# How big is a tiling’s return module?

Abigail Perryman  Department of Mathematics, University of Texas,
2515 Speedway, PMA 8.100 Austin, TX 78712, USA<EMAIL_ADDRESS>and
Lorenzo Sadun  Department of Mathematics, Univeristy of Texas,
2515 Speedway, PMA 8.100 Austin, TX 78712, USA<EMAIL_ADDRESS>
###### Abstract.

The rank of a tiling’s return module depends on the geometry of its tiles and
is not a topological invariant. However, the rank of the first Čech cohomology
$\check{H}^{1}(\Omega)$ gives upper and lower bounds for the size of the
return module. For all sufficiently large patches, the rank of the return
module is at most the same as the rank of the cohomology. For a generic choice
of tile shapes and an arbitrary reference patch, the rank of the return module
is at least the rank of $\check{H}^{1}(\Omega)$. Therefore, for generic tile
shapes and sufficiently large patches, the rank of the return module is equal
to the rank of $\check{H}^{1}(\Omega)$.

###### Key words and phrases:

Tiling cohomology, Tiling dynamics, return module, spectra

###### 2020 Mathematics Subject Classification:

37B52, also 37C79, 52C22, 52C23, 55N05

##  1\. Introduction and results

A basic objective in tiling theory is understanding which properties of a
tiling are topological (depending only on the tiling space up to
homeomorphism), which are combinatorial (depending on which tiles touch which
other tiles), and which are geometric (depending on the shapes and sizes of
the individual tiles). In this paper we relate a geometric object, namely the
rank of the return module for large patches $P$, to a topological object,
namely the rank of the first Čech cohomology of the associated tiling space.

Let $T$ be a tiling satisfying some basic axioms (see Section 2) and let $P$
be a patch. That is, $P$ is a finite set of tiles in $T$ whose relative
positions are fixed. The patch $P$ will appear in many different places in
$T$. The position of one occurrence of $P$ relative to another is called a
return vector of $P$. The additive group generated by the return vectors of
$P$ is called the return module of $P$ and is denoted $\mathrm{Ret}(P)$. The
rank of $\mathrm{Ret}(P)$ is the maximal number of return vectors that are
linearly independent over the integers. Equivalently, it is the dimension of
$\mathrm{Ret}(P)\otimes\mathbb{Q}$ as a vector space over the rational numbers
$\mathbb{Q}$.

If $P^{\prime}$ is a patch that contains $P$, then every return vector of
$P^{\prime}$ is also a return vector of $P$, so the rank of
$\mathrm{Ret}(P^{\prime})$ is at most equal to the rank of $P$. For any given
tiling, there is a limiting rank that applies to the return modules of all
sufficiently large patches.

If $T$ is any tiling, then the tiling space $\Omega_{T}$ associated with $T$,
also called the continuous hull of $T$, is the closure of the orbit of $T$
under translations in the “big ball” metric where two tilings are considered
close if they agree on a big ball around the origin up to a small translation.
There are many cohomology theories associated with $T$ and $\Omega_{T}$. For
our purposes, the most useful is the pattern equivariant cohomology [10, 11]
of $T$, which is isomorphic to the Čech cohomology
$\check{H}^{*}(\Omega_{T})$. The rank of $\check{H}^{1}(\Omega_{T})$ is the
dimension of
$\check{H}^{1}(\Omega_{T},\mathbb{Q})=\check{H}^{1}(\Omega)\otimes\mathbb{Q}$
as a vector space over $\mathbb{Q}$.

The first of our two main theorems bounds the rank of the return module of a
large patch by the rank of the cohomology.

######  Theorem 1.

Let $T$ be an aperiodic tiling that is repetitive and has finite local
complexity. Then for any sufficiently large patch $P$,

$\mathrm{Rank}(\mathrm{Ret}(P))\leq\mathrm{Rank}(\check{H}^{1}(\Omega_{T})).$

Note that this theorem only bounds the rank of $\mathrm{Ret}(P)$ for large
patches. This is what matters, as the return module for large patches
determines many of the dynamical properties of a tiling, such as its
diffraction spectrum. The return module for small patches does not have any
dynamical significance and can often be changed by adding local markings to a
tiling. See Section 5 for examples of this phenomenon.

Before stating our second main theorem, we must introduce the concept of shape
changes for tilings. It may happen that two tilings $T$ and $T^{\prime}$ have
identical combinatorics, with each tile in $T$ having a corresponding tile in
$T^{\prime}$ and with two tiles in $T$ touching if and only if the
corresponding tiles in $T^{\prime}$ touch. However, the shapes and sizes of
the tiles in $T^{\prime}$ may be different from those in $T$. In that case, we
say that $T^{\prime}$ is obtained by applying a shape change to $T$.

For instance, suppose that $T$ is a Thue-Morse tiling of the line by two kinds
of tiles, called $a$ and $b$, following the sequence $\ldots
abbabaabbaababba\ldots$. Suppose furthermore that the $a$ and $b$ tiles both
have length 1. Now let $T^{\prime}$ be a tiling by tiles $A$ and $B$ following
the analogous pattern $\ldots ABBABAABBAABABBA\ldots$, only with an $A$ tile
having length $\pi/4$ and a $B$ tile having length $\sqrt{2}$. The tilings $T$
and $T^{\prime}$ have identical combinatorics but different geometry.
Consequently, the spaces $\Omega_{T}$ and $\Omega_{T^{\prime}}$ are
homeomorphic, but differ as dynamical systems. The return modules of a patch
$P\subset T$ and a corresponding patch $P^{\prime}\subset T^{\prime}$ do not
necessarily have the same rank.

The shape changes to a tiling $T$ of $\mathbb{R}^{d}$, modulo a form of
equivalence called mutual local derivability (MLD), are parametrized by an
open subset of $\check{H}^{1}(\Omega_{T})\otimes\mathbb{R}^{d}$ [6]. We say
that a property of tilings in this family is generic if it occurs for all
shapes except for a set of measure zero. (The concept of measure zero is clear
when $\check{H}^{1}$ has finite rank, as we are then dealing with a Euclidean
space. If $\check{H}^{1}$ has infinite rank, then the set of possible shape
changes is a union of finite dimensional spaces. In that case, we say that a
property is generic if it applies on a set of full measure on each of these
spaces.)

Our second main theorem says that generic shape changes result in the largest
possible return modules.

######  Theorem 2.

Let $T$ be an aperiodic tiling that is repetitive and has finite local
complexity and suppose that $\ell=\mathrm{Rank}(\check{H}^{1}(\Omega_{T}))$.
Then, after applying a generic shape change, the rank of every return module
is at least $\ell$.

Combining the two theorems, we obtain

######  Corollary 3.

Let $T$ be an aperiodic tiling that is repetitive and has finite local
complexity and suppose that $\ell=\mathrm{Rank}(\check{H}^{1}(\Omega_{T}))$.
Then, after applying a generic shape change, the rank of the return module of
every sufficiently large patch is exactly $\ell$.

For simplicity, we have written Theorem 2 and Corollary 3 assuming that
$\check{H}^{1}(\Omega_{T})$ has finite rank. When $\check{H}^{1}$ has infinite
rank, we can restrict our attention to an $\ell$-dimensional subspace
$H^{1}_{\ell}\subset\check{H}^{1}$. Our proof of Theorem 2 actually shows
that, for a generic shape change with parameters in
$H^{1}_{\ell}\otimes\mathbb{R}^{d}$, the rank of the return module of every
patch will be at least $\ell$. Since we can choose $\ell$ to be as large as we
wish, there is no upper bound to the ranks of return modules, even for
arbitrarily large patches.

The dynamical spectrum of a tiling space is closely associated with return
modules of large patches. Suppose that a large patch $P$ occurs at two
locations $x$ and $y$. Then $T-x$ and $T-y$ agree on a large ball around the
origin and thus are close in the tiling metric (see Section 2 for precise
definitions). If $f$ is a continuous function on $\Omega_{T}$, then
$f(T-y)\approx f(T-y)$. In particular, if $f$ is a continuous eigenfunction of
translation with eigenvalue $\lambda$, then

$\exp(2\pi i\lambda\cdot(x-y))\approx 1,$

with the approximation getting better and better as $P$ gets bigger and
bigger. In most tilings of interest (in particular, in all primitive
substitution tilings), all measurable eigenfunctions can be chosen continuous,
so this constraint applies to all eigenvalues of translation.

Solomyak [19] used this observation to relate the point spectrum of a
substitution tiling to the set of return vectors and to the stretching factor.
Baake and Moody [5] went further, using the return module of a tiling with
pure point spectrum to reconstruct its cut-and-project structure. For a recent
generalization of the Baake-Moody construction, see [20].

##  2\. Definitions and notation

In this section we review the basics of tilings, tiling spaces and return
modules. For a more comprehensive review, see [3] and [14].

A tile is a (closed) topological ball that is the closure of its interior. In
addition to its geometry, a tile may carry a label to distinguish it from
other tiles of the same size and shape. If two tiles carry the same label,
then each must be a translate of the other. A tiling of $\mathbb{R}^{d}$ is a
collection of tiles whose union is all of $\mathbb{R}^{d}$ and whose interiors
are disjoint. A patch of a tiling is a finite subset of its tiles. A tiling is
said to have finite local complexity, or FLC, if for each $r>0$ there are only
finitely many patches, up to translation, of diameter up to $r$. Equivalently,
a tiling has FLC if two conditions are met:

  1. (1)

There are only finitely many tile types, up to translation, and

  2. (2)

There are only finitely many 2-tile patches, up to translation. That is, there
are only finitely many ways for one tile to touch another.

The group $\mathbb{R}^{d}$ of translations acts on tiles by shifting their
positions but leaving their labels unchanged. By extension, $\mathbb{R}^{d}$
acts on tilings and on patches by translating all of the tiles simultaneously.
The action of $x\in\mathbb{R}^{d}$ on a tiling $T$ is denoted $T+x$. A tiling
$T$ is said to be aperiodic if $T+x=T$ implies $x=0$. The orbit of a tiling
$T$ under translation is denoted $\mathcal{O}(T)$.

We will frequently consider a particular pattern of adjacent tiles, such as
the pattern $P=aaba$ in a Thue-Morse tiling, without specifying the location
of $P$. Strictly speaking, $P$ is an equivalence class of patches under
translation rather than a specific patch, but we will abuse notation and
terminology by calling $P$ a “patch” anyway. We can then talk about multiple
occurrences of $P$ in a tiling, i.e. multiple patches in the equivalence class
defined by $P$.

On the set of all tilings by a given set of tile types, we consider the
topology induced by the big ball metric, in which two tilings $T$ and
$T^{\prime}$ are considered $\epsilon$-close if they agree on a ball of radius
$1/\epsilon$ around the origin, up to translation of each by a distance
$\epsilon/2$ or less. A tiling space is a non-empty translation-invariant set
of tilings that is closed in this topology.

We can obtain a tiling space from any tiling $T$ by taking the closure of
$\mathcal{O}(T)$. This space is called the continuous hull of $T$ and is
denoted $\Omega_{T}$. It is the smallest tiling space that contains $T$. A
tiling $T^{\prime}$ is in $\Omega_{T}$ if and only if every patch of
$T^{\prime}$ is a translate of a patch in $T$.

A tiling $T$ is said to be repetitive if, for every patch $P\subset T$ there
exists a radius $r(P)$ such that every ball of radius $r(P)$ contains at least
one copy of $P$. This is equivalent to $\Omega_{T}$ being a minimal dynamical
system. That is, every orbit is dense, so for each $T^{\prime}\in\Omega_{T}$,
$\Omega_{T^{\prime}}=\Omega_{T}$. In that case, any two tilings
$T_{1},T_{2}\in\Omega_{T}$ have exactly the same patches (up to translation),
so any quantity based on those patches is the same for both tilings.

Suppose that $T$ and $T^{\prime}$ are two tilings with the following property:
There is a radius $r$ such that, for all $x,y\in\mathbb{R}^{d}$ such that
$T-x$ and $T-y$ agree on a ball of radius $r$ around the origin,
$T^{\prime}-x$ and $T^{\prime}-y$ agree on a ball of radius 1 around the
origin. This is a precise way of saying that the patterns of $T^{\prime}$ are
determined in a local way from the patterns of $T$ in the exact same
locations. If this condition is met, then we say that $T^{\prime}$ is locally
derivable, or LD, from $T$. If $T^{\prime}$ is LD from $T$ and $T$ is LD from
$T^{\prime}$, then $T$ and $T^{\prime}$ are mutually locally derivable, or
MLD. The local rule deriving $T$ from $T^{\prime}$ extends to a topological
conjugacy from $\Omega_{T}$ to $\Omega_{T^{\prime}}$ that we call an MLD
equivalence.

In principle, the tiles in a tiling may have very complicated (say, fractal)
boundaries. However, every FLC tiling is MLD to a tiling whose tiles are
convex polygons (or polytopes) that meet full edge to full edge. We can
therefore assume, with no loss of generality, that our tilings are of this
sort.

The procedure for doing this conversion is called the “Voronoi trick”. It
involves first picking a “control point” for each tile (or for each instance
of a more complicated patch $P$) and replacing the tiling with a point
pattern, and then replacing each point in the point pattern with its “Voronoi
cell” consisting of all points in $\mathbb{R}^{d}$ that are closer to the
given control point than to any other control point. For more information on
the Voronoi trick, see [3] or [14].

Now pick a patch $P$ and pick a control point within $P$ to represent the
patch. (For instance, we might pick the control point of the pattern $abaa$ to
be the left endpoint of the $b$ tile.) When we speak of the locations of $P$
in a tiling, we mean the locations of the control point. Let
$\\{x_{1},x_{2},\ldots\\}$ be all the locations of $P$ in $T$. The relative
positions $x_{i}-x_{j}$ are called return vectors for $P$ and do not depend on
the choice of control point. The span (over $\hskip 0.5pt\mathbb{Z}$) of the
the return vectors is called the return module of $P$ and is denoted
$\mathrm{Ret}(P)$. The fact that $x_{i}-x_{j}$ is a return vector is a
property of the ball of radius slightly larger than $|x_{i}-x_{j}|/2$ centered
at the point $(x_{i}+x_{j})/2$. If $T$ is repetitive, then a copy of this
large ball appears in every $T^{\prime}\in\Omega_{T}$, so $x_{i}-x_{j}$ is
also a return vector for $P$ in $T^{\prime}$. The set of return vectors and
the resulting return module, are thus quantities that we can associate with
the tiling space $\Omega_{T}$ rather than with just the specific tiling $T$.

If $P$ and $P^{\prime}$ are patches with $P\subset P^{\prime}$, then every
return vector of $P^{\prime}$ is also a return vector of $P$, so
$\mathrm{Ret}(P^{\prime})\subset\mathrm{Ret}(P)$ and
$\mathrm{Rank}(\mathrm{Ret}(P^{\prime}))\leq\mathrm{Rank}(\mathrm{Ret}(P))$.
We are interested in the limit of this rank as the patches grow to $\infty$.

Note that we are taking the limit of the rank, not the rank of the limit! In
some tilings, the return module for a large patch can be smaller, but of the
same rank, than the return module for a small patch. For instance, in the
Thue-Morse tiling with $a$ and $b$ tiles both having length 1, the return
module for the one-letter patch $P_{1}=a$ is $\hskip 0.5pt\mathbb{Z}$ while
the return module of $P_{1}=abb$ is $2\hskip 0.5pt\mathbb{Z}$, and there are
other patches whose return modules are $4\hskip 0.5pt\mathbb{Z}$, $8\hskip
0.5pt\mathbb{Z}$, etc. The limit of these modules is the rank-0 set $\\{0\\}$,
but the rank of $2^{n}\hskip 0.5pt\mathbb{Z}$ is 1 for all $n$, so the limit
of the rank is 1. (In other tilings, such as the Fibonacci tiling, all patches
have the same return module and there is no need to take a limit at all.)

If $T$ is repetitive, then the details of how we take the limit of large
patches are unimportant. Any patch $P_{i}$ has a repetitivity radius
$r(P_{i})$. Any patch $P_{j}$ that contains a ball of radius $r(P_{i})$ must
therefore contain a copy of $P_{i}$. In order to take a limit over all patches
with our partial ordering, it is sufficient to consider a single sequence
$P_{1}\subset P_{2}\cdots$ of patches such that the inner diameter of the
$P_{i}$’s goes to $\infty$. In particular, we could pick the $P_{i}$’s to come
from balls of increasing radius around the origin in a specific tiling $T$.

##  3\. Tiling cohomology

The precise definition of the Čech cohomology of a space is complicated,
involving open covers, nerves of said covers, the simplicial cohomology of
those nerves, and a limit over all open covers partially ordered by refinement
[9]. Fortunately, those details are not needed for a working understanding of
tiling cohomology. For our purposes, two facts are sufficient [15]:

  1. (1)

The Čech cohomology of a CW complex is isomorphic to the singular cohomology,
which in turn is isomorphic to many other cohomology theories. On a CW
complex, “all cohomologies are the same”.

  2. (2)

The Čech cohomology of an inverse limit space is the direct limit of the Čech
cohomologies of the approximants.

Tiling spaces are not CW complexes, but they are inverse limits of CW
complexes. Each approximant, called an Anderson-Putnam complex, describes the
tiling in a neighborhood of the origin. A point in the inverse limit of the
approximants is a set of consistent instructions for tiling bigger and bigger
portions of $\mathbb{R}^{d}$. The union of the neighborhoods is all of
$\mathbb{R}^{d}$, so a point in the inverse limit is a set of instructions for
tiling all of space, which is tantamount to a tiling itself. There are
numerous ways to construct approximants for tiling spaces, but they are all
qualitatively similar. The key ideas are due to Anderson and Putnam [1] and to
Gähler [7]. See [12] for a unification of their arguments and [14] for a
review.

The upshot is that the Čech cohomology of a tiling space is constructed from
data about the local structure of a tiling, where “local” can include
information out to any finite distance but not out to infinity. This idea was
codified by Kellendonk and Putnam [10, 11] as a new cohomology theory called
Pattern Equivariant (PE) cohomology.

Consider a reference tiling $T$ of $\mathbb{R}^{d}$ and a function
$f:\mathbb{R}^{d}\to\mathbb{R}$. We say that a function
$f:\mathbb{R}^{d}\to\mathbb{R}$ is PE with radius $r$ if its value at a point
$x$ depends only on the form of $T$ in a ball of radius $r$ around $x$. More
precisely, if $f$ is PE with radius $r$ and if $T-x_{1}$ and $T-x_{2}$ agree
on a ball of radius $r$ around the origin, then $f(x_{1})=f(x_{2})$. A
function is said to be PE (without any qualifiers) if it is PE with some
finite radius. We can likewise define PE differential forms. It is easy to
check that the exterior derivative of a PE form is PE. Kellendonk and Putnam
defined the (real-valued) PE cohomology of a tiling $T$ to be

$H^{k}_{PE}(T)=\frac{\hbox{Closed PE $k$-forms}}{d(\hbox{PE $k-1$-forms})}$

and proved that $H^{k}_{PE}(T)$ was isomorphic to
$\check{H}^{k}(\Omega_{T})\otimes\mathbb{R}$.

To obtain an integer version of PE cohomology, we can consider PE cochains. A
PE $k$-cochain with radius $r$ assigns an integer to every $k$-cell in the
tiling $T$ based on the neighborhood of size $r$ around that $k$-cell. (If $T$
does not have clearly defined vertices, edges, faces, etc., apply the Voronoi
trick to convert $T$ into a tiling that does.) That is, if the tiling is the
same within some fixed distance $r$ of two different $k$-cells, then our
function must assign the same value to both cells. A cochain is said to be PE
if it is PE with some radius.

The coboundary of a cochain is defined the same as with ordinary (not PE)
cochains. If $\alpha$ is a $k$-cochain and $c$ is a $(k+1)$-cell, then
$\delta\alpha$ is a $(k+1)$-cochain whose value on $c$ is

$\delta\alpha(c):=\alpha(\partial c),$

where $\partial c$ is the boundary of $c$. The coboundary of a PE cochain is
PE (although possibly with a slightly larger radius). Sadun [13] defined the
integer-valued PE cohomology of $T$ to be

$H^{k}_{PE}(T)=\frac{\hbox{Closed PE $k$-cochains}}{\delta(\hbox{PE
$(k-1)$-cochains})}$

and proved that this was isomorphic to $\check{H}^{k}(\Omega_{T})$. The same
construction works with values in any Abelian group, not just the integers
$\hskip 0.5pt\mathbb{Z}$.

We are interested in the rank of $\check{H}^{1}(\Omega_{T})$. This is the same
as the dimension over $\mathbb{Q}$ of
$\check{H}^{1}(\Omega_{T})\otimes\mathbb{Q}$, which is naturally isomorphic to
$H^{1}_{PE}(T,\mathbb{Q})$, the first PE cohomology of $T$ with values in
$\mathbb{Q}$. All of our calculations will be done in the PE setting, where we
represent cohomology classes with cochains on $T$.

The following theorem will allow us to restrict our attention to return
vectors of large patches.

######  Theorem 4.

Let $P$ be a patch in a repetitive tiling $T$ and suppose that the locations
of $P$ are $\\{x_{1},x_{2},\ldots\\}$. If $\alpha$ is a closed PE 1-cochain,
then the cohomology class of $\alpha$ depends only on the value of $\alpha$
applied to paths from $x_{i}$ to $x_{j}$. That is,

  1. (1)

If a closed PE 1-cochain $\alpha$ evaluates to zero on a path from $x_{i}$ to
$x_{j}$ for each pair $(i,j)$, then $\alpha$ represents the zero cohomology
class.

  2. (2)

If two closed PE 1-cochains $\alpha$ and $\beta$ give the same values on a
path from each $x_{i}$ to each other $x_{j}$, then $\alpha$ and $\beta$
represent the same cohomology class.

###### Proof.

We begin with the first statement. Let $r_{1}$ be the PE radius of $\alpha$
and let $r_{2}$ be the recognizability radius of $P$. For each vertex $z$ in
our tiling, let $f(z)=\alpha(c)$, where $c$ is a path from $x_{1}$ to $z$.
(Since $\alpha$ is closed, this does not depend on our choice of path.) We
manifestly have $\alpha=\delta f$.

We pick our path $c$ to be the concatenation of two paths $c_{1}$ and $c_{2}$,
where $c_{1}$ goes from $x_{1}$ to an $x_{i}$ that is close to $z$ and $c_{2}$
goes from $x_{i}$ to $z$. See Figure 1. We can also pick the path $c_{2}$ to
stay within a distance $r_{2}$ of $z$. Since $\alpha(c_{1})=0$,
$f(z)=\alpha(c_{2})$. However, $\alpha(c_{2})$ only depends on the pattern $T$
within a distance $r_{1}$ of all of the edges in $c_{2}$, and therefore within
a distance $r_{1}+r_{2}$ of $z$. This makes $f$ a PE function with radius
$r_{1}+r_{2}$. Since $\alpha$ is the coboundary of a PE function, $\alpha$
represents the zero class in cohomology.

Figure 1. The function $f(z)=\alpha(c_{1})+\alpha(c_{2})=\alpha(c_{2})$ is
pattern-equivariant with radius $r_{1}+r_{2}$.

For the second statement, we simply apply the first statement to
$\alpha-\beta$. Since $\alpha-\beta$ evaluates to zero on all paths from
$x_{i}$ to $x_{j}$, the class of $\alpha-\beta$ is zero, so the class of
$\alpha$ is the same as the class of $\beta$. ∎

In fact, we can go a step beyond Theorem 4. Using a dual version of the
Voronoi trick, we can construct a tiling $T^{\prime}$, MLD to the original
tiling $T$, whose vertices are the points $x_{1},x_{2},\ldots$. Since
$\Omega_{T}$ and $\Omega_{T^{\prime}}$ are homeomorphic, we can identify the
Čech cohomology of $\Omega_{T}$ with the Čech cohomology of
$\Omega_{T^{\prime}}$, which in turn is isomorphic to the PE cohomology of
$T^{\prime}$. As a result, we can represent classes in
$\check{H}^{1}(\Omega_{T})$ with closed PE 1-cochains on $T^{\prime}$. That
is, we can represent classes in $\check{H}^{1}(\Omega_{T})$ as functions on
edges in $T^{\prime}$ that connect pairs of locations of $P$ in $T$. In
particular, every linear function $L:\mathrm{Ret}(P)\to\hskip 0.5pt\mathbb{Z}$
defines a cohomology class in $\check{H}^{1}(\Omega_{T})$ that we denote
$\phi(L)$.

##  4\. Proofs of main theorems

###### Proof of Theorem 1.

Let $k$ be the minimum rank of the return module of any patch and let $P_{0}$
be a patch for which $\mathrm{Rank}(\mathrm{Ret}(P_{0}))=k$. Every
sufficiently large patch $P$ (e.g., any patch whose inner radius is greater
than the repetitivity radius of $P_{0}$) will contain a copy of $P_{0}$ and so
will have
$\mathrm{Rank}(\mathrm{Ret}(P))\leq\mathrm{Rank}(\mathrm{Ret}(P_{0}))=k$. But
$k$ is the minimum possible rank, so we must have
$\mathrm{Rank}(\mathrm{Ret}(P))=k$.

Let $P$ be any patch whose return module has rank $k$. As noted in the comment
after the proof of Theorem 4, every linear function
$L:\mathrm{Ret}(P)\to\hskip 0.5pt\mathbb{Z}$ defines a closed 1-cochain
$\alpha_{L}$ on an associated tiling $T^{\prime}$ whose vertices are the
locations of $P$ in $T$, and thus defines a cohomology class $\phi(L)\in
H^{1}_{PE}(T^{\prime})\simeq\check{H}^{1}(\Omega_{T})$.

We claim that the map $\phi$ from linear functions to cohomology classes is
injective. To see this, suppose that $\phi(L)$ is the zero class in
cohomology. Then the 1-cochain $\alpha_{L}$ on $T^{\prime}$ that is defined by
$L$ must be a coboundary:

$\alpha_{L}=\delta f,$

where $f$ is a PE function with some radius $r$. Let $P^{\prime}$ be a patch
of $T^{\prime}$ containing a ball of radius $r$ centered at a point $x_{i}$.
Then $\alpha$ applied to any return vector of $P^{\prime}$ must be zero,
insofar as $f$ takes on the same value at the endpoints of a chain connecting
two instances of $P^{\prime}$. Thus, $L$ restricted to
$\mathrm{Ret}(P^{\prime})\subset\mathrm{Ret}(P)$ is zero.

However, the rank of $\mathrm{Ret}(P)$ is already the minimum among all
patches of $T$. This implies that $\mathrm{Ret}(P^{\prime})$ is a submodule of
$\mathrm{Ret}(P)$ of full rank, so the only linear function on
$\mathrm{Ret}(P)$ that vanishes on $\mathrm{Ret}(P^{\prime})$ is the zero
function. That is, $\phi(L)=0$ implies that $L=0$.

The rank of $\mathrm{Ret}(P)$ is the same as the rank of the space of linear
functions $\mathrm{Ret}(P)\to\hskip 0.5pt\mathbb{Z}$, which (by the
injectivity of $\phi$) is the same as the rank of the image of $\phi$, which
is bounded by the rank of $\check{H}^{1}(\Omega_{T})$. Thus for all
sufficiently large patches $P$, the rank of $\mathrm{Ret}(P)$ is bounded by
the rank of $\check{H}^{1}(\Omega_{T})$. ∎

Before proving Theorem 2, we review the way that the shapes and sizes of all
tiles are parametrized by cochains. In any tiling $T$, there is a vector-
valued cochain $\mathcal{F}(T)$, called the Fundamental Shape Cochain of $T$,
that assigns to every edge the actual displacement along that edge. This
cochain is closed, since the net displacement along the boundary of any 2-face
is zero. The corresponding class $[\mathcal{F}(T)]\in
H^{1}_{PE}(T,\mathbb{R}^{d})$ is called the Fundamental Shape Class.

To obtain a tiling with the same combinatorics as $T$ but different geometry,
we deform the cochain $\mathcal{F}(T)$ into another cochain $S$. That is, we
construct a new tiling (denoted $S(T)$) whose vertices, edge, faces, etc. are
in 1–1 correspondence with those of $T$, carrying the same labels, such that
the relative position of any pair of vertices in $S(T)$ is given by $S$
applied to a path connecting the corresponding vertices of $T$. See Figure 2
for an example.

Figure 2. By changing the shape cochain to $S$, we convert a tiling by squares
into a tiling by parallelograms.

In order for the tiles of $S(T)$ to fit together and preserve finite local
complexity, we need three conditions:

  * •

The (vector-valued) cochain $S$ must be PE.

  * •

$S$ must be closed, and

  * •

The new shapes must not turn any of the tiles inside-out. If $d=2$, this is
equivalent to the boundary of each tile being a positively oriented simple
closed curve. (Figure 8’s are not allowed!). In higher dimensions the
condition is similar, although trickier to describe.

The third condition is open. That is, any closed PE cochain $S$ that is close
to $\mathcal{F}(T)$ will automatically satisfy it. As long as we concentrate
on small shape changes, we only need to keep track of the first two
conditions.

(While not needed for this paper, large shape change classes are indeed
possible. Given a cochain $S$ that turns some tiles inside-out, it is usually
possible to find another closed PE cochain $\tilde{S}$, cohomologous to $S$,
that satisfies the third requirement. The set of classes in
$H^{1}_{PE}(T,\mathbb{R}^{d})$ that can be realized as fundamental shape
classes is dense, open, and of full measure. See [8] for details.)

In general, if $A$ is any object associated with $T$, then we write $S(A)$ to
denote the corresponding object in $S(T)$, and if $B$ is any object in $S(T)$,
then we write $S^{-1}(B)$ to denote the corresponding object in $T$. Thus the
hull of $S(T)$ is $S(\Omega_{T})=\Omega_{S(T)}$, and for each patch $P$ in $T$
we are interested in the rank of $\mathrm{Ret}(S(P))$.

Note that $\Omega_{T}$ and $S(\Omega_{T})$ are homeomorphic and so have
identical cohomology groups. All statements about cohomology apply both to $T$
and to $S(T)$.

If the rank of $\check{H}^{1}(\Omega_{T})$ is $\ell$, then
$\check{H}^{1}(\Omega_{T},\mathbb{R}^{d})\simeq\mathbb{R}^{\ell d}$ is a
Euclidean space. On this space, the sets of zero (Lebesgue) measure are
clearly defined. The essence of Theorem 2 is that, after applying a shape
change with the class of $S$ avoiding a certain set of measure zero, we obtain
return modules of rank at least $\ell$.

###### Proof of Theorem 2.

Let $e_{1},\ldots,e_{d}$ be the standard basis for $\mathbb{R}^{d}$. Since
$\mathbb{R}^{d}$ has infinite dimension over $\mathbb{Q}$, we can find $\ell$
vectors $w_{1},\ldots,w_{\ell}\in\mathbb{R}^{d}$ such that
$(e_{1},\ldots,e_{d},w_{1},\ldots,w_{\ell})$ are linearly independent over
$\mathbb{Q}$. That is, there is no way to write 0 as a nontrivial linear
combination of these vectors with rational coefficients. Among all
$\ell$-tuples of vectors in $\mathbb{R}^{d}$, those that meet this condition
have full measure, since there are only countably many possible rational
linear combinations and each one is zero only on a set of codimension $d$. We
can also shrink the vectors $w_{i}$ by an integer factor to make them
arbitrarily small without affecting linear independence over $\mathbb{Q}$.

Since the rank of $\check{H}^{1}(\Omega_{T})$ is $\ell$, we can choose $\ell$
closed PE cochains $\alpha_{1},\ldots,\alpha_{\ell}$ whose classes form a
linearly independent set in
$H^{1}_{PE}(T,\mathbb{Q})\simeq\check{H}^{1}(\Omega_{T})\otimes\mathbb{Q}$.

We do our shape change in two steps. First we change to a new cochain $S_{0}$
that evaluates to a rational vector on each edge. The set of such rational
shape cochains is dense in the set of all shape cochains [16], so we can pick
$S_{0}$ arbitrarily close to our original $\mathcal{F}(T)$. We then define

$S=S_{0}+\sum_{i=1}^{\ell}\alpha_{i}\otimes w_{i}.$

What remains is picking an arbitrary patch $P$ in $T$ and showing that
$\mathrm{Rank}(\mathrm{Ret}(S(P)))\geq\ell$.

The displacement along any edge $e$ of $S(T)$, which is the same as $S$
applied to $S^{-1}(e)$, is a rational vector (namely the contribution of
$S_{0}$) plus a rational linear combination of the $w_{i}$’s. Let $I_{i}$,
applied to a rational linear combination of
$(e_{1},\ldots,e_{d},w_{1},\ldots,w_{\ell})$, give the coefficient of $w_{i}$.
This applies in particular to the return vectors of $S(P)$ and indeed to all
elements of $\mathrm{Ret}(S(P))$. Note that if $v_{1}$ and $v_{2}$ are
vertices in $T$, then $I_{i}$ applied to the displacement $S(v_{2})-S(v_{1})$
is the same as $c_{i}$ applied to a 1-chain from $v_{1}$ to $v_{2}$. The
collection $(I_{1},\ldots,I_{\ell})$ defines a linear map
$\varphi:\mathrm{Ret}(S(P))\to\mathbb{Q}^{\ell}$. We claim that the image of
$\varphi$ has rank $\ell$, which then implies that $\mathrm{Ret}(S(P))$ has
rank at least $\ell$, as claimed in Theorem 2.

To see that the image of $\varphi$ contains $\ell$ linearly independent
elements, suppose otherwise. Then there exists a nonzero vector
$b=(b_{1},\ldots,b_{\ell})\in\mathbb{Q}^{\ell}$ that is orthogonal to
$\varphi(v)$ for every return vector $v$ of $S(P)$. However,

$0=b\cdot v=\sum(b_{i}\alpha_{i})(S^{-1}(v)).$

This implies that $\sum b_{i}\alpha_{i}$ evaluates to zero on all return
vectors of $P$. By Theorem 4, this then implies that the cohomology class of
$\sum b_{i}\alpha_{i}$ is zero. However, the cochains $\alpha_{i}$ were
assumed to represent linearly independent cohomology classes and $b$ was
assumed to be nonzero. Contradiction.

We have obtained a return module of rank $\ell$ for any set of cochains
$\alpha_{i}$ whose cohomology classes form a basis for
$\check{H}^{1}(\Omega,\mathbb{Q})$ and for almost every set of vectors $w_{i}$
in a neighborhood of zero. This was done for a particular starting set of tile
shapes, but the same argument works for any starting set. As a result, the set
of shape classes for which all return modules have rank $\ell$ or greater has
full measure in the $\ell d$-dimensional space of possible shape classes. ∎

Note that, prior to the last paragraph, the proof never uses the assumption
that the dimension of $\check{H}^{1}(\Omega_{T},\mathbb{Q})$ was $\ell$. It
merely uses the existence of $\ell$ linearly independent cohomology classes
represented by closed PE cochains $c_{1},\ldots,c_{\ell}$. If
$\check{H}^{1}(\Omega_{T},\mathbb{Q})$ is infinite-dimensional, then we can
find such a set of classes (and cochains) for any positive integer $\ell$. A
shape change using generic shape classes in
$\check{H}^{1}_{\ell}\otimes\mathbb{Q}$, where $\check{H}^{1}_{\ell}$ is the
span of the cohomology classes of the $\alpha_{i}$’s, will then give us return
modules of rank at least $\ell$. This justifies the comment after the
statement of Corollary 3.

##  5\. Examples

###  5.1. One dimensional examples

######  Example 5 (Fibonacci).

The Fibonacci tiling is based on the substitution $a\to ab$, $b\to a$. The
first cohomology of the resulting tiling space has rank 2:
$\check{H}^{1}(\Omega)=\hskip 0.5pt\mathbb{Z}^{2}$. Let $P_{0}$ be a one-tile
patch consisting of an $a$ tile. The distance from each $P_{0}$ to the
subsequent $P_{0}$ is either the length of an $a$ tile (if the two $P_{0}$’s
are back-to-back, as in $aa$) or the length of an $a$ tile plus the length of
a $b$ tile (if the two $P_{0}$’s are separated by a $b$, as in $aba$). This
means that $\mathrm{Ret}(P_{0})$ is the span of the length of an $a$ tile and
the length of a $b$ tile. This has rank 1 if the ratio of the two lengths is
rational and has rank 2 if the ratio is irrational.

What about bigger patches? Every large patch $P$ must contain a supertile of
some order $m$ and must be contained in some supertile $M$. This means that we
must have

$\mathrm{Ret}(P_{2})\subset\mathrm{Ret}(P)\subset\mathrm{Ret}(P_{1}),$

where $P_{1}$ is a supertile of order $m$ and $P_{2}$ is a supertile of order
$M$. However, $\mathrm{Ret}(P_{1})$ and $\mathrm{Ret}(P_{2})$ are the spans of
the lengths of the two kinds of supertiles of order $m$ and $M$, respectively.
Since the substitution matrix $\left(\begin{smallmatrix}1&1\cr
1&0\end{smallmatrix}\right)$ is invertible over $\hskip 0.5pt\mathbb{Z}$,
having determinant $-1$, both of these are the same as the span of the lengths
of the $a$ and $b$ tiles. In other words, all patches $P$ have exactly the
same return module as $P_{0}$ and in particular have the same rank: 1 if
$|a|/|b|\in\mathbb{Q}$ and 2 if $|a|/|b|\not\in\mathbb{Q}$.

Note how these results fit with the general scheme of Theorems 1 and 2. The
cohomology has rank 2, so the return modules of large patches cannot have rank
greater than 2. With a generic choice of tile lengths, return modules have
rank equal to 2. Specifically, the return modules of large (and small) patches
have rank 2 unless the ratio $|a|/|b|$ lies in a countable union of
codimension-1 subsets of $\mathbb{R}$, namely the rational numbers
$\mathbb{Q}$.

So far we have only considered versions of the Fibonacci tiling where all $a$
tiles have length $|a|$ and all $b$ tiles have length $|b|$. We could also
describe the Fibonacci tiling using collared tiles, with several different
varieties of $a$ tile, each with its own length, and several different
varieties of $b$ tile. The return module of small patches could then have rank
greater than 2. For instance, the return vectors for vertices of arbitrary
type are spanned by the lengths of all the different collared tiles. By
collaring to a big enough radius and picking tile lengths that are rationally
independent, we can get this rank to be as large as we wish. However, the
return modules of large patches can only have rank 1 or 2.

Incidentally, every Fibonacci tiling space obtained by collaring and varying
the lengths of the collared tiles turns out to be mutually locally derivable
(MLD) to a Fibonacci tiling space involving uncollared tiles. (This is because
two shape classes in the same cohomology class give rise to MLD tilings and
because the generators of $\check{H}^{1}(\Omega)$ can be expressed in terms of
uncollared tiles.) Tilings that are MLD have exactly the same return modules
for large patches, but not necessarily for small patches.

######  Example 6 (Sturmian tilings).

Our analysis of the Fibonacci tiling relied on its substitutive structure, but
it is possible to derive similar results for any Sturmian tiling. A Sturmian
tiling (see e.g. [2]) is a canonical cut-and-project tiling from 2 dimensions
to 1. There are two kinds of tiles, which we call $a$ and $b$, with the ratio
of frequencies being an irrational number $\alpha$. Such a tiling can be
described via a substitution if and only if $\alpha$ is a quadratic
irrational. The Fibonacci tiling is the simplest such case, with $\alpha$
being the golden ratio $\phi=(1+\sqrt{5})/2$.

The first cohomology of any Sturmian tiling space is $\hskip
0.5pt\mathbb{Z}^{2}$, with generators that count $a$ and $b$ tiles,
respectively. As with the Fibonacci tiling, the return module for the smallest
possible patch (a single tile) is generated by the lengths of the two tiles.
This has rank 1 if $|a|/|b|$ is rational and rank 2 if $|a|/|b|$ is
irrational. If we collar the Sturmian tiling and vary the lengths of the
collared tiles, we can get the return modules of small patches to have
arbitrarily large rank, but the return modules of large patches always have
rank 2 for generic choices of the lengths of the $a$ and $b$ tiles and rank 1
for countably many values of $|a|/|b|$.

######  Example 7 (Thue-Morse).

The Thue-Morse tiling is based on the substitution $a\to ab$, $b\to ba$. Note
that both kinds of supertiles have one $a$ tile and one $b$ tile. No matter
what lengths we assign to the basic tiles, all supertiles have the same length
$|a|+|b|$.

Any patch $P$ of length 5 or greater must have two consecutive letters of the
same type, as the patterns $ababa$ and $babab$ never appear. However, $aa$ or
$bb$ can only appear when there is a supertile boundary in between the two
$a$’s or two $b$’s. As a result, there is a unique way to group the tiles in
$P$ into supertiles. This implies that any return vector between two instances
of $P$ must be a multiple of $|a|+|b|$, and hence that $\mathrm{Ret}(P)$ is an
infinite cyclic group.

Note that $\mathrm{Ret}(P)$ isn’t necessarily all of $(|a|+|b|)\hskip
0.5pt\mathbb{Z}$. If $P$ is long enough to determine the locations of the
order-$n$ supertiles, then $\mathrm{Ret}(P)$ is actually contained in
$2^{n-1}(|a|+|b|)\hskip 0.5pt\mathbb{Z}$. However, this does not affect the
rank of $\mathrm{Ret}(P)$, which is 1 for all $P$ of length 5 or greater.

At first glance this result is surprising, since the rank of
$\check{H}^{1}(\Omega)=\hskip 0.5pt\mathbb{Z}[1/2]\oplus\hskip
0.5pt\mathbb{Z}$ is 2, not 1. Theorem 2 says that a generic shape change
should result in return modules that have rank 2.

The answer is that part of the cohomology involves collared tiles and cannot
be expressed in terms of the basic uncollared tiles. As we will soon see, a
generic shape change among collared tiles does indeed result in return modules
of rank 2 or higher for arbitrary patches, and of rank 2 for large patches.

Since the patterns $aaa$ and $bbb$ never appear, there are six possible once-
collared tiles:

$\displaystyle a_{1}=(a)a(b),$ $\displaystyle b_{1}=(b)b(a),$ $\displaystyle
a_{2}=(b)a(a),$ $\displaystyle b_{2}=(a)b(b),$ $\displaystyle a_{3}=(b)a(b),$
$\displaystyle b_{3}=(a)b(a),$

where $(x)y(z)$ denotes a $y$ tile that is preceded by an $x$ tile and
followed by a $z$ tile.

The Anderson-Putnam complex that describes possible adjacencies between tiles
is shown in Figure 3.

Figure 3. The Anderson-Putnam complex for the once-collared Thue-Morse
substitution

A path between two instances of a tile defines a closed loop in this complex,
so the length of an arbitrary return vector is an integer combination of

$\displaystyle\gamma_{1}$ $\displaystyle=$
$\displaystyle|a_{1}|+|a_{2}|-|a_{3}|$ $\displaystyle\gamma_{2}$
$\displaystyle=$ $\displaystyle|b_{1}|+|b_{2}|-|b_{3}|$
$\displaystyle\gamma_{3}$ $\displaystyle=$ $\displaystyle|a_{3}|+|b_{3}|.$

The quantities $\\{\gamma_{1},\gamma_{2},\gamma_{3}\\}$ are rationally
independent given generic choices of tile lengths, so the return module for
individual collared tiles has (generic) rank 3.

However, return modules for larger patches involve closed loops in an
analogous complex built from supertiles, not just from individual tiles.
Substitution sends $\gamma_{1}$ and $\gamma_{2}$ to $\gamma_{3}$ and sends
$\gamma_{3}$ to $\gamma_{1}+\gamma_{2}+\gamma_{3}$, so the return vector for
any patch containing a well-defined collared supertile is a combination of
$\gamma_{1}+\gamma_{2}$ and $\gamma_{3}$. The quantities
$\gamma_{1}+\gamma_{2}$ and $\gamma_{3}$ transform under additional
substitution via the matrix $\left(\begin{smallmatrix}0&1\cr
2&1\end{smallmatrix}\right)$. This is invertible over $\mathbb{Q}$, so the
rank of the return module for a high-order supertile is the same as the rank
for an order-1 supertile. As a result, return modules of all large patches
have maximal (and generic) rank 2.

######  Example 8 (Three-e Morse).

Our final one-dimensional example is a generalization of Thue-Morse that we
call Three-e-Morse. Consider the substitution $a\to aab$, $b\to bba$. This has
features similar to both Thue-Morse and to the period-doubling substitution
$a\to ab$, $b\to aa$, only with a stretching factor of 3 instead of 2.

Working with basic tiles, there are only two lengths to play with, so the
return module of a patch cannot have rank greater than 2. The substitution
matrix $\left(\begin{smallmatrix}2&1\cr 1&2\end{smallmatrix}\right)$ is
invertible over $\mathbb{Q}$, so the rank of the return module of a high-order
supertile (or of any patch found within a high-order supertile, in other words
of any patch at all) has the same rank as the return module of a simple tile.
This rank is 1 if $|a|/|b|\in\mathbb{Q}$ and 2 if $|a|/|b|\not\in\mathbb{Q}$.

However, the rank of $\check{H}^{1}(\Omega)=\hskip
0.5pt\mathbb{Z}[1/3]\oplus\hskip 0.5pt\mathbb{Z}^{2}$ is 3, not 2, so a
generic shape change should give us rank-3 return modules. As with the Thue-
Morse tiling, part of the cohomology is invisible to uncollared tiles, so we
need to use collared tiles to achieve this. There are now eight collared
tiles:

$\displaystyle a_{1}=(a)a(a),$ $\displaystyle b_{1}=(b)b(b),$ $\displaystyle
a_{2}=(a)a(b),$ $\displaystyle b_{2}=(b)b(a),$ $\displaystyle a_{3}=(b)a(a),$
$\displaystyle b_{3}=(a)b(b),$ $\displaystyle a_{4}=(b)a(b),$ $\displaystyle
b_{4}=(a)b(a).$

Figure 4. The Anderson-Putnam complex for the once-collared Three-e Morse
tiling

The resulting Anderson-Putnam complex is shown in Figure 4.

We define the quantities

$\displaystyle\gamma_{1}$ $\displaystyle=$ $\displaystyle|a_{1}|,$
$\displaystyle\gamma_{2}$ $\displaystyle=$ $\displaystyle|b_{1}|,$
$\displaystyle\gamma_{3}$ $\displaystyle=$
$\displaystyle|a_{2}|+|a_{3}|-|a_{4}|$ $\displaystyle\gamma_{4}$
$\displaystyle=$ $\displaystyle|b_{3}|+|b_{3}|-|b_{4}|$
$\displaystyle\gamma_{5}$ $\displaystyle=$ $\displaystyle|a_{4}|+|b_{4}|.$

By picking generic tile lengths, we can get the $\gamma_{i}$’s to be
rationally independent, so the rank of the return module of a single tile may
have rank up to 5. Under substitution, the quantities
$\gamma_{1},\ldots,\gamma_{5}$ transform via the matrix

$\begin{pmatrix}0&0&0&0&1\cr 0&0&0&0&1\cr 1&0&1&0&1\cr 0&1&0&1&1\cr
1&1&1&1&1\end{pmatrix}.$

This matrix has rank 3, with eigenvalues 3, 1, -1, 0 and 0. Thus the return
module for order-1 supertiles (or for higher-order supertiles, or for any
large patch) will generically have rank 3. If the $\gamma_{i}$’s are not
rationally independent, the rank may be even less.

The lesson in all of these examples is that the rank of the cohomology gives
an upper bound on the ranks of return modules for large patches and gives the
rank of such return modules exactly for a generic choice of tile lengths. If
the cohomology can be expressed entirely using uncollared tiles, as in the
Fibonacci tiling and other Sturmian tilings, then we only need to vary the
lengths of the uncollared tiles to achieve this. If part of the cohomology can
only be expressed using collared tiles, as in the Thue-Morse and Three-e-Morse
tilings, then we need to vary the lengths of collared tiles to achieve generic
behavior. In all cases we can get return modules of small patches to be as
complicated as we want by collaring and varying a large number of tile
lengths. However, these complications disappear when we look at large patches,
typically by having a singular substitution matrix.

###  5.2. Higher dimensional examples

When it comes to return modules, there are several key differences between the
geometry of $\mathbb{R}$ and that of $\mathbb{R}^{d}$ with $d>1$. First, the
return module of a repetitive tiling in $d$ dimensions always has rank at
least $d$. Since every ball of sufficient radius contains a copy of the patch
$P$ that we are studying, the real span of the return vectors of $P$ is all of
$\mathbb{R}^{d}$, so there must be $d$ return vectors that are linearly
independent over $\mathbb{R}$, and thus over $\mathbb{Q}$.

Second, non-generic behavior is rarer in higher dimensions than in one
dimension, in the sense that it occurs on countably many copies of a
codimension-$d$ set instead of a codimension-1 set. (However, the word “rarer”
should be taken with a grain of salt, since in both cases the sets have
measure zero.) There are only countably many ways to have a rational linear
relation among return vectors. Each such relation reduces the number of free
vectors by one, thereby constraining us to a codimension-$d$ subset of our set
of possible shape parameters.

Finally, the maximal rank of a return module is the rank of $\check{H}^{1}$,
but the dimension of the space of shape parameters is $d$ times the rank of
$\check{H}^{1}$. There are many, many more ways to vary the shapes and sizes
of our tiles than there are generators of our return modules.

######  Example 9 (Chair).

There are two common versions of the chair tiling, each MLD to the other. In
one, the basic tiles are an L-shaped triomino and rotations of that triomino
by multiples of 90 degrees. The substitution rule for one tile is given in the
top portion of Figure 5. To substitute rotated versions of the tile, just
rotate the picture. In the other version of the chair tiling, a basic tile is
a unit square marked with an arrow pointing northeast, northwest, southwest,
or southeast. The substitution rule for arrow tiles is shown in the bottom
portion of Figure 5. To go from chairs to arrows, divide each chair-shaped
tile into three squares and draw arrows pointing towards the center. To go
from arrows to chairs, look for vertices with three arrows pointing in and one
pointing out. Glue the three squares with arrows pointing in to form a chair.

Figure 5. Substitutions for the classic and arrow versions of the chair tiling

In either version, all of the vertices have exactly the same $x$-coordinate
(mod 1) and the same $y$-coordinate (mod 1). The return module for any patch
is necessarily a subgroup of $\hskip 0.5pt\mathbb{Z}^{2}$ and so has rank at
most 2. (Depending on the size of the patch, the return module is typically
$2^{n}\hskip 0.5pt\mathbb{Z}^{2}$ for some integer $n$.) The chair tiling,
like all primitive substitution tilings, is repetitive, meaning that for each
patch $P$ there is a radius $r_{P}$ such that every ball of radius $r_{P}$
contains at least one copy of $P$. This implies that the rank of
$\mathrm{Ret}(P)$ is at least equal to the dimension of the ambient space, in
this case 2. Remarkably, it is no bigger.

It is reasonable to wonder whether that rank can be increased by applying a
shape change. For small patches the answer is “yes”, just as in our 1D
examples. But for large patches the answer is “no”.
$\check{H}^{1}(\Omega)=\hskip 0.5pt\mathbb{Z}[1/2]^{2}$, which only has rank
2. By Theorem 1, no matter how we change the shapes, the return modules of
large patches can only have rank 2. That is, the locations of an arbitrary
large patch $P$ must live on a lattice, up to an overall translation.

Another perspective on this comes from the classification of shape changes up
to MLD equivalence by $\check{H}^{1}(\Omega,\mathbb{R}^{2})$ [6]. There is a
4-parameter family of rigid linear transformations that can be applied to the
chair tiling. Since $\check{H}^{1}(\Omega,\mathbb{R}^{2})=\hskip
0.5pt\mathbb{Z}[1/2]^{2}\otimes\mathbb{R}^{2}=\mathbb{R}^{4}$ is
4-dimensional, all shape changes are MLD to rigid linear transformations, and
in particular respect any lattice structure on any length scale larger than
that of the MLD equivalence.

######  Example 10 (Hat).

In 2023, Smith et al [17] announced the discovery of a family of aperiodic
monotiles. Each element in the family is a shape that, together with rotated
and reflected versions of itself, can tile the plane but only nonperiodically.
Each tiling actually involves 12 tiles up to translation: the original shape
rotated by multiples of 60 degrees and the reflected shape rotated by
multiples of 60 degrees. (The same authors later displayed an aperiodic
monotile called the Spectre [18] that uses 12 rotations and no reflections. We
will not concern ourselves with the Spectre tilings here.)

Figure 6. The basic “Hat” tile. The lengths $\alpha,\beta$ of the solid and
dashed edges are arbitrary.

The basic tile, shown in Figure 6, is a degenerate 14-gon, with six edges of
length $\alpha$ (shown as solid lines) and eight edges of length $\beta$
(shown as dashed lines), and with two of the dashed edges laid end-to-end,
looking like a single edge of length $2\beta$. The construction works for all
positive values of $\alpha$ and $\beta$, although some care must be taken with
the special case $\alpha=\beta$ (to ensure that $\alpha$ edges cannot abut
$\beta$ edges) and with the limiting cases $\alpha=0$ and $\beta=0$ (to ensure
that the zero-length edges still line up). Several special cases have been
given names: $\beta=0$ is a “Chevron”, $\alpha/\beta=\sqrt{3}$ is a “Hat”,
$\alpha=\beta$ is a “Spectre”, $\alpha/\beta=\sqrt{3}/3$ is a “Turtle” and
$\alpha=0$ is a “Comet”. See Figure 7

Figure 7. The Chevron, Hat, Spectre, Turtle and Comet tiles, all rotated by
180 degrees.

In all cases, tiles must assemble into four shapes called “metatiles”. The
metatiles then assemble into larger metatiles, which assemble into larger
metatiles, and so on to infinity. The substitution involving metatiles has an
overall stretching factor of $\phi^{4}$, where $\phi$ is the golden mean.
Since $\phi^{4}$ is a unimodular Pisot number, this implies that the return
module for high-order metatiles (and other large patches) is the same as for
basic metatiles, just like what we saw with the Fibonacci tiling. We refer to
this as the return module of the tiling. While it is possible to investigate
the (possibly larger) return module for individual tiles, we will not do so.

The cohomology and return modules for tilings in the Hat family were computed
in [4]. $\check{H}^{1}(\Omega)=\hskip 0.5pt\mathbb{Z}^{4}$, so the biggest
rank the return module can have is 4. Our ambient space is 2-dimensional, so
the return module must have rank at least 2. 6-fold rotational symmetry
implies that the rank of the return module must be even. In short, the rank
must always be 2 or 4.

If we identify $\mathbb{R}^{2}$ with the complex numbers $\mathbb{C}$, then
the return module is closely related to the triangular lattice $\hskip
0.5pt\mathbb{Z}[\xi]$ spanned by 1 and $\xi=\exp(2\pi i/6)=(1+i\sqrt{3})/2$.
Specifically, the return lattice is the span of $(\alpha+i\beta)(1+\xi)\hskip
0.5pt\mathbb{Z}[\xi]$ and $2i\beta(1+\xi)\hskip 0.5pt\mathbb{Z}[\xi]$. This
has rank 2 if a rational multiple of $2i\beta/\alpha$ lies in $\hskip
0.5pt\mathbb{Z}[\xi]$ and rank 4 if it does not. Since the pure imaginary
elements of $\hskip 0.5pt\mathbb{Z}[\xi]$ are the multiples of $i\sqrt{3}$,
the return module has rank 2 if and only if $\beta\sqrt{3}/\alpha$ is rational
or $\alpha=0$.

The upshot is that almost every shape in the Hat family results in a
4-dimensional return module. However, four of the five named variants have
2-dimensional return modules. In two of these, namely the Chevron and the
Comet, all vertices lie on a triangular lattice, so the return module is
necessarily contained in that lattice. In two others, namely the Hat and the
Turtle, all vertices either lie on a triangular lattice or on its dual
honeycomb grid.

So far we have only considered the shapes discussed in [17], all of which
respect 6-fold rotational symmetry and reflectional symmetry. In [4], the
authors considered shape changes that broke reflectional symmetry while
preserving rotational symmetry, resulting in two distinct shapes, each
appearing in 6 orientations. This was done by allowing the parameters $\alpha$
and $\beta$ to be complex. As before, the return module has rank 2 exactly
when $i\beta/\alpha\in\mathbb{Q}[\xi]$, the set of rational multiples of
$\hskip 0.5pt\mathbb{Z}[\xi]$. The set $\mathbb{Q}[\xi]$ is countable and
dense in $\mathbb{C}$. Our space of shape parameters $(\alpha,\beta)$ is
$\mathbb{C}^{2}=\mathbb{R}^{4}$, most of which give rise to rank-4 return
modules. Only when $(\alpha,\beta)$ lie in a countable union of 2-dimensional
subsets do we get rank 2.

Expanding our horizons further, we can consider shape changes that break
6-fold rotational symmetry. The space of possible shapes (up to MLD
equivalence) is parametrized by $\check{H}^{1}(\Omega,\mathbb{R}^{2})=\hskip
0.5pt\mathbb{Z}^{4}\otimes\mathbb{R}^{2}=\mathbb{R}^{8}$. Within this
8-dimensional space, the return module has rank 4 everywhere except on a
countable union of 6-dimensional subsets, where it can have rank 2 or 3.

## Acknowledgements

It is our pleasure to thank Michael Baake, Franz Gähler and Jianlong Liu for
useful discussions. This work was supported by the National Science Foundation
under grant DMS-2113468.

## References

  * [1] J.E. Anderson and I.F. Putnam, Topological invariants for substitution tilings and their associated $C^{*}$-algebras, Ergod. Th. & Dynam. Syst. 18 (1998) 509–537.
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# Lorentz-violating electrodynamics in continuous matter

M.M. Ferreira Jr.1 P.D.S. Silva1 and M. Schreck1 1Departamento de Física,
Universidade Federal do Maranhão, Campus Universitário do Bacanga, São Luís
(MA), 65080-805, Brazil

###### Abstract

We review the propagation of electromagnetic waves in continuous matter in the
presence of Lorentz-violating terms. First, we briefly discuss classical
electrodynamics with regard to optical properties of a dielectric medium
exhibiting the Chiral Magnetic Effect (CME). Such a medium can be modeled by
Maxwell-Carroll-Field-Jackiw (MCFJ) theory. Second, we describe the impact of
CPT-odd terms of mass dimensions three and five, respectively, on
electromagnetic propagation in continuous matter. Birefringence and absorption
are analyzed in the scenarios investigated. Our findings provide new insights
into the interplay between classical field theory and optical effects in
matter.

##  1 Introduction

The minimal Standard-Model Extension (SME)[1] is a natural framework for
studying Lorentz violation (LV) in all particle sectors of the Standard Model.
Recently, the CPT-odd photon sector of the SME, represented by MCFJ
electrodynamics, was addressed in the context of condensed-matter systems
subject to the CME.[2] The latter effect, which arises due to an asymmetry in
the behavior between right- and left-handed fermions, describes the generation
of an electric current via an applied magnetic field. The material property
responsible for this current in a dielectric substrate was taken into account
at the classical level, revealing intriguing optical phenomena such as
isotropic birefringence.[3]

Incorporating field operators with mass dimensions greater than four
complemented the minimal SME and provided its nonminimal version.[4] Several
aspects of higher-derivative electrodynamics in vacuo were discussed in the
literature, including a derivation of the gauge field propagator and an
analysis of microcausality and unitarity at tree-level for different photon
modes.[5] More recently, the latter theory was adapted to the description of a
continuous medium, followed by an investigation of the optical properties in
this setting.[6] In the current work, we review some aspects of
electromagnetic propagation in continuous media in two scenarios: i) a
dielectric medium endowed with a magnetic conductivity; ii) a dielectric
medium affected by CPT-odd terms of mass dimensions three and five.

##  2 Magnetic-conductivity effects in continuous matter

We start from the Maxwell equations, supplemented by the usual constitutive
relations,

${\bf{D}}=\epsilon{\bf{E}},\quad{\bf{H}}=\mu^{-1}{\bf{B}}\,,$ (1)

with $\epsilon$ and $\mu$ being the electric permittivity and magnetic
permeability, respectively. Let us also consider an extension of Ohm’s law,

$J^{i}=\sigma^{B}_{ij}B^{j}\,,$ (2)

where $\sigma^{B}_{ij}$ is a general magnetic-conductivity tensor, describing
a material property of the medium. Using a plane-wave ansatz,
$({\bf{E}},{\bf{B}})\propto\exp[\mathrm{i}({\bf{k}}\cdot{\bf{r}}-\omega t)]$
the Maxwell equations together with Eqs. (1), (2) yield

$M_{ij}E^{j}=0\,,\quad
M_{ij}=k^{2}\delta_{ij}-k_{i}k_{j}-\mu\bar{\epsilon}_{ij}(\omega)\,,$ (3a)
with the effective electric permittivity given by
$\bar{\epsilon}_{ij}(\omega)=\epsilon\delta_{ij}+\frac{\mathrm{i}}{\omega}\sigma^{B}_{ia}\epsilon_{abj}n_{b}\,.$
(3b)

The dispersion equations, obtained through $\mathrm{det}[M_{ij}]=0$ and
${\bf{k}}=\omega{\bf{n}}$, determine the refractive indices
$|\mathbf{n}|\equiv n$ of the medium. For an isotropic conductivity,
$\sigma_{ij}^{B}=\Sigma\delta_{ij}$, being the common case for the CME, they
read

$n_{\pm}=\sqrt{\mu\epsilon+\left(\frac{\mu\Sigma}{2\omega}\right)^{2}}\pm\frac{\mu\Sigma}{2\omega}\,.$
(4)

The associated propagating modes are described by left- and right-handed
circularly polarized electric-field vectors, yielding birefringence measured
in terms of the rotatory power

$\delta=-\frac{\omega}{2}[\mathrm{Re}(n_{+})-\mathrm{Re}(n_{-})]\,.$ (5)

Applying the latter to the isotropic scenario implies $\delta=-{\mu\Sigma/2}$.
Since the refractive indices in Eq. (4) are real for all frequencies,
dichroism does not occur in this case.

We can also examine an antisymmetric current, whose conductivity
$\sigma^{B}_{ij}=\epsilon_{ijk}b_{k}$ is parameterized in terms of an
intrinsic three-vector ${\bf{b}}$. With the angle $\theta$ between ${\bf{n}}$
and ${\bf{b}}$, the corresponding refractive index is

$n=\sqrt{\mu\epsilon-\frac{\mu^{2}}{4\omega^{2}}b^{2}\cos^{2}\theta}+\mathrm{i}\frac{\mu}{2\omega}b\cos\theta\,,$
(6)

where $b\equiv|\mathbf{b}|$. The imaginary part in Eq. (6) acts like an
absorption coefficient, which implies that a propagating wave is subject to
attenuation. Thus, off-diagonal magnetic-conductivity terms ascribe a
conducting behavior to a dielectric substrate. The literature[3] can be
consulted for additional details on these kinds of phenomena.

##  3 MCFJ electrodynamics in continuous matter

In the following, we dedicate ourselves to CPT-odd MCFJ electrodynamics as
well as a dimension-5 extension of the latter. In continuous matter, these
types of theories are governed by

$\mathcal{L}=-\frac{1}{4}G^{\mu\nu}F_{\mu\nu}-\frac{1}{4}\epsilon^{\beta\lambda\mu\nu}V_{\beta}A_{\lambda}F_{\mu\nu}-A_{\mu}J^{\mu}\,,$
(7)

where $V^{\mu}=(V^{0},{\bf{V}})$ is a LV background vector,
$G^{\mu\nu}=(1/2)\chi^{\mu\nu\alpha\beta}F_{\alpha\beta}$ the field strength
tensor in matter, and $\chi^{\mu\nu\alpha\beta}$ the constitutive tensor. In
this case, the electromagnetic dispersion equation is[6]

$\overline{p}^{4}+\overline{p}^{2}\overline{V}^{2}-(\overline{p}\cdot\overline{V})^{2}=0\,,$
(8a) with a redefined four-momentum and background vector
$\overline{p}^{\mu}\equiv\left(\sqrt{{\epsilon}}\,\omega,\frac{\mathbf{k}}{\sqrt{\mu}}\right)\,,\quad\overline{V}^{\mu}\equiv\left(\sqrt{\mu}\,V^{0},\frac{\mathbf{V}}{\sqrt{{\epsilon}}}\right)\,.$
(8b)

For the purely timelike sector of the background, the refractive indices read

$n_{\pm}=\sqrt{\mu\epsilon+\frac{\mu^{2}V_{0}^{2}}{4\omega^{2}}}\pm\frac{\mu
V_{0}}{2\omega}\,,$ (9)

with $V_{0}$ playing the same role as does the isotropic conductivity $\Sigma$
in Eq. (4). The modes are described by circularly polarized waves with the
rotatory power of Eq. (5) given by $\delta=-{\mu V_{0}/2}$. For the purely
spacelike scenario, we distinguish between two special cases: i) a
${\bf{V}}$-perpendicular configuration with ${\bf{n}}\cdot{\bf{V}}=0$ and ii)
a ${\bf{V}}$-longitudinal configuration where ${\bf{n}}\cdot{\bf{V}}=\pm
n|{\bf{V}}|$. For the first, the refractive indices can be cast into the form

$n_{\pm}=\sqrt{\mu\epsilon+\frac{\mu{\bf{V}}^{2}}{2\epsilon\omega^{2}}(-1\pm
1)}\,.$ (10)

The propagating modes are linearly polarized and have an additional
longitudinal component. Birefringence is characterized by the phase shift
$\Delta=(2\pi/\lambda_{0})d(n_{+}-n_{-})$, with the vacuum wavelength
$\lambda_{0}$ of light and the thickness $d$ of the medium. For the second,
the refractive indices read

$\displaystyle n_{\pm}=\sqrt{\mu\epsilon\pm\frac{\mu|{\bf{V}}|}{\omega}}\,,$
(11)

where the associated propagating modes are circularly polarized. Thus, the
rotatory power is

$\frac{\Delta}{d}=\frac{2\pi}{\lambda_{0}}\sqrt{\mu\epsilon}\left(1-\sqrt{1-\frac{\mathbf{V}^{2}}{\epsilon^{2}\omega^{2}}}\right)\,.$
(12)

Furthermore, in the frequency range $0<\omega<|{\bf{V}}|/\epsilon$, the
refractive index $n_{-}$ of Eq. (11) becomes purely imaginary, so that one of
the two circularly polarized modes is absorbed. This behavior is suitably
characterized by the dichroism coefficient
$\delta_{\mathrm{d}}=-(\omega/2)[\mathrm{Im}(n_{+})-\mathrm{Im}(n_{-})]$, in
particular,

$\displaystyle\delta_{\mathrm{d}}$
$\displaystyle=\frac{\sqrt{\mu\epsilon}}{2}\omega\sqrt{-1+\frac{|{\bf{V}}|}{\omega\epsilon}}\,.$
(13)

Finally, we explore a higher-derivative dimension-five extension of MCFJ
electrodynamics in continuous matter based on[6]

$\mathcal{L}=-\frac{1}{4}G^{\mu\nu}F_{\mu\nu}+\frac{1}{2}\epsilon^{\beta\lambda\mu\nu}U_{\beta}A_{\lambda}\square
F_{\mu\nu}-A_{\mu}J^{\mu}\,,$ (14)

with the LV background vector $U^{\mu}=(U^{0},{\bf{U}})$. This theory gives
rise to the following dispersion equation:

$\overline{p}^{4}+4p^{4}\left[\overline{U}^{2}\overline{p}^{2}-(\overline{U}\cdot\overline{p})^{2}\right]=0\,,$
(15a) expressed in terms of Eq. (8b) and a properly redefined background,
$\overline{U}^{\mu}\equiv\left(\sqrt{\mu}\,{U}^{0},\frac{{\mathbf{U}}}{\sqrt{{\epsilon}}}\right)\,.$
(15b)

To analyze the purely spacelike scenario, it is again convenient to
distinguish between two cases: i) a ${\bf{U}}$-perpendicular configuration and
ii) a ${\bf{U}}$-longitudinal configuration, defined in analogy to above. For
the first,

$n_{0}=\sqrt{\mu\epsilon},\quad
n_{\pm}^{2}=1+\frac{\epsilon}{8\mu\omega^{2}{\bf{U}}^{2}}\left[-1\pm\sqrt{1+16\mu^{2}\omega^{2}{\bf{U}}^{2}\left(1-\frac{1}{\mu\epsilon}\right)}\,\right]\,.$
(16)

The refractive index $n_{0}$ corresponds the usual one of Maxwell’s
electrodynamics, while the remaining two, $n_{\pm}$, capture effects stemming
from the dimension-five term in Eq. (14). The refractive index $n_{+}$ is real
for the whole frequency range and exhibits anomalous dispersion, whereas
$n_{-}$ has a simple root, $\omega_{-}=\epsilon/(2|{\bf{U}}|)$. For
frequencies below the latter critical value, $n_{-}$ becomes purely imaginary,
which disqualifies this mode as a propagating one. The refractive index
$n_{0}$ is accompanied by a linearly polarized electric field. Linear
polarizations with additional longitudinal components are found for $n_{\pm}$.

For the second configuration of $\mathbf{U}$ we obtain

$n_{\pm}^{2}=\frac{\mu(\epsilon\pm 2\omega|{\bf{U}}|)}{1\pm
2\mu\omega|{\bf{U}}|}\,.$ (17)

The mode associated with $n_{+}$ propagates for any frequency and exhibits
anomalous dispersion. On the order hand, the behavior of the mode related to
$n_{-}$ is more intricate and one finds distinct behaviors in three different
regimes: i) for $0<\omega<\omega_{0}$, with $\omega_{0}=1/(2\mu|{\bf{U}}|)$,
the mode propagates and $n_{-}$ increases very fast; ii) for
$\omega_{0}<\omega<\omega_{-}$, with $\omega_{-}=\epsilon/(2|{\bf{U}}|)$, the
refractive index $n_{-}$ becomes purely imaginary, whereupon propagation is
prohibited; iii) for $\omega>\omega_{-}$, propagation is enabled again. As the
propagating waves are circularly polarized, we evaluate their rotatory power:

$\delta=-\frac{\sqrt{\mu\epsilon}\omega}{2}\left(\frac{g_{+}}{\sqrt{S_{+}}}-\frac{g_{-}}{\sqrt{S_{-}}}\right)\,,\quad
g_{\pm}=\sqrt{1\pm\frac{2\omega}{\epsilon}|{\bf{U}}|}\,,$ (18)

where $S_{\pm}=1\pm 2\mu\omega|{\bf{U}}|$. In total, there are three physical
modes in the dimension-five MCFJ theory defined in Eq. (14), since an
additional mode originates from the higher-derivative structure of the theory.
This is the reason why we find the three physical refractive indices of Eq.
(16). It is crucial to emphasize that the occurrence of more than two modes is
not indicative of a loss of U(1) gauge invariance.[6]

## Acknowledgments

The authors are grateful to the Brazilian research agencies CNPq, CAPES, and
FAPEMA for invaluable financial support.

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