, even in the case of locally Hausdorff and compact spaces.

For a proof of the first statement on stable uniqueness, see [20, Theorem
7.13]. The existence result in the second statement can be found in [6,
Corollary 5.5].

###  1.4. Correspondences

Results from this section and the next will not be needed until Section 5.
Furthermore, the reader only interested in the purely dynamical results of the
present paper can skip to Section 2 without issue.

A $C^{*}$-correspondence ${}_{A}E_{B}$ is a right Hilbert $B$-module equipped
with a left action given by a homomorphism
$\varphi_{E}:A\rightarrow\mathcal{L}(E)$, where $\mathcal{L}(E)$ denotes the
$C^{*}$-algebra of adjointable operators on $E$. In the literature one also
finds the term $A-B$-Hilbert $C^{*}$-module for a $C^{*}$-correspondence from
$A$ to $B$. A $C^{*}$-correspondence homomorphism
${}_{A}E_{B}\rightarrow{}_{A}F_{B}$ is a $B$-linear map $\Phi:E\rightarrow F$
satisfying

$\Phi(a\cdot\xi)=a\cdot\Phi(\xi)\hbox{ and
}\langle\xi,\nu\rangle_{C}=\langle\Phi(\xi),\Phi(\nu)\rangle_{C},$

for all $a\in A$, and $\xi,\nu\in E.$

######  Definition 1.18.

An $A-B$ bimodule $E_{0}$ is called a _pre-correspondence_ if it has a
$B$-valued semi-inner product satisfying

$\langle\xi,\nu\cdot b\rangle=\langle\xi,\nu\rangle b,\hskip
14.22636pt\langle\xi,\nu\rangle^{*}=\langle\nu,\xi\rangle$

and $\langle a\cdot\xi,a\cdot\xi\rangle\leq\lVert
a\rVert^{2}\langle\xi,\xi\rangle$ for all $a\in A,b\in B$ and $\xi,\nu\in
E_{0}$. Modding out by the elements of length $0$ and completing gives a
$C^{*}$-correspondence ${}_{A}E_{B}$ . We call ${}_{A}E_{B}$ the _completion_
of the pre-correspondence $E_{0}$.

######  Proposition 1.19.

[13, Lemma 1.23] Let $E_{0}$ be an $A-B$ pre-correspondence given with the
completion ${}_{A}E_{B}$ , and let $F$ be an $A-B$ correspondence. If there is
a map $\Phi:E_{0}\rightarrow F$ satisfying

$\Phi(a\cdot\xi)=\varphi_{Z}(a)\Phi(\xi)\qquad\text{and
}\qquad\langle\Phi(\xi),\Phi(\nu)\rangle_{B}=\langle\xi,\nu\rangle_{B},$

for all $a\in A$ and $\xi,\nu\in E_{0}$, then $\Phi$ extends uniquely to an
injective $A-B$ correspondence homomorphism $\tilde{\Phi}:E\rightarrow F.$

The _balanced tensor product_ $E\otimes_{B}F$ of an $A-B$ correspondence $E$
and a $B-C$ correspondence $F$ is formed as follows: the algebraic tensor
product $E\odot F$ is a pre-correspondence with the $A-C$ bimodule structure
satisfying

$a(\xi\otimes\nu)c=a\xi\otimes\nu c\qquad\text{for }a\in A,\xi\in E,\nu\in
F,c\in C,$

and the unique $C$-valued semi-inner product whose values on elementary
tensors are given by

$\langle\xi_{1}\otimes\nu_{1},\xi_{2}\otimes\nu_{2}\rangle_{C}=\langle\nu_{1},\langle\xi_{1},\xi_{2}\rangle_{B}\cdot\nu_{2}\rangle_{C}\qquad\text{for
}\xi_{1},\xi_{2}\in E,\nu_{1},\nu_{2}\in F.$

This semi-inner product defines a $C$-valued inner product on the quotient
$E{\odot}_{B}F$ of $E\odot F$ by the subspace generated by elements of form

$\xi\cdot b\otimes\nu-\xi\otimes\varphi_{Y}(b)\nu\qquad\text{($\xi\in E$,
$\nu\in F$, $b\in B$) }.$

The completion $E\otimes_{B}F$ of $E{\odot}_{B}F$ with respect to the norm
coming from the $C$-valued inner product is an $A-B$ correspondence, where the
left action is given by

$A\rightarrow\mathcal{L}(E\otimes_{B}F),\qquad\text{$a\mapsto\varphi_{E}(a)\otimes
1_{F},$ }$

for $a\in A.$ In other words, the $A-C$ correspondence $E\otimes_{B}F$ is the
completion of the pre-correspondence $E\odot F$ (as in Definition 1.18).

We denote the canonical image of $\xi\otimes\nu$ in $E\otimes_{B}F$ by
$\xi\otimes_{B}\nu$. The term _balanced_ refers to the property

$\xi\cdot b\otimes_{B}\nu=\xi\otimes_{B}b\cdot\nu\qquad\text{for }\xi\in
E,b\in B,\nu\in F,$

which is a consequence of the construction.

######  Definition 1.20.

[36, Definition 5.7] Hilbert modules $E_{A}$ and $F_{B}$ are Morita equivalent
if there exists an imprimitivity bimodule ${}_{B}M_{C}$ such that
$E\otimes_{B}M\cong F$ as Hilbert C-modules.

We now put Definition 1.20 in the setting of $C^{*}$-correspondences:

######  Definition 1.21.

$C^{*}$-correspondences ${}_{A}E_{B}$ and ${}_{A}F_{C}$ are Morita equivalent
if there exists an imprimitivity bimodule ${}_{B}M_{C}$ such that
$E\otimes_{B}M\cong F$ as $A-C$ correspondences.

###  1.5. Groupoid Actions and Equivalence

The correspondences reviewed in the last section will in this paper arise from
groupoids and their actions. Again, the reader only interested in the purely
dynamical results can skip to Section 2 without issue.

######  Definition 1.22.

Suppose $G$ is a groupoid and that $X$ is a set together with a map
$\rho:X\rightarrow G^{(0)}$ called the _moment map_. Then a left action of $G$
on $X$ is a map $(g,x)\mapsto g\cdot x$ from $G*X=\\{(g,x)\in G\times
X:s(g)=\rho(x)\\}$ to $X$ such that

  * •

$\rho(x)\cdot x=x$, for all $x\in X$ and

  * •

if $(g,g^{\prime})\in G^{(2)}$ and $(g^{\prime},x)\in G*X$, then
$(g,g^{\prime}\cdot x)\in G*X$ and $gg^{\prime}\cdot x=g\cdot(g^{\prime}\cdot
x).$

_Right actions are defined analogously, and we denote by $\sigma$ the moment
map for a right action._

######  Definition 1.23.

[25, Definition 1.2] Let $G_{1}$ and $G_{2}$ be second countable locally
compact Hausdorff groupoids and $Z$ a second countable locally compact
Hausdorff space. The space $Z$ is a groupoid correspondence from $G_{1}$ to
$G_{2}$ if it satisfies the following properties:

  1. (1)

there exists a left proper action of $G_{1}$ on $Z$ such that $\rho$ is an
open map;

  2. (2)

there exists a right proper action of $G_{2}$ on Z;

  3. (3)

the $G_{1}$ and $G_{2}$ actions commute;

  4. (4)

the map $\rho$ induces a bijection of $Z/G_{2}$ onto $G_{1}^{(0)}$.

######  Theorem 1.24.

[25, Theorem 1.4] Let $G_{1}$, $G_{2}$ be second countable locally compact
Hausdorff etale groupoids; and let $Z$ be a groupoid correspondence from
$G_{1}$ to $G_{2}$. Then the pre-correspondence
${}_{C_{c}(G_{2})}C_{c}(Z)_{C_{c}(G_{1})}$ extends to a correspondence from
$C^{*}(G_{2})$ to $C^{*}(G_{1})$ with the actions

$\displaystyle(\xi\cdot a)(z)$ $\displaystyle=\sum_{\begin{subarray}{c}g\in
G_{1}with\\\ s(g)=\rho(z)\end{subarray}}\xi(g\cdot z)a(g)$
$\displaystyle(b\cdot\xi)(z)$ $\displaystyle=\sum_{\begin{subarray}{c}g\in
G_{2}with\\\ \sigma(z)=r(g^{-1})\end{subarray}}b(g^{-1})\xi(z\cdot g^{-1})$

for $\xi\in C_{c}(Z),a\in C_{c}(G_{1}),b\in C_{c}(G_{2}),z\in Z.$ The inner
product is defined by

$\langle\xi_{1},\xi_{2}\rangle(g)=\sum_{\begin{subarray}{c}h^{-1}\in
G_{2}with\\\ r(h^{-1})=\sigma(z)\end{subarray}}\overline{\xi_{1}(z\cdot
h^{-1})}\xi_{2}(g^{-1}\cdot z\cdot h^{-1}),$

where $g\in G_{1},\xi_{1},\xi_{2}\in C_{c}(Z),$ and $z\in Z$ such that
$r(g)=\rho(z)$.

##  2\. $C^{*}$-algebras associated to a Wieler solenoid

The fine structure of the stable algebra of a Wieler solenoid was studied in
[11], extending results from [8] in the case of the pre-solenoid being defined
from a local homeomorphism. We here review and refine the relevant points of
[11], leading up to the stable algebra of a Wieler solenoid being a stationary
inductive limit of a Fell algebra defined from the dynamics. This Fell algebra
plays an important role in the paper, as its spectrum will be the non-
Hausdorff dynamical system $(X^{u}({\bf P}))/{\sim_{0}},\tilde{g})$ of main
interest in this paper.

###  2.1. The stable algebra of a Smale space

Following [32], we construct the stable groupoid of $(X,\varphi)$. Let ${\bf
P}$ denote a finite $\varphi$-invariant set of periodic points of $\varphi$
and define

$X^{u}({\bf P})=\\{x\in X\>|\>x\sim_{u}p\hbox{ for some }p\in{\bf P}\\}$

and

$G^{s}({\bf P}):=\\{(x,y)\in X^{u}({\bf P})\times X^{u}({\bf
P})\>|\>x\sim_{s}y\\}.$

Still following [32], a topology is defined on $G^{s}({\bf P})$ by
constructing a neighborhood base. Suppose $(x,y)\in G^{s}({\bf P})$. Then
there exists $k\in\mathbb{N}$ such that

$\varphi^{k}(x)\in X^{s}\left(\varphi^{k}(y),\frac{\epsilon_{X}}{2}\right).$

Since $\varphi$ is continuous there exists $\delta>0$ such that

$\varphi^{k}(X^{u}(y,\delta))\subseteq
X^{u}\left(\varphi^{k}(y),\frac{\epsilon_{X}}{2}\right).$

Using this data, we define a function
$h_{(x,y,\delta)}:X^{u}(y,\delta)\rightarrow X^{u}(x,\epsilon_{X})$ via

$z\mapsto\varphi^{-k}([\varphi^{k}(z),\varphi^{k}(x)])$

and have the following result from [32]:

######  Theorem 2.1.

The function $h=h_{(x,y,\delta)}$ is a homeomorphism onto its image and (by
letting $x$, $y$, and $\delta$ vary) the sets

$V(x,y,h,\delta):=\\{(h(z),z)\>|\>z\in X^{u}(y,\delta)\\}$

forms a neighborhood base for an étale topology on the groupoid $G^{s}({\bf
P})$. Moreover, the groupoid $G^{s}({\bf P})$ is amenable, second countable,
locally compact, and Hausdorff.

######  Definition 2.2.

The stable Ruelle groupoid is the groupoid $G^{s}({\bf P})\rtimes\mathbb{Z}$
where the $\mathbb{Z}$-actions is the one induced from $\varphi|_{X^{u}(P)}$;
the associated $C^{*}$-algebra is call the stable Ruelle algebra. It is worth
noting that this definition requires that ${\bf P}$ is $\varphi$-invariant (as
was assumed above).

###  2.2. The open subrelation, Fell algebra and its spectrum

We review the construction of $\sim_{0}$ and the associated groupoid
$C^{*}$-algebra studied in [11].

Let $\epsilon_{Y}>0$ be the global constant defined in [11] and
$\pi_{0}:X^{u}({\bf P})\rightarrow Y$ denote the map defined via

${\bf x}=(x_{n})_{n\in\mathbb{N}}\mapsto x_{0}.$

######  Definition 2.3.

Suppose ${\bf x}$ and ${\bf y}$ are in $X^{u}({\bf P})$. Then ${\bf
x}\sim_{0}{\bf y}$ if

  1. (1)

$\pi_{0}({\bf x})=\pi_{0}({\bf y})$ (i.e., $x_{0}=y_{0}$);

  2. (2)

there exists $0<\delta_{{\bf x}}<\epsilon_{Y}$ and open set $U\subseteq
X^{u}({\bf y},\epsilon_{Y})$ such that

$\pi_{0}(X^{u}({\bf x},\delta_{{\bf x}}))=\pi_{0}(U).$

Let $G_{0}({\bf P})=\\{({\bf x},{\bf y})\>|\>{\bf x}\sim_{0}{\bf y}\\}$.

To parse the requirements of Definition 2.3, the reader can return to Theorem
1.7 for a description of the local unstable sets. Results in [11] imply that
$G_{0}({\bf P})$ is an open subgroupoid of $G^{s}({\bf P})$ and hence that
$C^{*}(G_{0}({\bf P}))$ is a subalgebra of $C^{*}(G^{s}({\bf P}))$.
Furthermore, we can define

$G_{k}({\bf P})=\\{({\bf x},{\bf y})\>|\>\varphi^{k}({\bf
x})\sim_{0}\varphi^{k}({\bf y})\\}$

and one of the main results of [11] is the following:

######  Theorem 2.4.

Using the notation above, there is a nested sequence of étale subgroupoids

$G_{0}({\bf P})\subset G_{1}({\bf P})\subset G_{2}({\bf P})\subset\ldots$

of $G^{s}({\bf P})$ such that $G^{s}({\bf P})=\bigcup_{k=0}^{\infty}G_{k}({\bf
P})$ and each $G_{k}({\bf P})$ is isomorphic to $G_{0}({\bf P})$ in the
natural way $({\bf x},{\bf y})\mapsto(\varphi^{k}({\bf x}),\varphi^{k}({\bf
y}))$.

In practical terms, this theorem reduces the study of $G^{s}({\bf P})$ to
$G_{0}({\bf P})$ and likewise the study of the $C^{*}$-algebra
$C^{*}(G^{s}({\bf P}))$ to $C^{*}(G_{0}({\bf P}))$. Since $C^{*}(G_{0}({\bf
P}))$ is a type I $C^{*}$-algebra many of its properties can be determined
from its spectrum, $X^{u}({\bf P}))/{\sim_{0}}$.

Furthermore, again following [11], we define $\tilde{g}:X^{u}({\bf
P}))/{\sim_{0}}\rightarrow X^{u}({\bf P}))/{\sim_{0}}$ via

$[{\bf x}]\mapsto[(g(x_{0}),g(x_{1}),\ldots)]$

and $r:X^{u}({\bf P}))/{\sim_{0}}\rightarrow Y$ via

$[{\bf x}]\mapsto x_{0}.$

Proofs that $\tilde{g}$ and $r$ are well-defined can be found in [11].
Moreover, there is a commutative diagram

$\begin{CD}X^{u}({\bf P})@>{\varphi}>{}>X^{u}({\bf P})\\\
@V{q}V{}V@V{q}V{}V\\\ X^{u}({\bf P})/{\sim_{0}}@>{\tilde{g}}>{}>X^{u}({\bf
P})/{\sim_{0}}\\\ @V{r}V{}V@V{r}V{}V\\\ Y@>{g}>{}>Y\end{CD}$

######  Proposition 2.5.

The maps $q:X^{u}({\bf P})\rightarrow X^{u}({\bf P})/{\sim_{0}}$ and
$\tilde{g}:X^{u}({\bf P}))/{\sim_{0}}\rightarrow X^{u}({\bf P}))/{\sim_{0}}$
are each a local homeomorphisms (but not in general covering maps). In
particular, $q:X^{u}({\bf P})\rightarrow X^{u}({\bf P})/{\sim_{0}}$ is a
Hausdorff resolution of $X^{u}({\bf P})/{\sim_{0}}$.

Next, we discuss $G_{0}({\bf P})$ and $X^{u}({\bf P})/{\sim_{0}}$ for the
three examples considered in Section 1.2.

######  Example 2.6.

Recall the setup of Example 1.11. The space $Y$ is the unit circle,
$S^{1}\subseteq\mathbb{C}$ and (with fixed $n>1$) $g:S^{1}\rightarrow S^{1}$
is defined via $z\mapsto z^{n}$. In this case, we take ${\bf P}$ to be the set
containing the single point $(1,1,1,\ldots)$. Then $X^{u}({\bf P})$ is
homeomorphic to $\mathbb{R}$, $\sim_{0}$ can be identified with the
equivalence relation $x\sim y$ when $x-y\in\mathbb{Z}$ and $(X^{u}({\bf
P})/{\sim_{0}},\tilde{g})$ can be identified with the original system
$(S^{1},g)$. In fact the results in this example generalize to the case when
$g:Y\rightarrow Y$ is a local homeomorphism, in which case $(X^{u}({\bf
P})/{\sim_{0}},\tilde{g})=(Y,g)$, see [11] for details.

######  Example 2.7.

The relations $\sim_{0}$ and $\sim_{1}$ associated to the $aab/ab$-solenoid
defined in Example 1.13 can be viewed as in Figures 3 and 4.

The details for $G_{0}({\bf P})$ (i.e., $\sim_{0}$), which is illustrated in
Figure 3, are as follows. We take ${\bf P}$ to be the set containing the fixed
point associated to the wedge point (that is, $(p,p,p,\ldots)$ see Figure 2)
and then have that $X^{u}({\bf P})$ is homeomorphic to the real line.
Intervals labelled with $a$ (resp. $b$) are mapped by $\pi_{0}=g\circ q$ to
the outer (resp. inner) circle in $Y$. Identifying the endpoints of these
intervals as $\mathbb{Z}$, we have that two integer points are equivalent if
and only if the intervals to the left and right are labelled the same. While
non-integer points are equivalent if and only if they are in intervals with
the same label and their difference is in $\mathbb{Z}$.

Figure 4 illustrates $G_{1}({\bf P})$ in a similar way. The reader can find
more details in both cases in [11].

$\ldots$$a$$b$$\mathbf{p}$$a$$\mathbf{q}$$a$$b$$a$$\ldots$ Figure 3.
$\sim_{0}$ for $aab/ab$ solenoid
$\ldots$$a$$a$$b$$a$$b$$\mathbf{p}$$a$$a$$b$$\mathbf{q}$$a$$a$$b$$a$$b$$a$$a$$b$$\ldots$
Figure 4. $\sim_{1}$ for $aab/ab$ solenoid

We will show that in general $G_{0}({\bf P})$ is not closed in $G_{1}({\bf
P})$ using this example. Notice that the points $p$ and $q$ are equivalent
with respect to $\sim_{1}$ but not with respect to $\sim_{0}$. However there
is a sequence of points (namely $p+\frac{1}{n}\sim_{0}q+\frac{1}{n}$ where
$(p+\frac{1}{n},q+\frac{1}{n})$ converge to $(p,q)$). It follows that
$\sim_{0}$ is not closed in $\sim_{1}$.

Continuing with this example, we discuss $(X^{u}({\bf
P})/{\sim_{0}},\tilde{g})$. Notice that the map $g:Y\rightarrow Y$ is not a
local homeomorphism. Furthermore, the discussion of $G_{0}({\bf P})$ above
shows that $X^{u}({\bf P})/{\sim_{0}}$ is given as follows. The point $p\in Y$
splits into three non-Hausdorff points, denoted $ab,ba,aa$. This space is
illustrated in Figure 5. These points correspond to the three different
$\sim_{0}$-equivalence classes for “integer” points, as seen in Figure 3. Open
neighborhoods of the three points (that is, $ab,ba,aa$) are pictured in Figure
6.

$ab$$ba$$aa$$q_{1}$$q_{2}$$q_{3}$ Figure 5. $X^{u}({\bf P})/{\sim_{0}}$ for
$aab/ab$ solenoid.

$($$)$$ab$$ba$$aa$

$($$)$$aa$$ab$$ba$

$($$)$$aa$$ab$$ba$

Figure 6. Open neighborhoods of the three non-Hausdorff points in $X^{u}({\bf
P})/{\sim_{0}}$ for the $aab/ab$ solenoid.

Next, we discuss the map $\tilde{g}:X^{u}({\bf P})/{\sim_{0}}\rightarrow
X^{u}({\bf P})/{\sim_{0}}$. The map $\tilde{g}$ takes points labelled $ab$,
$ba$, and $aa$ to $ba$. It takes the point labelled $q_{1}$ to $aa$, the point
labelled $q_{2}$ to $ab$, and the point labelled $q_{3}$ to $ab$. The other
points (i.e., the ones without labels) are maps in the same way as
$g:Y\rightarrow Y$.

Some important properties to notice about $\tilde{g}:X^{u}({\bf
P})/{\sim_{0}}\rightarrow X^{u}({\bf P})/{\sim_{0}}$ are the following:

  1. (1)

$\tilde{g}$ is a local homeomorphisms, in particular it is open;

  2. (2)

$\tilde{g}(r^{-1}(p))=\\{ba\\}$ where $r:X^{u}({\bf P})/{\sim_{0}}\rightarrow
Y$ and $p$ is the wedge point of $Y$;

  3. (3)

informally, $\tilde{g}$ is locally expanding.

The importance of the second and third properties will be seen in Section 3.
In particular, $\tilde{g}$ will be shown to be “forward orbit expansive” in
Section 3, which makes precise the third item in the list above.

######  Example 2.8.

When $(Y,g)$ is the ab/ab-solenoid the map $g$ is not a local homeomorphism.
We take the set ${\bf P}$ to be the set containing the single element
$(p,p,\ldots)$ where $p$ is the wedge point. Then, using a similar method to
the one in the previous example, one can show that $X^{u}({\bf P})$ is
homeomorphic to $\mathbb{R}$, $X^{u}({\bf P})/{\sim_{0}}$ is homeomorphic to
the circle and $\tilde{g}$ is the two-fold cover of the circle. The main point
of this example is that $X^{u}({\bf P})/{\sim_{0}}$ can be Hausdorff even when
$g$ is not a local homeomorphism.

We restate [11, Lemma 4.4] here for ease of the reader; it will be used a
number of times below.

######  Lemma 2.9.

There exists constant $K_{0}>0$ such that if ${\bf x}$ and ${\bf y}$ are in
$X^{u}(P)$ and $x_{i}=y_{i}$ for $0\leq i\leq K_{0}$, then ${\bf
x}\sim_{0}{\bf y}$.

######  Theorem 2.10.

Suppose that $K_{0}$ is as in the previous lemma, $[{\bf y}]_{0}\in X^{u}({\bf
P})/{\sim_{0}}$ and there exists $V$ open in $Y$ such that

  1. (1)

$\pi_{0}({\bf y})=y_{0}\in V$ and

  2. (2)

$g^{K_{0}}|_{V}$ is injective.

Then $r^{-1}(y_{0})=\\{[{\bf y}]_{0}\\}$.

###### Proof.

Take $\delta>0$ such that $B(y_{0},\delta)\subseteq V$.

Suppose that $\hat{{\bf y}}=(y_{0},\hat{y_{1}},\hat{y_{2}},\ldots)\in
X^{u}({\bf P})$. We must show ${\bf y}\sim_{0}\hat{{\bf y}}$. By the previous
lemma,

$(g^{K_{0}}(y),\ldots,g(y),y,y_{1},\ldots)\sim_{0}(g^{K_{0}}(y),\ldots,g(y),y,\hat{y}_{1},\ldots).$

By the definition of $\sim_{0}$, there exists open sets in $X^{u}(P)$,
$U\subseteq X^{u}(\varphi^{K_{0}}({\bf y}),\delta)$ and $\hat{U}\subseteq
X^{u}(\varphi^{K_{0}}(\hat{{\bf y}}),\delta)$, such that

  1. (1)

$\varphi^{K_{0}}({\bf y})=(g^{K_{0}}(y),\ldots,g(y),y,y_{1},\ldots)\in U$;

  2. (2)

$\varphi^{K_{0}}(\hat{{\bf
y}})=(g^{K_{0}}(y),\ldots,g(y),y,\hat{y}_{1},\ldots)\in\hat{U}$;

  3. (3)

$\pi_{0}(U)=\pi(\hat{U})$.

Since $\varphi|_{X^{u}({\bf P})}$ is a homeomorphism, both
$\varphi^{-K_{0}}(U)$ and $\varphi^{-K_{0}}(\hat{U})$ are open in $X^{u}({\bf
P})$. Furthermore, ${\bf y}\in\varphi^{-K_{0}}(U)$ and $\hat{{\bf
y}}\in\varphi^{-K_{0}}(\hat{U})$. Notice that $\varphi^{-K_{0}}(U)\subseteq
X^{u}({\bf y},\delta)$ and $\varphi^{-K_{0}}(\hat{U})\subseteq X^{u}(\hat{{\bf
y}},\delta)$.

We show that
$\pi_{0}(\varphi^{-K_{0}}(U))=\pi_{0}(\varphi^{-K_{0}}(\hat{U}))$. Take ${\bf
z}\in\varphi^{-K_{0}}(U)$. Then

$(g^{K_{0}}(z_{0}),\ldots,g(z_{0}),z_{0},\ldots)\in U$

and by the third item in the list above, there exists
$(\bar{z}_{0},\bar{z}_{1},\ldots)\in\hat{U}\subseteq X^{u}(\hat{{\bf
y}},\delta)$ with $\bar{z}_{0}=g^{K_{0}}(z_{0})$. Since
$(\bar{z}_{0},\bar{z}_{1},\ldots)\in X^{u}(\varphi^{K_{0}}(\hat{{\bf
y}}),\delta)$, $\mathrm{d}_{Y}(z_{K_{0}},y_{0})<\delta$. However, $g^{K_{0}}$
is injective on $B(y_{0},\delta)\subseteq V$ hence $z_{K_{0}}=z_{0}$ (i.e.,
$z_{0}$ is the unique preimage of $g^{K_{0}}(z_{0})$ inside $V$). Hence
$\pi_{0}({\bf z})=\pi_{0}(z_{K_{0}},z_{K_{0}+1},\ldots)$ and the result
follows. ∎

######  Corollary 2.11.

If $(Y,g)$ is a Williams presolenoid, then $r$ is one-to-one on a dense open
set of $X^{u}({\bf P})/{\sim_{0}}$.

###### Proof.

The conditions of the previous theorem hold for any non-branched point in $Y$.
Since the set of branched points is closed and nowhere dense the result
follows. (The reader can find the precise definition of branched point in
[41]. This is the only proof in the present paper that uses this term.) ∎

To summarize, we believe the results of this section along with the three
examples gives sufficient motivation to consider the properties of the
dynamical systems $(X^{u}({\bf P}))/{\sim_{0}},\tilde{g})$ and its connection
to $(X,\varphi)$ and $(Y,g)$.

##  3\. Expansive dynamics in the non-Hausdorff setting

Since $X^{u}({\bf P})/{\sim_{0}}$ is a non-Hausdorff space and we are
interested in the dynamics on it, we discuss the general situation for
expansive maps in the non-Hausdorff setting. Much of the general theory is
based on work of Achigar, Artigue, and Monteverde [1], who consider the case
of expansive homeomorphisms in the non-Hausdorff setting. The work in this
section differs from [1] in that we relax the homeomorphism assumption and
consider a pair $(\tilde{Y},\tilde{g})$ of a compact topological space
$\tilde{Y}$ and a continuous surjective map
$\tilde{g}:\tilde{Y}\rightarrow\tilde{Y}$; additional conditions on
$\tilde{g}$ (such as expansiveness and being a local homeomorphism) will be
explicitly stated as required. We emphasize that $\tilde{Y}$ need not be
Hausdorff. It might be useful for the reader to have a copy of [1] while
reading the present section, so they compare results here to those in [1].

###  3.1. Expansive maps on non-Hausdorff spaces

######  Definition 3.1.

Suppose that $\mathcal{U}=\\{U_{i}\\}_{i\in I}$ is an open cover of a
topological space and $(x_{n})_{n\in\mathbb{N}}$ and
$(y_{n})_{n\in\mathbb{N}}$ are sequences in the space. Then we write

$\\{x_{n},y_{n}\\}_{n\in\mathbb{N}}\prec\mathcal{U}$

if for each $n\in\mathbb{N}$, there exists $i_{n}\in I$ such that

$x_{n}\hbox{ and }y_{n}\hbox{ are elements of }U_{i_{n}}.$

We use similar notation for two-sided sequences (that is, for
$(x_{n})_{n\in\mathbb{Z}}$ and $(y_{n})_{n\in\mathbb{Z}}$)

In [1, Definition 2.1], the reader can find a notion of expansiveness for
homeomorphisms of compact, non-Hausdorff spaces. Upon replacing the conditions
of [1] involving the $\mathbb{Z}$-action associated to a homeomorphism, with
an $\mathbb{N}$-action we arrive at a completely analogous definition of orbit
expansive maps that need not be invertible.

######  Definition 3.2.

We say that $(\tilde{Y},\tilde{g})$ is forward orbit expansive if there exists
a finite open cover of $\tilde{Y}$, $\mathcal{U}=\\{U_{i}\\}_{i=1}^{l}$, such
that if $x$, $y$ are in $Y$ with

$\\{\tilde{g}^{n}(x),\tilde{g}^{n}(y)\\}_{n\in\mathbb{N}}\prec\mathcal{U},$

then $x=y$.

######  Proposition 3.3.

Suppose that $(\tilde{Y},\tilde{g})$ is forward orbit expansive and
$(\tilde{X},\tilde{\varphi})$ is the associated solenoid. Then
$(\tilde{X},\tilde{\varphi})$ is orbit expansive in the sense of [1,
Definition 2.1].

###### Proof.

Take an open cover of $\tilde{Y}$,
$\mathcal{U}_{\tilde{Y}}=\\{U_{i}\\}_{i=1}^{l}$, as in the definition of
forward orbit expansive. Form

$\mathcal{U}_{X}=\\{U_{i}\times\tilde{Y}\times\tilde{Y}\times\ldots\\}_{i=1}^{l},$

which is an open cover of $\tilde{X}$.

Suppose ${\bf x}=(x_{0},x_{1},\ldots)$ and ${\bf y}=(y_{0},y_{1},\ldots)$ are
in $\tilde{X}$ with

$\\{\varphi^{n}({\bf x}),\varphi^{n}({\bf
y})\\}_{n\in\mathbb{Z}}\prec\mathcal{U}_{X}.$

Then there exists $i_{0}\in\\{1,\ldots l\\}$ such that ${\bf x}$ and ${\bf y}$
are in $U_{i_{0}}\times\tilde{Y}\times\tilde{Y}\times\ldots$. Hence, $x_{0}$
and $y_{0}$ are in $U_{i_{0}}$. Using the fact that $\tilde{\varphi}({\bf
x})=(\tilde{g}(x_{0}),\tilde{g}(x_{1}),\ldots)$ and an induction argument give
us

$\\{\tilde{g}^{n}(x_{0}),\tilde{g}^{n}(y_{0})\\}\prec\mathcal{U}_{Y}.$

The definition of forward orbit expansive then implies that $x_{0}=y_{0}$.
Using the fact that $\tilde{\varphi}^{-1}({\bf x})=(x_{1},x_{2},\ldots)$
allows us to repeat the above argument to get $x_{1}=y_{1}$. The proof is then
completed by showing that $x_{n}=y_{n}$ for each $n\in\mathbb{N}$ in a similar
way; this implies that ${\bf x}={\bf y}$. ∎

######  Proposition 3.4.

(compare with [1, Proposition 2.5]) Suppose that $(\tilde{Y},\tilde{g})$ is
forward orbit expansive, then $\tilde{Y}$ is $T_{1}$.

###### Proof.

Take an open cover of $\tilde{Y}$, $\mathcal{U}=\\{U_{i}\\}_{i=1}^{l}$, as in
the definition of forward orbit expansive. Suppose $x\neq y$ are in
$\tilde{Y}$. Then, using the definition of forward orbit expansive, there
exists $n\in\mathbb{N}$ and $i_{x}\neq i_{y}$ such that $\tilde{g}^{n}(x)\in
U_{i_{x}}$ but $\tilde{g}^{n}(x)\not\in U_{i_{y}}$ and $\tilde{g}^{n}(y)\in
U_{i_{y}}$ but $\tilde{g}^{n}(y)\not\in U_{i_{x}}$. Since $\tilde{g}$ is
continuous, the sets $\tilde{g}^{-n}(U_{i_{x}})$ and
$\tilde{g}^{-n}(U_{i_{y}})$ are open and the properties in the previous
sentence imply that these sets are a $T_{1}$-separation of $x$ and $y$. ∎

######  Proposition 3.5.

(compare with [1, Theorem 2.7]) Suppose that $Y$ is compact and Hausdorff,
$(Y,g)$ is forward orbit expansive, and $g$ is an open map. Then $Y$ is
metrizable and $(Y,g)$ is forward expansive.

###### Proof.

To begin, we note that $g$, in addition to being open, is also a closed map.
Let $\mathcal{U}$ be an open cover of $Y$ as in the definition of forward
orbit expansive. Given $U\in\mathcal{U}$ and $y\in U$, there exists $V_{y}$
open such that

$y\in V_{y}\subseteq\overline{V}_{y}\subseteq U.$

Using this fact and the compactness of $Y$, there exists an open
$\mathcal{V}=\\{V_{1},\ldots,V_{m}\\}$ such that for each $V_{i}$ there exists
$U\in\mathcal{U}$ with $\overline{V}_{i}\subseteq U$. This property and
forward orbit expansiveness imply that

${\rm card}\left(\bigcap_{i\geq 0}g^{i}(\overline{V}_{k_{i}})\right)\leq 1$

for any $(k_{i})_{i\geq 0}\in\\{1,\ldots m\\}^{\mathbb{N}}$.

Now suppose that $y\in Y$ and $W$ is an open set with $y\in W$. Since
$\mathcal{V}$ is a cover and $g$ is onto, there exists $(k_{i})_{i\geq
0}\in\\{1,\ldots m\\}^{\mathbb{N}}$ such

$y\in\bigcap_{i\geq 0}g^{i}(V_{k_{i}}).$

It follows that $\bigcap_{i\geq 0}g^{i}(\overline{V}_{k_{i}})=\\{y\\}$ and
hence that

$W^{c}\cap\left(\bigcap_{i\geq
0}g^{i}(\overline{V}_{k_{i}})\right)=\emptyset.$

By the reformulation of compactness via the finite intersection property (this
is also where we use the fact that $g$ is a closed map) there exists
$N\in\mathbb{N}$ such that

$W^{c}\cap\left(\bigcap_{i\geq
0}^{N}g^{i}(\overline{V}_{k_{i}})\right)=\emptyset$

and hence $\bigcap_{i\geq 0}^{N}g^{i}(V_{k_{i}})\subseteq W$.

It now follows that the collection

$\left\\{\bigcap_{i\geq 0}^{N}g^{i}(V_{k_{i}})\mid N\in\mathbb{N}\hbox{ and
}k_{i}\in\\{1,\ldots,m\\}\right\\}$

is a basis for the topology on $Y$. Since this collection is countable (and
$Y$ is compact and Hausdorff), the topology on $Y$ is metrizable.

For the second part of the theorem, again let $\mathcal{U}$ be an open cover
of $Y$ as in the definition of forward orbit expansive. Fix a metric on $Y$
that induces the given topology. Let $\delta>0$ be the Lebesgue number of
$\mathcal{U}$ with respect to $d$. One can show that $\delta$ is an expansive
constant for $(Y,g)$; the details are omitted. ∎

######  Proposition 3.6.

(compare with [1, Proposition 2.14]) Suppose that $(\tilde{Y},\tilde{g})$ is
forward orbit expansive, and $\tilde{g}$ is an open map. Then, for each
$n\in\mathbb{N}$, $\tilde{g}^{n}$ is forward orbit expansive.

###### Proof.

Take an open cover of $\tilde{Y}$, $\mathcal{U}=\\{U_{i}\\}_{i=1}^{l}$, as in
the definition of forward orbit expansive for $\tilde{g}$. Using the openness
of the map $\tilde{g}$ we have that

$\mathcal{U}_{n}=\\{U_{i_{1}}\cap\tilde{g}(U_{i_{2}})\cap\ldots\cap\tilde{g}^{n-1}(U_{i_{n-1}})\mid
i_{1},i_{2},\ldots,i_{n-1}\in\\{1,\ldots,l\\}\\}$

is an open cover of $\tilde{Y}$. Moreover, the fact that $\mathcal{U}$ is
forward orbit expansive for $\tilde{g}$ implies that $\mathcal{U}_{n}$ is
forward orbit expansive for $\tilde{g}^{n}$; the details are omitted. ∎

######  Proposition 3.7.

Suppose that $(\tilde{Y},\tilde{g})$ is forward orbit expansive with
$\tilde{g}$ open, $(\tilde{Y}_{{\rm Haus}},\tilde{g}_{{\rm Haus}})$ denotes
its Hausdorffization (see [28]), and
$\tilde{r}:\tilde{Y}\rightarrow\tilde{Y}_{{\rm Haus}}$ denotes the natural
map. Furthermore suppose that there exists $L\in\mathbb{N}$ such that for each
$y\in\tilde{Y}_{{\rm Haus}}$, $\tilde{g}^{L}(\tilde{r}^{-1}(y))$ is a
singleton. Then $\tilde{Y}$ is locally Hausdorff.

###### Proof.

By Proposition 3.6 and replacing $\tilde{g}$ with $\tilde{g}^{L}$, we can
assume that $L=1$. Take $\mathcal{U}=\\{U_{i}\\}_{i=1}^{l}$, as in the
definition of forward orbit expansive and $x\in\tilde{Y}$. Let $U=U_{i_{0}}$
where $x\in U_{i_{0}}$.

We will show that $U$ with the subspace topology is Hausdorff. Suppose $y_{1}$
and $y_{2}$ in $U$ cannot be separated. We will show that $y_{1}=y_{2}$. Since
$\tilde{g}$ is continuous, $\tilde{g}(y_{1})$ and $\tilde{g}(y_{2})$ cannot be
separated, but then $\tilde{g}(y_{1})=\tilde{g}(y_{2})$ (since $L=1$). Hence
$\\{\tilde{g}^{n}(x),\tilde{g}^{n}(y)\\}\prec\mathcal{U}$ and by the
definition of forward expansive, $y_{1}=y_{2}$ as required. ∎

The next example shows that an additional condition (e.g., the assumption that
$\tilde{g}^{L}(\tilde{r}^{-1}(y))$ is a singleton in the previous result) is
required to ensure the solenoid is Hausdorff.

######  Example 3.8.

Let $\tilde{Y}$ be the unit circle in $\mathbb{C}$ with two “1”s, see Figure
7.

$1^{a}$$1^{b}$ Figure 7. Circle with two 1s pre-solenoid

We define $\tilde{g}:\tilde{Y}\rightarrow\tilde{Y}$ to be the two fold
covering map for points other than $-1$, $1^{a}$ and $1^{b}$; for those points
we define

$-1\mapsto 1^{a}\hbox{ and }1^{a}\mapsto 1^{b}\hbox{ and }1^{b}\mapsto 1^{a}.$

One can check that $\tilde{g}$ is continuous, onto, and open. Furthermore, the
solenoid associated to $(\tilde{Y},\tilde{g})$ is not Hausdorff since the
points

$(1^{a},1^{b},1^{a},\ldots)\hbox{ and }(1^{b},1^{a},1^{b},\ldots)$

cannot be separated.

######  Theorem 3.9.

Suppose that $(\tilde{Y},\tilde{g})$ is forward orbit expansive with
$\tilde{g}$ open (and continuous and onto as usual in this section) and there
exists $(Y,g)$ and $r:\tilde{Y}\rightarrow Y$ with

  1. (1)

$Y$ compact and Hausdorff;

  2. (2)

$g$ is continuous and onto;

  3. (3)

$r$ continuous, onto, and $r\circ\tilde{g}=g\circ r$;

  4. (4)

there exists $L\in\mathbb{N}$ such that for each $y\in Y$, $g^{L}(r^{-1}(y))$
is a singleton.

Then the solenoids associated to $(\tilde{Y},\tilde{g})$ and $(Y,g)$ are
conjugate. In particular, the solenoid associated to $(\tilde{Y},\tilde{g})$
is Hausdorff.

###### Proof.

Following [41], we will define a shift equivalence between
$(\tilde{Y},\tilde{g})$ and $(Y,g)$. In addition to the map
$r:\tilde{Y}\rightarrow Y$ in the statement of the theorem, we have the map
$s:Y\rightarrow\tilde{Y}$ defined via

$y\mapsto\tilde{g}^{L}(r^{-1}(y))$

where we have abused notation in the sense that $\tilde{g}^{L}(r^{-1}(y))$ is
really a set that contains a single element. The map $s$ is onto since $r$ and
$\tilde{g}$ are onto. One can also show that $s$ is continuous.

To show that $r$ and $s$ define a shift equivalence, we must show that

  1. (1)

$r\circ\tilde{g}=g\circ r$;

  2. (2)

$s\circ g=\tilde{g}\circ s$;

  3. (3)

$s\circ r=\tilde{g}^{L}$;

  4. (4)

$r\circ s=g^{L}$.

The first item is an assumption of the theorem. The fourth is similar to the
third so only the proofs of the second and third will be considered in detail.

To see that $s\circ g=\tilde{g}\circ s$, we have that

$(s\circ g)(y)=\tilde{g}^{L}(r^{-1}(g(y)))\hbox{ and }(\tilde{g}\circ
s)(y)=\tilde{g}^{L+1}(r^{-1}(y)).$

Using $r\circ\tilde{g}=g\circ r$, it follows that
$\tilde{g}(r^{-1}(y))\subseteq r^{-1}(g(y))$. Apply $\tilde{g}^{L}$ to both
sides, we find that $\tilde{g}^{L}(r^{-1}(g(y)))$ is a singleton by
assumption. Hence $\tilde{g}^{L+1}(r^{-1}(y))$ is the same singleton because
it is non-empty and contained in the singleton set
$\tilde{g}^{L}(r^{-1}(g(y)))$.

To see that $s\circ r=\tilde{g}^{L}$, we have

$(s\circ r)(\tilde{y})=\tilde{g}^{L}(r^{-1}(r(\tilde{y}))).$

Now, $\tilde{y}\in r^{-1}(r(y))$, so
$\tilde{g}^{L}(\tilde{y})\in\tilde{g}^{L}(r^{-1}(r(\tilde{y})))$. The set
$\tilde{g}^{L}(r^{-1}(r(\tilde{y})))$ is a singleton, so it must be equal to
$\\{\tilde{g}^{L}(\tilde{y})\\}$ as required.

Using work in [41], it follows that the map $S:X\rightarrow\tilde{X}$ defined
via

$(y_{0},y_{1},y_{2},\ldots)\mapsto(s(y_{0}),s(y_{1}),s(y_{2}),\ldots)$

is a conjugacy from $(X,\varphi)$ (the solenoid associated to $(Y,g)$) to
$(\tilde{X},\tilde{\varphi})$ (the solenoid associated to
$(\tilde{Y},\tilde{g})$). Its inverse is given by $R:\tilde{X}\rightarrow X$
defined via

$(\tilde{y}_{0},\tilde{y}_{1},\tilde{y}_{2},\ldots)\mapsto(r(\tilde{y}_{0}),r(\tilde{y}_{1}),r(\tilde{y}_{2}),\ldots).$

For the final part of the theorem, $X$ is Hausdorff because $Y$ is Hausdorff.
Hence, the solenoid associated to $\tilde{X}$ is Hausdorff as well because $X$
and $\tilde{X}$ are homeomorphic. ∎

###  3.2. The dynamics on the locally Hausdorff quotient space

We now come to the main application of Subsection 3.1 on Wieler-Smale spaces.

######  Theorem 3.10.

The dynamical system $(X^{u}(P)/{\sim_{0}},\tilde{g})$ is forward orbit
expansive. Moreover, $\tilde{g}$ is a local homeomorphism and there exists
$L\in\mathbb{N}$ such that for each $y\in\tilde{Y}_{{\rm Haus}}$,
$\tilde{g}^{L}(r^{-1}(y))$ is a singleton.

###### Proof.

Recall the constants $\beta>0$ and $0<\lambda<1$ in the definition of a Wieler
presolenoid, see Definition 1.4. Using the nature of the local stable and
unstable sets for a Wieler solenoid (see Definition 1.5) there exists
$0<\delta<\frac{\beta}{2}$ such that if ${\bf x}\neq{\bf y}$ are in
$X^{u}({\bf z},\delta)$ (for some ${\bf z}$) then $x_{0}\neq y_{0}$. Using the
compactness of $X^{u}(P)/{\sim_{0}}$, we have an open cover of the form

$\mathcal{U}=\\{\pi_{0}(X^{u}({\bf z}_{i},\delta))\cap
r^{-1}(B(w_{j},\frac{\beta}{2}))\\}_{i\in I,j\in J}.$

where ${\bf z}_{i}\in X^{u}(P)$, $w_{j}\in Y$, $I$ and $J$ are finite index
sets, and the fact that $\pi_{0}$ is an open map and $r$ is continuous ensures
that the given sets are open.

Let $[{\bf x}]_{0}\neq[{\bf y}]_{0}$ be in $X^{u}(P)/{\sim_{0}}$. We will find
$N\in\mathbb{N}$ such that for each $i$, the two element set
$\\{\tilde{g}^{N}([{\bf x}]_{0}),\tilde{g}^{N}([{\bf y}]_{0})\\}$ is not a
subset of $\pi_{0}(X^{u}({\bf z}_{i},\delta))\cap
r^{-1}(B(w_{j},\frac{\beta}{2}))$.

We are done unless there exists $i_{0}$ such that

$[{\bf x}]_{0}\hbox{ and }[{\bf y}]_{0}\hbox{ are both in }\pi_{0}(X^{u}({\bf
z}_{i_{0}},\delta)).$

Let ${\bf x}$ and ${\bf y}$ be points in $X^{u}({\bf z}_{i_{0}},\delta)$
representing $[{\bf x}]_{0}$ and $[{\bf y}]_{0}$ respectively. By assumption,
$x_{0}\neq y_{0}$ and

$\mathrm{d}_{Y}(x_{K},y_{K})<2\delta<\beta$

where $K$ is the constant in Wieler’s Axioms. Using the first of Wieler’s
Axioms,

$0<\mathrm{d}_{Y}(x_{0},y_{0})\leq\gamma^{K}\mathrm{d}_{Y}(g^{K}(x_{0}),g^{K}(y_{0})).$

Using this inequality and the fact that $x_{0}\neq y_{0}$, we have that
$g^{K}(x_{0})\neq g^{K}(y_{0})$. If
$\mathrm{d}_{Y}(g^{K}(x_{0}),g^{K}(y_{0}))>\beta$, then we stop. Otherwise,
since

$\mathrm{d}_{Y}(x_{0},y_{0})<2\delta<\beta,$

we can again use the first of Wieler’s Axioms. We get that

$\mathrm{d}_{Y}(g^{K}(x_{0}),g^{K}(y_{0}))\leq\gamma^{K}\mathrm{d}_{Y}(g^{2K}(x_{0}),g^{2K}(y_{0})).$

Using this inequality and the previous one, we obtain

$0<\mathrm{d}_{Y}(x_{0},y_{0})\leq\gamma^{2K}\mathrm{d}_{Y}(g^{2K}(x_{0}),g^{2K}(y_{0})).$

Using the fact that $0<\gamma<1$ and possibly repeating this process, there
exists $N\in\mathbb{N}$ such that $d(g^{N}(x_{0}),g^{N}(y_{0}))>\beta$. Now,

$r(\tilde{g}^{N}([{\bf x}]_{0}))=g^{N}(x_{0})\hbox{ and }r(\tilde{g}^{N}([{\bf
y}]_{0}))=g^{N}(y_{0}).$

Therefore, $\tilde{g}^{N}([{\bf x}]_{0})$ and $\tilde{g}^{N}([{\bf y}]_{0})$
cannot both be in $r^{-1}(B(w_{j},\frac{\beta}{2}))$ for any $j$. Thus,
$\tilde{g}$ is forward orbit expansive.

For the second part of the theorem, it was already noted that $\tilde{g}$ is a
local homeomorphism and the existence of the required $L$ follows from Lemma
2.9. ∎

###  3.3. Inverse limit space associated to the spectrum

The main goal of this section is to show that the inverse limit formed from
$(X^{u}(P)/{\sim_{0}},\tilde{g})$ also gives the Wieler solenoid associated to
$(Y,g)$. This is perhaps somewhat surprising in light of the fact that
$X^{u}(P)/{\sim_{0}}$ is often non-Hausdorff.

To fix notation, $(Y,g)$ is assumed to satisfy Wieler’s axioms, $(X,\varphi)$
is the associated solenoid and $(X^{u}(P)/{\sim_{0}},\tilde{g})$ is as in the
previous section. We let

$\tilde{X}:=\varprojlim(X^{u}(P)/{\sim_{0}},\tilde{g})=\\{(\tilde{y}_{n})_{n\in\mathbb{N}}=(\tilde{y}_{0},\tilde{y}_{1},\tilde{y}_{2},\ldots)\>|\>\tilde{g}(\tilde{y}_{i+1})=\tilde{y}_{i}\hbox{
for each }i\geq 0\\}$

with the map $\tilde{\varphi}:\tilde{X}\rightarrow\tilde{X}$ be defined via

$\tilde{\varphi}(\tilde{y}_{0},\tilde{y}_{1},\tilde{y}_{2},\ldots)=(\tilde{g}(\tilde{y}_{0}),\tilde{g}(\tilde{y}_{1}),\tilde{g}(\tilde{y}_{2}),\ldots)=(\tilde{g}(\tilde{y}_{0}),\tilde{y}_{0},\tilde{y}_{1},\ldots).$

Again we will make use of Lemma 4.4 in [11], which we restate here for the
reader.

We have the two maps. The first one was defined in [11]; it is

$r:X^{u}(P)/{\sim_{0}}\rightarrow Y\hbox{ defined via }[{\bf x}]_{0}\mapsto
x_{0}$

where we note that the definition of $\sim_{0}$ implies that $r$ is well-
defined. Furthermore, it is continuous and surjective.

The second map is

$s:Y\rightarrow X^{u}(P)/{\sim_{0}}\hbox{ defined via
}y\mapsto[g^{K_{0}}(y),g^{K_{0}-1}(y),\ldots,g(y),y,\ldots]_{0}$

where we note that the Lemma 4.4 in [11] implies that $s$ is well-defined.

######  Theorem 3.11.

Using the notation in the previous paragraphs, the maps $r$ and $s$ define a
shift equivalence. That is, they satisfy

$r\circ\tilde{g}=g\circ r,s\circ g=g\circ s,r\circ s=g^{K_{0}}\hbox{ and
}s\circ r=\tilde{g}^{K_{0}}.$

Moreover, the map $S:X\rightarrow\tilde{X}$ defined via

$(y_{0},y_{1},y_{2},\ldots)\mapsto(s(y_{0}),s(y_{1}),s(y_{2}),\ldots)$

is a conjugacy from $(X,\varphi)$ to $(\tilde{X},\tilde{\varphi})$.

###### Proof.

This follows from the statement and proof of Theorem 3.9. We note that the
assumptions of Theorem 3.9 hold by Theorem 3.10. Also, the reader can check
that the map $s$ defined just before the statement of the current theorem is
equal to the map $s$ consider in Theorem 3.9. ∎

######  Corollary 3.12.

If $(X,\varphi)$ is an irreducible Smale space with totally disconnected
stable sets, then there exists $(\tilde{Y},\tilde{g})$ such that $\tilde{g}$
is a surjective local homeomorphism that is forward orbit expansive and
$(X,\varphi)$ is conjugate the solenoid associated to $(\tilde{Y},\tilde{g})$.

###### Proof.

This follows from the previous result and Wieler’s theorem. ∎

###  3.4. The relationship between $(X,\varphi)$, $(X^{u}({\bf
P})/{\sim_{0}},\tilde{g})$ and $(Y,g)$

It follows from Theorem 3.11 that there is a continuous surjection from $X$ to
$X^{u}(P)/{\sim_{0}}$. However, we can write it is explicitly without
reference to the conjugacy in Theorem 3.11:

######  Theorem 3.13.

Define $p:X\rightarrow X^{u}(P)/{\sim_{0}}$ via

$x\mapsto[y]_{0}$

where $y\in X^{u}(P)\cap X^{s}(x,\frac{\epsilon_{X}}{2})$. Then $p$ is a
continuous surjection and for each $z\in X^{u}(P)/{\sim_{0}}$, $p^{-1}(z)$ is
a Cantor set.

###### Proof.

To begin, we must show that $p$ is well-defined. Suppose that $y^{\prime}$ is
another element in $X^{u}(P)\cap X^{s}(x,\frac{\epsilon_{X}}{2})$. Firstly,
since both $y$ and $y^{\prime}$ are in $X^{s}(x,\frac{\epsilon_{X}}{2})$ we
have that $\pi_{0}(x)=\pi_{0}(y)=\pi_{0}(y^{\prime})$. Moreover, properties of
the bracket implies that the map $h:X^{u}(y,\epsilon_{X})\rightarrow
X^{u}(y^{\prime},\epsilon_{X})$ defined via

$z\mapsto[z,y^{\prime}]$

is well defined and that $\pi_{0}(z)=\pi_{0}(h(z))$ for each $z\in
X^{u}(y,\epsilon_{X})$. This implies that $[y]_{0}=[y^{\prime}]_{0}$.

An equivalent definition of $p$ is the following: Let
$U(x,\frac{\epsilon_{X}}{2})$ denote the image of the bracket of the set
$X^{s}(x,\frac{\epsilon_{X}}{2})\times X^{u}(x,\frac{\epsilon_{X}}{2})$. Given
$x\in X$ and $w\in X^{u}(P)\cap U(x,\frac{\epsilon_{X}}{2})$ then
$p(x)=[x,w]$. To see this is the same as the previous definition, we need only
check that $[x,w]$ is an element of $X^{u}(P)\cap
X^{s}(x,\frac{\epsilon_{X}}{2})$ but this follows from the definitions of the
bracket and the set $U(x,\frac{\epsilon_{X}}{2})$.

∎

######  Remark 3.14.

If $g$ is a local homeomorphism satisfying Wieler’s axioms, then
$X^{u}(P)/{\sim_{0}}=Y$ and the map $p:X\to Y$ is a locally trivial bundle of
Cantor sets. This fact follows from [8, Theorem 3.12]. Already for Williams
solenoids, local triviality of $p$ fails to hold in general [14].

Moving forward we will use an abuse of notation to identify sets of the form
$X^{s}(x,\delta_{1})\times X^{u}(x,\delta_{2})$ with their image in $X$ under
the bracket map. For example, using this convention, there would be no
reference to the set $U(x,\frac{\epsilon_{X}}{2})$ in the proof of the
previous theorem.

######  Lemma 3.15.

Suppose $x\in X$ and $0<\delta<\delta^{\prime}<\epsilon^{\prime}$. Then

$p(X^{s}(x,\delta)\times
X^{u}(x,\epsilon^{\prime}))=p(X^{s}(x,\delta^{\prime})\times
X^{u}(x,\epsilon^{\prime})).$

###### Proof.

Let $z\in X^{s}(x,\delta^{\prime})\times X^{u}(x,\epsilon^{\prime})$. Then
$[z,x]\in X^{u}(x,\epsilon^{\prime})\subseteq X^{s}(x,\delta)\times
X^{u}(x,\epsilon^{\prime})$. Furthermore, taking $w\in X^{u}(P)\cap
U(x,\frac{\epsilon_{X}}{2})$, we have that

$p(z)=[z,w]=[[z,x],w]=p([z,x])$

as required. ∎

######  Theorem 3.16.

Using the notation in the past few paragraphs, we have that the following
diagram commutes:

$\begin{CD}X@>{\varphi}>{}>X\\\ @V{p}V{}V@V{p}V{}V\\\
X^{u}(P)/{\sim_{0}}@>{\tilde{g}}>{}>X^{u}(P)/{\sim_{0}}\\\
@V{r}V{}V@V{r}V{}V\\\ Y@>{g}>{}>Y\end{CD}$

where $r$ is defined above (see Section 2.2) and $r\circ p$ is equal to the
projection map $\pi_{0}:X\rightarrow Y$.

###### Proof.

The result follows directly from the definitions of the relevant maps; the
details are omitted. ∎

######  Theorem 3.17.

The map $p$ induces a bijection between periodic points of $\varphi$ and
periodic points of $\tilde{g}$ and hence $\varphi$ and $\tilde{g}$ have the
same zeta function. Furthermore, if the set of periodic points with respect to
$\varphi$ is dense in $X$, then the set of periodic points with respect to
$\tilde{g}$ is dense in $X^{u}(P)/{\sim_{0}}$.

###### Proof.

The first statement using Theorem 3.11 and applying the proof of [41, Lemma
5.3] to our situation. The second follows since $p$ is onto so the image of a
dense set in $X$ under $p$ is dense in $X^{u}(P)/{\sim_{0}}$. ∎

######  Theorem 3.18.

The map $r:X^{u}(P)/{\sim_{0}}\rightarrow Y$ induces a bijection between the
periodic points of $\tilde{g}$ and the periodic points of $g$. Hence
$\tilde{g}$ and $g$ have the same zeta function.

###### Proof.

This follows from Theorem 3.11 and [41, Theorem 5.2 Part (3)], see in
particular the argument just before Lemma 5.3 on page 189 of [41]. ∎

######  Theorem 3.19.

The following are equivalent

  1. (1)

$(X,\varphi)$ is mixing,

  2. (2)

$(X^{u}(P)/{\sim_{0}},\tilde{g})$ is mixing,

  3. (3)

$(Y,g)$ is mixing.

###### Proof.

We only prove the case $(X,\varphi)$ is mixing implies
$(X^{u}(P)/{\sim_{0}},\tilde{g})$ is mixing in detail. Let $U$ and $V$ be non-
empty open sets in $X^{u}(P)/{\sim_{0}}$. Then $p^{-1}(U)$ and $p^{-1}(V)$ are
non-empty open sets in $X$ and since $(X,\varphi)$ is mixing, there exists
$N\in\mathbb{N}$ such that

$\varphi^{n}(p^{-1}(U))\cap p^{-1}(V)\neq\emptyset\hbox{ for each }n\geq N$

Then, using the fact that $p$ is onto,

$\displaystyle p(\varphi^{n}(p^{-1}(U))\cap p^{-1}(V))$
$\displaystyle\subseteq(p(\varphi^{n}(p^{-1}(U))))\cap p(p^{-1}(V))$
$\displaystyle\subseteq(\tilde{g}^{n}(p(p^{-1}(U))))\cap p(p^{-1}(V))$
$\displaystyle=\tilde{g}^{n}(U)\cap V$

This implies the result.

Notice that we have only used the following properties: $p$ is onto,
continuous and $\tilde{g}\circ p=p\circ\varphi$. Thus to see that
$(X^{u}(P)/{\sim_{0}},\tilde{g})$ is mixing implies that $(Y,g)$ is mixing,
one replaces $p$ with $r$ in the previous argument. ∎

##  4\. Full projections

The main goal of this section is to prove that $C^{*}(G_{0}({\bf P}))$
contains a full projection. This result is used in Section 5 to construct
unital Cuntz-Pimsner models. In light of [10], the existence of a full
projection in $C^{*}(G_{0}({\bf P}))$ puts restrictions on the type of Fell
algebras that appear as $C^{*}(G_{0}({\bf P}))$.

###  4.1. Dynamical results

######  Lemma 4.1.

Suppose $(X^{u}(P)/{\sim_{0}},\tilde{g})$ is mixing and $U$ is a non-empty
open set in $X^{u}(P)/{\sim_{0}}$ such that there exists $L\in\mathbb{N}$ that
satisfies $\tilde{g}^{L}(U)\subseteq U$. Then $U$ is dense in
$X^{u}(P)/{\sim_{0}}$.

###### Proof.

The result will follow by showing that if $V$ is a non-empty open set in
$X^{u}(P)/{\sim_{0}}$ then $U\cap V\neq\emptyset$. Since $\tilde{g}$ is
mixing, there exists $N\in\mathbb{N}$ such that for each $n\geq N$,

$\tilde{g}^{n}(U)\cap V\neq\emptyset$

Thus $\emptyset\neq\tilde{g}^{N\cdot L}(U)\cap V\subseteq U\cap V$. ∎

######  Lemma 4.2.

Suppose that $(X,\varphi)$ is mixing and $\emptyset\neq U\subseteq
X^{u}(P)/{\sim_{0}}$ is open. Then there exists $\emptyset\neq V\subseteq U$
open and $N\in\mathbb{N}$ such that $V\subseteq\tilde{g}^{N}(V)$.

###### Proof.

Since periodic points of $(X,\varphi)$ are dense, there exists $x\in
p^{-1}(U)$ and $N\in\mathbb{N}$ such that $\varphi^{N}(x)=x$. Since $\varphi$
and $p$ are both continuous, there exists $\delta>0$ and
$0<\delta^{\prime}<\epsilon^{\prime}$ such that
$\varphi^{N}(X^{u}(x,\delta))\subseteq X^{u}(x,\delta^{\prime})$ and
$p(X^{s}(x,\delta)\times X^{u}(x,\delta))\subseteq U$.

The set $X^{s}(x,\delta)\times X^{u}(x,\delta)$ is open in $X$ and so
$p(X^{s}(x,\delta)\times X^{u}(x,\delta))$ is open in $X^{u}(P)/{\sim_{0}}$.
We also have that $X^{u}(x,\delta)\subseteq\varphi^{N}(X^{u}(x,\delta))$.

By Lemma 3.15 and the fact that $\varphi$ contracts in the local stable
direction

$p(\varphi^{N}(X^{s}(x,\delta)\times
X^{u}(x,\delta)))=p(X^{s}(x,\delta)\times\varphi^{N}(X^{u}(x,\delta)))$

and using $X^{u}(x,\delta)\subseteq\varphi^{N}(X^{u}(x,\delta))$, we have that

$p(X^{s}(x,\delta)\times X^{u}(x,\delta))\subseteq
p(\varphi^{N}(X^{s}(x,\delta)\times
X^{u}(x,\delta)))=\tilde{g}^{N}(p(X^{s}(x,\delta)\times X^{u}(x,\delta)))$

∎

######  Theorem 4.3.

Suppose that $(X,\varphi)$ is mixing and $U\subseteq X^{u}(P)/{\sim_{0}}$ is a
non-empty, open set. Then there exists $N\in\mathbb{N}$ such that
$\tilde{g}^{N}(U)=X^{u}(P)/{\sim_{0}}$.

###### Proof.

By Lemma 4.2, there exists a nonempty open set $V$ such that $V\subseteq U$
and $V\subseteq\tilde{g}^{K}(V)$. Moreover, the $V$ constructed in Lemma 4.2
depended on $\delta>0$ and so we denote the set associated to $\delta>0$ as
$V_{\delta}$. In more detail, $V_{\delta}=p(X^{s}(z,\delta)\times
X^{u}(z,\delta))$ where $z$ is a periodic point of period $K$ in $U$.

Form

$G_{\delta}=\bigcup_{n\in\mathbb{N}}\tilde{g}^{n\cdot K}(V_{\delta}).$

Notice that

$G_{\delta}=p\left(\bigcup_{n\in\mathbb{N}}\varphi^{n\cdot
K}(X^{s}(z,\delta)\times X^{u}(z,\delta))\right)$

and that since $(X,\varphi)$ is mixing

$\bigcup_{n\in\mathbb{N}}\varphi^{n\cdot K}(X^{s}(z,\delta)\times
X^{u}(z,\delta))$

is dense in $X$. Since $p$ is onto, it follows that $G_{\delta}$ is dense for
each valid $\delta>0$. In particular, $G_{\delta/2}$ is dense.

Next, suppose that $\tilde{y}\in X^{u}(P)/{\sim_{0}}$ is a limit point of the
set $G_{\delta/2}$. We show $\tilde{y}\in G_{\delta}$. From this, it follows
that $G_{\delta}=X^{u}(P)/{\sim_{0}}$. Using the compactness of $X$ and the
fact that

$\bigcup_{n\in\mathbb{N}}\varphi^{n\cdot K}(X^{s}(z,\delta)\times
X^{u}(z,\delta))$

is dense in $X$, there exists a sequence $(x_{k})_{k\in\mathbb{N}}$ in $X$
converging to $x\in X$ such that

$x_{k}\in\bigcup_{n\in\mathbb{N}}\varphi^{n\cdot K}(X^{s}(z,\delta/2)\times
X^{u}(z,\delta/2))$

for each $n$ and $p(x)=\tilde{y}$. There exists $N\in\mathbb{N}$ such that
$x_{k}\in X^{s}(x,\frac{\delta}{2})\times X^{u}(x,\frac{\delta}{2})$ for
$k\geq N$. In particular, $[x,x_{N}]$ is well-defined and is an element in
$X^{u}(x_{N},\frac{\delta}{2})$. Lemma 3.15 implies that
$p([x,x_{N}])=p(x)=\tilde{y}$. Since
$x_{N}\in\bigcup_{n\in\mathbb{N}}\varphi^{n\cdot K}(X^{s}(z,\delta/2)\times
X^{u}(z,\delta/2))$ there exists $l\in\mathbb{N}$ such that

$x_{N}\in\varphi^{l\cdot K}(X^{s}(z,\delta/2)\times X^{u}(z,\delta/2))$

which implies that

$\varphi^{-l\cdot K}(x_{N})\in X^{s}(z,\delta/2)\times X^{u}(z,\delta/2).$

Since $[x,x_{N}]$ is an element in $X^{u}(x_{N},\frac{\delta}{2})$, we have
that

$\varphi^{-l\cdot K}(x)\in X^{u}(\varphi^{-l\cdot
K}(x_{N}),\frac{\delta}{2}).$

The triangle inequality and the previous two statements imply that

$\varphi^{-l\cdot K}(x)\in X^{s}(z,\delta)\times X^{u}(z,\delta).$

and hence that

$x\in\varphi^{l\cdot K}(X^{s}(z,\delta)\times X^{u}(z,\delta))\hbox{ and
}\tilde{y}=p(x)\in G_{\delta}.$

In summary, we have shown that $G_{\delta}=X^{u}(P)/{\sim_{0}}$. Using this
along with the fact that $X^{u}(P)/{\sim_{0}}$ is compact and
$V_{\delta}\subseteq\tilde{g}^{K}(V_{\delta})$ imply that there exists
$N\in\mathbb{N}$ such that $\tilde{g}^{N}(V_{\delta})=X^{u}(P)/{\sim_{0}}$.
The required result follows since $V_{\delta}\subseteq U$.

∎

###  4.2. Existence of a full projection

######  Lemma 4.4.

Let $V$ be a basic set of $G_{0}({\bf P})$. Then, there exists
$N\in\mathbb{N}$ such that for any $f\in C_{c}(G_{0}({\bf P}))$ supported in
another basic set and nonzero on $V$, we have that the following two
properties hold:

  1. (1)

For each $x\in X^{u}(P)$ there exists $(\hat{x},\hat{w})\in{\rm
supp}(\alpha^{N}(f))$ such that $x\sim_{0}\hat{x}$;

  2. (2)

likewise for each $b\in X^{u}(P)$ there exists $(\hat{a},\hat{b})\in{\rm
supp}(\alpha^{N}(f))$ such that $b\sim_{0}\hat{b}$.

###### Proof.

This follows by applying Theorem 4.3 to the open set $s(V)$ and $r(V)$ and
taking the max of the two $N$s. ∎

######  Lemma 4.5.

Let $V$ be a basic set of $G_{0}({\bf P})$. Then, there exists
$N\in\mathbb{N}$ such that for any $f\in C_{c}(G_{0}({\bf P}))$ supported in
another basic set with $||f|_{V}||>0$, we have that for each $k\in
C_{c}(G_{0}({\bf P}))$ supported in yet another basic set, there exists
$f_{1}$, $f_{2}\in C_{c}(G_{0}({\bf P}))$ such that

$f_{1}\alpha^{K}(f)f_{2}=k\hbox{ and
}||f_{1}||\leq||k||,||f_{2}||=\frac{1}{||f|_{V}||}$

Moreover, in this case, we have that $\overline{C^{*}(G_{0}({\bf
P}))\alpha^{N}(f)C^{*}(G_{0}({\bf P}))}=C^{*}(G_{0}({\bf P}))$.

###### Proof.

Let $A=C^{*}(G_{0}({\bf P}))$. The second part of the theorem follows from the
first part. To see this note that $\overline{A\alpha^{N}(f)A}$ denotes the
closed linear span of $A\alpha^{N}(f)A$ and that closed linear span of the set
of functions of compact support with support contained in a single basic set
is $A$.

We now prove the first part. Recall that $q:X^{u}({\bf P})\rightarrow
X^{u}({\bf P})/{\sim_{0}}$ denotes the quotient map. A basic set, $U$, is of
the form

$\\{(h_{U}(x),x)\mid x\in X^{u}(z,\delta)\\}$

where $z\in X^{u}({\bf P})$, $\delta>0$ is small, and $h_{U}$ is a local
homeomorphism onto its image. Because we are dealing with a very specific
groupoid, we have that $h_{U}=(q|_{r(U)})^{-1}\circ q|_{s(U)}$. This in
particular holds for the basic set $U_{k}$ that contains the support of $k$.
We likewise let $U_{f}$ denote the basic set containing the support of $f$. By
the previous lemma, there exists $N\in\mathbb{N}$ such that

  1. (1)

$q(s(\varphi^{N}\times\varphi^{N}(U_{f})))=X^{u}({\bf P})/{\sim_{0}}$ and

  2. (2)

$q(r(\varphi^{N}\times\varphi^{N}(U_{f})))=X^{u}({\bf P})/{\sim_{0}}$.

It follows that we have $V_{1}\subseteq
s(\varphi^{N}\times\varphi^{N}(U_{f}))$ and $V_{2}\subseteq
r(\varphi^{N}\times\varphi^{N}(U_{f}))$ such that

  1. (1)

$q|_{s(U_{k})}$ is a homeomorphism from $s(U_{k})$ to $V_{1}$ and

  2. (2)

$q|_{V_{2}}$ is a homeomorphism from $V_{2}$ to $r(U_{k})$.

Since $q(r(U_{k}))=q(s(U_{k}))$ and all the relevant maps are homeomorphism
(because the domains have been restricted) we have

$(q|_{r(U_{k})})^{-1}\circ q|_{s(U_{k})}=(q|_{r(U_{k})})^{-1}\circ
q|_{V_{2}}\circ(q|_{V_{2}})^{-1}\circ q|_{V_{1}}\circ(q|_{V_{1}})^{-1}\circ
q|_{s(U)}.$

Using this setup we can define $f_{1}$ and $f_{2}$ as follows. For $y\in
s(U_{k})$, let

$f_{1}(((q|_{V_{1}})^{-1}\circ q|_{s(U_{k})})(y),y)=k(h_{U_{k}}(y),y)$

and otherwise define $f_{1}$ to be zero. For $z\in V_{2}$, let

$f_{2}(((q|_{r(U_{k})})^{-1}\circ
q|_{V_{2}})(z),z)=\frac{1}{f(\varphi^{-N}(z),\varphi^{-N}(((q|_{V_{1}})^{-1}\circ
q|_{V_{2}})(z)))}$

and then extend $f_{2}$ so that its support is contained in a slightly large
basic set. We note that $f_{2}$ is well-defined because $f$ is non-zero on
$V$. A short computation using the convolution product shows that $f_{1}$ and
$f_{2}$ satisfy requirements in the statement of the theorem. ∎

######  Theorem 4.6.

Suppose that $a\in A=C^{*}(G_{0}({\bf P}))$ is nonzero, then there exists
$N\in\mathbb{N}$ such that $\alpha^{N}(a)$ is full in A. In particular,
$C^{*}(G_{0}({\bf P}))$ contains a full projection.

###### Proof.

Let $a\in A=C^{*}(G_{0}({\bf P}))$ be nonzero. Without loss of generality we
can and will assume that $||a||=1$.

There exists a basic set $U\in\mathcal{G}$ with the following property. If
$c\in C_{c}(\mathcal{G})$ with $||c||=1$ and

$||a-c||<\frac{1}{4}$

then $|c(x,y)|>\frac{1}{2}$ for each $(x,y)\in U$.

Take a basic set such that $V\subseteq\overline{V}\subseteq U$ and also take
$\tilde{f}$ a continuous bump function with support contained in $U$ that is
one on $V$. Let $N\in\mathbb{N}$ be as in the previous lemma where $N$ depends
only on $V$.

Let $0\neq k\in C_{c}(\mathcal{G})$ be supported in a basic set and
$\epsilon>0$. Take $b\in C_{c}(\mathcal{G})$ such that

$||a-b||<{\rm min}\left\\{\frac{1}{4},\frac{\epsilon}{||k||}\right\\}\hbox{
and }||b||=1$

By construction, the previous lemma can be applied to $\tilde{f}b\tilde{f}$.
We obtain $f_{1}$ and $f_{2}$ such that

$f_{1}\alpha^{N}(\tilde{f}b\tilde{f})f_{2}=k$

Therefore

$\displaystyle||f_{1}\alpha^{N}(\tilde{f})\alpha^{N}(a)\alpha^{N}(\tilde{f})f_{2}-k||$
$\displaystyle=||f_{1}\alpha^{N}(\tilde{f})\alpha^{N}(a)\alpha^{N}(\tilde{f})f_{2}-f_{1}\alpha^{N}(\tilde{f})\alpha^{N}(b)\alpha^{N}(\tilde{f})f_{2}||$
$\displaystyle\leq||f_{1}||||\alpha^{N}(\tilde{f})||||a-b||||\alpha^{N}(\tilde{f})||||f_{2}||$
$\displaystyle<\epsilon$

where we have used the estimates for $||f_{i}||$ from the previous lemma and
the fact that $||\tilde{f}||=1$. In summary we have shown that the set of
compactly supported functions with support in a single basic set is contained
in $\overline{A\alpha^{N}(a)A}$ and this implies the result.

For the second part, by the main result of [9], $C^{*}(G_{0}({\bf P}))$
contains a non-zero projection. Using the first part it therefore contains a
full projection. ∎

######  Corollary 4.7.

There exists a projection $p\in C_{c}(G_{0}({\bf P}))$ which is full in
$C^{*}(G_{0}({\bf P}))$. In particular, the subalgebra
$A_{p}:=pC^{*}(G_{0}({\bf P}))p$ is a unital Fell algebra with spectrum
$X^{u}({\bf P})/{\sim_{0}}$ and the dense unital subalgebra $pC_{c}(G_{0}({\bf
P}))p\subseteq A_{p}$ is closed under holomorphic functional calculus.

###### Proof.

By Theorem 4.6, there exists a full projection $p_{0}\in C^{*}(G_{0}({\bf
P}))$. The $*$-subalgebra $C_{c}(G_{0}({\bf P}))\subseteq C^{*}(G_{0}({\bf
P}))$ is closed under holomorphic functional calculus by [11, Proposition
7.4], so by standard density and functional calculus arguments there exists a
projection $p\in C_{c}(G_{0}({\bf P}))$ and a unitary $u\in 1+C^{*}(G_{0}({\bf
P}))$ such that $p=up_{0}u^{*}$. Since $p_{0}$ is full in $C^{*}(G_{0}({\bf
P}))$, it follows that $p$ is full in $C^{*}(G_{0}({\bf P}))$. It is clear
that $pC_{c}(G_{0}({\bf P}))p$ is closed under holomorphic functional calculus
from [11, Proposition 7.4]. ∎

######  Remark 4.8.

The authors were surprised by the length of the proof of Theorem 4.6 for the
following reason. The stable algebra $S=\varinjlim(C^{*}(G_{0}({\bf
P})),\alpha)$ is a simple $C^{*}$-algebra arising as an inductive limit, and
by [9] $S$ admits a full projection. The existence of projections in $S$
readily produces projections in $C^{*}(G_{0}({\bf P}))$, but fullness is not
immediate even if it is conceivably provable from some soft argument. By the
results of [10], there is no gain from $C^{*}(G_{0}({\bf P}))$ being a Fell
algebra. As the following example shows, something extra is needed beyond the
inductive limit, e.g., the proof we gave above relied on expansiveness.

For a stationary inductive limit $S=\varinjlim(A,\alpha)$ where $\alpha$ is an
injective, nondegenerate $*$-homomorphism, such that $S$ is simple and admits
a projection (which is automatically full), one can ask if there is a full
projection in $A$? The following example due to Jamie Gabe answers the
question in the negative. Consider $A:=C_{0}(\mathbb{N},\mathbb{K}(H))$ for an
infinite-dimensional, separable Hilbert space $H$. Take a pair of Cuntz
isometries $s$ and $t$ on $H$, i.e. $s$ and $t$ are isometries such that
$ss^{*}+tt^{*}=1$. We define $\alpha:A\to A$ by

$\alpha(f)(n):=\begin{cases}sf(0)s^{*}+tf(1)t^{*},\;&n=0,\\\
f(n+1),\;&n>0.\end{cases}$

It is clear from the construction that $\alpha$ is an injective, nondegenerate
$*$-homomorphism. Moreover, consider two non-zero elements $f_{1},f_{2}\in S$
in the image of any of the functorial embeddings
$C_{c}(\mathbb{N},\mathbb{K}(H))\hookrightarrow S$. Then for sufficiently
large $N$, $\alpha^{N}(f_{1}),\alpha^{N}(f_{2})\in S$ are both embeddings of
non-zero elements in $A$ supported in $\\{0\\}\subseteq\mathbb{N}$. Since
$\mathbb{K}(H)$ is simple, we have that
$\overline{Sf_{1}S}=\overline{Sf_{2}S}$. From this observation, one can show
that $S$ is simple. In fact, one can show that $S\cong\mathbb{K}(H)$. Since
$A$ admits no full projection, we can conclude our negative answer to the
above question. One can note that in contrast to the expansive scenario of
Theorem 4.6, the $*$-homomorphism in this counterexample is contracting on the
spectrum of $A$.

##  5\. Cuntz-Pimsner models for stationary inductive limits

In this section we consider a purely $C^{*}$-algebraic set up. However, the
prototypical example is the case of the $C^{*}$-algebras associated to a
Wieler solenoid. The general set up is as follows. We consider a non-
degenerate self-$*$-monomorphism $\alpha:A\to A$ of a $C^{*}$-algebra $A$. The
prototypical example is $A=C^{*}(G_{0}({\bf P}))$ with $\alpha$ given by the
composition of $C^{*}(G_{0}({\bf P}))\subseteq C^{*}(G_{1}({\bf P}))$ and the
canonical isomorphism $C^{*}(G_{1}({\bf P}))\cong C^{*}(G_{0}({\bf P}))$ from
Theorem 2.4.

Before proceeding to Cuntz-Pimsner models, we make a remark about the
stationary inductive limits.

######  Proposition 5.1.

Write $S:=\varinjlim(A,\alpha)$ and let $\phi:S\to S$ denote the right shift
in the direct limit. Then $\phi$ is a well defined $*$-automorphism of $S$.

###### Proof.

The proof of this proposition is trivial once having parsed its statement. So
parsing the statement we shall. By construction,
$S:=\bigoplus_{k\in\mathbb{N}}A/I$ where $I$ is the ideal generated by
$a\delta_{k}-\alpha(a)\delta_{k+1}$ for $a\in A$. Here we write
$a\delta_{k}\in\bigoplus_{k\in\mathbb{N}}A$ for the element $a$ placed in
position $k$. In this notation $\phi(a\delta_{k}\mod I)=a\delta_{k+1}\mod I$.
The map $\phi$ is well defined since it is induced from the right shift
mapping on $\bigoplus_{k\in\mathbb{N}}A$ that preserves $I$. The map $\phi$ is
an inverse to the left shift mapping that coincides with the mapping
$\alpha(a\delta_{k}+I):=\alpha(a)\delta_{k}+I$. Thus $\phi$ is a well defined
$*$-automorphism. ∎

Associated with the $*$-monomorphism $\alpha:A\to A$ there is an $A$-Hilbert
bimodule $E:={}_{\alpha}A_{\rm id}$. That is $E=A$ is a vector space with the
action $a.\xi.b:=\alpha(a)\xi b$ for $a,b\in A$ and $\xi\in E$. The right
$A$-inner product is given by

$\langle\xi_{1},\xi_{2}\rangle:=\xi_{1}^{*}\xi_{2}\in A.$

It is readily verified that $E$ is a right $A$-Hilbert module and that $A$
acts as adjointable operators on the left. Consider the associated Cuntz-
Pimsner algebra $O_{E}$ and its core $\mathcal{C}_{E}:=O_{E}^{U(1)}$ for the
standard gauge action.

######  Proposition 5.2.

The functorial property of inductive limits and the inclusion
$A\hookrightarrow\mathcal{C}_{E}$ induces isomorphisms $S\cong\mathcal{C}_{E}$
and $S\rtimes_{\phi}\mathbb{Z}\cong O_{E}$.

###### Proof.

By [30] it holds that

$\mathcal{C}_{E}=\varinjlim\mathbb{K}_{A}(E^{\otimes k}).$

Since $\mathbb{K}_{A}(E^{\otimes k})=A$ and $\mathbb{K}_{A}(E^{\otimes
k})\to\mathbb{K}_{A}(E^{\otimes k+1})$ coincides with $\alpha$, it follows
that $S\cong\mathcal{C}_{E}$. The construction of $S\cong\mathcal{C}_{E}$ is
by definition the map functorially constructed from the inclusion
$A\hookrightarrow\mathcal{C}_{E}$.

Following the standard procedure of “extension of scalars” (see [30] or [3,
Section 3.1]) let $E_{\infty}:=E\otimes_{A}\mathcal{C}_{E}$ denote the self-
Morita equivalence of $\mathcal{C}_{E}$ induced by $E$. For the details of the
left action of $\mathcal{C}_{E}$ on $E_{\infty}$, see [3, Proposition 3.2].
There are natural isomorphisms

$O_{E}\cong O_{E_{\infty}}\cong\mathcal{C}_{E}\ltimes E_{\infty}.$

For details, see [3, Proposition 3.2]. Here $\mathcal{C}_{E}\ltimes
E_{\infty}$ denotes the generalized crossed product by the the self-Morita
equivalence $E_{\infty}$. In terms of the isomorphism $S\cong\mathcal{C}_{E}$
we can identify the $\mathcal{C}_{E}$-self Morita equivalence $E_{\infty}$
with the $S$-self Morita equivalence ${}_{\phi}S_{\rm id}$. Therefore

$\mathcal{C}_{E}\ltimes E_{\infty}\cong S\rtimes{}_{\phi}S_{\rm
id}=S\rtimes_{\phi}\mathbb{Z}.$

∎

Proposition 5.2 has implications for the stable Ruelle algebra of a Wieler
solenoid. It was proven in [11, Theorem 1.2] that $C^{*}(G^{s}({\bf
P}))\cong\varinjlim(C^{*}(G_{0}({\bf P})),\alpha)$ for the Fell subalgebra
$C^{*}(G_{0}({\bf P}))\subseteq C^{*}(G^{s}({\bf P}))$ with spectrum
$X^{u}({\bf P})/{\sim_{0}}$. The next result extends this result to the stable
Ruelle algebra, and is an immediate consequence of Proposition 5.2 and [11,
Theorem 1.2].

######  Corollary 5.3.

Consider a Wieler solenoid, and write $E$ for the $C^{*}(G_{0}({\bf
P})-C^{*}(G_{0}({\bf P}))$-correspondence defined as $E:=C^{*}(G_{0}({\bf
P}))$ as a right Hilbert module with left action defined from $\alpha$. The
stable Ruelle algebra $C^{*}(G^{s}({\bf P}))\rtimes\mathbb{Z}$ is isomorphic
to the Cuntz-Pimsner algebra $O_{E}$.

We are also interested in unital Cuntz-Pimsner models. We first describe the
general setup and return to the specifics of Wieler solenoids at the end of
the section. They can be constructed from choosing a full projection $p\in A$.
For such a projection, define the unital $C^{*}$-algebra $A_{p}:=pAp$. Set
$E_{p,k}:=\alpha^{k}(p)Ap$ equipped with the structure of an $A_{p}$-Hilbert
bimodule defined by $a.\xi.b:=\alpha^{k}(a)\xi b$ for $a,b\in A$ and $\xi\in
E$. The right $A_{p}$-inner product on $E_{p,k}$ is given by

$\langle\xi_{1},\xi_{2}\rangle:=\xi_{1}^{*}\xi_{2}\in A_{p}.$

We set $E_{p}:=E_{p,1}$.

######  Proposition 5.4.

Let $p\in A$ be a full projection and define the unital $C^{*}$-algebra
$A_{p}:=pAp$. It then holds that

  1. (1)

$E_{p,k}=E_{p}^{\otimes_{A}k}$

  2. (2)

$\mathbb{K}_{A_{p}}(E_{p}^{\otimes_{A}k})=A_{\alpha^{k}(p)}$

  3. (3)

$\mathcal{C}_{E_{p}}=\varinjlim(A_{\alpha^{k}(p)},\alpha)$ is a unital
$C^{*}$-algebra which is stably isomorphic to $S$

  4. (4)

$O_{E_{p}}$ is a unital $C^{*}$-algebra which is $U(1)$-equivariantly stably
isomorphic to $S\rtimes_{\phi}\mathbb{Z}$.

###### Proof.

For (1), we have for any $k,l\in\mathbb{N}$ that

$E_{p,k}\otimes_{A}E_{p,l}=\overline{\alpha^{l}(\alpha^{k}(p)Ap)\alpha^{l}(p)Ap}=\alpha^{k+l}(p)Ap=E_{p,k+l}.$

For (2), we have that

$\mathbb{K}_{A_{p}}(E_{p}^{\otimes_{A}k})=E_{p}^{\otimes_{A}k}\otimes_{A}(E_{p}^{\otimes_{A}k})^{*}=\alpha^{k}(p)A\alpha^{k}(p)=A_{\alpha^{k}(p)}.$

Item (3) follows from (2) and the fact that
$\varinjlim(A_{\alpha^{k}(p)},\alpha)$ coincides with $pSp$ which is Morita
equivalent to $S$. Item (4) follows from (3) and Proposition 5.2. ∎

We also note the following consequence on traces and KMS-weights that relies
on [24].

######  Theorem 5.5.

Let $(A,\alpha)$ be a stationary inductive system, $p\in A$ a full projection
and $\beta\in\mathbb{R}$. There are bijections between the sets of following
objects:

  1. (1)

Tracial weights $\tau$ on $A$ such that
$\tau\circ\alpha=\mathrm{e}^{\beta}\tau$ normalized by $\tau(p)=1$.

  2. (2)

Tracial states $\tau_{p}$ on $A_{p}$ such that
$\mathrm{Tr}_{\tau_{p}}^{E_{p}}=\mathrm{e}^{\beta}\tau_{p}$.

  3. (3)

KMSβ-weights $\Phi$ on $S\rtimes_{\phi}\mathbb{Z}$ normalized by $\Phi(p)=1$.

The bijection from the set of objects in (1) to the set of objects in (2) is
given by restriction along $A_{p}\hookrightarrow A$. The bijection from the
set of objects in (3) to the set of objects in (1) is given by restriction
along $A\hookrightarrow S\rtimes_{\phi}\mathbb{Z}$.

It is worth noting that the results from the previous section on the existence
of full projections in the case of the Fell algebra associated to a Wieler
solenoid imply that the previous theorem can be applied to the case of
$(C^{*}(G_{0}({\bf P})),\alpha)$ where $\alpha$ is the composition of
$C^{*}(G_{0}({\bf P}))\subseteq C^{*}(G_{1}({\bf P}))$ with the isomorphism
$C^{*}(G_{1}({\bf P}))\cong C^{*}(G_{0}({\bf P}))$. In the case of Wieler
solenoids, we can summarize the results of this section in the following
corollary. Recall that Corollary 4.7 ensures the existence of a projection
$p\in C_{c}(G_{0}({\bf P}))$ which is full in $C^{*}(G_{0}({\bf P}))$.

######  Corollary 5.6.

Let $(Y,\mathrm{d}_{Y},g)$ be a Wieler pre-solenoid with associated Wieler
solenoid $X:=\varprojlim(Y,g)$. We fix a projection $p\in C_{c}(G_{0}({\bf
P}))$ which is full in $C^{*}(G_{0}({\bf P}))$. Consider the unital Fell
algebra $A_{p}:=pC^{*}(G_{0}({\bf P}))p$ and define the $A_{p}$-bimodule
$E_{p}:=\alpha(p)C^{*}(G_{0}({\bf P}))p$. It then holds that

  1. (1)

The Cuntz-Pimsner algebra $O_{E_{p}}$ is a unital $C^{*}$-algebra which is
$U(1)$-equivariantly stably isomorphic to the stable Ruelle algebra
$C^{*}(G^{s}({\bf P}))\rtimes\mathbb{Z}$

  2. (2)

The core of the Cuntz-Pimsner algebra $\mathcal{C}_{E_{p}}=(O_{E_{p}})^{U(1)}$
is a unital $C^{*}$-algebra which is stably isomorphic to the stable algebra
$C^{*}(G^{s}({\bf P}))$.

  3. (3)

If $(Y,\mathrm{d}_{Y},g)$ is mixing, the equation of traces on $A_{p}$

$\mathrm{Tr}_{\tau}^{E_{p}}=\mathrm{e}^{\beta}\tau$

only has a solution when $\beta$ is the topological entropy of $X$, in which
case there exists a one-dimensional space of solutions determined from the
Bowen measure on $X$ and the correspondence in Theorem 5.5 combined with item
1).

For more details on the Bowen measure and how it determines traces and weights
on the algebras associated with a Smale space, see [9, 23, 31].

##  6\. The $K$-theory as a functor for compact, locally Hausdorff spaces

The goal of this section is to extend the $K$-theory functor from compact,
Hausdorff spaces to compact, locally Hausdorff spaces. As mentioned in Remark
1.17, the process of taking spectra of Fell algebras does not define a functor
when the morphisms between compact, locally Hausdorff spaces are taken to be
continuous maps. Nevertheless, the $K$-theory of the associated Fell algebra
is better behaved than the algebra itself.

In addition to the contravariant functor associated to (at least some)
continuous maps between compact, locally Hausdorff space, there is also a
wrong-way functor for self local homeomorphisms. Wrong way functoriality is
particularly important for our study of the Fell algebra associated to a
Wieler solenoid because it appears both in the inductive limits structure of
the stable algebra and the Cuntz–Pimsner model of the stable Ruelle algebra.

We start by defining $K$-theory of a compact, locally Hausdorff space
$\tilde{Y}$. As in Subsection 1.3, we choose a Hausdorff resolution
$\psi:X\to\tilde{Y}$. The associated groupoid $R(\psi)$ is defined in Example
1.15. We define

$K^{*}(\tilde{Y}):=K_{*}(C^{*}(R(\psi))).$

This definition is a priori depending on the choice of $\psi$. We shall in the
next subsection study functoriality and in the subsequent subsection show that
up to canonical isomorphism, $K^{*}(\tilde{Y})$ is independent on the choice
of $\psi$.

###  6.1. Correspondences and maps between locally Hausdorff spaces

We will in this subsection study how maps between compact, locally Hausdorff
spaces give rise to correspondences between the related Fell algebras
$C^{*}(R(\psi))$. Recall the terminology of a Hausdorff resolution of a
compact, locally Hausdorff space $\tilde{Y}$ from Definition 1.14. We shall
make use of another terminology.

######  Definition 6.1.

Let $p_{1}:X_{1}\to\tilde{Y}_{1}$ and $p_{2}:X_{2}\to\tilde{Y}_{2}$ be
Hausdorff resolutions of two compact, locally Hausdorff spaces. A proper
morphism $(\Pi,\pi):p_{1}\to p_{2}$ is a pair of continuous maps fitting into
the commuting diagram

$X_{1}$$\tilde{Y}_{1}$$\tilde{Y}_{2}$$X_{2}$$\Pi$$p_{1}$$p_{2}$$\pi$

such that $\Pi:X_{1}\to X_{2}$ is a proper mapping. We say that $\pi$ lifts to
a proper morphism if there is a proper mapping $\Pi:X_{1}\to X_{2}$ making
$(\Pi,\pi)$ into a proper morphism.

The results of this subsection can be summarized in the following theorem.

######  Theorem 6.2.

Let $\tilde{Y}_{1}$ and $\tilde{Y}_{2}$ be compact, locally Hausdorff spaces,
and assume that

$\pi:\tilde{Y}_{1}\to\tilde{Y}_{2},$

is a continuous mapping. Fix Hausdorff resolutions
$p_{1}:X_{1}\to\tilde{Y}_{1}$ and $p_{2}:X_{2}\to\tilde{Y}_{2}$.

  1. (1)

Associated with this data there is a canonically associated
$C^{*}(R(p_{2}))-C^{*}(R(p_{1}))$-correspondence
$\operatorname{Corr}(\pi,p_{1},p_{2})$, see Definition 6.7.

  2. (2)

The left action of $C^{*}(R(p_{2}))$ on $\operatorname{Corr}(\pi,p_{1},p_{2})$
is via $C^{*}(R(p_{1}))$-compact operators if $\pi$ lifts to a proper morphism
for some Hausdorff resolutions of $\tilde{Y}_{1}$ and $\tilde{Y}_{2}$, for
instance if $\pi$ is a local homeomorphism. In fact, if $\pi$ is a
homeomorphism then $\operatorname{Corr}(\pi,p_{1},p_{2})$ is a Morita
equivalence.

  3. (3)

If $\pi^{\prime}:\tilde{Y}_{2}\to\tilde{Y}_{3}$ is another continuous mapping
to a space with a Hausdorff resolution $p_{3}:X_{3}\to\tilde{Y}_{3}$, then
there is a unitary isomorphism of
$C^{*}(R(p_{3}))-C^{*}(R(p_{1}))$-correspondences

$\operatorname{Corr}(\pi^{\prime},p_{2},p_{3})\otimes_{C^{*}(R(p_{2}))}\operatorname{Corr}(\pi,p_{1},p_{2})\cong\operatorname{Corr}(\pi^{\prime}\circ\pi,p_{1},p_{3}).$

We shall start with an easy result for proper morphisms of Hausdorff
resolutions.

######  Proposition 6.3.

Given a proper morphism $(\Pi,\pi):p_{1}\to p_{2}$ of Hausdorff resolutions
$p_{1}:X_{1}\to\tilde{Y}_{1}$ and $p_{2}:X_{2}\to\tilde{Y}_{2}$, the pullback
map

$\Pi^{*}:C_{c}(R(p_{2}))\to C_{c}(R(p_{1})),$

along the proper groupoid homomorphism
$(x,x^{\prime})\mapsto(\Pi(x),\Pi(x^{\prime}))$, is a well defined
$*$-homomorphism that induces a $*$-homomorphism

$\Pi^{*}:C^{*}(R(p_{2}))\to C^{*}(R(p_{1})).$

Recall the notion of a groupoid correspondence from Definition 1.23.

######  Definition 6.4.

For Hausdorff resolutions $p_{1}:X_{1}\to\tilde{Y}_{1}$ and
$p_{2}:X_{2}\to\tilde{Y}_{2}$, we consider a diagram $\mathfrak{X}$ of the
form

$X_{1}$$\tilde{Y}_{1}$$\tilde{Y}_{2}$$X_{2}$$p_{1}$$p_{2}$$\pi$

where $\pi$ is a continuous map. From the diagram $\mathfrak{X}$, we define
the subspace $Z_{\mathfrak{X}}\subseteq X_{1}\times X_{2}$ by

$Z_{\mathfrak{X}}=\\{(x,y)\in X_{1}\times X_{2}:\pi\circ
p_{1}(x)=p_{2}(y)\\}.$

######  Lemma 6.5.

The space $Z_{\mathfrak{X}}$ defined above is a groupoid correspondence from
$R(p_{1})$ to $R(p_{2})$.

###### Proof.

The moment map for the right action will be defined as
$\sigma:Z_{\mathfrak{X}}\rightarrow G_{2}^{(0)}$, $(x,y)\mapsto(y,y)$. The map
$\sigma$ is an open map. Define the right action of $R(p_{2})$ on
$Z_{\mathfrak{X}}$ by $(x,y)\cdot(y,y^{\prime})=(x,y^{\prime})$ for $(x,y)\in
Z_{\mathfrak{X}}$ and $(y,y^{\prime})\in R(p_{2})$. This action is free and
proper.

Define the moment map $\rho:Z_{\mathfrak{X}}\rightarrow R(p_{1})^{(0)}$ for
the left action by $(x,y)\mapsto(x,x)$, which is open and surjective. Define
the action of $R(p_{1})$ on $Z_{\mathfrak{X}}$ by
$(x_{1},x_{2})\cdot(x_{2},y)=(x_{1},y),$ for $(x_{1},x_{2})\in
R(p_{1}),(x_{2},y)\in Z_{\mathfrak{X}}$ which is free and proper. Moreover,
$\rho$ induces a homemorphism
$\overline{\rho}:Z_{\mathfrak{X}}/R(p_{2})\rightarrow R(p_{1})^{(0)}.$ ∎

The following lemma follows from Theorem 1.24.

######  Proposition 6.6.

Let $Z_{\mathfrak{X}}$ be the groupoid correspondence defined from a diagram
$\mathfrak{X}$ as in Definition 6.4. Then, the space $C_{c}(Z_{\mathfrak{X}})$
becomes a pre-correspondence from $C_{c}(R(p_{2}))$ to $C_{c}(R(p_{1}))$ with
the operations given as in Theorem 1.24.

######  Definition 6.7.

Given a diagram $\mathfrak{X}$ as in Definition 6.4, we define the
correspondence $\operatorname{Corr}(\pi,p_{1},p_{2})$ from $C^{*}(R(p_{2}))$
to $C^{*}(R(p_{1}))$ as the completion of the pre-correspondence
$C_{c}(Z_{\mathfrak{X}})$. In other words
$\operatorname{Corr}(\pi,p_{1},p_{2})$ is a
$C^{*}(R(p_{2}))-C^{*}(R(p_{1}))$-Hilbert $C^{*}$-module.

######  Lemma 6.8.

Let $Z_{\mathfrak{X}}$ be the groupoid correspondence defined from a diagram
$\mathfrak{X}$ as in Definition 6.4. If $\pi$ is a continuous bijection then
$\operatorname{Corr}(\pi,p,q)$ is an imprimitivity bimodule from
$C^{*}(R(p_{2}))$ to $C^{*}(R(p_{1}))$ in the left $C^{*}(R(p_{2}))$-valued
inner product defined as follows. The $C_{c}(R(p_{2}))$-valued inner product
$\langle\langle\xi_{1},\xi_{2}\rangle\rangle\in C_{c}(R(p_{2}))$ of
$\xi_{1},\xi_{2}\in C_{c}(Z_{\mathfrak{X}})$ as

$\langle\langle\xi_{1},\xi_{2}\rangle\rangle(y,y^{\prime})=\sum_{\begin{subarray}{c}x\in
Xwith\\\ (x,x^{\prime})\in
R(p_{1})\end{subarray}}\xi_{1}(x,y^{\prime})\overline{\xi_{2}(x,y)},\quad(y,y^{\prime})\in
R(p_{2}),$ (6.1)

for $x^{\prime}\in X$ with $(x^{\prime},y)\in Z_{\mathfrak{X}}$, and this left
inner product is extended to a left $C^{*}(R(p_{2}))$-valued inner product on
$\operatorname{Corr}(\pi,p_{1},p_{2})$ by continuity.

###### Proof.

Notice that $\sigma:Z_{\mathfrak{X}}\rightarrow R(p_{2})^{(0)}$ is an open
map; and since $\pi$ is a bijection, the map $\sigma$ induces a bijection from
$R(p_{1})\backslash Z_{\mathfrak{X}}$ onto $R(p_{1})^{(0)}$.
Moreover,$Z_{\mathfrak{X}}$ is a left principal $R(p_{1})$-space and a right
principal $R(p_{2})$-space. Now, as in [27], the operation (6.1) defines a
left semi-inner product, and the correspondence
${}_{C_{c}(R(p_{2}))}C_{c}(Z_{\mathfrak{X}})_{C_{c}(R(p_{1}))}$ becomes a pre-
imprimitivity bimodule. ∎

######  Example 6.9.

Let $X_{1},X_{2}$ be compact Hausdorff spaces and let $\pi:X_{1}\rightarrow
X_{2}$ be a continuous map. The map $\pi$ induces a unital homomorphism via
pulback

$\pi^{*}:C(X_{2})\rightarrow C(X_{1}),\hskip
28.45274pt\pi^{*}(g)(x)=f\left(\pi(x)\right),$

for $g\in C(X_{2}),$ $x\in X_{1}$. The homomorphism $\pi^{*}$ allows one to
view $C(X_{1})$ as a $C(X_{2})-C(X_{1})$ correspondence. We show that this
correspondence is isomorphic to the correspondence
${}_{C(X_{2})}C(Z_{\mathfrak{X}})_{C(X_{1})}$ associated to the following
diagram.

$X_{1}$$X_{1}$$X_{2}$$X_{2}$$\operatorname{id}_{X_{1}}$$\operatorname{id}_{X_{2}}$$\pi$

We have $Z_{\mathfrak{X}}=\\{(x,\pi(x))\in X_{1}\times X_{2}:x\in X_{1}\\}$.
Notice that $R(\operatorname{id}_{X_{j}})=\\{(x,x):x\in X_{j}\\}$ for $j=1,2$.
Therefore, we may identify the groupoid $R(\operatorname{id}_{X_{j}})$ with
the trivial groupoid $X_{j}$, for $j=1,2$. The diagram above gives us the
correspondence ${}_{C(X_{2})}C(Z_{\mathfrak{X}})_{C(X_{1})}$ with the
operations

$\displaystyle\xi\cdot f\left(x,\pi(x)\right)$
$\displaystyle=\xi\left(x,\pi(x)\right)f(x)$ $\displaystyle
g\cdot\xi(x,\pi(x))$
$\displaystyle=g\left(\pi(x)\right)\xi\left(x,\pi(x)\right)$
$\displaystyle\langle\xi_{1},\xi_{2}\rangle(x)$
$\displaystyle=\overline{\xi_{1}\left(x,\pi(x)\right)}\xi_{2}\left(x,\pi(x)\right)$

for $\xi,\xi_{1},\xi_{2}\in C(Z_{\mathfrak{X}}),g\in C(X_{2}),$ and $f\in
C(X_{1})$. We can define the isomorphism of correspondences
$\phi:C(Z_{\mathfrak{X}})\rightarrow C(X)$ by

$\phi(\xi)(x)=\xi(x,\pi(x)).$

Indeed, it is clear that $\phi$ has the inverse $\phi^{-1}(f)(x,\pi(x))=f(x)$
and a short computation shows that $\phi$ is compatible with the
correspondence structure.

######  Lemma 6.10.

Assume we have the following diagram

$X_{1}$$\tilde{Y}_{1}$$\tilde{Y}_{2}$$\tilde{Y}_{3}$$X_{2}$$X_{3}$$p_{1}$$p_{2}$$p_{3}$$\pi_{2}$$\pi_{1}$

where $p_{1}:X_{1}\to\tilde{Y}_{1}$, $p_{2}:X_{2}\to\tilde{Y}_{2}$ and
$p_{3}:X_{3}\to\tilde{Y}_{3}$ are Hausdorff resolutions; and $\pi_{1}$ and
$\pi_{2}$ are continuous maps. Write $\mathfrak{X}_{1}$ for the left part of
the diagram and $\mathfrak{X}_{2}$ for the right part and $\mathfrak{X}_{3}$
for the diagram

$X_{1}$$\tilde{Y}_{1}$$\tilde{Y}_{3}$$X_{3}$$p_{1}$$p_{3}$$\pi_{2}\circ\pi_{1}$

For $f\in C_{c}(Z_{\mathfrak{X}_{2}})$, $\xi\in C_{c}(Z_{\mathfrak{X}_{1}})$,
the map $F(f,\xi)$ defined by

$F(f,\xi)(x,r)=\sum_{\begin{subarray}{c}y\in Ywith\\\ \pi\circ
p_{1}(x)=p_{2}(y)\end{subarray}}\xi(x,y)f(y,r)$

is an element of $C_{c}(Z_{\mathfrak{X}_{3}})$.

###### Proof.

Consider the closed subset

$Z_{\mathfrak{X}_{1}}*Z_{\mathfrak{X}_{2}}=\left\\{\big{(}(x,y),(y,r)\big{)}:(x,y)\in
Z_{\mathfrak{X}_{1}}\hskip 5.69046ptand\hskip 5.69046pt(y,r)\in
Z_{\mathfrak{X}_{2}}\right\\}$

of $Z_{\mathfrak{X}_{1}}\times Z_{\mathfrak{X}_{2}}$, and the continuous
surjective map

$V:Z_{\mathfrak{X}_{1}}*Z_{\mathfrak{X}_{2}}\rightarrow
Z_{\mathfrak{X}_{3}},\hskip 14.22636pt\left((x,y),(y,r)\right)\mapsto(x,r).$

Give $f$ and $\xi$ as in the statement of the lemma, we define the map
$g:Z_{\mathfrak{X}_{1}}*Z_{\mathfrak{X}_{2}}\rightarrow\mathbb{C}$ by

$g\left((x,y),(y,r)\right)=\xi(x,y)f(y,r).$

Then $\mathrm{supp}(g)$ is contained in the compact
$\mathrm{supp}(\xi)\times\mathrm{supp}(f)$, and thus $g$ is compactly
supported. We now show that $\mathrm{supp}(F(f,\xi))\subset
V(\mathrm{supp}g)$. Assume $(x,r)\notin V(\mathrm{supp}g)$. Then, we have
$g(m)=0$ for any $m\in V^{-1}\left((x,r)\right).$ Since

$F(f,\xi)(x,r)=\sum_{m\in V^{-1}\left((x,r)\right)}g(m),$

we have $F(f,\xi)(x,r)=0,$ which completes the proof. ∎

######  Theorem 6.11.

In the notation of Lemma 6.10, the map $\Phi$ defined by

$C_{c}(Z_{\mathfrak{X}_{2}})\otimes C_{c}(Z_{\mathfrak{X}_{1}})\rightarrow
C_{c}(Z_{\mathfrak{X}_{3}}),\hskip 14.22636pt\text{$f\otimes\xi\mapsto
F(f,\xi)$}$

is a $C_{c}(R(p_{3}))-C_{c}(R(p_{1}))$ pre-correspondence isomorphism. The map
$\Phi$ induces an isomorphism of
$C^{*}(R(p_{3}))-C^{*}(R(p_{1}))$-correspondences

$\operatorname{Corr}(\pi_{2},p_{2},p_{3})\otimes_{C^{*}(R(p_{2}))}\operatorname{Corr}(\pi_{1},p_{1},p_{2})\cong\operatorname{Corr}(\pi_{2}\circ\pi_{1},p_{1},p_{3}).$

The proof is omitted as it only consists of long computations verifying that
the module structure and inner products are respected by $\Phi$. The following
immediate corollary of Theorem 6.11 shows that the correspondence
$\operatorname{Corr}(\pi,p_{1},p_{2})$ only depends on $\pi$ up to canonical
Morita equivalence.

######  Corollary 6.12.

Assume that $\pi:\tilde{Y}\to\tilde{Y}^{\prime}$ is a continuous mapping
between compact, locally Hausdorff spaces. Let
$p_{j}:X_{j}\rightarrow\tilde{Y}$ and
$p_{j}^{\prime}:X_{j}^{\prime}\rightarrow\tilde{Y^{\prime}}$ be surjective
local homeomorphisms from locally compact, Hausdorff spaces, for $j=1,2$.
Then, for $j=1,2$, $\operatorname{Corr}(\pi,p_{1},p_{j}^{\prime})$ and
$\operatorname{Corr}(\pi,p_{2},p_{j}^{\prime})$ are Morita equivalent via
$\operatorname{Corr}(\operatorname{id}_{\tilde{Y}},p_{1},p_{2})$ and
$\operatorname{Corr}(\pi,p_{j},p_{1}^{\prime})$ and
$\operatorname{Corr}(\pi,p_{j},p_{2}^{\prime})$ are Morita equivalent via
$\operatorname{Corr}(\operatorname{id}_{\tilde{Y}^{\prime}},p_{1}^{\prime},p_{2}^{\prime})$.

######  Lemma 6.13.

In the notation of Lemma 6.10, if $\pi_{1}$ lifts to a proper morphism
$(\Pi_{1},\pi_{1}):p_{1}\to p_{2}$ then

$\operatorname{Corr}(\pi_{2},p_{2},p_{3})\otimes_{\Pi_{1}^{*}}C^{*}(R(p_{2}))\cong\operatorname{Corr}(\pi_{2}\circ\pi_{1},p_{1},p_{3}).$

Similarly, if $\pi_{2}$ lifts to a proper morphism $(\Pi_{2},\pi_{2}):p_{2}\to
p_{3}$ then

$C^{*}(R(p_{3}))\otimes_{\Pi_{2}^{*}}\operatorname{Corr}(\pi_{1},p_{1},p_{2})\cong\operatorname{Corr}(\pi_{2}\circ\pi_{1},p_{1},p_{3}).$

In particular, if $(\Pi,\pi):p_{1}\to p_{2}$ is a proper morphism then there
is a Morita equivalence of correspondences from
$\operatorname{Corr}(\pi,p_{1},p_{2})$ the
$C^{*}(R(p_{2}))-C^{*}(R(p_{1}))$-correspondence
${}_{\Pi^{*}}C^{*}(R(p_{1}))$, i.e. the right $C^{*}(R(p_{1}))$-Hilbert module
$C^{*}(R(p_{1}))$ with left action defined from $\Pi^{*}$.

###### Proof.

The proof of the first two isomorphisms is analogous to Lemma 6.10 and Theorem
6.11 and is omitted. To prove the final statement, we note that
$\operatorname{Corr}(\operatorname{id}_{\tilde{Y}_{1}},p_{1},p_{1})=C^{*}(R(p_{1}))$
and so

${}_{\Pi^{*}}C^{*}(R(p_{1}))=C^{*}(R(p_{2}))\otimes_{\Pi^{*}}\operatorname{Corr}(\operatorname{id}_{\tilde{Y}_{1}},p_{1},p_{1})\cong\operatorname{Corr}(\pi,p_{1},p_{2}).$

∎

######  Lemma 6.14.

Consider a diagram as in Definition 6.4. The left action of $C^{*}(R(p_{2}))$
on $\operatorname{Corr}(\pi,p_{1},p_{2})$ is via $C^{*}(R(p_{1}))$-compact
operators if $\pi$ lifts to a proper morphism for some Hausdorff resolutions
of $\tilde{Y}_{1}$ and $\tilde{Y}_{2}$.

###### Proof.

The statement that the left action of $C^{*}(R(p_{2}))$ on
$\operatorname{Corr}(\pi,p_{1},p_{2})$ is via $C^{*}(R(p_{1}))$-compact
operators is Morita invariant, so by Lemma 6.8 it suffices to prove the lemma
for particular choices of Hausdorff resolutions $p_{1}$ and $p_{2}$. If it is
the case that $\pi$ lifts to a proper morphism for some Hausdorff resolutions
of $\tilde{Y}_{1}$ and $\tilde{Y}_{2}$, we can therefore assume that $\pi$
lifts to a proper morphism $p_{1}\to p_{2}$. In this case, the left action of
$C^{*}(R(p_{2}))$ on $\operatorname{Corr}(\pi,p_{1},p_{2})$ is via
$C^{*}(R(p_{1}))$-compact operators by the final statement of Lemma 6.13. ∎

###  6.2. Functoriality in $K$-theory of compact, locally Hausdorff spaces

In this section we consider compact, locally Hausdorff spaces. We are
interested in their $K$-theory. First we show that the $K$-theory of compact,
locally Hausdorff spaces is well-defined.

######  Proposition 6.15.

Consider a compact, locally Hausdorff space $\tilde{Y}$. Defining the
$K$-theory of $\tilde{Y}$ as

$K^{*}(\tilde{Y}):=K_{*}(C^{*}(R(\psi))),$

for some Hausdorff resolution $\psi:X\to\tilde{Y}$, produces a group uniquely
determined up to canonical isomorphism.

###### Proof.

If $\psi_{1}$ and $\psi_{2}$ are two different Hausdorff resolutions of
$\tilde{Y}$, Lemma 6.8 shows that
$\operatorname{Corr}(\operatorname{id}_{\tilde{Y}},\psi_{1},\psi_{2})$ is a
Morita equivalence producing the sought after isomorphism
$K_{*}(C^{*}(R(\psi_{1})))\cong K_{*}(C^{*}(R(\psi_{2})))$. ∎

Next, we study the contravariant properties of $K$-theory for a sub-class of
continuous mappings. In compliance with the results of the last subsection, we
say that a continuous map

$\pi:\tilde{Y}_{1}\to\tilde{Y}_{2},$

is HRP (Hausdorff Resolution Proper) if $\pi$ lifts to a proper morphism
$p_{1}\to p_{2}$ for some Hausdorff resolutions $p_{1}:X_{1}\to\tilde{Y}_{1}$
and $p_{2}:X_{2}\to\tilde{Y}_{2}$. We note the following consequence of Lemma
6.13 and 6.14.

######  Theorem 6.16.

If the continuous map of compact, locally Hausdorff spaces

$\pi:\tilde{Y}_{1}\to\tilde{Y}_{2},$

is HRP, there is an associated class $[\pi]\in
KK_{0}(C^{*}(R(p_{2})),C^{*}(R(p_{1})))$ for any Hausdorff resolutions
$p_{1}:X_{1}\to\tilde{Y}_{1}$ and $p_{2}:X_{2}\to\tilde{Y}_{2}$. Moreover, if
$\pi^{\prime}:\tilde{Y}_{2}\to\tilde{Y}_{3}$ is another HRP-map, then we have
the following Kasparov product

$[\pi^{\prime}]\otimes_{C^{*}(R(p_{2}))}[\pi]=[\pi^{\prime}\circ\pi]\in
KK_{0}(C^{*}(R(p_{3})),C^{*}(R(p_{1}))),$

for any Hausdorff resolution $p_{3}:X_{3}\to\tilde{Y}_{3}$.

Contravariance of $K$-theory of compact, locally Hausdorff spaces under HRP-
maps is now immediate.

######  Corollary 6.17.

Taking $K$-theory of compact, locally Hausdorff spaces defines a contravariant
functor from the category of compact, locally Hausdorff spaces with morphisms
being HRP-maps to the category of $\mathbb{Z}/2$-graded abelian groups.

We now turn to wrong way functoriality. Assume that
$\tilde{g}:\tilde{Y}_{1}\rightarrow\tilde{Y}_{2}$ is a surjective local
homeomorphism. Our goal is the definition of a wrong way map from the
$K$-theory of $\tilde{Y}_{1}$ to that of $\tilde{Y}_{2}$. Take a Hausdorff
resolution $p:X\rightarrow\tilde{Y}_{1}$. The following diagram is clearly
commutative:

$\begin{CD}X@>{\operatorname{id}_{X}}>{}>X\\\ @V{p}V{}V@V{}V{\tilde{g}\circ
p}V\\\ \tilde{Y}_{1}@>{\tilde{g}}>{}>\tilde{Y}_{2}\\\ \end{CD}.$

Since the vertical maps are both surjective local homeomorphisms,
$\tilde{g}\circ p:X\rightarrow\tilde{Y}_{2}$ is a Hausdorff resolution. We can
form the étale groupoids $R(p)$ and $R(\tilde{g}\circ p)$ over $X$, whose
assocated groupoid algebras have spectrum $\tilde{Y}_{1}$ and $\tilde{Y}_{2}$,
respectively. Furthermore, $R(p)\subseteq R(\tilde{g}\circ p)$ as an open
subgroupoid. This inclusion at the groupoid level leads to an inclusion of
$C^{*}$-algebras, $C^{*}(R(p))\subseteq C^{*}(R(\tilde{g}\circ p))$. The
induced map on $K$-theory will be denoted by

$\tilde{g}!:K^{*}(\tilde{Y}_{1})=K_{*}(C^{*}(R(p)))\rightarrow
K_{*}(C^{*}(R(\tilde{g}\circ p)))=K^{*}(\tilde{Y}_{2}).$

Following the same argument as in the proof of Proposition 6.15, we arrive at
the following.

######  Proposition 6.18.

If $\tilde{g}:\tilde{Y}_{1}\rightarrow\tilde{Y}_{2}$ is a surjective local
homeomorphism of compact, locally Hausdorff spaces, then the map

$\tilde{g}!:K^{*}(\tilde{Y}_{1})\rightarrow K^{*}(\tilde{Y}_{2}).$

is well-defined, i.e. independent of Hausdorff resolution.

The next proposition shows that we indeed have wrong way functoriality for
surjective local homeomorphisms of compact, locally Hausdorff spaces.

######  Proposition 6.19.

If $\tilde{g}_{1}:\tilde{Y}_{1}\rightarrow\tilde{Y}_{2}$ and
$\tilde{g}_{2}:\tilde{Y}_{2}\rightarrow\tilde{Y}_{3}$ are surjective local
homeomorphisms of compact, locally Hausdorff spaces, then

$\tilde{g}_{2}!\circ\tilde{g}_{1}!=(\tilde{g}_{2}\circ\tilde{g}_{1})!:K^{*}(\tilde{Y}_{1})\rightarrow
K^{*}(\tilde{Y}_{3}).$

###### Proof.

The claim follows from that both sides can be defined from the chain of
inclusions

$C^{*}(R(p))\subseteq C^{*}(R(\tilde{g}_{1}\circ p))\subseteq
C^{*}(R(\tilde{g}_{2}\circ\tilde{g}_{1}\circ p))$

where $p:X\to\tilde{Y}_{1}$ is a Hausdorff resolution. The first inclusion
defines $\tilde{g}_{1}!$, the second inclusion defines $\tilde{g}_{2}!$ and
the composed inclusion defines $(\tilde{g}_{2}\circ\tilde{g}_{1})!$. ∎

######  Remark 6.20.

We note that also for wrong way maps, we could also work at the level of
$KK$-theory. Associated with a surjective local homeomorphism
$\tilde{g}:\tilde{Y}_{1}\rightarrow\tilde{Y}_{2}$, we can for any Hausdorff
resolutions $p_{1}:X_{1}\to\tilde{Y}_{1}$ and $p_{2}:X_{2}\to\tilde{Y}_{2}$
define a wrong way class

$[g!]:=\iota^{*}\operatorname{Corr}(\operatorname{id}_{\tilde{Y}_{2}},p_{2},\tilde{g}\circ
p_{1})\in KK_{0}(C^{*}(R(p_{1})),C^{*}(R(p_{2}))),$

where $\iota$ denotes the ideal inclusion $C^{*}(R(p_{1}))\subseteq
C^{*}(R(\tilde{g}\circ p_{1}))$. We can then identify $g!$ with the Kasparov
product from the right

$\cdot\otimes_{C^{*}(R(p_{1}))}[g!]:KK_{*}(\mathbb{C},C^{*}(R(p_{1})))\to
KK_{*}(\mathbb{C},C^{*}(R(p_{2}))),$

under the natural isomorphisms

$KK_{*}(\mathbb{C},C^{*}(R(p_{1})))\cong
K_{*}(C^{*}(R(p_{1})))\quad\mbox{and}\quad
KK_{*}(\mathbb{C},C^{*}(R(p_{2})))\cong K_{*}(C^{*}(R(p_{2}))).$

In this setting, the analogue of Proposition 6.19 is the Kasparov product
statement that

$[g_{1}!]\otimes_{C^{*}(R(p_{2}))}[g_{2}!]=[(g_{2}\circ g_{1})!]\in
KK_{0}(C^{*}(R(p_{1})),C^{*}(R(p_{3}))),$

which can be verified using Theorem 6.11 and Lemma 6.13.

######  Example 6.21.

If we consider the non-Hausdorff dynamical system $(X^{u}({\bf
P})/{\sim_{0}},\tilde{g})$ associated with a Wieler solenoid, and the
Hausdorff resolution $q:X^{u}({\bf P})\to X^{u}({\bf P})/{\sim_{0}}$, then as
an element of $KK_{0}(C^{*}(G_{0}({\bf P})),C^{*}(G_{0}({\bf P})))$ we have
that

$[g!]:=\iota^{*}\operatorname{Corr}(\operatorname{id}_{X^{u}({\bf
P})/{\sim_{0}}},q,\tilde{g}\circ q)=[E],$

where $E$ is the $C^{*}(G_{0}({\bf P}))-C^{*}(G_{0}({\bf P}))$-correspondence
defined as $E:=C^{*}(G_{0}({\bf P}))$ as a right Hilbert module with left
action defined from $\alpha$. The correspondence $E$ was considered in
Corollary 5.2 and will play a role in $K$-theory computations below.

######  Example 6.22.

Let $g:Y_{1}\to Y_{2}$ be a surjective local homeomorphism of compact,
Hausdorff spaces. Write $E_{g}$ for the finitely generated projective
$C(Y_{2})$-module $C(Y_{1})$, with right action defined from $g^{*}$. We can
equipp $E_{g}$ with the $C(Y_{2})$-valued right inner product

$\langle f_{1},f_{2}\rangle(x):=\sum_{y\in
g^{-1}(x)}\overline{f_{1}(y)}f_{2}(y),\quad x\in Y_{2},\;f_{1},f_{2}\in
E_{g}=C(Y_{1}).$

The algebra $C(Y_{1})$ acts adjointably from the left on $E_{g}=C(Y_{1})$ by
pointwise multiplication. With these structures, $E_{g}$ is a
$C(Y_{1})-C(Y_{2})$-correspondence in which $C(Y_{1})$ acts as
$C(Y_{2})$-compact operators.

It is immediate from the definition of the inner product on $E_{g}$ that
$\mathbb{K}_{C(Y_{2})}(E_{g})=C^{*}(R(g))$. By taking
$p=\operatorname{id}_{Y_{1}}$ in the constructions above, it follows that

$[g!]=[E_{g}]\in KK_{0}(C(Y_{1}),C(Y_{2})).$

In the case that $Y=Y_{1}=Y_{2}$, the class $[E_{g}]\in KK_{0}(C(Y),C(Y))$
plays an important role in the $K$-theory, or more generally $KK$-theory, of
the Cuntz-Pimsner algebra associated with the module $E_{g}$ [16, 30]. For
Wieler solenoids this is of interest due to the results of the next
subsection.

###  6.3. Wrong way maps and $K$-theory of the stable and unstable Ruelle
algebra

In Corollary 5.2 we saw that the stable Ruelle algebra $C^{*}(G^{s}({\bf
P}))\rtimes\mathbb{Z}$ of a Wieler solenoid is a Cuntz-Pimsner algebra over
$C^{*}(G_{0}({\bf P}))$. From the general theory of Cuntz-Pimsner algebras,
the $K$-theory of the stable Ruelle algebra can therefore be computed from
$K^{*}(X^{u}({\bf P})/{\sim_{0}})$ – the $K$-theory of $C^{*}(G_{0}({\bf
P}))$. Using Kaminker-Putnam-Whittaker duality result [21], the $K$-theory of
the unstable Ruelle algebra can similarly be computed from the $K$-homology
group $K_{*}(X^{u}({\bf P})/{\sim_{0}}):=K^{*}(C^{*}(G_{0}({\bf P})))$.
Throughout the subsection, we tacitly identify the stable Ruelle algebra
$C^{*}(G^{s}({\bf P}))\rtimes\mathbb{Z}$ with the Cuntz-Pimsner algebra
$O_{E}$ over $C^{*}(G_{0}({\bf P}))$ in Corollary 5.2.

######  Theorem 6.23.

Assume that $(X,\phi)$ is a Wieler solenoid and write $(X^{u}({\bf
P})/{\sim_{0}},\tilde{g})$ for the associated non-Hausdorff dynamical system.
The $K$-theory of the stable Ruelle algebra $K_{*}(C^{*}(G^{s}({\bf
P}))\rtimes\mathbb{Z})$ fits into a six term exact sequence with the
$K$-theory of $X^{u}({\bf P})/{\sim_{0}}$:

$\begin{CD}K^{0}(X^{u}({\bf
P})/{\sim_{0}})@>{1-\tilde{g}!}>{}>K^{0}(X^{u}({\bf
P})/{\sim_{0}})@>{j_{S}}>{}>K_{0}(C^{*}(G^{s}({\bf P}))\rtimes\mathbb{Z})\\\
@A{\beta_{S}}A{}A@V{}V{\beta_{S}}V\\\ K_{1}(C^{*}(G^{s}({\bf
P}))\rtimes\mathbb{Z})@<{j_{S}}<{}<K^{1}(X^{u}({\bf
P})/{\sim_{0}})@<{1-\tilde{g}!}<{}<K^{1}(X^{u}({\bf P})/{\sim_{0}})\end{CD}$

where $j_{S}:C^{*}(G_{0}({\bf P}))\hookrightarrow C^{*}(G^{s}({\bf
P}))\rtimes\mathbb{Z}$ denotes the inclusion and $\beta_{S}$ is the Pimsner
boundary map in $KK_{1}(C^{*}(G^{s}({\bf
P}))\rtimes\mathbb{Z},C^{*}(G_{0}({\bf P})))$, cf. [3, 16, 30].

###### Proof.

By standard results for Cuntz-Pimsner algebras [3, 16, 30], we have a six term
exact sequence

$\begin{CD}K_{0}(C^{*}(G_{0}({\bf P})))@>{1-\cdot\otimes_{C^{*}(G_{0}({\bf
P}))}[E]}>{}>K_{0}(C^{*}(G_{0}({\bf P})))@>{j_{S}}>{}>K_{0}(C^{*}(G^{s}({\bf
P}))\rtimes\mathbb{Z})\\\ @A{\beta_{S}}A{}A@V{}V{\beta_{S}}V\\\
K_{1}(C^{*}(G^{s}({\bf
P}))\rtimes\mathbb{Z})@<{j_{S}}<{}<K_{1}(C^{*}(G_{0}({\bf
P})))@<{1-\cdot\otimes_{C^{*}(G_{0}({\bf P}))}[E]}<{}<K_{1}(C^{*}(G_{0}({\bf
P})))\end{CD}$

where $E$ is the $C^{*}(G_{0}({\bf P}))-C^{*}(G_{0}({\bf P}))$-correspondence
from Corollary 5.2. The theorem now follows from the identity
$[\tilde{g}!]=[E]\in KK_{0}(C^{*}(G_{0}({\bf P})),C^{*}(G_{0}({\bf P})))$ in
Example 6.21. ∎

Using Kaminker-Putnam-Whittaker duality result [21], we can derive an
analogous result for the $K$-theory of the unstable Ruelle algebra
$C^{*}(G^{u}({\bf P}))\rtimes\mathbb{Z}$. Here one uses $K$-homology of
$X^{u}({\bf P})$ instead of $K$-theory, and the theory for $K$-homology of
compact, locally Hausdorff spaces can be developed in the same way as that of
$K$-theory in Subsection 6.2.

######  Theorem 6.24.

Assume that $(X,\phi)$ is a Wieler solenoid and write $(X^{u}({\bf
P})/{\sim_{0}},\tilde{g})$ for the associated non-Hausdorff dynamical system.
The $K$-theory of the unstable Ruelle algebra $K_{*}(C^{*}(G^{u}({\bf
P}))\rtimes\mathbb{Z})$ fits into a six term exact sequence the $K$-homology
of $X^{u}({\bf P})/{\sim_{0}}$:

$\begin{CD}K_{0}(X^{u}({\bf
P})/{\sim_{0}})@>{1-[\tilde{g}!]\otimes\cdot}>{}>K_{0}(X^{u}({\bf
P})/{\sim_{0}})@>{\beta_{U}}>{}>K_{0}(C^{*}(G^{u}({\bf
P}))\rtimes\mathbb{Z})\\\ @A{j_{U}}A{}A@V{}V{j_{U}}V\\\ K_{1}(C^{*}(G^{u}({\bf
P}))\rtimes\mathbb{Z})@<{\beta_{U}}<{}<K_{1}(X^{u}({\bf
P})/{\sim_{0}})@<{1-[\tilde{g}!]\otimes\cdot}<{}<K_{1}(X^{u}({\bf
P})/{\sim_{0}})\end{CD}$

where $j_{U}$ is Kaminker-Putnam-Whittaker dual to $j_{S}$ and $\beta_{U}$ is
Kaminker-Putnam-Whittaker dual to the Pimsner boundary map in
$KK_{1}(C^{*}(G^{s}({\bf P}))\rtimes\mathbb{Z},C^{*}(G_{0}({\bf P})))$, cf.
[3, 16, 30].

###  6.4. Groupoid homology

Although we have only discussed the $K$-theory of stable and stable Ruelle
groupoid $C^{*}$-algebras, one can also use the results in this paper to
compute the homology of these groupoids. For the stable groupoid, one uses the
main result of [11] (stated above in Theorem 2.4) and [15, Proposition 4.7 ].
Again, determining the map (now on groupoid homology) associated to the open
inclusion $G_{0}({\bf P})\subseteq G_{1}({\bf P})$ is the key to explicit
computations. For the stable Ruelle groupoid, one uses the homology of the
stable groupoid and [29, Lemma 1.3] (also see [7, Theorem 3.8]).

###  6.5. Examples

In this section of the paper we discuss a few explicit examples. In the first
few, the original map $g$ is a local homeomorphism and hence $X^{u}({\bf
P})/{\sim_{0}}=Y$ and $\tilde{g}=g$. Nevertheless, these examples illustrate
the importance of computing the wrong-way map in $K$-theory computation. We
also summarize the case of the aab/ab-solenoid where $X^{u}({\bf
P})/{\sim_{0}}\neq Y$ and $\tilde{g}\neq g$, which is discussed in much more
detail in [11]. In fact, typically $g:Y\rightarrow Y$ is not a local
homeomorphism, so that $X^{u}({\bf P})/{\sim_{0}}\neq Y$ and $\tilde{g}\neq
g$, see for example [8, Page 14] where tiling spaces are discussed.

######  Example 6.25.

Suppose $Y$ is a manifold and $g:Y\rightarrow Y$ is an expanding endomorphism
as defined by Shub [35]. In this case, $g$ is a covering map and hence a local
homeomorphism. In [5] it is shown that the map $g!$ is the transfer map
associated to the cover, compare to Example 6.22 above. In particular, $g!$ is
a rational isomorphism in the case of $K$-theory and in the case of homology
has even better properties, see [5] for details.

The case when $Y$ is the circle and $g$ is the two-fold cover from the circle
to itself is a special case of this situation. It is well-known or one can
check directly that $g!$ is given by multiplication by 2 in degree zero and
the identity in degree one. Hence,

$K_{*}(C^{*}(G^{s}({\bf
P})))\cong\left\\{\begin{array}[]{cc}\mathbb{Z}\left[\frac{1}{2}\right]&*=0\\\
\mathbb{Z}&*=1\end{array}\right.$

and

$K_{*}(C^{*}(G^{s}({\bf
P}))\rtimes\mathbb{Z})\cong\left\\{\begin{array}[]{cc}\mathbb{Z}&*=0\\\
\mathbb{Z}&*=1\end{array}\right.$

The reader can see [5] for other explicit computations of the transfer map in
the case of flat manifolds.

The unstable and unstable Ruelle algebras associated to a Wieler solenoid are
relevant in the study of the HK-conjecture. However, the stable and stable
Ruelle algebras are not relevant for the HK-conjecture because the unit space
of the relevant groupoids are not totally disconnected (except for the case of
shifts of finite type). For example, in the case of a flat manifold $Y$, we
have that the unit space is $\mathbb{R}^{{\rm dim}(Y)}$.

######  Example 6.26.

When the relevant Smale space is a two-sided shifts of finite type, the
relevant Wieler pre-solenoid is the one-sided shifts of finite type, see [39].
In this case, the map $g$ is the shift map and $g!$ is not a rational
isomorphism even though $g$ is a local homeomorphism. This fact is in contrast
with the previous example.

######  Example 6.27.

We will discuss the case of the $p/q$-solenoid [4] briefly. The details will
be published elsewhere and were obtained at the same time as [5]. As such, we
give a short summary of the results. Let $S^{1}$ be the unit circle in the
complex plane and $1<q<p$ be positive integers with gcd($p$, $q$)=1. The
$p/q$-solenoid can be realized as an inverse limit where $Y$ is the
$q$-solenoid. That is,

$Y=S_{q}=\\{(z_{0},z_{1},z_{2},\ldots)\mid z_{i}\in S^{1}\hbox{ and
}z_{i+1}^{q}=z_{i}\\}$

The map is defined via

$g(z_{0},z_{1},z_{2},\ldots)=(z_{1}^{p},z_{2}^{p},\ldots)$

where $(z_{0},z_{1},z_{2},\ldots)\in Y=S_{q}$.

Since $g$ is a local homeomorphism in this example, $K_{*}(C^{*}(G_{0}({\bf
P}))\cong K^{*}(S_{q})$. Furthermore, the $K$-theory of $S_{q}$ is known and
given by

$K^{*}(S_{q})\cong\left\\{\begin{array}[]{cc}\mathbb{Z}&*=0\\\
\mathbb{Z}\left[\frac{1}{q}\right]&*=1\end{array}\right.$

Then one computes that $g!$ is multiplication by $p$ in degree zero and the
identity in degree one. Hence,

$K_{*}(C^{*}(G^{s}({\bf
P})))\cong\left\\{\begin{array}[]{cc}\mathbb{Z}\left[\frac{1}{p}\right]&*=0\\\
\mathbb{Z}\left[\frac{1}{q}\right]&*=1\end{array}\right.$

and

$K_{*}(C^{*}(G^{s}({\bf
P}))\rtimes\mathbb{Z})\cong\left\\{\begin{array}[]{cc}\mathbb{Z}\left[\frac{1}{q}\right]&*=0\\\
\mathbb{Z}\left[\frac{1}{q}\right]\oplus\mathbb{Z}/(p-1)\mathbb{Z}&*=1\end{array}\right.$

######  Example 6.28.

When $(Y,g)$ is the aab/ab-solenoid the map $g$ is not a local homeomorphism.
The $K$-theory of $C^{*}(G_{0}({\bf P}))$ and the induced map $\tilde{g}!$
were computed in [11]. A summary is as follows. We have that

$K_{*}(C^{*}(G_{0}({\bf
P})))\cong\left\\{\begin{array}[]{cc}\mathbb{Z}\oplus\mathbb{Z}&*=0\\\
\mathbb{Z}&*=1\end{array}\right.$

with $\tilde{g}!$ given by $\begin{bmatrix}2&1\\\ 1&1\end{bmatrix}$ in degree
zero and the identity in degree one. Hence,

$K_{*}(C^{*}(G^{s}({\bf
P})))\cong\left\\{\begin{array}[]{cc}\mathbb{Z}\oplus\mathbb{Z}&*=0\\\
\mathbb{Z}&*=1\end{array}\right.$

and

$K_{*}(C^{*}(G^{s}({\bf
P}))\rtimes\mathbb{Z})\cong\left\\{\begin{array}[]{cc}\mathbb{Z}&*=0\\\
\mathbb{Z}&*=1\end{array}\right.$

Other similar computations, both for one dimensional solenoids and more
general constructions from tiling spaces, can be found in [17, 18, 33, 37, 38,
42].

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