1]Instituto de Matemáticas, Universidad Nacional Autónoma de México,

2]Departmento de Matemáticas, Centro Universitario de Ciencias Exactas e
Ingenierías, Universidad de Guadalajara,

# Tiling spaces are covering spaces over irrational tori

Darío Alatorre<EMAIL_ADDRESS>Diego Rodríguez-Guzmán
<EMAIL_ADDRESS>[  [

###### Abstract

We study tiling spaces in the diffeological context. We prove some basic
diffeological properties for tiling spaces and analyze two different fiber
bundle structures of one-dimensional tiling spaces over the irrational torus.
We use the diffeological classification of irrationatl tori which captures
their arithmetical escence in order to inherit the diffeological equivalence
in the context of tiling spaces.

###### keywords:

Tiling Spaces, Diffeology, Fiber Bundles, Aperiodic Tilings

###### pacs:

[

MSC Classification]37B50, 58A40, 52C23

##  1 Introduction

Motivation comes from the naive question: what is the aperiodic analog of the
orbifolds that characterize the symmetries of periodic tilings of the plane?
Conway’s Magic Theorem [9] classifies the symmetries of periodic tilings of
the plane through the orbifolds obtained as quotients
$\mathcal{O}=\mathbb{R}^{2}/G$, where $G$ is the symmetry group of the tiling.
In the crystallographic case, $G$ has a subgroup of translations isomorphic to
$\mathbb{Z}^{2}$. Thus the aforementioned orbifolds are quotients of the torus
$\mathbb{T}^{2}$ by a discrete subgroup of $G$. In the aperiodic one-
dimensional case, however, one will find that a plausible analog of the
translations subgroups $\mathbb{Z}$ could be (a subset of)
$\mathbb{Z}+\alpha\mathbb{Z}$, with $\alpha$ an irrational number. So one
should look at
$\mathbb{T}_{\alpha}:=\mathbb{R}/(\mathbb{Z}+\alpha\mathbb{Z})$, which is
quite a strange object, its natural topology is coarse. It is nevertheless
non-trivial when endowed with a structure of diffeological space, it is called
the _irrational torus_ of slope $\alpha$ and it was extensively studied by
Iglesias-Zemmour in [19, 17]. On the other hand, another possible analog for
$\mathcal{O}$ could be the tiling space $\Omega$, also known as the
_continuous hull_ of a tiling. The tiling space of a periodic tiling is a
torus obtained by identification by translations, if further symmetries are to
be considered then one will end up with Conway’s orbifolds.

Our approach is to define a structure of diffeological space for $\Omega$ and
analyze its relationships with the irrational torus $\mathbb{T}_{\alpha}$. We
verify some basic diffeological properties for $\Omega$ and explore two
different fiber bundles constructions of $\Omega$ over $\mathbb{T}_{\alpha}$.
The first one is a variant of the fiber bundle over the torus with Cantor set
fibers first constructed by Williams [36] and later on generalized by Williams
himself and Sadun [34]. This variant turns out to be more natural in the sense
that there is no need to deform the tiles in order to construct it, so it
holds up to stronger equivalence – we include a discussion of different sorts
of equivalence in geometrical and combinatorial contexts. This result is
stated as

######  Theorem A.

Let $\Omega$ be an unidimensional tiling space of primitive substitution
tilings with a rule $\omega$ and expansion factor $\alpha$. Then $\Omega$ is a
covering space over the irrational torus $\mathbb{T}_{\alpha}$.

The second bundle is closely related to the irrational flows on tori. We
exploit the diffeological formalism in this case by using the classification
of irrational tori in order to spot an equivalence of symbolic sequences
called _strong orbital equivalence_ which is inherited by tiling spaces:

######  Theorem B.

Let $\Omega_{\alpha}$ a tiling space of canonical projection of dimension 2 to
1 and irrational slope $\alpha$. Then $\Omega_{\alpha}$ is a
$\mathbb{R}$-principal bundle over the double pointed irrational torus
$\mathbb{T}_{\alpha}^{\bullet}$.

######  Theorem C.

Let $\Omega_{\alpha}$, $\Omega_{\beta}$ tiling spaces satisfying the
conditions of Theorem B. Then the following conditions are equivalent:

$\Omega_{\alpha}$ and $\Omega_{\beta}$ are strong orbit equivalent.

Their associated sturmian spaces $X_{\alpha},X_{\beta}$ are strong orbit
equivalent.

$\mathbb{T}_{\alpha}$ and $\mathbb{T}_{\beta}$ are diffeomorphic.

$\alpha$ and $\beta$ are conjugated under the modular group
$\mathrm{GL}(2,\mathbb{Z})$.

These results imply some other nice properties. On the one hand, the existence
of such a bundle allows us to define and compute a fundamental group for
tiling spaces thanks to the long exact sequence of homotopy groups – this
appears as Corollary 2. On the other, by using previous work on the existence
of substitution rules for projection method tilings [27, 16] and the
arithmetic implications of the mentioned equivalence, we prove the following

######  Theorem D.

Within the class of strong orbit equivalent one-dimensional tiling spaces
associated to a quadratic irrational slope $\alpha$ there is a substitution
tiling space.

This kind of problems have been extensively studied from different approaches
as topological [24, 6, 22, 29, 33], dynamical [31, 1, 4, 3, 11], and algebraic
[8, 10, 12, 13, 35, 25, 30], among many others. Our results are in principle
similar in many cases. It is worth mentioning that there are some recent
advances about the links of the diffeological theory and the Non-commutative
geometry in more general settings [20, 21, 23]. Our work could provide a
series of examples for these recent advances.

The rest of the article is organized as follows. In Section 2 we introduce the
basic notions of both tiling spaces and diffeology, we provide in each case
several definitions and state the results that we use throughout the text. In
Section 3 we define and study a diffeological version of tilings spaces, we
construct the first of the mentioned bundles
$\hat{p}:\Omega\to\mathbb{T}_{\alpha}$ and investigate its _structure
groupoid_ , in order to prove Theorem A. Section 4 is devoted to investigate
the strong orbital equivalence between tiling spaces and to the proof of the
remaining results.

##  2 Preliminars

###  2.1 Background on Tiling Spaces

####  2.1.1 Tilings

######  Definition 1.

A _tiling_ $T=\\{t_{i}\\}_{i=1}^{\infty}$ of $\mathbb{R}^{d}$ is a family of
closed subsets $t_{i}$ of $\mathbb{R}^{d}$ called _tiles_ satisfying the
following conditions:

  * i)

Each $t_{i}$ is a translated copy of some element of a finite set of _proto-
tiles_ $\mathcal{P}=\\{\tau_{1},\ldots,\tau_{2}\\}$.

  * ii)

Each $\tau_{i}$ is homeomorphic to a closed $d$-disk.

  * iii)

$\cup_{i=1}^{\infty}t_{i}=\mathbb{R}^{d}$.

  * iv)

For each $i,j$, if $i\neq j$ then
$\mathrm{int}(t_{i})\cap\mathrm{int}(t_{j})=\emptyset$.

A tiling $T$ is _periodic_ if its symmetry group has an infinite cyclic
subgroup and _aperiodic_ in other case. There are at least three methods for
constructing aperiodic tilings: substitution, projection, and local rules. We
will consider only the first two classes.

A _substitution_ rule is a function from the set of proto-tiles onto the set
of patterns. It consists of a subdivision rule for the proto-tile followed by
a homothety, resulting in a pattern of the original set of proto-tiles. By
iterating a substitution rule over a finite initial configuration of tiles,
arbitrarily large portions of an euclidean space can be covered and thus a
tiling can be constructed.

The _projection_ method, also known as the _cut-and-project_ method, consists
in selecting a strip from a higher dimensional periodic tiling and projecting
it onto a totally irrational subspace.

Each of these methods has its own structural implications. We will explain
them further in the following sections.

####  2.1.2 Tiling spaces

There is a natural action by translation of $\mathbb{R}^{d}$ on every tiling
$T$ given by

$T+x=\\{t+x:t\in T\\}$

for $x\in\mathbb{R}^{d}$. Let

$O(T)=\\{T+x:x\in\mathbb{R}^{d}\\}$

be the orbit of $T$ under this action. Define a metric $d$ on $O(T)$ where two
tilings $T,T^{\prime}$ are $\varepsilon$-close to each other if there are
vectors $x,x^{\prime}$ in $\mathbb{R}^{d}$ with norm less than
$\frac{\varepsilon}{2}$ such that

$B_{\frac{1}{\varepsilon}}(0)\cap T+x=B_{\frac{1}{\varepsilon}}(0)\cap
T^{\prime}+x^{\prime}.$

This means that two tilings are close to each other if they coincide in a big
euclidean ball around the origin up to a small translation.

######  Definition 2.

Define the tiling space (also known as the _hull_ of $T$) $\Omega_{T}$ of the
tiling $T$ as the completion of $O(T)$ for the metric $d$ .

By definition, $\Omega_{T}$ contains every tiling $T^{\prime}$ which is
locally the same as $T$, meaning that every _patch_ of $T^{\prime}$ may be
found on $T$. By patch we mean a finite and connected union of tiles. In
general, a tiling space $\Omega$ is a translation invariant set of tilings
which is complete in the metric $d$.

Two tilings $T$ and $T^{\prime}$ are locally indistinguishable $(LI)$ if every
patch of $T$ is a patch of $T^{\prime}$ and viceversa. Local
indistinguishability is obviously an equivalence relation whose equivalence
clases $LI$ are contained in the hull $\Omega_{T}$. In fact, the $LI$ class
$LI(T)$ of a tiling $T$ is a dense subset of $\Omega_{T}$. In particular, if
$T^{\prime}$ is $LI$ from $T$ then $\Omega_{T}=\Omega_{T^{\prime}}$. Moreover,
under additional assumptions, the set of $LI(T)$ classes coincides with the
hull $\Omega_{T}$, as stated in Proposition 1.

A substitution rule $\omega$ is called _primitive_ if every proto-tile appears
in some iteration of the substitution rule applied to any other proto-tile.
Primitive substitutions generate an unique LI class, thus every tiling
generate the same hull and so we may speak about a tiling space
$\Omega_{\omega}$ associated to $\omega$.

A tiling $T$ has finite local complexity (FLC) if for every real number $R>0$
there is a finite number of patches, up to translation, that fit inside the
ball of radius $R$; and $T$ is called repetitive if for every pattern $P$ of
$T$ there is a radius $R>0$ such that for every $x\in\mathbb{R}^{d}$ the ball
$B_{R}(x)$ contains a translated copy of $P$.

The following is a folklore result that may be found in [5], together with the
previous discussion.

######  Proposition 1.

Let $T$ be an $FLC$ tiling. Then $\Omega_{T}$ is compact and the following
conditions are equivalent:

  1. 1.

$T$ is repetitive,

  2. 2.

The dynamical system $(\Omega_{T},\mathbb{R}^{d})$ is minimal (every orbit is
dense),

  3. 3.

$LI(T)$ is closed,

  4. 4.

$LI(T)=\Omega_{T}$.

This fairly large class of tilings produces nice structures on the tiling
spaces. The hulls $\Omega$ of such tilings are locally homeomorphic to a
cartesian product of an euclidean ball $D$ times a totally disconnected set
$K$ (a Cantor set under additional assumptions); they are a rich source of
examples for hyperbolic dynamics, they are known to be laminated spaces [15],
inverse limits of branched manifolds [2, 32, 7] and fiber bundles over tori
[36, 34]. For a deeper introduction to the topology of tiling spaces see [33].
In the following paragraphs, we will state the theorems and briefly explain
the intuition behind the last couple of results enlisted.

######  Theorem 2 ([2, 32, 7]).

Tiling spaces are inverse limits of branched manifolds.

Intuition behind Theorem 2. The iteration of a substitution rule assures that
arbitrarily large euclidean balls can be covered. This procedure can be
encoded in the structure of an inverse limit. Regard each iteration of a
substitution rule both as a finite pattern and a cylinder set, meaning that
the finite pattern also represents all the tilings showing that pattern in
some particular position (the canonical choice being around the origin). This
cylinder set is identified with a branched manifold in the following way:
consider the disjoint union of the set of prototiles and identify two of them
by their borders if they appear next to each other somewhere in the tiling.
Let $\Omega_{0}$ be the space obtained after the identification. It is a
$CW$-complex, and it is actually a branched manifold. Then let $\Omega_{n}$ be
the branched manifold obtained by applying to $\Omega_{0}$ the substitution
rule $n$ times (see Fig. 1). The theorem by Anderson and Putnam [2] asserts
that the tiling space $\Omega$ is topologically conjugate to the inverse limit
of ${\Omega_{n},f_{n}}$ where the $f_{n}$ maps consist in forgetting the extra
information from $\Omega_{n}$ to fit in $\Omega_{n-1}$. In other words, we can
regard $\Omega$ as a subset of the infinite product of branched manifolds
$\Pi_{n\in\mathbb{N}}\Omega_{n}$, and

$\Omega=\lim_{\leftarrow
i}(\Omega_{n},f_{n})=\\{(\bar{x}_{n})\in\Pi_{n\in\mathbb{N}}\Omega_{n}:f_{n+1}(\bar{x}_{n+1})=\bar{x}_{n},\forall
n\in\mathbb{N})\\}.$

This result was later generalized in [32] and [7] for general tiling spaces
with FLC without the need of a substitution rule.

Figure 1: The branched 1-dimensional manifolds $\Omega_{0}$, $\Omega_{1}$ and
$\Omega_{2}$ associated to the subtitution rule $a\mapsto b$, $b\mapsto ab$.

######  Theorem 3 (Sadun-Williams, 2001).

Tiling spaces are fiber bundles over torus with totally disconnected fiber.

Intuition behind Theorem 3. The first versions of this theorem are due to R.
Williams [36] for Penrose, Ammann-Beenker and the stepped plane tiling spaces.
It was generalized later on by himself and L. Sadun [34] for every tiling
space with FLC.

The idea for constructing such a bundle uses the previous result and the
canonical projection $p_{0}$ from the inverse limit (which can be identified
with) $\Omega$ onto the branched manifold $\Omega_{0}$, and then constructing
a map from $\Omega_{0}$ which identifies each proto-tile with a torus of the
corresponding dimension. The construction of the later map relies on the fact
that tiles can be deformed in order for their vertices to have rational
coordinates. Such a deformation works only up to homeomorphism of tiling
spaces.

Regarding the fiber, in the cases of substitution or projection tilings, it is
a Cantor set. Whereas for periodic tilings it is a finite collection of
points.

One of the results of this article is the construction of a variation of this
bundle in the diffeological setting.

####  2.1.3 Equivalence of tiling spaces

There are several different sorts of equivalence among tiling spaces. It is
rather common to find at least three of them in the literature:

  * •

Homeomorphism, in the usual topological sense. This is the weaker equivalence
of this list. Homeomorphisms map translational orbits into translational
orbits, but not much more than that.

  * •

Topological conjugacy or simply conjugacy is stronger than homeomorphism. This
equivalence preserves most of the dynamical properties of the spaces.

  * •

Mutual local derivability is the strongest of these equivalences. It means
that it is possible to obtain different tilings by local manipulation. For
example, Penrose’s Kite and Dart tilings are MLD with Penrose’s rhombi.

######  Example 1.

Topological conjugacy is a rather rigid property among tilings. To exemplify
this we reproduce the following example from [33]. Consider a one-dimensional
tiling space $\Omega$ coming from the Fibonacci substitution $a\mapsto b$,
$b\mapsto ab$, with tile lengths $|a|=1$ and
$|b|=\varphi=\frac{1+\sqrt{5}}{2}$. Let $a_{n},b_{n}$ denote the patches
(words) obtained by applying the substitution $n$ times to $a$ and $b$,
respectively. Let $\Omega^{\prime}$ be a suspension of the Fibonacci subshift
with tile lengths $|a^{\prime}|=|b^{\prime}|=(2+\varphi)/(1+\varphi)$, and let
$a^{\prime}_{n}$ and $b_{n}^{\prime}$ the analogs of $a_{n},b_{n}$. The spaces
$\Omega$ and $\Omega^{\prime}$ are known to be topologically conjugate by a
mapping which preserves the sequences of tiles. This happens because the
differences in lengths $|a_{n}|-|a^{\prime}_{n}|$ and
$|b_{n}|-|b_{n}^{\prime}|$ go to zero as $n$ tends to infinity.

The first kind of equivalence between tiling spaces that we will encounter in
sintony with diffeology equivalences is the following:

######  Definition 3.

Two tiling spaces $\Omega$ and $\Omega^{\prime}$ are _orbit equivalent_ if
there exist a homeomorphism $h:\Omega\to\Omega^{\prime}$ mapping the orbit
$O(T)$ in $\Omega$ onto the orbit $O(h(T))$ in $\Omega^{\prime}$ for every
tiling $T\in\Omega$.

We will see in Section 4 that there is a stronger equivalence between tiling
spaces that fits into the diffeological machinery: the so-called _strong orbit
equivalence_.

####  2.1.4 Geometrical and combinatorial aspects of tiling spaces

######  Definition 4.

A cut-and-project set (or CPS) is a triple $(E,H,\mathcal{L})$ where
$E\cong\mathbb{R}^{d}$ is the physical space of the tiling, $H$ is a locally
compact abelian group called internal space, a lattices
$\mathcal{L}\subset\mathbb{R}^{d}\times H$, and the canonical projections
$\pi_{1}:\mathbb{R}^{d}\times H\to\mathbb{R}^{d}$ and
$\pi_{2}:\mathbb{R}^{d}\times H\to H$ such that $\pi_{1}|_{\mathcal{L}}$ is
injective and $\pi_{2}(\mathcal{L})$ is dense in $H$.

Figure 2: Cut and project tiling from dimension 2 to 1.

We will assume that $H=E^{\perp}=\mathbb{R}^{n}$, thus we regard
$(\mathbb{R}^{d},\mathbb{R}^{n},\mathcal{L})$ as a CPS from dimension $n+d$ to
$d$. Given a window $W\subset H$, the CPS returns a point set: the projection
onto $E$ of the subset $\mathcal{L}^{\prime}\subset\mathcal{L}$ of lattice
points whose projection $\pi_{2}(\mathcal{L}^{\prime})$ onto $H$ lies within
$W$. That is

$\Lambda(W)=\pi_{1}(\\{y\in\mathcal{L}:\pi_{2}(y)\in W\\}).$

We regard this point set as the set of vertices of a tiling. The injectivity
of $\pi_{1}|_{\mathcal{L}}$ implies the non-periodicity of the configuration.
When the window $W$ is the projection of the unit cube of the lattice
$\mathbb{Z}^{n+d}$ onto the internal space $\mathbb{R}^{n}$, the CPS is called
a canonical projection.

######  Remark 1.

The tiling space of a $(\mathbb{R}^{d},\mathbb{R}^{n},\mathcal{L})$-CPS is
closely related to a torus of dimension $n+d$. We explain further the case of
projection from dimension 2 to 1. Consider the line $L_{\alpha}=\\{y=\alpha
x\\}$ with slope the irrational number $\alpha$. Thus, the
$(L_{\alpha},\mathbb{R},\mathbb{Z}^{2})$-CPS of canonical projection is an
aperiodic tiling $T(\alpha)$ with two tiles labelled with $0,1$ of lengths
$\frac{1}{\sqrt{1+\alpha^{2}}},\frac{\alpha}{\sqrt{1+\alpha^{2}}}$
respectively. Observe that each point $p$ in $L_{\alpha}$ corresponds to an
element of $O(T(\alpha))$ by the relation

$p=x\frac{(1,\alpha)}{\sqrt{1+\alpha^{2}}}\sim T(\alpha)-x$ (1)

Note that there exist an ambiguity whenever the line $L_{\alpha}$ intersects a
point with integer coordinates, say $(0,0)$ for simplicity, as both
projections $\pi_{2}(0,1)$ and $\pi_{2}(1,0)$ fall in the border of the
canonical window $W$. Selecting either of them give us a different tiling, and
these two tilings only differ in the tiles located on their origins, so they
are at maximun distance in the tiling metric.

Thus the tiling space $\Omega_{\alpha}=\Omega_{T(\alpha)}$ (Definition 2) is
homeomorphic to the 2-torus

$\pi:\mathbb{R}^{2}\to\mathbb{R}^{2}/\mathbb{Z}^{2}=\mathbb{T}^{2},$ (2)

with a double solenoid
$\mathcal{S}_{\alpha}^{\prime}=\pi(L_{\alpha})^{(0,1)}\cup\pi(L_{\alpha})^{(1,0)}$
and every translational orbit is represented by the projection $\pi(y=\alpha
x+\rho)$ with $\rho\in(0,1)$ in $\mathbb{T}^{2}$. The double solenoid
$\mathcal{S}_{\alpha}^{\prime}$ comes from the ambiguity mentioned above. See
Section 1.4 in [33], and [12] for a more complete account.

For the line $L_{\frac{1}{\varphi}}$ with $\varphi=\frac{1+\sqrt{5}}{2}$, the
canonical projection tiling obtained by this method coincides with a resize of
the Fibonacci tiling described in Example 1.

Given a tiling $T=\\{t_{i}\\}_{i\in\mathbb{Z}}$ of $\mathbb{R}$ we can
identify its set of proto-tiles $\\{\tau_{j}\\}$ with numbers $\\{a_{j}\\}$ so
as to obtain a sequence $S=\\{s_{i}\\}_{i\in\mathbb{Z}}$ which take values in
$\\{a_{j}\\}$. The sequence $S$ is the combinatorial aspect of the tiling $T$.
This kind of sequences, endowed with the shift map

$\begin{array}[]{ccccc}\sigma:&\\{a_{j}\\}^{\mathbb{Z}}&\to&\\{a_{j}\\}^{\mathbb{Z}}&\\\
&S&\mapsto&\sigma(S)&\quad\text{with}\quad\sigma(S)_{i}=s_{i+1},\end{array}$
(3)

are studied as symbolic dynamical systems within the more general context of
_Cantor minimal systems_ (see for example [28]).

The symbolic sequences we are mainly interested in are the _sturmian
sequences_. In fact, the combinatorial aspect of the Fibonacci tiling (called
in that context the Fibonacci word) is an example of a sturmian sequence.

######  Definition 5.

The _sturmian sequence_ of slope $\alpha$ and intercept $\rho\in[0,1)$ is the
sequence $S=\\{s_{i}\\}_{i\in\mathbb{Z}}$ of 0 and 1 defined by
$s_{n}(\rho)=\lceil n\alpha+\rho\rceil-\lceil(n-1)\alpha+\rho\rceil$.

The sturmian word of slope $\alpha$ and intercept $\rho$ can also be encoded
as the cutting sequence111 The cutting sequence of a line $L_{\alpha}$ is
constructed by writing 1 anytime the line $L_{\alpha}$ crosses a horizontal
line between the parallels lines $x=n-1$ and $x=n$ and 0 when it doesn’t cross
any horizontal line. of the line $y=\alpha x+\rho$. In terms of their
complexity, sturmian sequences are characterized as the words containing
exactly $n+1$ different subwords of lenght $n$. The literature for sturmian
sequences is vast (see for example [26]). We will only focus on the types of
equivalences among sturmian sequences that turn out to be relevant in our
context.

######  Definition 6.

Let $S_{\alpha}$ be a sturmian sequence. The _sturmian system_
$(X_{\alpha},\sigma)$ asociated to $S_{\alpha}=\\{s_{j}\\}_{j\in\mathbb{Z}}$
is defined as

$X_{\alpha}:=\overline{\\{\\{s_{j}(\rho)\\}_{j\in\mathbb{Z}}:\rho\in[0,1)\\}}\subset\\{0,1\\}^{\mathbb{Z}}$

and the shift map $\sigma:X_{\alpha}\to X_{\alpha}$ (3).

Note the similitudes of the constructions of tiling spaces and sturmian
spaces. We can regard the latter as “symbolic” or “discretized” tiling spaces.

######  Definition 7.

Two sturmian spaces $(X,\sigma)$ and $(Y,\sigma)$ are _orbit equivalent_ if
there exist a homeomorphism $h:X\to Y$ mapping the orbit $O_{\sigma}(x)$ of
$x$ under $\sigma$ in $X$ onto the orbit $O_{\sigma}(h(x))$ under $\sigma$ of
$h(x)$ in $Y$ for every $x\in X$.

This implies that there is an unique _cocycle map_ $n:X\to\mathbb{Z}$ such
that $h(\sigma(x))=\sigma^{n(x)}(h(x))$.

######  Definition 8.

Two sturmian spaces $(X,\sigma)$ and $(Y,\sigma)$ are _strong orbit
equivalent_ if they are orbit equivalent and the cocycle map has an unique
point of discontinuity.

######  Definition 9.

We will say that two irrational number $\alpha$ and $\beta$ are _equivalent_
if there exists a matrix $M\in\mathrm{GL}(2,\mathbb{Z})$ such that

$\alpha=\frac{c+\beta d}{a+\beta b}=M\beta\quad\text{with}\quad ad-bc=\pm 1.$
(4)

######  Lemma 1.

[8, Proposition 3.3] Let $(X_{\alpha},\sigma)$ and $(X_{\beta},\sigma)$ be two
sturmian spaces. The following are equivalent:

  * (1)

$\alpha\sim\beta$ (equivalence of irrationals);

  * (2)

$X_{\alpha}$ and $X_{\beta}$ are strong orbit equivalent;

###  2.2 Background on Diffeology

Diffeological spaces are a generalization of differential manifolds. One of
its attributes is that it is possible to deal with pathological topological
spaces such as quotients by dense subspaces, the case we are interested in.

All of the content of this section may be found fully developed in [18]. We
will adopt most of its terminology. By _euclidean domain_ it should be
understood an open subset of some euclidean space $\mathbb{R}^{n}$, and by
_parametrization_ any function from an euclidean domain to a set $X$.

####  2.2.1 Diffeological Spaces

######  Definition 10.

Given a set $X$, a _diffeology_ $\mathcal{D}$ for $X$ is a set of
parametrizations $P:U\to X$, where $U$ is an open subset of some
$\mathbb{R}^{d}$, subject to the following axioms:

  * D1.

Every constant parametrization of every euclidean domain on $X$ belongs to
$\mathcal{D}$.

  * D2.

If $P:U\to X$ is a parametrization such that every $u\in U$ has a neighborhood
$V\subset U$ such that $P|_{V}\in\mathcal{D}$, then $P\in\mathcal{D}$.

  * D3.

If $P:U\to X$ belongs to $\mathcal{D}$ and $f:V\to U$ is a smooth map between
euclidean domains then $P\circ f\in\mathcal{D}$.

A diffeological space is a pair $(X,\mathcal{D})$ where $\mathcal{D}$ is a
diffeology for the set $X$. The elements of $\mathcal{D}$ are called plots.

######  Example 2.

Every $\mathbb{R}^{d}$ is a diffeological space whose structure is inherited
from that of a differentiable manifold. Consider the set of parametrizations

$\mathcal{C}^{\infty}_{*}=\bigcup_{n\in\mathbb{N}}\\{P:U\rightarrow\mathbb{R}^{d}:U\subset\mathbb{R}^{n}\text{
and }P\text{ is }C^{\infty}\\}_{n}.$

The pair $(\mathbb{R}^{d},\mathcal{C}^{\infty}_{*})$ is a diffeological space
and $\mathcal{C}^{\infty}_{*}$ is called _the standard diffeology_ for
$\mathbb{R}^{d}$. In the same fashion, every differentiable manifold is a
diffeological space.

A function $f:X\to Y$ between diffeological spaces is smooth if, for every
plot $P:U\to X$ of $X$, the composition $f\circ P$ is a plot of $Y$.

In a similar manner as in topology, diffeologies can be ordered, and there is
a minimum and a maximum for this order. The minimum is the the _fine_ or
_discrete_ diffeology whose plots are all locally constant; and the maximum is
the _coarse_ or _indiscrete_ diffeology and is defined as the diffeology that
contain every parametrization.

Diffeological spaces and smooth maps between them form a category which we
shall denote by $\bf{Diff}$. It is a well-behaved category as it is closed
under taking quotients, subobjects, limits, colimits and spaces of functions
$C^{\infty}(\star,\star)$, where $C^{\infty}$ denotes the set of smooth maps
between spaces in the diffeological sense. Natural structures are given to
them via the basic constructions pull-back and push-forward of diffeologies
from a space $(X,\mathcal{D})$ and maps $f:Y\to X$ and $g:X\to Z$,
respectively. The pull-back diffeology $f^{*}(\mathcal{D})$ along $f$ (resp.
push-forward $g_{*}(\mathcal{D})$ through $g$) are defined as the coarsest
diffeology on $Y$ (resp. the finest diffeology on $Z$) such that $f$ (resp.
$g$) is smooth. The plots for these diffeologies are specified as follows:

  * i.

A parametrization $Q:V\to Y$ is a plot in $f^{*}(\mathcal{D})$ if and only if
$f\circ Q$ is a plot for $X$. As is shown in the following diagram:

$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Q}$$\scriptstyle{f\circ
Q}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X}$

  * ii.

A parametrization $R:W\to Z$ is a plot in $g_{*}(\mathcal{D})$ if and only if,
in every point $w$ of $W$, either $R$ is locally constant or there is a
neighborhood $U$ of $w$ and a plot $P:U\to X$ such that $R|_{U}=g\circ P$. In
the latter case we shall say that $P$ _locally lifts_ $R$ along $g$. As shown
in the following diagram:

$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Z}$

The monomorphisms of $\bf{Diff}$ are called inductions: injective maps $i$
where the diffeology of the domain coincides with the pull-back along $i$ of
the diffeology of the range; and the epimorphisms subductions: surjective maps
$p$ where the diffeology of the range coincides with the push-forward through
$p$ of the diffeology of the domain. The isomorphisms of $\bf{Diff}$ are
called _diffeomorphisms_ , they are smooth invertible maps whose inverse
function is again smooth.

The _subspace diffeology_ is defined as the pull-back diffeology along the
inclusion, the _quotient diffeology_ as the push-forward diffeology through
the canonical projection, and diffeologies for products and coproducts (=
diffeological sum) as the finest (resp. the coarsest) diffeology such that all
the projections (resp. inclusions) are smooth.

The _functional diffeology_ for the set of smooth maps
$C^{\infty}(X,X^{\prime})$ between two diffeological spaces may be defined as
the coarsest diffeology such that the evaluation map

$ev:C^{\infty}(X,X^{\prime})\times X\to X^{\prime}\text{ \hskip 8.5359pt given
by \hskip 8.5359pt }ev(f,x)=f(x)$

is smooth.

For a smooth surjection $\pi:X\to B$ between diffeological spaces $X,B$ it is
possible to define the _structure groupoid_ $K_{\pi}$ of the surjection $\pi$
as the groupoid whose objets are $\mathrm{Obj}(K_{\pi})=B$ and its arrows are
$\mathrm{Mor}(K_{\pi})=\bigcup_{b,b^{\prime}\in
B}C^{\infty}(X_{b},X_{b^{\prime}})$, where $X_{b}$ denotes the fiber
$\pi^{-1}(b)$. See [18, Postulate 8.2].

Let $s$ and $t$ denote the usual _source_ and _target_ maps of an arrow
$\varphi\in\mathrm{Mor}(K_{\pi})$ given by $t(\varphi)=\pi(rank(\varphi))$ and
$s(\varphi)=\pi(dom(\varphi))$. A diffeology for the groupoid $K_{\pi}$ is
defined as the coarsest diffeology such that the evaluation map $ev:X_{s}\to
X$, given by $ev(\varphi,x)=\varphi(x)$, is smooth, where

$X_{s}=\\{(\varphi,x)\in\mathrm{Mor}(K_{\pi})\times X:x\in dom(\varphi)\\}$

denotes the total space of the pull-back of $\pi$ along the map $s$.

The groupoid structure comes with a smooth map called the _characteristic map_
:

$\begin{array}[]{cccc}\chi:&\mathrm{Mor}(K_{\pi})&\to&B\times B\\\
&\varphi&\mapsto&(s(\varphi),t(\varphi))\end{array}$ (5)

which, as we shall see in the next section, encodes the properties of the
fibers of $\pi$.

######  Remark 2.

The diffeological structure over a space induces naturally a topology, the so-
called $D$-topology, whose open sets $A\subset X$ are precisely the subsets
whose pre-images $P^{-1}(A)$ under every plot $P$ are open subsets of some
$\mathbb{R}^{n}$. The $D$-topology over a diffeological space is the coarsest
topology such that every plot is continuous.

The existence of the $D$-topology allows us to talk about local diffeological
properties, defined in $D$-neighborhoods. A map $f:X\to X^{\prime}$ between
diffeological spaces is said to be _locally smooth_ in $x\in X$ if there is a
$D$-open subset $A$ containing $x$ such that the map $f|_{A}$ is smooth. That
is, if for every plot $P$ of $X$ the parametrization $f\circ P|_{P^{-1}(A)}$
is a plot of $X^{\prime}$. The map $f:X\to X^{\prime}$ is smooth if and only
if it is locally smooth in every point of $X$. See [18, Postulate 2.3].

####  2.2.2 Diffeological Fiber Bundles

######  Definition 11.

Let $\pi:X\to B$ be a smooth surjection between diffeological spaces. Then
$\pi$ is a _diffeological fibration_ if and only if the characteristic map
$\chi$ of the structure groupoid $K_{\pi}$ (see Eq. 5) is a subduction. In
such case $K_{\pi}$ is called _fibrant_.

A morphism between surjections $\pi:X\to B$ and $\pi^{\prime}:X^{\prime}\to
B^{\prime}$ is a pair of smooth maps $(\Phi,\phi)$ making commutative the
following diagram.

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\Phi}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\prime}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{B^{\prime}}$

Whenever $\Phi$ and $\phi$ are diffeomorphisms we call $\pi$ and
$\pi^{\prime}$ _equivalent_. In particular, $\pi$ is _trivial_ with
diffeological fiber $F$ if it is equivalent to the first factor projection
$\mathrm{pr}_{1}:B\times F\to B$. And, we say that $\pi$ is _locally trivial_
with fiber $F$ if there exists a $D$-open cover $\\{U_{i}\\}_{i\in\Lambda}$ of
$B$, such that the restriction $\pi|_{U_{i}}:\pi^{-1}(U_{i})\to U_{i}$ is a
trivial surjection with fiber $F$.

When $\pi:X\to B$ is a diffeological fibration the surjectivity of $\chi$
implies that the preimages $X_{b}=\pi^{-1}(b)$ are all diffeomorphic: they are
the fibers $F=X_{b}$.

######  Lemma 2.

[18, Postulate 8.9] Let $X$ and $B$ diffeological spaces. A smooth map
$\pi:X\to B$ is a fibration if and only if there exists a diffeological spaces
$F$ such that the pullback $P^{*}(X)$ of $\pi$ by any plot $P:U\to B$ of $B$
is locally trivial with fiber $F$, where

$P^{*}(X):=\\{(u,x)\in U\times X|P(u)=\pi(x)\\}.$

Thus $\pi$ is _locally trivial along the plots_ of $B$.

A group $G$ is a diffeological group if the multiplication and the inverse
mappings are smooth:

$\begin{array}[]{cccc}\mathrm{mul}&=&[(g,g^{\prime})\to g\cdot g^{\prime}]&\in
C^{\infty}(G\times G,G)\\\ \mathrm{inv}&=&[g\to g^{-1}]&\in
C^{\infty}(G,G).\end{array}$

We say that the group $G$ acts smoothly on a diffeological space $X$ if the
group homomorphism $h:G\to{\bf Diff}(X)$ is smooth. The _action map_ of $G$
over $X$ is given by:

$\begin{array}[]{ccccc}\mathcal{A}_{X}^{G}:&X\times G&\to&X\times X&\\\
&(x,g)&\mapsto&(x,g_{X}(x))&\quad\text{with}\quad g_{X}=h(g).\end{array}$

######  Proposition 4.

[18, Postulate 8.11] Consider the projection $\pi:X\to X/G$. If
$\mathcal{A}_{X}^{G}$ is an induction then $\pi$ is a diffeological fibration
with fiber $G$ and the action of $G$ on $X$ is called _principal_. Moreover,
any equivalent fiber bundle $\pi^{\prime}:X\to Q$ to $\pi$ is a $G$-_principal
fiber bundle_ or $G$-_principal fibration_.

####  2.2.3 The irrational torus

The covering
$\mathrm{pr}:\mathbb{R}^{2}\to\mathbb{T}^{2}=\mathbb{R}^{2}/\mathbb{Z}^{2}$ is
clearly a $\mathbb{Z}^{2}$-principal fibration and the torus $\mathbb{T}^{2}$
is a diffeological group. Then any line $L_{\alpha}$ of slope $\alpha$ in
$\mathbb{R}^{2}$ through the origin is projected onto a diffeological one-
parameter subgroup $\mathcal{S}_{\alpha}$ of $\mathbb{T}^{2}$. Then it induces
a group action whose action map
$\mathcal{A}_{\mathbb{T}^{2}}^{\mathcal{S}_{\alpha}}$ is an induction. When
$\alpha$ is an irrational number we get that $\mathcal{S}_{\alpha}$ is
isomorphic to $\mathbb{R}$ and it is called the _irrational solenoid_ of slope
$\alpha$. Thus, the projection

$\pi_{\alpha}:\mathbb{T}^{2}\to\mathbb{T}^{2}/\mathcal{S}_{\alpha}=\mathbb{T}_{\alpha}$
(6)

is a $\mathbb{R}$-principal fiber bundle over the _irrational torus_
$\mathbb{T}_{\alpha}$ known as the Kronecker flow of slope $\alpha$ in
$\mathbb{T}^{2}$.

Consider the subgroup $\mathbb{Z}+\alpha\mathbb{Z}$ in $\mathbb{R}$ with
$\alpha$ irrational. It is a diffeological group and its action map
$\mathcal{A}_{\mathbb{R}}^{\mathbb{Z}+\alpha\mathbb{Z}}$ is an induction.
Then, the quotient

$\mathrm{pr}_{\alpha}:\mathbb{R}\to\mathbb{R}/\mathbb{Z}+\alpha\mathbb{Z}=\mathbb{T}_{\alpha}$
(7)

is a diffelogical space and it is diffeomorphic to
$\mathbb{T}^{2}/\mathcal{S}_{\alpha}$ (see [17, Proposition 3]). The density
of $\mathbb{Z}+\alpha\mathbb{Z}$ in $\mathbb{R}$ implies that $\pi_{\alpha}$
is locally trivial along the plots of $\mathbb{T}_{\alpha}$ (see Lemma 2) but
it is not locally trivial.

For two irrational numbers $\alpha,\beta$ the following lemma dictates the
conditions for the irrational tori $\mathbb{T}_{\alpha},\mathbb{T}_{\beta}$ to
be diffeomorphic.

######  Lemma 3.

Let $\alpha$ and $\beta$ be irrational numbers. The following assertions hold:

  * i)

There is a non-trivial smooth function $f\in
C^{\infty}(\mathbb{T}_{\alpha},\mathbb{T}_{\beta})$ if and only if there are
integers $a,b,c,d$ such that

$\beta=\frac{a\alpha-c}{d-b\alpha},\quad\text{with}\quad ad-bc\neq 0.$

  * ii)

$f$ lifts through the projections $\mathrm{pr}_{\alpha},\mathrm{pr}_{\beta}$
(Eq. 7) to an affine map $F(x)=\lambda x+\mu$ with $\lambda=a+b\beta$ and
$\lambda\alpha=c+d\beta$.

  * iii)

$f$ is a diffeomorphism if and only if $ad-cb=\pm 1$.

  * iv)

The diffeomorphism $f$ induces an equivalence between the
$\mathbb{R}$-principal fibrations $\pi_{\alpha}$ and $\pi_{\beta}$.

###### Proof.

Let $f:\mathbb{T}_{\alpha}\to\mathbb{T}_{\beta}$ be a smooth map. Then for
every $x_{0}\in\mathbb{R}$ there exists an open interval $V$ centered at
$x_{0}$ and a smooth map $F:V\subset\mathbb{R}\to\mathbb{R}$ such that

$\mathrm{pr}_{\beta}\circ F=f\circ\mathrm{pr}_{\alpha}.$ (8)

Consider $\epsilon>0$ and an interval $U\subset V$ centered at $x_{0}$ such
that for every $x\in U$ and for all $n,m\in\mathbb{Z}$ satisfying that
$n+m\alpha\in(-\epsilon,\epsilon)$ then $x+n+m\alpha\in V$. By Equation 8 we
have that

$F(x+n+m\alpha)=F(x)+n^{\prime}+\beta m^{\prime},$ (9)

for $x\in U$ and $n+m\alpha$ fixed. The irrationality of $\beta,\alpha$
implies that $n^{\prime}$ and $m^{\prime}$ are unique. But $F$ is smooth, then
its derivative with respect to $x$ at $x_{0}$ gives us

$F^{\prime}(x_{0}+n+m\alpha)=F^{\prime}(x_{0}).$ (10)

But $\mathbb{Z}+\alpha\mathbb{Z}\cap(-\epsilon,\epsilon)$ is dense in
$(-\epsilon,\epsilon)$ and $F^{\prime}$ is continuous, then
$F^{\prime}(x)=F^{\prime}(x_{0})$ for every $x\in U$. Thus $F|_{U}$ is affine,
that is, there exist $\lambda,\mu\in\mathbb{R}$ such that $F(x)=\lambda x+\mu$
for every $x$ in $U$. As $\mathbb{Z}+\alpha\mathbb{Z}$ is dense in
$\mathbb{R}$, $\mathrm{pr}_{\alpha}(U)=\mathbb{T}_{\alpha}$. Therefore the map
$F$ completely defines $f$ and extends to the covers.

Using Equation 9, we get that

$\lambda(x+n+\alpha m)+\mu=\lambda x+\mu+n^{\prime}+\beta m^{\prime},$ (11)

for every $n+\alpha m\in(-\epsilon,\epsilon)$. Thus, we get that
$\lambda(\mathbb{Z}+\alpha\mathbb{Z})\subset\mathbb{Z}+\beta\mathbb{Z}$.
Therefore $\lambda=a+\beta b,\lambda\alpha=c+\beta d$, and

$\alpha=\frac{c+\beta d}{a+\beta b}.$

Considering $\mathbb{Z}+\alpha\mathbb{Z},\mathbb{Z}+\beta\mathbb{Z}$ as
$\mathbb{Z}$ modules, the smooth map $F$ is a isomorphism if and only if $ad-
cb=\pm 1$.

For the quotients
$\mathbb{T}^{2}/\mathcal{S}_{\alpha},\mathbb{T}^{2}/\mathcal{S}_{\beta}$, it
is easy to see that the matrix element

$M=\begin{pmatrix}d&-b\\\ -c&a\end{pmatrix}$ (12)

is a linear transformation of $\mathbb{R}^{2}$ that fixes the $\mathbb{Z}^{2}$
lattice and maps the line $L_{\alpha}$ onto $L_{\beta}$. And whenever
$\det(M)=\pm 1$, the linear transformation $M$ induces an equivalence between
$\mathbb{R}$-principal fiber bundles

$\textstyle{\mathbb{T}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\alpha}}$$\scriptstyle{\Phi_{M}}$$\textstyle{\mathbb{T}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\beta}}$$\textstyle{\mathbb{T}_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\mathbb{T}_{\beta}}$

For a more complete treatment we refer the reader to [17, Propositions 4-7].

∎

In particular, the orbit of $L_{\alpha}$ under the action of the group
$\mathrm{GL}(2,\mathbb{Z})$ is a family of irrational tori diffeomorphic to
$\mathbb{T}_{\alpha}$. The group
$C^{\infty}(\mathbb{T}_{\alpha},\mathbb{T}_{\alpha})$ is a subgroup of
$\mathrm{GL}(2,\mathbb{Z})$ and it is different from the group
$\\{\mathrm{id},\mathrm{-id}\\}$ if and only if $\alpha$ is a quadratic
irrational.

The diffeological fiber bundles satisfy the _exact sequence of homotopy_ (see
[18, Postulate 8.21]), which in our case gives us the following exact
sequence:

$\cdots\longrightarrow\pi_{1}(\mathbb{R})\longrightarrow\pi_{1}(\mathbb{T}^{2})\longrightarrow\pi_{1}(\mathbb{T}_{\alpha})\longrightarrow\pi_{0}(\mathbb{R})\longrightarrow\cdots.$
(13)

The line $\mathbb{R}$ as diffeological space is connected and simple
connected, then the fundamental groups of $\mathbb{T}^{2}$ and
$\mathbb{T}_{\alpha}$ are isomorphic.

##  3 Diffeological Tiling Spaces

In the present section we introduce a structure of diffeological space for the
tiling spaces $\Omega$. For now, we let $\Omega$ be any space of tilings of
$\mathbb{R}^{d}$.

###  3.1 The Diffeological Space $\Omega$

######  Definition 12.

Define a diffeology $\mathcal{D}$ for $\Omega$ as the generated by the family
of parametrizations

$\mathcal{F}=\\{F_{T}:U\subset\mathbb{R}^{d}\to\Omega:T\in\Omega\text{ and
}F_{T}(x)=T+x\\}.$

By definition of generating family [18, Postulate 1.68], the plots of
$\mathcal{D}$ are specified as the parametrizations $P:U\to\Omega$ such that
for every $u\in U$ there exists an open neighborhood $V\subset U$ of $u$ such
that either $P|_{V}$ is a constant parametrization or there exist an element
$F_{T}:W\to\Omega$ of $\mathcal{F}$ and a smooth map $Q:V\to W$ such that
$P|_{V}=F_{T}\circ Q$. As shown in the following diagram.

$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Q}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{T}}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P}$$\textstyle{\Omega}$

Note that the dimension of this generating family, defined as the supremum of
the dimensions of its domains, is $d$. And, as no family of dimension less
than $d$ may generate this diffeology, we deduce that the dimension of
$\Omega$ is $d$ [18, Postulate 1.78].

The following couple of results verify that the natural actions by translation
and substitution over $\Omega$ are smooth.

######  Lemma 4.

Let $\mathbf{Diff}(\Omega)$ be the group of diffeomorphisms of $\Omega$
endowed with the functional diffeology. Then the translation action
$\rho:\mathbb{R}^{d}\to\mathbf{Diff}(\Omega)$ of $\mathbb{R}^{d}$ over
$\Omega$, given by $\rho(x)(T)=T+x$, is smooth.

###### Proof.

We must show that the evaluation map

$ev_{\rho}:\Omega\times\mathbb{R}^{d}\to\Omega\text{ \hskip 0.56905pt given by
\hskip 0.56905pt }ev_{\rho}(T,x)=\rho(x)(T)=T+x$ (14)

is smooth, _i.e._ for every plot $P$ of $\Omega\times\mathbb{R}^{d}$, the map
$ev_{\rho}\circ P$ is a plot of $\Omega$. The latter condition happens if and
only if every element $u\in U$ has a neighbourhood $V\subset U$ and there is a
smooth map $f:V\to W$ between euclidian domains and an element
$F_{T}:W\to\Omega$ of the generating family $\mathcal{F}$ such that

$(ev_{\rho}\circ P)|_{V}=F_{T}\circ f,$

that is, $ev_{\rho}\circ P$ is locally a translation up to pre-composing with
a smooth map.

Let $P:U\to\Omega\times\mathbb{R}^{d}$ a plot in the product diffeology. This
means that $\pi_{1}\circ P$ is a plot of $\Omega$ and $\pi_{2}\circ P:=h$ is a
$C^{\infty}$ map from $U$ to $\mathbb{R}^{d}$, where $\pi_{i}$ denotes the
projection onto the $i$-th factor. But being $\pi_{1}\circ P$ a plot for
$\Omega$ means that it is, locally in a neighborhood $V$, a translation up to
pre-composing with a smooth map, say $g$.

Then for every $u\in V\subset U$ we have that

$P(u)=(\pi_{1}\circ P(u),\pi_{2}\circ P(u))=(F_{T}\circ
g(u),h(u))=(T+g(u),h(u))$

and replacing $P(u)$ in Equation 14 we get that

$ev_{\rho}\circ P(u)=\rho(h(u))(T+g(u))=T+g(u)+h(u)=F_{T}\circ(g+h)(u),$

where $g+h$ is a smooth map because both summands are smooth. Thus
$ev_{\rho}\circ P$ is a plot for $\Omega$. Hence $ev_{\rho}$ is smooth and
then $\rho$ acts smoothly.

∎

Note that there are at least two natural topologies over $\Omega$: the
$D$-topology and the topology induced by the metric $d$.

Let $A\subset\Omega$ be an open subset, assume without loss of generality that
$A=\mathcal{B}_{\varepsilon}(T)$ for some $T\in\Omega$. Let $T^{\prime}\in A$,
pick an $\varepsilon^{\prime}>0$ such that all translates of $T^{\prime}$
within radius $\varepsilon^{\prime}$ remain inside $A$. Let
$B=\\{T^{\prime}+x:\|x\|<\varepsilon^{\prime}\\}\subset A$, we claim it is
$D$-open. Let $P:U\to\Omega$ be a plot. By definition of a generating family
(see discussion below Definition 12), for every $u\in U$, there exist a
neighborhood $V_{u}\subset U$ of $u$ and a smooth map $Q_{u}:V_{u}\to U$ such
that $P|_{V_{u}}=F_{T}\circ Q_{u}$. Then

$P^{-1}(B)=\bigcup_{u\in U}(P|_{V_{u}})^{-1}(B)=\bigcup_{u\in U}(F_{T}\circ
Q_{u})^{-1}(B)=\bigcup_{u\in U}(Q_{u}^{-1}(F_{T}^{-1}(B)).$

But $F_{T}^{-1}(B)$ coincides with the euclidean open ball
$B_{\varepsilon^{\prime}}(0)$ and $Q_{u}$ is a smooth map between euclidean
domains therefore it is continuos. Thus $P^{-1}(B)$ is open in some
$\mathbb{R}^{n}$ and $B$ is $D$-open. Thus the $D$-topology is finer than the
topology induced by the metric $d$.

######  Proposition 5.

If $\omega$ is a primitive substitution rule for $\Omega$ then the map
$\omega:\Omega\to\Omega$ is smooth.

###### Proof.

Let $P:U\to\Omega$ be a plot for $\Omega$. We must show that $\omega\circ P$
is also a plot. By definition, for every point $u\in U$ there is a
neighborhood $V\subset U$, an element $F_{T}:W\to\Omega$ of the generating
family of $\mathcal{D}$ and a smooth map $Q:V\to W$ such that
$P|_{V}=F_{T}\circ Q$ (assuming $P$ non constant, as if it were the result
would follow).

---
$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P}$$\textstyle{\Omega\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$$\textstyle{\Omega}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Q}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{T}}$$\scriptstyle{F_{\omega(T)}}$

Then for every $v\in V$ and some $T\in\Omega$ we have that

$\omega\circ P(v)=\omega\circ F_{T}\circ Q(v)=\omega\circ
F_{T}(x)=\omega(T+x)=\omega(T)+\lambda\cdot x=F_{\omega(T)}(\lambda\cdot x),$

where $x=Q(v)$ and $\lambda$ is the expansion factor of $\omega$. The map
$F_{\omega(T)}$ is an element of $\mathcal{F}$ because $\omega(T)\in\Omega$.
It follows that every $u$ in $U$ has a neighborhood $V$ such that $\omega\circ
P|_{V}=F_{\omega(T)}\circ Q$. Then $\omega\circ P$ is a plot for $\Omega$. ∎

Recall that the topological space $\Omega$ is locally homeomorphic to a
cartesian product of a ball $D$ times totally disconnected set $K$. Note that
Lemma 4 asserts that it is possible to move smoothly within the factor $D$,
while the previous proposition means that there is also a smooth dynamic
within the factor $K$.

With regard to equivalence we note that diffeomorphism of tiling spaces map
orbits onto orbits, and viceversa, an _orbital equivalence_ between tiling
spaces is a diffeomorphism.

######  Proposition 6.

Let $(\Omega,\rho)$ and $(\Omega^{\prime},\rho^{\prime})$ be two
$d$-dimensional tiling spaces endowed with their respective
$\mathbb{R}^{d}$-actions $\rho$ and $\rho^{\prime}$ by translation. Then, they
are orbitally equivalent if and only if they are diffeomorphic.

###### Proof.

Let $\varphi:\Omega\to\Omega^{\prime}$ an orbital equivalence between tiling
spaces and $P:U\to\Omega$ a plot. Up to pre-composing with a smooth map $f$,
$P$ is locally a translation. This means that for every $u\in U$ there exists
an open neighborhood $V\subset U$ of $u$, a tiling $T\in\Omega$ and a smooth
map $f:V\to W$ such that $P|_{V}=F_{T}\circ f$. Hence

$\varphi\circ P(u)=\varphi\circ F_{T}\circ
f(u)=\varphi(T+f(u))=\varphi(T)+y=F^{\prime}_{\varphi(T)}(y),$

for some $y\in\mathbb{R}^{d}$. The third equality is given by hypothesis and
$F^{\prime}_{\varphi(T)}$ is an element of the generating family
$\mathcal{F}^{\prime}$ for the diffeology of $\Omega^{\prime}$. Thus
$\varphi\circ P$ is a plot of $\Omega^{\prime}$ and then $\varphi$ is smooth.
The same argument works for proving the smoothness of $\varphi^{-1}$. Hence
$\varphi$ is a diffeomorphism.

Conversely, let $\varphi:\Omega\to\Omega^{\prime}$ be a diffeomorphism. Let
$T\in\Omega$ and $x\in\mathbb{R}^{d}$. Consider the path
$\gamma:\mathbb{R}\to\Omega$ given by $\gamma(s)=(1-s)T+s(T+x)$. This path
joins $T$ to $T+x$. Then, as $\varphi$ is smooth, $\varphi\circ\gamma$ is a
path in $\Omega^{\prime}$ such that $\varphi\circ\gamma(0)=\varphi(T)$ and
$\varphi\circ\gamma(1)=\varphi(T+x)$. Thus $\varphi(T)$ and $\varphi(T+x)$ lie
in the same path component, so $\varphi(T+x)$ must be a translate of
$\varphi(T)$. So $\varphi$ takes orbits in $\Omega$ onto orbits in
$\Omega^{\prime}$.

∎

###  3.2 The bundle $\hat{p}:\Omega\to\mathbb{T}_{\alpha}$

In the following we restrict to one-dimensional tilings. We will discuss on
how to bring Sadun-Williams’ construction into the diffeological setting.
Consider an unidimensional tiling space $\Omega$ of primitive substitution
tilings with a rule $\omega$ on a two-tile set $\\{0,1\\}$. Recall from [2]
that $\Omega$ is conjugate – and hence diffeomorphic following Proposition 6–
to the inverse limit of branched curves $\Omega_{n}$

$\Omega=\lim_{\leftarrow n}\\{\Omega_{n},f_{n}\\}.$

The first approximant222We omit the technicality of whether the substitution
rule _forces the border_ because both complexes $\Omega_{0}$ and $\Omega_{1}$
from Anderson-Putnam construction [2] work for our purpose. $\Omega_{0}$ is
the wedge of two circles with lengths 1 and $\alpha$, one for each letter of
the alphabet $\\{0,1\\}$, the following approximants $\Omega_{n}$ are the same
wedge of circles whose radii are $\alpha$ times the radii of the previous and
subdivided by the substitution rule, where $\alpha$ is the expansion factor of
the substitution rule $\omega$ (see Theorem 2 and Fig. 1). The transition
functions $f_{n}$ forget the tiles from the $n$-th level. The general case for
substitution rules with more than two letters is treated in the same fashion.

The space $\Omega$ is a (topological) fiber bundle over the one-dimensional
torus $\mathbb{S}^{1}$ and its fibers are homeomorphic to the Cantor set $K$
(see Theorem 3). Consider such fibration

$K\hookrightarrow\Omega\stackrel{{\scriptstyle p}}{{\to}}\mathbb{S}^{1}.$

The map $p$ is the composition of maps

$\Omega\stackrel{{\scriptstyle\pi_{0}}}{{\longrightarrow}}\Omega_{0}\stackrel{{\scriptstyle
q}}{{\longrightarrow}}\mathbb{R}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}\mathbb{S}^{1},$

where $\pi_{0}:\Omega\to\Omega_{0}$ is the projection onto the first factor of
$\Omega$ as an inverse limit, $q$ is the function that maps the cells of
$\Omega_{0}$ onto intervals $[n,n+1]\subset\mathbb{R}$, and $\pi$ is the
quotient map $\pi:\mathbb{R}\to\mathbb{R}/\mathbb{Z}$. Thus the function $p$
maps the tiling $T$ to the position of its origin in the tile it belongs to,
projected onto $\mathbb{S}^{1}$.

In the diffeological context, it is perhaps more natural to consider a similar
map $\hat{p}:\Omega\to\mathbb{T}_{\alpha}$ over the irrational torus
$\mathbb{T}_{\alpha}=\mathbb{R}/(\mathbb{Z}+\alpha\mathbb{Z})$, which may be
defined by mapping the vertices of a tiling $T$ to
$\mathbb{Z}+\alpha\mathbb{Z}$ instead of $\mathbb{Z}$. We define $\hat{p}$ as
the composition of maps

$\Omega\stackrel{{\scriptstyle\pi_{0}}}{{\longrightarrow}}\Omega_{0}\stackrel{{\scriptstyle\hat{q}}}{{\longrightarrow}}\mathbb{R}\stackrel{{\scriptstyle\mathrm{pr}_{\alpha}}}{{\longrightarrow}}\mathbb{T}_{\alpha},$

in this case $\pi_{0}:\Omega\to\Omega_{0}$ is the same projection as before,
$\hat{q}$ is the function that maps the circles forming $\Omega_{0}$ onto the
intervals of the form $[n+m\alpha,n+1+m\alpha]$ or $[n+m\alpha,n+(m+1)\alpha]$
for $n,m\in\mathbb{Z}$, and $\mathrm{pr}_{\alpha}$ is the quotient map
$\mathrm{pr}_{\alpha}:\mathbb{R}\to\mathbb{R}/(\mathbb{Z}+\alpha\mathbb{Z})$.
Thus $\hat{p}$ is the function that maps a tiling $T$ to the location of its
origin into intervals like $[-1,0],[0,\alpha]\subset\mathbb{R}$, followed by
the quotient map $\mathrm{pr}_{\alpha}$.

We endow $\Omega_{0}$ with the push-forward diffeology
${\pi_{0}}_{*}(\mathcal{D})$ of $\mathcal{D}$ through the map $\pi_{0}$,
$\mathbb{R}$ with the standard diffeology and $\mathbb{T}_{\alpha}$ with the
quotient diffeology for $\mathbb{R}/(\mathbb{Z}+\alpha\mathbb{Z})$.

######  Lemma 5.

With the aforementioned diffeologies the map
$\hat{p}:\Omega\to\mathbb{T}_{\alpha}$ is a smooth surjection.

###### Proof.

The map $\hat{p}$ is clearly surjective as, for a given
$z\in\mathbb{T}_{\alpha}$ represented by $t+n+\alpha m$ with
$t\in\mathbb{R},n,m\in\mathbb{Z}$, there is a tiling $T\in\Omega$ whose origin
lies in $t\mod 1$ that is mapped to $z$. For proving smoothness note that the
maps $\pi_{0}$ and $\mathrm{pr}_{\alpha}$ are smooth by definition and
$\hat{q}\circ\pi_{0}$ is locally smooth in every tiling $T$ whose origin is
not located in an end-point of a tile, because there are $D$-open intervals
$A\subset\Omega$ where the plots parametrize translations within $A$. That is,
given a plot $P$ of $\Omega$, $\hat{q}\circ\pi_{0}\circ P$ is a translation in
$\mathbb{R}$, which is smooth and thus a plot. Hence
$\hat{p}=\mathrm{pr}_{\alpha}\circ\hat{q}\circ\pi_{0}$ is locally smooth in
every tiling $T$ whose origin is not located in the border of a tile.

However $\hat{p}$ is also locally smooth also in the rest of the tilings
because the endpoints of the tiles are smoothly identified in
$\mathbb{T}_{\alpha}$, by definition of its diffeology. Thus $\hat{p}$ is
locally smooth in every point of $\Omega$ so it is smooth.

∎

####  3.2.1 The structure groupoid for $\hat{p}:\Omega\to\mathbb{T}_{\alpha}$.

We proceed to investigate the structure groupoid associated to the smooth
surjection $\hat{p}:\Omega\to\mathbb{T}_{\alpha}$. Recall from 2.2.2. that
$G_{0}=\mathbb{T}_{\alpha}$ and

$G_{1}=\bigcup_{z,z^{\prime}\in
G_{0}}\mathbf{Diff}(\Omega_{z},\Omega_{z^{\prime}}),$

for $\Omega_{z}:=\hat{p}^{-1}(z)$. The maps _source_ $s:G_{1}\to G_{0}$ and
_target_ $t:G_{1}\to G_{0}$ are defined by $s(\varphi)=\hat{p}(dom(\varphi))$
and $t(\varphi)=\hat{p}(ran(\varphi))$.

A diffeology for the groupoid $(G_{1},G_{0})$ is defined as the coarsest
diffeology such that the evaluation map $ev:X_{s}\to\Omega$, given by
$ev(\varphi,T)=\varphi(T)$, from the pull-back

$X_{s}=\\{(\varphi,T)\in G_{1}\times\Omega:T\in dom(\varphi)\\}$

along the map $s$, is smooth and makes $(G_{1},G_{0})$ a diffeological
groupoid.

######  Proposition 7.

The map $\hat{p}:\Omega\to\mathbb{T}_{\alpha}$ is a diffeological fibration.

###### Proof.

We must show that the structure groupoid $(G_{1},G_{0})$ associated to
$\hat{p}$ is fibrant, _i.e._ the characteristic map
$\chi:G_{1}\to\mathbb{T}_{\alpha}\times\mathbb{T}_{\alpha}$ is a
subduction333See discussion after Example 2.. In order to conclude that, we
have to show that for every plot
$Q:U\to\mathbb{T}_{\alpha}\times\mathbb{T}_{\alpha}$ and every point $u\in U$
there exists a neighborhood $V$ of $u$ such that either $Q|_{V}$ is constant
or there exists a plot $P:V\to G_{1}$ that locally lifts $Q$.

The plots of $G_{1}$ are specified as those parametrizations $P$ such that the
maps $\chi\circ P$ and

$P_{s}:\Omega_{s\circ P}\to\Omega\text{ given by }P_{s}(u,T)=P(u)(T)$

$P_{t}:\Omega_{t\circ P}\to\Omega\text{ given by }P_{t}(u,T)=P(u)^{-1}(T),$

where $\Omega_{s\circ P}$ stands for the pull-back of $\hat{p}$ along $s\circ
P$, and analogously $\Omega_{t\circ P}$, are smooth.

Let $Q:U\to\mathbb{T}_{\alpha}\times\mathbb{T}_{\alpha}$ be a plot for the
product diffeology on $\mathbb{T}_{\alpha}\times\mathbb{T}_{\alpha}$, then
$Q(u)=(Q_{1}(u),Q_{2}(u))$ where $Q_{i}$ a plot of $\mathbb{T}_{\alpha}$.
Moreover each $Q_{i}$ lifts locally (say in neighborhoods $V_{i}$, with
$V_{1}\times V_{2}:=V\subset U$) to a plot $\bar{Q}_{i}$ of $\mathbb{R}$
because the diffeology on $\mathbb{T}_{\alpha}$ coincides with the quotient
diffeology $\mathbb{R}/(\mathbb{Z}+\alpha\mathbb{Z}$).

Let $u\in U$, $z_{s}=\bar{Q}_{1}(u)$,
$z_{t}=\bar{Q}_{2}(u),\bar{z}_{s}=\mathrm{pr}_{\alpha}(z_{s})$ and
$\bar{z}_{t}=\mathrm{pr}_{\alpha}(z_{t})$. Then $z_{s},z_{t}\in\mathbb{R}$ and
$\bar{z}_{s},\bar{z}_{t}\in\mathbb{T}_{\alpha}$. Let
$\varphi_{z_{s},z_{t}}:\Omega\to\Omega$ be the map given by

$\varphi_{z_{s},z_{t}}(T)=T-z_{s}+z_{t}.$

Note that this map translates the fiber
$\hat{p}^{-1}(\bar{z}_{s}):=\Omega_{\bar{z}_{s}}$ to
$\hat{p}^{-1}(\bar{z}_{t}):=\Omega_{\bar{z}_{t}}$. It is moreover a
diffeomerphism between these fibers: it is smooth because it is a translation
as well as its inverse. Then it is an element of $G_{1}$ satisfying
$s(\varphi_{z_{s},z_{t}})=\bar{z}_{s}$ and
$t(\varphi_{z_{s},z_{t}})=\bar{z}_{t}$. Thus we define a parametrization
$P:V\to G_{1}$ given by $P(u)=\varphi_{z_{s},z_{t}}$ and assert that it is the
plot we needed.

The maps $P_{s}$, $P_{t}$ are immediately seen to be smooth because they are
translations of tilings, and $\chi\circ P$ equals
$(\mathrm{pr}_{\alpha}\times\mathrm{pr}_{\alpha})\circ\bar{Q}$ which is a
composition of smooth maps, and so it is also smooth. Hence $P$ is a plot of
$G_{1}$ and it lifts the plot
$Q:U\to\mathbb{T}_{\alpha}\times\mathbb{T}_{\alpha}$ by construction.
Consequently $\chi$ is a subduction and the groupoid $(G_{1},G_{0})$ is
fibrant.

$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P}$$\scriptstyle{\bar{Q}}$$\textstyle{G_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Q}$$\textstyle{\mathbb{R}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{pr}_{\alpha}^{2}}$$\textstyle{\mathbb{T}_{\alpha}\times\mathbb{T}_{\alpha}}$

∎

######  Theorem A.

Let $\Omega$ be an unidimensional tiling space of primitive substitution
tilings with a rule $\omega$ and expansion factor $\alpha$. Then $\Omega$ is a
covering space over the irrational torus $\mathbb{T}_{\alpha}$.

###### Proof.

The result is essentially the previous proposition plus the fact that, just as
in the topological case, a _covering space_ in diffeology is a fiber bundle
with discrete fiber.

∎

This map has the advantage over Williams’ and Sadun-Williams constructions
that the proportion of the length of the tiles is preserved. So it provides a
necessary condition for diffeomorphism between tiling spaces and thus for
orbital equivalence. This due to the classification of irrational tori [19],
as we shall see in the following section.

######  Remark 3.

This result is fairly general within the class of one-dimensional
substitutional tiling spaces. In this case, there is not too much room for
distinction of tilings (as in their higher dimensional counterparts) other
than the combinatorics and lenghts of the tiles. This information is pretty
much encoded in the dynamics of the substitution matrix. There is a close
relation between these dynamics with the arithmetics of the eigenvalues of
this matrix and the possibility of obtaining the same tilings by means of a
cut-and-project scheme.

In fact, one-dimensional tilings lie within the intersection of the classes of
substitution and projection tilings if the irrational number $\alpha$ is a
quadratic irrational Pisot number444A real number $\alpha$ is a Pisot-
Vijayaraghavan number, also called PV-number or just Pisot number, if it is
the root of a monic polynomial with integer coeficients, $\alpha>1$, and all
of its Galois conjugates are less than 1 in absolute value. [16, 27] and the
points in the border of the window $W$ are elements of $\mathbb{Q}[\alpha]$.
In the next section we will turn our attention to the tiling spaces arising
from the cut-and-project method.

##  4 Diffeological and sturmian equivalences for Tiling spaces

In the present section we study another bundle structure from projection
method tiling spaces $\Omega_{\alpha}$ over $\mathbb{T}_{\alpha}$. We will see
how to bring notions of equivalences between sturmian systems (Lemma 1) into
the context of tiling spaces through their diffeological structures.

As discussed in Section 2.1.4 above, we can relate explicitly the sturmian
sequences of $(X_{\alpha},\sigma)$ with canonical projection tilings in
$\Omega_{\alpha}$. We define a function $\Phi:\Omega_{\alpha}\to X_{\alpha}$
which“forgets” the length of the tiles but preserves the combinatorics of
every tiling, _i.e._ the funcion $\Phi$ maps the tiling $T$ onto the sequence
$S\in\\{0,1\\}^{\mathbb{Z}}$ encoding the tiles of $T$, where $\\{0,1\\}$
represent the proto-tile set of $T$.

This function is not invertible but there is a canonical section
$\Psi:X_{\alpha}\to\Omega_{\alpha}$ which assigns to a sturmian sequence $S$
the tiling $\Psi(S)=T$ in the tile set $\\{0,1\\}$ with lenghts
$\frac{1}{\sqrt{1+\alpha^{2}}}$ and $\frac{\alpha}{\sqrt{1+\alpha^{2}}}$,
respectively, appearing with the same combinatorics as in $S$, and with its
origin placed on the left vertex of the tile associated to $s_{0}$. We define
a diffeology for $X_{\alpha}$ as the pull-back diffeology of $\Omega_{\alpha}$
along the map $\Psi$. Thus the map $\Psi$ is automatically smooth.

The following lemmas verify that we can bring notions of sturmian systems into
the diffeological context.

######  Lemma 6.

With the aforementioned diffeology, the map $\Phi$ is smooth.

###### Proof.

Let $P:U\to\Omega_{\alpha}$ a plot of $\Omega_{\alpha}$. We must show that
$\Phi\circ P:U\to X_{\alpha}$ is a plot of $X_{\alpha}$. But this is the case
if and only if $\Psi\circ\Phi\circ P$ is a plot of $\Omega_{\alpha}$. As $P$
is a plot of $\Omega_{\alpha}$, by definition of the generating family
$\mathcal{F}$ (see Definition 12) for the diffeology of $\Omega_{\alpha}$, for
every $u\in U$ there exists a neighborhood $V\subset U$ containing $u$, a
smooth map $g:V\to W\subset\mathbb{R}$, and an element
$F_{T}:W\to\Omega_{\alpha}$ of the generating family, with
$T\in\Omega_{\alpha}$, such that $P|_{V}=F_{T}\circ g$. Then

$(\Psi\circ\Phi\circ P)|_{V}(v)=(\Psi\circ\Phi)\circ F_{T}\circ
g(v)=(\Psi\circ\Phi)(T+g(v)).$ (15)

Note that both $T+g(v)$ and $(\Psi\circ\Phi)(T+g(v))$ are elements of
$\Omega_{\alpha}$ with exactly the same combinatorics, so they must be
translates of each other. So $\Psi\circ\Phi\circ P$ is locally a translation,
and thus a plot of $\Omega_{\alpha}$. Then $\Phi$ is smooth. ∎

######  Lemma 7.

With the aforementioned diffeology for $X_{\alpha}$ the shift action
$\sigma:X_{\alpha}\to X_{\alpha}$ is a diffeomorphism.

###### Proof.

We must show first that the composition of every plot of $X_{\alpha}$ with the
shift map is a plot of $X_{\alpha}$ itself. Let $P:U\to X_{\alpha}$ be a plot
of $X_{\alpha}$. By definition of the pull-back diffeology along the map
$\Psi$ defined above, the composition $\Psi\circ P$ is a plot of
$\Omega_{\alpha}$. Then, by definition of the generating family $\mathcal{F}$
(see Definition 12) for the diffeology of $\Omega_{\alpha}$, for every $u\in
U$ there exists a neighborhood $V\subset U$ containing $u$, a smooth map
$g:V\to W\subset\mathbb{R}$, and an element $F_{T}:W\to\Omega_{\alpha}$ of the
generating family, with $T\in\Omega_{\alpha}$, such that

$(\Psi\circ P)|_{V}=F_{T}\circ g.$ (16)

This particularly means that a translate of the tiling $T$ is in the image of
the map $\Psi$. Let $v\in V$, $P(v)=s\in X_{\alpha}$ and $\Psi(s)=T+x$, with
$x=g(v)$. Then

$\Psi(\sigma(s))=T+x+\delta_{\alpha},$

where

$\delta_{\alpha}=\left\\{\begin{array}[]{lll}\frac{1}{\sqrt{1+\alpha^{2}}}&&\mbox{
if }\sigma(s)_{0}=0\\\ \\\ \frac{\alpha}{\sqrt{1+\alpha^{2}}}&&\mbox{ if
}\sigma(s)_{0}=1.\end{array}\right.$ (17)

Thus,

$(\Psi\circ\sigma\circ P)|_{V}(v)=F_{T+\delta_{\alpha}}+g(v),$

and $\Psi\circ\sigma\circ P$ is a plot of $\Omega_{\alpha}$. Which means that
$\sigma\circ P$ is a plot of $X_{\alpha}$. Therefore $\sigma$ is smooth.

The same argument works for proving the smoothness of $\sigma^{-1}$, whose
action is carried onto $\Omega_{\alpha}$ as a translation in the opposite
direction. So the $\delta_{\alpha}$ from above is negative in the present
case. We then conclude that $\sigma$ is a diffeomorphism.

∎

An inductive argument gives us the following

######  Corollary 1.

The $\mathbb{Z}$ action on $X_{\alpha}$ given by

$(n,S)\mapsto\sigma^{n}(S)$

is smooth.

######  Lemma 8.

Let $X_{\alpha}$ and $X_{\beta}$ sturmian spaces endowed with the diffeology
from above. They are orbit equivalent (Definition 7) if and only if they are
diffeomorphic.

###### Proof.

The result is a direct consequence of Proposition 6, Lemmas 6 and 7, and the
definition of the mappings $\Psi$ and $\Phi$. The actions may be seen in the
following diagram

---
$\textstyle{X_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{\alpha}}$$\scriptstyle{\sigma}$$\scriptstyle{f}$$\textstyle{\Omega_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu_{\alpha}}$$\textstyle{X_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{X_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{\Omega_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu_{\beta}}$$\textstyle{X_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{\beta}}$$\textstyle{\Omega_{\beta}}$

where $\nu_{\alpha}:\Psi_{\alpha}\circ\sigma\circ\Phi_{\alpha}$,
$\nu_{\beta}:\Psi_{\beta}\circ\sigma\circ\Phi_{\beta}$ and
$g=\Psi_{\beta}\circ f\circ\Phi_{\alpha}$.

∎

######  Theorem B.

Let $\Omega_{\alpha}$ a tiling space of canonical projection of dimension 2 to
1 and irrational slope $\alpha$. Then $\Omega_{\alpha}$ is a
$\mathbb{R}$-principal bundle over the double pointed irrational tori
$\mathbb{T}_{\alpha}^{\bullet}$.

###### Proof.

The function $\Psi$ is a bijection between the orbits of the shift $\sigma$ in
$X_{\alpha}$ and the orbits by continuous translation in $\Omega_{\alpha}$.
Thus we will prove that the the surjection $X_{\alpha}\to\mathbb{T}_{\alpha}$
given by $S(\rho)\mapsto\mathrm{pr}_{\alpha}(\rho)$, where
$\mathrm{pr}_{\alpha}$ is defined by 7. Consider $\rho,\rho^{\prime}\in(0,1)$
such that

$\rho-\rho^{\prime}=e+f\alpha\in\mathbb{Z}+\alpha\mathbb{Z},$

and that $S(\rho),S(\rho^{\prime})$ represent the sequences
$\\{s_{n}\\}_{n\in\mathbb{Z}},\\{s^{\prime}_{n}\\}_{n\in\mathbb{Z}}$
respectively. Thus, we have the following equalities

$\displaystyle s_{n}$ $\displaystyle=\lceil
n\alpha+\rho\rceil-\lceil(n-1)\alpha+\rho\rceil$ (18) $\displaystyle s_{n}$
$\displaystyle=\lceil(n+f)\alpha+\rho^{\prime}+e\rceil-\lceil(n+f-1)\alpha+\rho^{\prime}+e\rceil$
$\displaystyle s_{n}$
$\displaystyle=\lceil(n+f)\alpha+\rho^{\prime}\rceil-\lceil(n+f-1)\alpha+\rho^{\prime}\rceil$
$\displaystyle s_{n}$ $\displaystyle=\sigma^{f}(s^{\prime}_{n}).$

Then each orbit in $X_{\alpha}$ is projected onto a class of
$\mathbb{T}_{\alpha}$. But recall that there is a double orbit corresponding
to sequences differing in two consecutive letters. Including both of these
orbits, we obtain a $\mathbb{Z}$-principal fiber bundle
$X_{\alpha}\to\mathbb{T}_{\alpha}$ with a singular point with a double fiber.
In order to avoid this singularity, we consider the double pointed irrational
torus

$\mathbb{T}^{\bullet}_{\alpha}=\mathbb{T}_{\alpha}\times\\{0,1\\}/\sim,\quad\text{with}\quad([x],j)\sim([y],i)\quad\text{iff}\quad[x]=[y]\quad\text{and}\quad[x]\neq[0].$
(19)

Where $[x]$ is the equivalent class of $x\in\mathbb{R}$ through the projection
$\mathrm{pr}_{\alpha}$ (7). Now we can extend through $\Psi$ to obtain the
conmutative diagram of smooth maps (it should be clear by now that the
contruction of $\mathbb{T}_{\alpha}^{\bullet}$ is smooth)

$\textstyle{X_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\scriptstyle{p}$$\textstyle{\Omega_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\alpha}^{\bullet}}$$\textstyle{\mathbb{T}_{\alpha}^{\bullet}.}$
(20)

We have seen already in Lemma 2 and Section 2.2.3 that the Kronecker flow
$\mathcal{K}_{\alpha}$ is a $\mathbb{R}$-principal bundle over
$\mathbb{T}_{\alpha}$. So there is only left to prove that the new action map
$\mathcal{A}^{\mathbb{R}}_{\Omega_{\alpha}}$ is still an induction. But by
construction the extra fiber of $\Omega_{\alpha}$ is projected onto the extra
point of $\mathbb{T}_{\alpha}^{\bullet}$, so if
$\mathcal{A}^{\mathbb{R}}_{\mathcal{K}_{\alpha}}$ is an induction then
$\mathcal{A}^{\mathbb{R}}_{\Omega_{\alpha}}$ must also be. Thus the surjection
$\pi_{\alpha}^{\bullet}$ is $\mathbb{R}$-principal fibration.

∎

An importat consequence of these results is that we get to define a
fundamental group for $\Omega$.

######  Corollary 2.

The fundamental group of $\Omega_{\alpha}$ is

$\pi_{1}(\Omega)=\mathbb{Z}+\mathbb{Z}+\alpha\mathbb{Z}.$

###### Proof.

The proof relies on relative homotopy and the long exact sequence of homotopy
of a pair [18, Postulate 5.19] – we omit here the basepoint because we are
dealing with homogeneus spaces. We consider the relative homotopy of the pair
$\mathcal{K}_{\alpha}\subset\Omega_{\alpha}$:

$\ldots\longrightarrow\pi_{2}(\Omega_{\alpha},K_{\alpha})\longrightarrow\pi_{1}(\mathcal{K}_{\alpha})\longrightarrow\pi_{1}(\Omega_{\alpha})\longrightarrow\pi_{1}(\Omega_{\alpha},\mathcal{K}_{\alpha})\longrightarrow\pi_{0}(\mathcal{K}_{\alpha})\longrightarrow\pi_{0}(\Omega_{\alpha})$

We can straightforward check that
$\pi_{2}(\Omega_{\alpha},\mathcal{K}_{\alpha})=0$, because there is no room
for 2-dimensional paths; also
$\pi_{1}(\mathcal{K}_{\alpha})=\mathbb{Z}+\alpha\mathbb{Z}$, see the exact
sequence 13. As $\pi_{1}(\Omega_{\alpha},\mathcal{K}_{\alpha})$ is isomorphic
to the fundamental group of the line with double origin, which is
$\mathbb{Z}$. And $\pi_{0}(\mathcal{K}_{\alpha})=0$ because it is connected.

This means that

$\pi_{1}(\Omega_{\alpha})=\pi_{1}(\mathcal{K}_{\alpha})+\pi_{1}(\Omega_{\alpha},\mathcal{K}_{\alpha})=\mathbb{Z}+\mathbb{Z}+\alpha\mathbb{Z}$

∎

Theorem B makes it possible to bring the concept of strong orbit equivalence
efectivelly into the context of tiling spaces. Note that the identification
$I_{\bullet}:\mathbb{T}^{\bullet}_{\alpha}\to\mathbb{T}_{\alpha}$ induces a
surjective morphism $\hat{I}_{\bullet}$ between $\mathbb{R}$-principal fiber
bundles
$\pi_{\alpha}^{\bullet}:\Omega_{\alpha}\to\mathbb{T}^{\bullet}_{\alpha}$ and
$\pi_{\alpha}:\mathcal{K}_{\alpha}\to\mathbb{T}_{\alpha}$, besides the smooth
map $\hat{I}_{\bullet}\circ\Psi_{\alpha}:X_{\alpha}\to\mathcal{K}_{\alpha}$.

If $X_{\alpha}$ and $X_{\beta}$ are strong orbit equivalent there is a cocycle
map $n:X_{\alpha}\to\mathbb{Z}$ and a diffeomorphism $h:X_{\alpha}\to
X_{\beta}$ such that $h(\sigma(S))=\sigma^{n(S)}(h(S))$. Then, it is possible
lift $h$ through the smooth maps $\hat{I}_{\bullet},\Psi_{\alpha}$ and
$\Psi_{\beta}$ so as to obtain the following commutative diagram.

---
$\textstyle{X_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{\alpha}}$$\scriptstyle{h}$$\textstyle{\Omega_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{I_{\bullet}\circ\pi^{\bullet}_{\alpha}}$$\textstyle{\mathcal{K}_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{T}_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{I}_{\bullet}\circ\Psi_{\beta}}$$\textstyle{\Omega_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{K}_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\beta}}$$\textstyle{\mathbb{T}_{\beta}}$

The lift $\tilde{h}:\Omega_{\alpha}\to\Omega_{\beta}$ is more than an orbital
equivalence. To see this, we define another cocycle
$m:X_{\alpha}\times\mathbb{R}\to\mathbb{Z}$ such that
$\sigma^{m(T,x)}(\Phi_{\alpha}(T))=\Phi_{\alpha}(T+x)$. Then we have the
following relation

$\tilde{h}(T+x)=\tilde{h}(T)+\sum_{i=0}^{m(T,x)}\sum_{j=0}^{n(\sigma^{i}(S))}\delta_{\beta}(\sigma^{j}(h(S)))+d(\tilde{h}(T),\Psi_{\beta}(h(S))),$
(21)

with $S=\Phi_{\alpha}(T)$ and $d$ the distance function of the metric of
$\Omega_{\beta}$, and $\delta_{\beta}$ as in 17.

Then we define this stronger equivalence as follows and summarize the results
in another theorem.

######  Definition 13.

Two tiling spaces $\Omega_{\alpha}$, $\Omega_{\beta}$ are _strong orbit
equivalent_ if their associated sturmian spaces
$(X_{\alpha},\sigma),(X_{\beta},\sigma)$ are.

######  Theorem C.

Let $\Omega_{\alpha}$, $\Omega_{\beta}$ tiling spaces satisfying the
conditions of Theorem B. Then the following conditions are equivalent:

$\Omega_{\alpha}$ and $\Omega_{\beta}$ are strong orbit equivalent.

$(X_{\alpha},\sigma),(X_{\beta},\sigma)$ are strong orbit equivalent.

$\mathbb{T}_{\alpha}$ and $\mathbb{T}_{\beta}$ are diffeomorphic.

$\alpha$ and $\beta$ are conjugated under the modular group
$\mathrm{GL}(2,\mathbb{Z})$.

###### Proof.

The equivalence $i)\Leftrightarrow ii)$ is by definition, $ii)\Leftrightarrow
iv)$ is Lemma 1, and $iii)\Leftrightarrow iv)$ Lemma 3.

∎

Our results provide a classification of one-dimensional tiling spaces as
conjugacy classes of irrational numbers under the modular group
$\mathrm{GL}(2,\mathbb{Z})$, which implies what we have called _strong orbit
equivalence_ for tiling spaces. Our last result shows that there is a special
representative within this classes which is a substitution tiling space.

######  Theorem D.

Within the class of strong orbit equivalent one-dimensional tiling spaces
associated to a quadratic irrational slope $\alpha$ there is a substitution
tiling space.

###### Proof.

Recall from Remark 3 and results from [27, 16] that a cut-and-project tiling
admit a substitution if the irrational slope $\alpha$ is a quadratic Pisot
number. Moreover, a theorem by Galois [14] asserts that quadratic Pisot
numbers have purely periodic continued fraction. This, together with the well
known fact that the continued fraction of conjugates under the modular group
differ in a finite number of coeficients, implies that there is a Pisot number
within this class of conjugation, provided the irrational number $\alpha$ is
quadratic. This Pisot number is expresed as the continued fraction of the
periodic tail of $\alpha$.

∎

Acknowledgments

We thank Ana Rechtman, Alberto Verjovsky, Ángel Zaldívar, José Aliste-Prieto,
Andrés Navas and Fabien Durand for their support in different stages of this
research.

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