# Spectral cocycle for substitution tilings

Boris Solomyak  Boris Solomyak
Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel
<EMAIL_ADDRESS>and  Rodrigo Treviño  Rodrigo Treviño
Department of Mathematics
University of Maryland, College Park<EMAIL_ADDRESS>
###### Abstract.

The construction of spectral cocycle from the case of 1-dimensional
substitution flows [10] is extended to the setting of pseudo-self-similar
tilings in ${\mathbb{R}}^{d}$, allowing expanding similarities with rotations.
The pointwise upper Lyapunov exponent of this cocycle is used to bound the
local dimension of spectral measures of deformed tilings. The deformations are
considered, following [39], in the simpler, non-random setting. We review some
of the results on quantitative weak mixing from [39] in this special case and
illustrate them on concrete examples.

The research of B. S. was supported by the Israel Science Foundation grant
911/19.

Dedicated to the memory of Uwe Grimm

##  1\. Introduction

We extend the construction of spectral cocycle and partially extend the
results, from the case of 1-dimensional substitution flows treated in [10], to
higher dimensions. Another motivation for us is to make the results of [39]
more accessible by presenting them in the simplest possible nontrivial case of
a single self-similar tiling (corresponding to a stationary Bratteli diagram
in [39]). We also indicate how the results of [39], where it is assumed that
the expansion map is a pure dilation, may be extended to the case of general
expanding similarities with rotations, which necessitates dealing with fractal
boundaries and passing from self-similar to pseudo-self-similar tiling spaces.
We do not repeat the rather technical proofs of [39], but illustrate the
results on concrete examples (Kenyon’s tilings defined via free group
endomorphisms [24] and a “square” tiling).

The spectral cocycle was introduced in [10] for translation flows and $S$-adic
systems; here we are concerned with the case of a single substitution.
Briefly, given a primitive aperiodic substitution $\zeta$ on $m$ symbols with
a non-singular substitution matrix ${\sf S}_{\zeta}$, the spectral cocycle
from [10] is a complex matrix $m\times m$ cocycle over the endomorphism of the
$m$-torus induced by the transpose ${\sf S}_{\zeta}^{\sf T}$. (In fact, in the
case when ${\sf S}_{\zeta}$ is singular, one can restrict the underlying
system to a lower-dimensional sub-torus on which non-singularity holds, see
[41].) Implicitly, the spectral cocycle appeared already in [8] as a
generalized matrix Riesz product. In [10] it was shown that the (top,
pointwise upper) Lyapunov exponents of this cocycle in some sense control the
local spectral behavior of substitution ${\mathbb{R}}$-actions – suspension
flows over the usual substitution ${\mathbb{Z}}$-action, with a piecewise-
constant roof function. By “control” we mean lower and upper bounds for the
local dimension of spectral measures of cylindrical functions, see [8] for
details. Independently, Baake et al [4, 5] introduced a Fourier matrix
cocycle, which coincides with the spectral cocycle on the one-dimensional
${\sf S}_{\zeta}^{\sf T}$-invariant manifold on the torus corresponding to the
Perron-Frobenius eigenvector; it is tailored for the spectral analysis of
(geometrically) self-similar tiling flows and was used to prove singularity of
the diffraction spectrum for a class of non-Pisot substitution systems. In
[39], it appeared as the application of traces applied to elements of certain
AF algebras.

The tools of spectral cocycle (without calling it such) provided a framework
for the proof of almost sure Hölder regularity of translation flows on higher
genus flat surfaces, first in [9] for genus 2 and then in [11] for an
arbitrary genus greater than 1, including many surfaces of infinite genus and
finite area. (The last paper appeared after the preprint of Forni [15] who
used a different technique.)

The work [39] extended almost sure Hölder regularity results of [11] to the
case of (globally) random substitution tilings in ${\mathbb{R}}^{d}$. In a
sense, this is a higher-dimensional version of an $S$-adic system, in which
self-similar tile substitutions are applied randomly, according to an
underlying ergodic dynamical system, resulting in a tiling
${\mathbb{R}}^{d}$-action. A crucial distinction with the one-dimensional case
is that one needs to define what is meant by a “deformation” of the tiling
dynamical system. Whereas for $d=1$ it was rather natural to vary the tile
lengths, or equivalently, view the system as a suspension with a piecewise-
constant roof function and vary the “heights”, this issue becomes more
complicated for $d\geq 2$. In this setting admissible deformations were
studied, first by Clark and Sadun [13] and then by Kellendonk [22] and Julien
and Sadun [21]; this required dealing with the (e.g., Čech) cohomology group
$\check{H}^{1}(X,{\mathbb{R}}^{d})$, where $X$ is the tiling space. The
spectral cocycle for $d\geq 2$ is still a complex matrix $m\times m$ cocycle,
where $m$ is the number of prototiles, but the cocycle is over the
endomorphism induced by the inflation map on the first cohomology group.
Incidentally, this point of view offers an advantage even for $d=1$.

The Fourier matrix cocycle has been introduced for self-similar tilings in
[5], see also [6]; it is a cocycle over the inflation map of the entire
${\mathbb{R}}^{d}$ and it coincides with the spectral cocycle on a measure
zero subset. In this case it was also used to prove singularity for some
systems, such as the Godrèche-Lançon-Billard tiling.

The paper is organized as follows. In Section 2 we recall the background on
tilings and tiling dynamical systems, which is mostly standard and appeared in
many articles and books. We also discuss pseudo-self-similar tilings, which
are needed in order to define deformations when the prototiles of the original
self-similar tiling have fractal boundaries. Section 3 deals with cohomology
and deformations. It also includes a treatment of geometric properties of
deformed tiling spaces and their consequences, in particular, unique
ergodicity and a formula for measures of cylinder sets. In Section 4 we define
the spectral cocycle and state the main result on local dimension estimates.
In Section 5 we first state the result saying that for deformed tiling spaces
weak mixing is equivalent to topological weak mixing, which was characterized
in [13]. The proof is a minor variation of the argument from [34]; we sketch
it in Section 7. Next we explain how a natural quantitative strengthening of
the condition, which we call the “quantitative Host-Veech criterion”, yields
Hölder regularity of spectral measures and quantitative weak mixing. Further,
we state a theorem on quantitative weak mixing, which is essentially a special
case of [39, Theorem 1], but we believe it is more accessible and easier to
apply. A brief proof outline is included. Finally, we revisit the case of one-
dimensional substitution tilings and show how to extend [8, Th. 4.1] to the
case of reducible substitution matrix. Section 6 is devoted to examples. There
is a detailed discussion of a family of pseudo self-similar planar tilings due
to Kenyon [24], to which our results apply. An example with of “square
tilings” is included as well. Section 7 contains the remaining proofs.

##  2\. Background

###  2.1. Tilings and tiling spaces

A tiling ${\mathcal{T}}$ of ${\mathbb{R}}^{d}$ is a covering by compact sets,
called tiles, such that their interiors don’t intersect. The tiles need not be
homeomorphic to balls or even connected. We assume that every tile is a
closure of its interior. We will consider the translation
${\mathbb{R}}^{d}$-action on tiling spaces (defined below in 2.1): the
translation of ${\mathcal{T}}$ by $t\in{\mathbb{R}}^{d}$ is $\\{T-t:\
T\in{\mathcal{T}}\\}$. Strictly speaking, a tile is a pair $T=(A,j)$, where
$A$ is the support of $T$ and $j$ is its type, color, or label. This is
needed, since frequently we need to distinguish geometrically equivalent
tiles. However, to avoid cluttered notation, we will often think of tiles as
sets with colors, or labels. The colors are assumed to be preserved under
translations, linear maps, etc.

A patch of the tiling ${\mathcal{T}}$ is a finite subset of ${\mathcal{T}}$.
For two patches ${\mathcal{P}},{\mathcal{P}}^{\prime}\subset{\mathcal{T}}$ we
write ${\mathcal{P}}\sim{\mathcal{P}}^{\prime}$ if there exists
$t\in{\mathbb{R}}^{d}$ such that ${\mathcal{P}}^{\prime}={\mathcal{P}}-t$;
such patches are called translation-equivalent. For a bounded
$A\subset{\mathbb{R}}^{d}$ define the patches:

${\mathcal{O}}^{-}_{\mathcal{T}}(A)\ =\ \mbox{the largest patch of
${\mathcal{T}}$ completely contained in $A$};$
${\mathcal{O}}^{+}_{\mathcal{T}}(A)\ =\ \mbox{the smallest patch of
${\mathcal{T}}$ containing $A$}.$

We will always assume that all our tilings have (translational) finite local
complexity, or FLC: for any $R>0$ there are finitely many patches of the form
${\mathcal{O}}_{\mathcal{T}}^{-}(B_{R}(x))$ up to translation equivalence. In
particular, there are finitely many tiles up to translation. We fix a
collection of representatives ${\mathcal{A}}=\\{T_{1},\ldots,T_{m}\\}$ and
call them prototiles. Another standing assumption will be that ${\mathcal{T}}$
is repetitive, i.e., for any patch ${\mathcal{P}}\subset{\mathcal{T}}$ there
exists $R_{\mathcal{P}}>0$ such that any ball $B_{R_{\mathcal{P}}}(x)$, with
$x\in{\mathbb{R}}^{d}$, contains a ${\mathcal{T}}$-patch
${\mathcal{P}}^{\prime}\sim{\mathcal{P}}$.

The tiling space (or “hull”) determined by ${\mathcal{T}}$ is the closure of
the translation orbit:

(2.1) $X_{\mathcal{T}}:={\rm clos}\\{{\mathcal{T}}-x:\
x\in{\mathbb{R}}^{d}\\}$

in the standard “local” metric, in which ${\mathcal{T}}$ and
${\mathcal{T}}^{\prime}$ are ${\varepsilon}$-close for ${\varepsilon}>0$
sufficiently small if and only if there exists $t\in B_{\varepsilon}(0)$ such
that ${\mathcal{T}}$ and ${\mathcal{T}}^{\prime}-t$ coincide in
$B_{1/{\varepsilon}}(0)$. The tiling dynamical system
$(X_{\mathcal{T}},{\mathbb{R}}^{d})$ is the ${\mathbb{R}}^{d}$-action by
translations. If ${\mathcal{T}}$ is repetitive, then
$X_{\mathcal{T}}=X_{{\mathcal{T}}^{\prime}}$ for any
${\mathcal{T}}^{\prime}\in X_{{\mathcal{T}}}$, so the tiling dynamical system
is minimal. We will often omit the subscript ${\mathcal{T}}$ and simply write
$X$ for a tiling space. An alternative way to define the tiling space is via
the atlas of admissible patches (analogous to a language in symbolic
dynamics).

####  2.1.1. Functions

Let ${\mathcal{T}}$ be a tiling of ${\mathbb{R}}^{d}$. A function
$h:X_{\mathcal{T}}\to{\mathbb{C}}$ is transversally locally constant (TLC) if
there exists $R>0$ such that

${\mathcal{O}}^{-}_{{\mathcal{T}}^{\prime}}(B_{R}(0))={\mathcal{O}}^{-}_{{\mathcal{T}}^{\prime\prime}}(B_{R}(0))\implies
h({\mathcal{T}}^{\prime})=h({\mathcal{T}}^{\prime\prime}).$

A function $f:{\mathbb{R}}^{d}\to{\mathbb{C}}$ is ${\mathcal{T}}$-equivariant
if there exists $R>0$ such that

(2.2)
${\mathcal{O}}^{-}_{{\mathcal{T}}}(B_{R}(x))={\mathcal{O}}^{-}_{{\mathcal{T}}}(B_{R}(y))+(x-y)\implies
f(x)=f(y).$

Given a TLC-function $h:X_{\mathcal{T}}\to{\mathbb{C}}$, we get a
${\mathcal{T}}$-equivariant function via $f(x):=h({\mathcal{T}}-x)$.

####  2.1.2. Mutual local derivability

For two tilings ${\mathcal{T}}_{1}$ and ${\mathcal{T}}_{2}$, we say that
${\mathcal{T}}_{2}$ is locally derivable (LD) from ${\mathcal{T}}_{1}$ with
radius $R>0$ if for all $x,y\in\mathbb{R}^{d}$,

${\mathcal{O}}^{-}_{{\mathcal{T}}_{1}}(B_{R}(x))={\mathcal{O}}^{-}_{{\mathcal{T}}_{1}}(B_{R}(y))+(x-y)\Rightarrow{\mathcal{O}}^{+}_{{\mathcal{T}}_{2}}(\\{x\\})={\mathcal{O}}^{+}_{{\mathcal{T}}_{2}}(\\{y\\})+(x-y).$

If ${\mathcal{T}}$ is LD from ${\mathcal{S}}$ and vice-versa, then we say that
tilings ${\mathcal{T}}$ and ${\mathcal{S}}$ are mutually locally derivable
(MLD). Note that this is similar in spirit to ${\mathcal{T}}$-equivariance.
MLD implies conjugacy of tiling dynamical systems, but not the other way
round, see [31, 32]. We will say that two minimal tiling spaces $X$ and
$X^{\prime}$ are MLD if there exist ${\mathcal{T}}\in X$ and
${\mathcal{T}}^{\prime}\in X^{\prime}$ that are MLD.

####  2.1.3. Frequency of patches and unique ergodicity

Denote $Q_{R}=[-R,R]^{d}$. The following is well-known and may be considered
folklore. We refer the reader to [27], which is written in the language of
Delone multisets; passing to the tiling setting is routine.

Given an FLC tiling ${\mathcal{T}}$, we say that ${\mathcal{T}}$ has uniform
patch frequencies (UPF) if for any patch ${\mathcal{P}}\subset{\mathcal{T}}$,
the limit

${\rm
freq}({\mathcal{P}},{\mathcal{T}})=\lim_{n\to\infty}\frac{\\#\\{t\in{\mathbb{R}}^{d}:\
t+{\mathcal{P}}\subset Q_{R}+x\\}}{(2R)^{d}}\geq 0$

exists uniformly in $x\in{\mathbb{R}}^{d}$. For a repetitive tiling space, the
UPF property and the uniform frequencies do not depend on the choice of the
tiling.

######  Theorem 2.1 ([27, Theorem 2.7]).

Let ${\mathcal{T}}$ be an FLC tiling in ${\mathbb{R}}^{d}$. The dynamical
system $(X_{\mathcal{T}},{\mathbb{R}}^{d})$ is uniquely ergodic if and only if
${\mathcal{T}}$ has the UPF property.

Under the UPF assumption, we also have an explicit formula for the measure of
“cylinder sets”. For a tiling space $X$ we define the transversal of a patch
${\mathcal{P}}$ by

$\Upsilon({\mathcal{P}})=\\{{\mathcal{T}}\in X:\
{\mathcal{P}}\subset{\mathcal{T}}\\}.$

For a patch ${\mathcal{P}}$ and a Borel set $U\subset{\mathbb{R}}^{d}$, the
corresponding cylinder set is $\Upsilon({\mathcal{P}})+U:=\\{{\mathcal{T}}+x:\
{\mathcal{T}}\in\Upsilon({\mathcal{P}}),\ x\in U\\}.$

######  Proposition 2.2 ([27, Corollary 2.8]).

Let $X$ be a uniquely ergodic FLC tiling space, with the unique invariant
Borel probability measure $\mu$. Let $\eta=\eta(X)$ be the diameter of the
largest ball contained in every prototile. Then for any patch ${\mathcal{P}}$
and a Borel set $U$ with ${\rm diam}(U)<\eta(X)$,

(2.3) $\mu(\Upsilon({\mathcal{P}})+U)={\mathcal{L}}^{d}(U)\cdot{\rm
freq}({\mathcal{P}},{\mathcal{T}}).$

###  2.2. Self-similar tilings

Now we define a self-similar tiling space, in the sense of [38, 24, 34]. Here
we start with a substitution rule. As before, suppose that we have a finite
prototile set $\\{{\mathfrak{T}}_{1},\ldots,{\mathfrak{T}}_{m}\\}$, where each
${\mathfrak{T}}_{j}\subset{\mathbb{R}}^{d}$ is the closure of its interior,
possibly with a fractal boundary. Let $\varphi$ be an expanding similarity on
${\mathbb{R}}^{d}$, with expansion constant $\theta>1$ (we do not assume that
$\varphi$ is a pure dilation; in general, $\varphi=\theta{\mathcal{O}}$, where
${\mathcal{O}}$ is an orthogonal transformation). Assume that there is a tile
substitution

(2.4) $\omega({\mathfrak{T}}_{j})=\bigcup_{k\leq
m}({\mathfrak{T}}_{k}+{\mathcal{D}}_{jk}),\ \ j\leq m,$

where ${\mathcal{D}}_{jk}$ is a finite set of translations and the right-hand
side represents a patch such that

(2.5) $\varphi A_{j}=\bigcup_{k}(A_{k}+{\mathcal{D}}_{jk}),\ \ j\leq m,\ \
A_{j}={\rm supp}({\mathfrak{T}}_{j}).$

For a translated prototile the substitution acts by

(2.6) $\omega({\mathfrak{T}}_{j}+x)=\omega({\mathfrak{T}}_{j})+\varphi(x),\ \
j\leq m,\ x\in{\mathbb{R}}^{d}.$

This and the property (2.5) imply that the substitution map can be iterated,
resulting in larger and larger patches. In particular,

(2.7)
$\omega^{2}({\mathfrak{T}}_{j})=\\{{\mathfrak{T}}_{s}+\varphi{\mathcal{D}}_{jk}+{\mathcal{D}}_{ks}\\}_{s\leq
m,k\leq m},\ \ 1\leq j\leq m,$

etc. The substitution tiling space $X_{\omega}$ corresponding to $\omega$ is,
by definition, the collection of all tilings ${\mathscr{T}}$ of
${\mathbb{R}}^{d}$ such that every ${\mathscr{T}}$-patch is a sub-patch of
$\omega^{n}({\mathfrak{T}}_{j})$ for some $j\leq m$ and $n\in{\mathbb{N}}$. We
will sometimes omit the subscript $\omega$ and simply write $X$ for the tiling
space. By definition, one can pass from $\omega$ to $\omega^{k}$ for $k\geq
2$, without affecting the substitution tiling space. The substitution
naturally extends to a continuous self-map of the tiling space
$\omega:\,X_{\omega}\to X_{\omega}$. This map is surjective, but not
necessarily injective. Note that

(2.8) $\omega({\mathscr{T}}-t)=\omega({\mathscr{T}})-\varphi(t),\ \
{\mathscr{T}}\in X_{\omega},\ t\in{\mathbb{R}}^{d}.$

The substitution matrix is defined by

(2.9) ${\sf S}_{\omega}(j,k)=\\#{\mathcal{D}}_{kj}.$

We will assume that the substitution is primitive, i.e., some power of ${\sf
S}_{\omega}$ has all entries strictly positive. Repetitivity of
${\mathscr{T}}\in X_{\omega}$ implies primitivity. A tiling is called self-
similar if it is a fixed point of the substitution:
$\omega({\mathscr{T}})={\mathscr{T}}$. Such a tiling always exists, after
passing to a higher power of $\omega$, if necessary. FLC repetitive self-
similar tiling spaces are known to be uniquely ergodic, see [34, 27].

####  2.2.1. Aperiodicity and recognizability; hierarchical structure

We will assume that our tiling spaces are aperiodic, i.e.,
${\mathscr{T}}={\mathscr{T}}-t$ implies $t=0$. It follows from (2.8) that
aperiodicity of $X_{\omega}$ is necessary for $\omega$ to be injective. It
turns out that it is also sufficient [35]. Then $\omega$ is a homeomorphism,
and we can “desubstitute” any ${\mathscr{T}}\in X_{\omega}$ in a unique way;
this property is usually called “recognizability”. Equivalently, there is a
unique way to compose the tiles of ${\mathscr{T}}$ into a collection of
patches of the form $\omega({\mathfrak{T}}_{j}),\ j\leq m$, in such a way that
we get a tiling from the space $\varphi(X_{\omega})$. These patches are called
supertiles of order 1. This procedure can be iterated; thus any
${\mathfrak{T}}\in{\mathscr{T}}$ is contained in a unique increasing sequence
of supertiles of order $n$.

####  2.2.2. Substitution Delone set associated with a self-similar tiling

Let ${\mathscr{T}}$ be a self-similar tiling,
$\omega({\mathscr{T}})={\mathscr{T}}$, which we assume exists, without loss of
generality. Specify the location of each prototile ${\mathfrak{T}}_{j}$ in
${\mathscr{T}}$. Then $\omega({\mathfrak{T}}_{j})$ is a ${\mathscr{T}}$-patch,
with the support $\varphi({\rm supp}({\mathfrak{T}}_{j}))$. The definition
(2.4) implies that each element of ${\mathcal{D}}_{jk}$ is translation vector
between two occurrences of equivalent tiles in ${\mathscr{T}}$, namely,
${\mathfrak{T}}_{k}$ and its translate. This means that ${\mathcal{D}}_{jk}$
is a set of return vectors, which will be important in what follows. We can
write

(2.10)
${\mathscr{T}}=\bigcup_{k=1}^{m}\bigl{(}{\mathfrak{T}}_{k}+{\mathscr{L}}_{k}({\mathscr{T}})\bigr{)},$

where ${\mathscr{L}}_{k}({\mathscr{T}})$ represents the set of locations of
tiles of type $k$ in ${\mathscr{T}}$ (relative to the prototiles). Each
${\mathscr{L}}_{k}({\mathscr{T}})$ is a Delone set, that is, a uniformly
discrete relatively dense set in ${\mathbb{R}}^{d}$. By convention,
$0\in{\mathscr{L}}_{k}({\mathscr{T}})$ for each $k$. Note that (2.10) and
(2.4) yield a “dual system of equations” for the Delone sets:

(2.11) ${\mathscr{L}}_{k}({\mathscr{T}})=\coprod_{j\leq
m}\bigl{(}\varphi{\mathscr{L}}_{j}({\mathscr{T}})+{\mathcal{D}}_{jk}\bigr{)},\
\ k\leq m.$

This means that $\bigl{(}{\mathscr{L}}_{k}({\mathscr{T}})\bigr{)}_{k\leq m}$
is a substitution Delone multiset in the sense of [26, 27].

######  Definition 2.3.

[27]. A family of Delone sets $\bigl{(}{\mathscr{L}}_{k}\bigr{)}_{k\leq m}$ is
called a substitution Delone multiset with expansion $\varphi$ if there exist
finite sets ${\mathcal{D}}_{jk}$ such that

(2.12) ${\mathscr{L}}_{k}=\coprod_{j\leq
m}\bigl{(}\varphi{\mathscr{L}}_{j}+{\mathcal{D}}_{jk}\bigr{)},\ \ k\leq m.$

This notion was introduced by Lagarias and Wang [26], except that they allowed
each $\Lambda_{k}$ to be a set “with multiplicities”.

We will also need a set of control points for the tiling ${\mathscr{T}}$ and
other tilings in $X_{\omega}$. The sets ${\mathscr{L}}_{k}$ are not
convenient, since they represent locations of tiles of type $k$ in
${\mathscr{T}}$ relative to the prototiles
${\mathfrak{T}}_{k}\in{\mathscr{T}}$. To this end, pick a point
$c({\mathfrak{T}}_{k})\in{\rm int}({\mathfrak{T}}_{k})$ for each prototile,
and let $c({\mathfrak{T}}_{k}+x)=c({\mathfrak{T}}_{k})+x$ for translated
tiles. Then

$\Lambda_{k}({\mathscr{T}}):={\mathscr{L}}_{k}({\mathscr{T}})+c({\mathfrak{T}}_{k})$

represents the set of control points in all the tiles of type $k$, and

(2.13) $\Lambda({\mathscr{T}}):=\coprod_{k\leq m}\Lambda_{k}({\mathscr{T}})$

is a Delone set of all the control points for the tiling. Note that
$(\Lambda_{k}({\mathscr{T}}))_{k\leq m}$ is a substitution Delone multiset as
well, satisfying (2.12), with ${\mathcal{D}}_{jk}$ replaced by
${\mathcal{D}}_{jk}+c({\mathfrak{T}}_{k})-\varphi c({\mathfrak{T}}_{j})$.

###  2.3. Pseudo self-similar (PSS) tilings

Our main object in this paper is tiling deformations, which are more
conveniently defined for tilings whose tiles are convex polytopes meeting
face-to-face. When the tile boundaries are fractal, which is necessarily the
case if the expansion map $\varphi$ involves an irrational rotation, an extra
step is needed. We are going to use (a variant of) the construction of Derived
Voronoi (DV) tilings introduced by N. Priebe Frank [16], which turns a self-
similar tiling into a pseudo-self-similar one, but with “nice” convex polytope
tiles.

######  Definition 2.4.

Let $\varphi:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ be an expanding
similarity map. A repetitive tiling $\mathcal{T}$ of finite local complexity
is called a _pseudo-self-similar tiling_ , or PSS, with expansion $\varphi$ if
$\mathcal{T}$ is locally derivable from $\varphi\mathcal{T}$.

In fact, in [16] the map $\varphi$ is allowed to be any expanding linear map,
but here we restrict ourselves to similitudes.

Let $\Lambda$ be a Delone set in ${\mathbb{R}}^{d}$. The Voronoi cell of a
point $x\in\Lambda$ is, by definition,

$V(x)=\bigl{\\{}{\bf t}\in{\mathbb{R}}^{d}:\ \|{\bf t}-x\|\leq\|{\bf t}-y\|,\
\forall\,y\in\Lambda\setminus\\{x\\}\bigr{\\}},$

and the corresponding Voronoi tiling (tesselation) is

${\mathcal{T}}_{\Lambda}:=\\{V(x):\ x\in\Lambda\\}.$

The tiles of ${\mathcal{T}}_{\Lambda}$ are convex polytopes meeting face-to-
face. Applying this procedure to $\Lambda({\mathscr{T}})$, we want to make
sure that the cells corresponding to equivalent tiles are also translationally
equivalent. This will usually require increasing the set of labels, by
“decorating” each point with the translation equivalence class of a
sufficiently large neighborhood. It is always possible to achieve this, see
[16, Section 4]. Below we assume that this has already been done and the
equations (2.4),(2.5) still apply, possibly with a larger $m$. Now let
$\Lambda=\Lambda({\mathscr{T}})$ be from (2.13) and consider the tiling
${\mathcal{T}}_{\Lambda}$, where the tiles “inherit” the labels from the
corresponding control points. It can be shown, as in [16], that, assuming that
the labels carry the information about a sufficiently large neighborhood, the
tiling ${\mathcal{T}}_{\Lambda}$ is MLD with the original ${\mathscr{T}}$ and
hence the translation dynamics are conjugate. If ${\mathscr{T}}$ is self-
similar, then ${\mathcal{T}}_{\Lambda}$, with
$\Lambda=\Lambda({\mathscr{T}})$, is pseudo-self-similar with the same
expansion map. We have

${\mathcal{T}}_{\Lambda}=\bigcup_{k=1}^{m}\bigl{(}T_{k}+{\mathscr{L}}_{k}({\mathscr{T}})\bigr{)},$

where

${\rm supp}(T_{k})={\rm
supp}\bigl{(}V_{\Lambda-x}(c({\mathfrak{T}}_{k})\bigr{)}\ \ \mbox{for}\
x\in{\mathscr{L}}_{k}({\mathscr{T}}),$

and consistency (independence of $x$) is guaranteed by construction. For the
pseudo-self-similar tiling ${\mathcal{T}}_{\Lambda}$ we have a substitution,
“inherited” from $\omega$, which we denote by the same letter by abuse of
notation. In fact, similarly to (2.4) and (2.6),

(2.14) $\omega(T_{j})=\bigcup_{k\leq
m}\bigl{(}T_{k}+{\mathcal{D}}_{jk}\bigr{)},\ \ \ \
\omega(T_{j}+x)=\omega(T_{j})+\varphi(x),\ \ \ j\leq m,\
x\in{\mathbb{R}}^{d},$

but the analog of (2.5) does not hold exactly – only approximately. Then the
pseudo-self-similar (PSS) tiling satisfies
$\omega({\mathcal{T}}_{\Lambda})={\mathcal{T}}_{\Lambda}$. The atlas of
patches of ${\mathcal{T}}_{\Lambda}$, or equivalently, its orbit closure in
the natural topology, defines the PSS tiling space. By construction, the PSS
tiling space is MLD with the initial self-similar tiling space.

We summarize this discussion in the following proposition, but first we need
some terminology.

######  Definition 2.5.

A tiling of ${\mathbb{R}}^{d}$ will be called polytopal if its tiles are
convex polytopes meeting face-to-face, and this induces a structure of a CW-
complex.

######  Definition 2.6.

A repetitive FLC PSS tiling ${\mathcal{T}}$, with the prototile set
$\\{T_{j}\\}_{j\leq m}$, an expansion map $\varphi$, and a (combinatorial-
geometric) substitution $\omega$, will be called an L-PSS tiling (L is for
“lucky”) if it is polytopal and there exists a substitution Delone multi-set
$\bigl{(}{\mathscr{L}}_{k}\bigr{)}_{k\leq m}$ with expansion $\varphi$,
satisfying (2.12), such that
${\mathcal{T}}=\bigcup_{k=1}^{m}\bigl{(}T_{k}+{\mathscr{L}}_{k}\bigr{)}=\omega({\mathcal{T}})$
and $\omega$ acts by (2.14).

We have proved the following

######  Proposition 2.7.

If ${\mathcal{T}}^{\prime}$ is a self-similar tiling with an expansion map
$\varphi$, then there exists an L-PSS tiling ${\mathcal{T}}$ with the
expansion map $\varphi$, such that ${\mathcal{T}}$ and
${\mathcal{T}}^{\prime}$ are MLD.

######  Remark 2.8.

(a) Although the procedure described above is general, for specific examples
there is frequently a direct and relatively simple way to obtain a PSS tiling
from a self-similar one without increasing the number of prototiles. In fact,
often a planar PSS tiling is given directly, equipped with a “substitution-
with-amalgamation”, see [13], and a procedure of “redrawing the boundary” is
used to show that it is MLD to a genuine self-similar one [18]. This is the
case for the class of Kenyon’s tilings [24] which we analyze in Section 6.

(b) The step of passing from a self-similar tiling to a PSS tiling is not
needed if the self-similar tiling is polytopal to begin with.

(c) It is proved in [18] for tilings in the plane and in [36] in the general
case, that any PSS tiling with an expansion map $\varphi$ is MLD with a
genuine self-similar tiling, with the expansion map $\varphi^{n}$, for some
$n\in{\mathbb{N}}$. Thus in the last proposition we could start with an
arbitrary PSS tiling, at the cost of raising the associated expansion map to a
power.

##  3\. Cohomology and deformations

###  3.1. Pattern equivariant cohomology

Let $\mathcal{T}$ be an aperiodic repetitive tiling of finite local complexity
and recall the definition of a $\mathcal{T}$-equivariant function (2.2). A
$\mathcal{T}$-equivariant $k$-form is a $k$-form $\alpha$ such that its
coefficients are $\mathcal{T}$-equivariant functions. We denote the set of
$C^{\infty}$, $\mathcal{T}$-equivariant $k$-forms by
$\Delta_{\mathcal{T}}^{k}$, which is a subspace of the set of smooth $k$-forms
on $\mathbb{R}^{d}$. As such the restriction of the usual de Rham differential
operator gives a differential operator on the complex
$\\{\Delta_{\mathcal{T}}^{k}\\}_{k}$ of $\mathcal{T}$-equivariant forms.

######  Definition 3.1.

The cohomology of the complex of smooth $\mathcal{T}$ equivariant forms

$H^{k}(X;\mathbb{R}):=\frac{\ker
d:\Delta_{\mathcal{T}}^{k}\rightarrow\Delta_{\mathcal{T}}^{k+1}}{\mathrm{im}\,d:\Delta_{\mathcal{T}}^{k-1}\rightarrow\Delta_{\mathcal{T}}^{k}}$

is called the _$\mathcal{T}$ -equivariant cohomology_.

We denoted the cohomology as $H^{k}(X;\mathbb{R})$ since it is independent of
which tiling $\mathcal{T}\in X$ we used to define it [23].

###  3.2. Čech cohomology

Let $\mathcal{T}$ be an aperiodic, repetitive tiling of finite local
complexity, which we now assume has a CW structure. More specifically, we
assume that all the tiles of $\mathcal{T}$ are $d$-cells of the CW complex
$\mathcal{T}$ where all tiles meet face-to-face. (In practice, we will work
with polytopal tilings, see Definition 2.5.) For any tile $t\in\mathcal{T}$ we
define $\mathcal{T}(t)={\mathcal{T}}^{(1)}$, called the 1-corona of $t$, to be
the set of all tiles in $\mathcal{T}$ which intersect $t$. Continue
recursively as follows: Given a $(k-1)$-corona of a tile
$\mathcal{T}^{(k-1)}(t)$, the $k$-corona of the tile $t$ is the patch

$\mathcal{T}^{(k)}(t)=\\{t^{\prime}\in\mathcal{T}:t^{\prime}\cap\mathcal{T}^{(k-1)}(t)\neq\varnothing\\}.$

A $k$-collaring of a tile $t$ is the tile with the same support as $t$, but
the label being the translation-equivalence class of ${\mathcal{T}}^{(k)}(t)$.
This is a useful tool to keep track of bigger neighborhoods of tiles by
increasing the set of labels.

####  3.2.1. The Anderson-Putnam complex

Following Anderson-Putnam [2], we define a cell complex ${AP}_{0}(X)$ by
gluing together the prototiles along their faces in all ways in which they can
be adjacent in the tiling space. This can be done equally well for
$k$-collared tiles (this just increases the set of labels). The resulting
space is denoted $AP_{k}(X)$. Here is a formal definition:

######  Definition 3.2.

Let $X$ be a tiling space of repetitive, aperiodic tilings of $\mathbb{R}^{d}$
of finite local complexity. We define on $X\times\mathbb{R}^{d}$ an
equivalence relation $\sim_{1}$, where $X$ carries the discrete topology and
$\mathbb{R}^{d}$ the usual topology, as
$(\mathcal{T}_{1},v_{1})\sim_{1}(\mathcal{T}_{2},v_{2})$ if and only if
$\mathcal{T}_{1}(t_{1})-v_{1}=\mathcal{T}_{2}(t_{2})-v_{2}$ for some tiles
$t_{1},t_{2}$ with $v_{1}\in t_{1}\in\mathcal{T}_{1}$ and $v_{2}\in
t_{2}\in\mathcal{T}_{2}$. The space $X\times\mathbb{R}^{d}/\sim_{1}$ is the
_Anderson-Putnam complex_ of $X$, denoted by $AP(X)$. For $k\geq 0$ the
$k^{th}$-collared Anderson-Putnam complex $AP_{k}(X)$ is similarly obtained by
using $k$-collared tiles instead of $1$-collared tiles, where 0-collared tiles
are just tiles. Note that $AP(X)=AP_{1}(X)$ by definition.

Assuming a tile substitution rule as in (2.14) is a cellular map, it defines a
map $\gamma:AP(X)\rightarrow AP(X)$. The important observation of Anderson and
Putnam is that, as defined,

$X\cong\lim_{\longleftarrow}(AP(X),\gamma),$

that is, the tiling space $X$ is homeomorphic to the set of all infinite
sequences $\\{x_{i}\\}_{i\in\mathbb{N}}\in AP(X)^{\infty}$ with the property
that $\gamma(x_{i})=x_{i-1}$. What this result gives is the easy calculation
of the Čech cohomology of $X$ using the induced maps $\gamma^{*}$ on the
cohomology of $AP(X)$:

(3.1)
$\check{H}^{*}(X;\mathbb{Z})\cong\lim_{\longrightarrow}\left(\check{H}^{*}(AP(X);\mathbb{Z}\right),\gamma^{*}).$

####  3.2.2. Cohomology for PSS tilings

Let $\mathcal{T}$ be a polytopal pseudo-self-similar tiling. The latter means
that there exists an $R>0$ and expansive map $\varphi$ such that for all
$x,y\in\mathbb{R}^{d}$,

$\mathcal{O}^{-}_{\mathcal{\varphi
T}}(B_{R}(x))=\mathcal{O}^{-}_{\mathcal{\varphi
T}}(B_{R}(x))+(x-y)\Longrightarrow\mathcal{O}^{-}_{\mathcal{T}}(B_{1}(x))=\mathcal{O}^{-}_{\mathcal{T}}(B_{1}(x))+(x-y).$

Let $X$ be the tiling space of $\mathcal{T}$ and $X_{\varphi}$ the tiling
space of $\varphi\mathcal{T}$. If $\varphi$ is pure dilation, then
$AP(X_{\varphi})$ is a rescaled copy of $AP(X)$ by
$\theta:=|\det\varphi|^{1/d}$, since linear expansive maps do not affect the
process of collaring. If $\varphi$ is not pure dilation, then
$AP(X_{\varphi})$ is a rescaled copy of $AP(X)$ by $\theta$, but where all the
cells have also been rotated.

######  Proposition 3.3.

Let $X$ be the tiling space of a polytopal pseudo-self-similar tiling
$\mathcal{T}$. Then there exists a $\kappa\in\mathbb{N}$ and a map
$\gamma:AP_{\kappa}(X)\rightarrow AP_{\kappa}(X)$ such that

(3.2) $X\cong\lim_{\longleftarrow}\left(AP_{\kappa}(X),\gamma\right).$

###### Proof.

Since $\mathcal{T}$ is PSS with expanding map $\varphi$, there exists an $R>0$
such that for any $x\in\mathbb{R}^{d}$ the $R$-neighborhood of $x$ in
$\varphi\mathcal{T}$ determines the tile(s) to which $x$ belongs in
$\mathcal{T}$. Let $k^{\prime}\in\mathbb{N}$ be the smallest integer $k$ such
that the $R$-neighborhood of any tile $t\in\varphi\mathcal{T}$ is contained in
$(\varphi\mathcal{T})^{(k)}(t)$. We will first show that any $\ell\geq
k^{\prime}$ allows us to define a map $\gamma_{\ell}:AP_{\ell}(X)\rightarrow
AP_{\ell}(X)$.

Let $X_{\varphi}$ be the tiling space of $\varphi\mathcal{T}$. Since the
pseudo self-similarity $\varphi$ maps $n$-cells of $\mathcal{T}$ to $n$-cells
of $\varphi\mathcal{T}$ and respects collaring, it determines a bijection
$r_{\ell}:AP_{\ell}(X)\rightarrow AP_{\ell}(X_{\varphi})$ for every
$\ell\in\mathbb{N}$. We now define a map
$s_{\ell}:AP_{\ell}(X_{\varphi})\rightarrow AP_{\ell}(X)$ for any $\ell\geq
k^{\prime}$ as follows. Let $[x]_{\varphi}\in AP_{\ell}(X_{\varphi})$ and let
$x\in t\in\varphi\mathcal{T}\in X_{\varphi}$ be a representative on the
tiling. Then the $R$ neighborhood of $x$ in $\varphi\mathcal{T}$ determines a
point in $\mathcal{T}$ whereon $x$ lies, and thus a point $[x]\in
AP_{\ell}(X)$. Since $[x]$ was completely determined by the $R$-neighborhood
of $x$ and this neighborhood is completely contained in
$(\varphi\mathcal{T})^{(\ell)}$, then the map $s_{\ell}([x]_{\varphi})=[x]\in
AP_{\ell}(X)$ is well-defined (that is, independent of representatives of
classes). Let $\gamma_{\ell}:=s_{\ell}\circ r_{\ell}:AP_{\ell}(X)\rightarrow
AP_{\ell}(X)$.

Now the inverse limit of $(AP_{k^{\prime}}(X),\gamma_{k^{\prime}})$ is well
defined, but it may or may not be homeomorphic to $X$. By the argument of
Anderson and Putnam, collaring once more guarantees that the map forces the
border. Therefore, picking $\kappa=k^{\prime}+1$ ensures that and we obtain
(3.2). ∎

######  Corollary 3.4.

Let $X$ be the tiling space of a polytopal pseudo-self-similar tiling
$\mathcal{T}$ and $\gamma:AP_{\kappa}(X)\rightarrow AP_{\kappa}(X)$ the map
from Proposition 3.3. Then

(3.3)
$\check{H}^{*}(X;\mathbb{Z})\cong\lim_{\longrightarrow}\left(\check{H}^{*}(AP_{\kappa}(X);\mathbb{Z}),\gamma^{*}\right).$

Finally, we note that for polytopal tilings of finite local complexity in
which cells meet face to face, we have that the $\mathcal{T}$-equivariant and
Čech cohomologies are isomorphic [23], that is,

(3.4) $H^{*}(X;\mathbb{R})\cong\check{H}^{*}(X;\mathbb{R}).$

###  3.3. Deformations

In this section we go over deformations of tiling spaces, following [13, 21].
Let $\mathcal{T}$ be a repetitive, aperiodic polytopal tiling of finite local
complexity and let $X$ be the corresponding tiling space. Observe that for
L-PSS tiling spaces $H^{1}(X;\mathbb{R})$ is finite dimensional by (3.3). Each
class $[\alpha]\in H^{1}(X;\mathbb{R})$ is represented by a
$\mathcal{T}$-equivariant smooth 1-form $\alpha:\mathbb{R}^{d}\rightarrow
T^{*}\mathbb{R}^{d}$ (up to a $\mathcal{T}$-equivariant exact form). Now
consider the space
$H^{1}(X;\mathbb{R}^{d})=H^{1}(X;\mathbb{R})\otimes\mathbb{R}^{d}$. Each class
$[{\mathfrak{f}}]\in H^{1}(X;\mathbb{R}^{d})$ is represented by a
$\mathcal{T}$-equivariant smooth 1-form
${\mathfrak{f}}:\mathbb{R}^{d}\rightarrow
T^{*}\mathbb{R}^{d}\otimes\mathbb{R}^{d}\cong M_{d\times d}$ (up to a
$\mathcal{T}$-equivariant exact form). That is, the representative
${\mathfrak{f}}$ is a $\mathcal{T}$-equivariant choice of linear
transformation of $\mathbb{R}^{d}$.

A representative ${\mathfrak{f}}$ of a class in $H^{1}(X;\mathbb{R}^{d})$ is
called _shape deformation_ as it defines a deformation of $\mathcal{T}$ as
follows. (We essentially quote the beginning of [21, Section 8] here.) Suppose
that ${\mathfrak{f}}$ is ${\mathcal{T}}$-equivariant with some radius $R$. Let
$H_{\mathfrak{f}}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ be defined as
$H_{\mathfrak{f}}(x)=\int_{0}^{x}{\mathfrak{f}}$. We will deform the vertices
of tiles, assuming that one of the vertices is at the origin. Each of these
vertices is decorated with the equivalence class of the pattern of radius
$R_{0}>R$ around it, where $R_{0}$ is greater than $R$ plus the greatest
distance between the adjacent vertices (i.e., connected by an edge). If $x$ is
a vertex, we let $H_{\mathfrak{f}}(x)$ be the deformed vertex (with the same
label). If $v_{1}v_{2}$ was an edge, then the displacement vector
$v_{2}-v_{1}$ has been changed to $\int_{v_{1}}^{v_{2}}{\mathfrak{f}}$. The
label of $v_{1}$ determines the pattern of ${\mathcal{T}}$ out to distance $R$
along the entire edge and hence determines
$\int_{v_{1}}^{v_{2}}{\mathfrak{f}}$. Thus the local patterns of
${\mathcal{T}}^{\mathfrak{f}}$ are determined from local patterns of
${\mathcal{T}}$, and ${\mathcal{T}}^{\mathfrak{f}}$ has FLC.

Since we deform the tiles by deforming their edges, for $d>2$ we need to make
sure that higher-dimensional faces are well-defined. For instance, for
2-dimensional faces we would need the side edges to remain co-planar. In order
to avoid imposing such a condition, we can simply triangulate all the tiles to
begin with. It is easy to see that there exists a triangulation which is
equivariant and preserves the face-to-face property. When all the faces are
simplices, they stay being simplices after perturbing the edges. The new
simplex sub-tiles should carry a label which contains both the label of the
tile it came from, and its location in the triangulation; then the
triangulated tiling is MLD to the original one, and we can work with it from
the start without loss of generality.

What is not clear at this point is whether the deformation is in fact a
homeomorphism or not. As proved by [21], what determines the answer to this
question is invertibility of the Ruelle-Sullivan cycle applied to
$[{\mathfrak{f}}]$.

######  Definition 3.5.

Let $\mathcal{T}$ be a polytopal, repetitive, aperiodic tiling of finite local
complexity, with uniform patch frequency. The _Ruelle-Sullivan map_ is the map
$C:H^{1}(X;\mathbb{R}^{d})\rightarrow M_{d\times d}$ defined, for a class
$[{\mathfrak{f}}]\in H^{1}(X;\mathbb{R}^{d})$, as

$C([{\mathfrak{f}}])=\lim_{R\rightarrow\infty}\frac{1}{\mathrm{Vol}(B_{R})}\int_{B_{R}}{\mathfrak{f}}(t)\,dt,$

which is independent of the representative ${\mathfrak{f}}$.

We can now recall one of the main results of [21]: if $C([{\mathfrak{f}}])$ is
invertible, then one can choose a representative ${\mathfrak{f}}$ so that
$H_{\mathfrak{f}}$ is a homeomorphism of ${\mathbb{R}}^{d}$. Then the shape
deformation ${\mathfrak{f}}$ induces an orbit equivalence between the tiling
spaces $X$ and $X^{\mathfrak{f}}$. In particular, the spaces $X$ and
$X^{\mathfrak{f}}$ are homeomorphic. As such, the set

$\mathcal{M}(X):=\left\\{[{\mathfrak{f}}]\in H^{1}(X;\mathbb{R}^{d}):\det
C([{\mathfrak{f}}])\neq 0\right\\}$

parametrizes deformations of the tiling space $X$, up to deformations given by
coboundaries. Deformations given by representatives of classes in
$\mathcal{M}(X)$ are all orbit equivalences. Moreover, ${\mathfrak{f}}$ can be
chosen so that the homeomorphism $H_{\mathfrak{f}}$ will be arbitrarily close
to the identity, see the proof of [21, Theorem 8.1]. This, however, will come
at the cost of collaring to a possibly very large radius.

It is important for us that the deformation preserves the combinatorial
structure of the tiling, in the sense neighbor graph and faces of all
dimensions. This can be achieved either by collaring, or by working only with
super-tiles of sufficiently large size and ignoring the “small-scale”
combinatorics.

######  Definition 3.6.

A shape deformation of an FLC polytopal tiling space $X$ will be called
admissible if it preserves the local combinatorial structure of the tilings
and it defines a homeomorphism via the map $H_{{\mathfrak{f}}}$ introduced
above. In practical terms this means that we will consider deformations that
are small compared to the size of the prototiles.

Now consider the tiling space $X_{\omega}$ of an L-PSS tiling, see Definition
2.6. Picking $[{\mathfrak{f}}]\in\mathcal{M}(X_{\omega})$ and applying an
admissible ${\mathfrak{f}}$ to ${\mathscr{S}}\in X_{\omega}$ having a vertex
at the origin, we obtain a tiling ${\mathscr{S}}^{\mathfrak{f}}$ and the
${\mathfrak{f}}$-deformed substitution tiling space
$X^{\mathfrak{f}}_{\omega}$ (the translation orbit closure), whose prototiles
will be denoted by $T_{j}^{\mathfrak{f}}$. To be precise, we need to specify
the location of $T_{j}^{\mathfrak{f}}$ in ${\mathbb{R}}^{d}$. By construction,
the L-PSS tiling ${\mathcal{T}}$ has distinguished prototiles
$T_{k}\in{\mathcal{T}}$ and is also a fixed point of the substitution
$\omega({\mathcal{T}})={\mathcal{T}}$. Choose any vertex $v$ in
${\mathcal{T}}$ and consider the shifted tiling ${\mathcal{T}}-v$, with a
vertex at the origin. Then the deformation $({\mathcal{T}}-v)^{\mathfrak{f}}$
is well-defined. We then let the deformed prototiles to be

(3.5) $T_{j}^{\mathfrak{f}}:=(T_{j}-v)^{\mathfrak{f}},\ \ j\leq m.$

Further, $\omega^{n}(T_{j})-v\subset{\mathcal{T}}-v$ for all
$n\in{\mathbb{N}}$, so we can define deformed higher order super-prototiles by

(3.6) $T_{j}^{{\mathfrak{f}},n}=(\omega^{n}(T_{j})-v)^{\mathfrak{f}}.$

(Strictly speaking, these are patches rather than individual tiles. The exact
location is not that important, but we want their supports to be subsets of
${\mathbb{R}}^{d}$, e.g., in order to be able to perform integration over
them.) The assumption that ${\mathfrak{f}}$ is admissible implies that the
tilings in the deformed tiling space $X_{\omega}^{\mathfrak{f}}$ will have a
hierarchical structure combinatorially equivalent to those of $X_{\omega}$.

###  3.4. Recurrences, return vectors and recurrence vectors

Let ${\mathcal{T}}$ be a repetitive FLC polytopal tiling and $X$ the
corresponding tiling space. Following [13], we say that a pair of points
$(z_{1},z_{2})$ in a tiling ${\mathcal{T}}$ is a recurrence of size $r$, if

(3.7)
${\mathcal{O}}^{-}_{\mathcal{T}}(B_{r}(z_{2}))={\mathcal{O}}^{-}_{\mathcal{T}}(B_{r}(z_{1}))+(z_{2}-z_{1}),$

and $r$ is maximal possible. The vector $z_{2}-z_{1}$ is called a return
vector of size $r$. We will always assume that $r\geq 2D_{\max}$, where
$D_{\max}$ is the diameter of the largest prototile, then (3.7) implies that
${\mathcal{O}}^{+}_{\mathcal{T}}(\\{z_{2}\\})={\mathcal{O}}^{+}_{\mathcal{T}}(\\{z_{1}\\})+(z_{2}-z_{1})$.
We can assume, without loss of generality, that $z_{1}$ and $z_{2}$ are
vertices of the tiling ${\mathcal{T}}$. Each path along edges from $z_{1}$ to
$z_{2}$ projects to a closed loop in $AP(X)$, and hence to a closed chain in
$C_{1}(AP(X))$. Different paths from $z_{1}$ to $z_{2}$ project to homologous
chains. The class in $H_{1}(AP(X),{\mathbb{Z}})$ of a recurrence is called a
recurrence class. Recurrences of size greater than $(k+1)D_{\max}$ project to
closed chains in $AP_{k}(X)$ and define classes in
$H_{1}(AP_{k}(X),{\mathbb{Z}})$.

######  Definition 3.7.

Let ${\mathcal{T}}$ be a polytopal tiling. A pair of points $(z_{1},z_{2})$ in
${\mathcal{T}}$ will be called an elementary recurrence if there exists a tile
$T\in{\mathcal{T}}$ such that $z_{1}\in{\rm supp}(T)$ and
$T+(z_{2}-z_{1})\in{\mathcal{T}}$. Similarly to the above, we can associate to
$(z_{1},z_{2})$ a closed loop in ${AP}_{0}(X)$ which defines a class in
$H_{1}({AP}_{0}(X),{\mathbb{Z}})$, called an elementary recurrence class.

######  Definition 3.8.

Let $[{\mathfrak{f}}]\in\mathcal{M}(X)\subset H^{1}(X;{\mathbb{R}}^{d})$. We
will call it elementary if it is a pull-back of a class in
$H^{1}({AP}_{0}(X),{\mathbb{R}}^{d})$ under the “forgetful” map. In
particular, it is a class of a shape deformation which acts on elementary
recurrences. A representative ${\mathfrak{f}}$ of an elementary class
$[{\mathfrak{f}}]$ will also be called elementary.

In the rest of the paper we will restrict ourselves to elementary shape
deformations. This is not an essential restriction, since we can always
“collar” the tiles of ${\mathcal{T}}$ to the level $\kappa$, given in
Proposition 3.3, and then an elementary deformation defines a class in
$H^{1}(AP_{\kappa}(X);{\mathbb{R}}^{d})$, hence in
$H^{1}(X;{\mathbb{R}}^{d})$, by Proposition 3.3.

We continue with the construction, following [13]. Let $X$ be an L-PSS tiling
space. The set of integer linear combinations of elementary recurrence classes
is a subgroup $\Gamma<H_{1}({AP}_{0}(X),{\mathbb{Z}})$, which is a finitely
generated free ${\mathbb{Z}}$-module. Let $\\{a_{1},\ldots,a_{s}\\}$ be a
basis (set of free generators) for $\Gamma$. For each elementary recurrence
$(z_{1},z_{2})$, which defines a class $[(z_{1},z_{2})]$, the corresponding
recurrence vector in ${\mathbb{Z}}^{s}$ is the decomposition of the class in
this basis; it will be denoted $\alpha(z_{1},z_{2})$. The transformation
$\alpha:\Gamma\to{\mathbb{Z}}^{s}$ is sometimes called the address map. For a
shape parameter ${\mathfrak{f}}$, define

(3.8) ${\bf
L}_{\mathfrak{f}}=({\mathfrak{f}}(a_{1}),\ldots,{\mathfrak{f}}(a_{s})),$

which can be thought of as a ${\mathbb{R}}^{d}$-valued row vector that gives
the displacements of the deformed vectors in the basis. Then the deformation
of the displacement vector $z_{2}-z_{1}$, corresponding to the recurrence
vector ${\bf v}=\alpha(z_{1},z_{2})$ is equal to ${\bf L}_{\mathfrak{f}}{\bf
v}$.

Given an elementary recurrence $(z_{1},z_{2})$ in ${\mathcal{T}}$, we have
that $(\varphi(z_{1}),\varphi(z_{2}))$ is an elementary recurrence as well,
because ${\mathcal{T}}$ is a PSS tiling with expansion $\varphi$, see (2.14).
(In fact, applying the substitution increases the size of the recurrence, but
is still an elementary too.) It follows that the expansion map $\varphi$
induces an endomorphism of the ${\mathbb{Z}}$-module generated by elementary
recurrence vectors. Thus there exists an integer $s\times s$ matrix $M$
satisfying

(3.9) $\alpha(\varphi(z_{1}),\varphi(z_{2}))=M\alpha(z_{1},z_{2}),\ \
z_{1},z_{2}\in\Lambda({\mathcal{T}}).$

Let ${\mathfrak{f}}$ be an elementary admissible deformation. Then, combining
(2.14) and (3.6) (for $n=1$) yields

(3.10) $T_{j}^{{\mathfrak{f}},1}=\bigcup_{j\leq
m}\bigl{(}T_{k}^{\mathfrak{f}}+{\mathfrak{f}}({\mathcal{D}}_{jk})\bigr{)},$

where the right-hand side is a patch of a tiling in
$X_{\omega}^{\mathfrak{f}}$. Here we view the elements of ${\mathcal{D}}_{jk}$
as elementary recurrences, since they represent the prototile
$T_{k}\in{\mathcal{T}}$ and its translated copy in
$\omega({\mathcal{T}})\subset{\mathcal{T}}$. Using the address map, as above,
we get an associated set of recurrence vectors, denoted
$\alpha({\mathcal{D}}_{jk})$. By definition,

${\mathfrak{f}}({\mathcal{D}}_{jk})={\bf
L}_{\mathfrak{f}}\alpha({\mathcal{D}}_{jk}).$

We can iterate the procedure and obtain the decomposition of higher order
deformed super-tiles as well. For instance,

(3.11) $\displaystyle T_{j}^{{\mathfrak{f}},2}$ $\displaystyle=$
$\displaystyle\bigcup_{s\leq m,k\leq
m}\bigl{(}T_{s}^{\mathfrak{f}}+{\mathfrak{f}}(\varphi{\mathcal{D}}_{jk})+{\mathfrak{f}}({\mathcal{D}}_{ks})\bigr{)},$
$\displaystyle=$ $\displaystyle\bigcup_{s\leq m,k\leq
m}\bigl{(}T_{s}^{\mathfrak{f}}+{\bf
L}_{\mathfrak{f}}M\alpha({\mathcal{D}}_{jk})+{\bf
L}_{\mathfrak{f}}\alpha({\mathcal{D}}_{ks})\bigr{)},\ \ 1\leq j\leq m.$

###  3.5. Geometric properties of deformed tilings and consequences

The following geometric lemma will be useful. We will write $\asymp$ to
indicate that the equality holds up to a uniformly bounded from $0$ and
$\infty$ multiplicative constant.

######  Lemma 3.9.

Let $\omega$ be an L-PSS tile substitution with expansion $\varphi$ and
$X_{\omega}$ the corresponding tiling space. Fix an elementary admissible
shape deformation ${\mathfrak{f}}$, and consider the corresponding deformed
tiling space $X_{\omega}^{\mathfrak{f}}$. For the ${\mathfrak{f}}$-deformed
supertiles $T_{j}^{{\mathfrak{f}},n},\ j\leq m,$ let $R_{n}^{{\mathfrak{f}}}$
be the radius of the smallest ball containing (a translate of) every
$T_{j}^{{\mathfrak{f}},n}$, and let $r_{n}^{{\mathfrak{f}}}$ be the radius of
largest ball contained in a (translate of) any $T_{j}^{{\mathfrak{f}},n}$.
Then there exists $C_{\mathfrak{f}}>1$ depending only on ${\mathfrak{f}}$ such
that

(3.12) $r_{n}^{\mathfrak{f}}\geq C_{\mathfrak{f}}^{-1}\theta^{n}$

and

(3.13) $R^{\mathfrak{f}}_{n}\leq C_{\mathfrak{f}}\theta^{n},$

where $\theta=\|\varphi\|$.

###### Proof.

We use the fact that, combinatorially, the hierarchical structure of the
tilings in $X_{\omega}^{\mathfrak{f}}$ is the same as in $X_{\omega}$. The
L-PSS tile substitution was obtained from a self-similar, “geometric” tile
substitution, so that ${\mathcal{T}}$ is MLD with a self-similar tiling
${\mathscr{T}}$ having the expansion $\varphi$. For the tiles and super-tiles
of ${\mathscr{T}}$ the properties (3.12) and (3.13) are immediate by self-
similarity. It follows that the same properties hold the for the PSS tiling,
since it was built using a derived Voronoi tesselation construction, using
$\Lambda=\Lambda({\mathscr{T}})$.

Let us call a sequence of tiles $S_{1},\ldots,S_{k}$ (of arbitrary type) a
chain if $S_{j}\cap S_{j+1}\neq\emptyset$ for $j=1,\ldots,k-1$ (recall that
our tiles are compact sets which share faces).

Claim 1. Let $T_{j}$ be a prototile of the L-PSS tiling ${\mathcal{T}}$, with
$c(T_{j})$ (control point) in its interior, and consider the
${\mathcal{T}}$-patch $\omega^{n}(T_{j})$, as well as $A_{j}^{n}:={\rm
supp}(\omega^{n}(T_{j}))$. Let $S$ be any ${\mathcal{T}}$-tile in the patch
$\omega^{n}(T_{j})$ containing the point $\varphi^{n}(c(T_{j}))$. Then any
chain of ${\mathcal{T}}$-tiles in $\omega^{n}(T_{j})$ connecting $S$ to
$\partial A_{j}^{n}$ contains at least $c_{1}\theta^{n}$ tiles, where
$c_{1}>0$ depends only on ${\mathcal{T}}$.

The claim is immediate since $A^{n}_{j}$, being a bounded displacement of
$\varphi^{n}A_{j}$, with $A_{j}={\rm supp}(T_{j})$, contains a ball of radius
$\asymp\theta^{n}$ centered at $\varphi^{n}(c(T_{j}))$, by the (3.12) property
for ${\mathcal{T}}$.

Now consider the ${\mathfrak{f}}$-deformed super-tile
$T_{j}^{{\mathfrak{f}},n}=(\omega^{n}(T_{j})-v)^{\mathfrak{f}}$, see (3.6),
with the deformed tile $(S-v)^{\mathfrak{f}}$ inside. Denote
$A_{j}^{{\mathfrak{f}},n}:={\rm supp}(T_{j}^{{\mathfrak{f}},n})$.

Claim 2. Let $d((S-v)^{\mathfrak{f}},\partial A_{j}^{{\mathfrak{f}},n})$ be
the usual Euclidean distance between the compact boundary of the deformed
super-tile and the compact $(S-v)^{\mathfrak{f}}$ in its interior. Then there
exists a chain of ${\mathfrak{f}}$-deformed tiles in the patch
$T_{j}^{{\mathfrak{f}},n}$ connecting $(S-v)^{\mathfrak{f}}$ to $\partial
A_{j}^{{\mathfrak{f}},n}$ of cardinality $N_{\mathfrak{f}}(n)$, such that

(3.14) $N_{\mathfrak{f}}(n)\cdot V_{\min}^{\mathfrak{f}}\leq
c_{d-1}\cdot\bigl{[}d((S-v)^{\mathfrak{f}},\partial
A_{j}^{{\mathfrak{f}},n})+2{\sf d}^{\mathfrak{f}}_{\max}\bigr{]}\cdot({\sf
d}^{\mathfrak{f}}_{\max})^{d-1},$

where $V^{\mathfrak{f}}_{\min}$ is the minimal volume of an
${\mathfrak{f}}$-deformed tile, ${\sf d}^{\mathfrak{f}}_{\max}$ is the maximal
diameter of an ${\mathfrak{f}}$-deformed tile, and $c_{d-1}$ is the volume of
a unit ball in ${\mathbb{R}}^{d-1}$.

Combining Claim 2 with Claim 1 yields (3.12). Indeed, for every
${\mathfrak{f}}$-chain in $T_{j}^{{\mathfrak{f}},n}$ there is a
combinatorially equivalent chain in the pseudo-self-similar
$\omega^{n}(T_{j})$, and this yields a lower bound $\asymp\theta^{n}$ for
$d((S-v)^{\mathfrak{f}},\partial A_{j}^{{\mathfrak{f}},n})$ for large enough
$n$. (For small $n$ both (3.12) and (3.13) are trivial.)

For the proof of Claim 2, consider the shortest straight line segment $J$ from
$(S-v)^{\mathfrak{f}}$ to $\partial A_{j}^{{\mathfrak{f}},n}$ and consider the
union of deformed tiles in $T_{j}^{{\mathfrak{f}},n}$ intersecting $J$. One
can certainly form a chain of ${\mathfrak{f}}$-deformed tiles connecting
$(S-v)^{\mathfrak{f}}$ to $\partial A_{j}^{{\mathfrak{f}},n}$ out of them, and
their total volume is at least the expression in the left-hand side of (3.14).
On the other hand, these tiles are all contained in the ${\sf
d}^{\mathfrak{f}}$-neighborhood of $J$, whose volume is less that the right-
hand side of (3.14).

The proof of (3.13) is similar. Indeed, the maximal distance from a point in
$S$ to a point on the boundary $\partial A^{n}_{j}$ is at most
$C_{2}\theta^{n}$ by self-similarity, hence for any point on $\partial
A^{n}_{j}$ there exists a chain of ${\mathcal{T}}$-tiles connecting that point
to $S$, of cardinality at most $\widetilde{C}_{2}\theta^{n}$. For any such
chain there is a corresponding deformed chain of the same cardinality, showing
that the diameter of $A_{j}^{{\mathfrak{f}},n}$ is bounded above by ${\rm
const}\cdot\theta^{n}$, with the constant depending only on the deformation.
This completes the proof of the lemma. ∎

For each prototile $T_{j}^{\mathfrak{f}}$ of the deformed tiling space
$X_{\omega}^{\mathfrak{f}}$ we also fix a control point
$c(T_{j}^{\mathfrak{f}})$ in the interior and then in all tiles at the same
relative location. For a tiling ${\mathscr{S}}^{\mathfrak{f}}\in
X_{\omega}^{\mathfrak{f}}$ let

$\Lambda({\mathscr{S}}^{\mathfrak{f}}):=\\{c(T^{\mathfrak{f}}):\
T^{\mathfrak{f}}\in{\mathscr{S}}^{\mathfrak{f}}\\}.$

######  Lemma 3.10.

Let ${\mathcal{T}}\in X_{\omega}$ be an L-PSS tiling, ${\mathfrak{f}}$ and
admissible elementary deformation, and
${\mathscr{S}}^{\mathfrak{f}}=({\mathcal{T}}-v)^{\mathfrak{f}}\in
X_{\omega}^{\mathfrak{f}}$, where $v$ is a vertex of ${\mathcal{T}}$. Then
$\Lambda({\mathcal{T}})$ and $\Lambda({\mathscr{S}}^{\mathfrak{f}})$ are
quasi-isometric; in fact, there exists $C_{\omega,{\mathfrak{f}}}$
(independent of ${\mathcal{T}}$) such that for any
$z_{1},z_{2}\in\Lambda({\mathcal{T}})$ and the corresponding
$z_{1}^{\mathfrak{f}},z_{2}^{\mathfrak{f}}\in\Lambda({\mathscr{S}}^{\mathfrak{f}})$,
holds

(3.15)
$C_{\omega,{\mathfrak{f}}}^{-1}|z_{1}-z_{2}|\leq|z_{1}^{\mathfrak{f}}-z_{2}^{\mathfrak{f}}|\leq
C_{\omega,{\mathfrak{f}}}|z_{1}-z_{2}|.$

###### Proof.

Consider the graph ${\rm Graph}({\mathcal{T}})$ associated with a tiling
${\mathcal{T}}$, in which the vertices are the tiles and graph edges connect
neighboring tiles (tiles whose compact supports intersect). By construction,
this graph is stable under the deformations that we consider. It follows from
the proof of Lemma 3.9 that for the deformed tiling
${\mathscr{S}}^{\mathfrak{f}}$, the set
$\Lambda({\mathscr{S}}^{\mathfrak{f}})$ is quasi-isometric to the set of
vertices of the graph ${\rm Graph}({\mathscr{S}}^{\mathfrak{f}})\cong{\rm
Graph}({\mathcal{T}})$, endowed with the graph metric. This implies the
desired claim. ∎

######  Definition 3.11.

A tiling ${\mathcal{T}}$ is called linearly repetitive if there exists $C>0$
such that for every patch ${\mathcal{P}}\subset{\mathcal{T}}$ a
translationally equivalent ${\mathcal{T}}$-patch may be found in every ball of
radius $C{\rm diam}({\mathcal{P}})$.

######  Corollary 3.12.

Let $X_{\omega}$ be an FLC primitive aperiodic L-PSS tiling space. Fix an
admissible elementary shape deformation ${\mathfrak{f}}$ and consider the
corresponding deformed tiling space $X_{\omega}^{\mathfrak{f}}$. Then

(i) all the tilings in $X^{\mathfrak{f}}_{\omega}$ are linearly repetitive;

(ii) the deformed tiling dynamical system
$(X^{\mathfrak{f}}_{\omega},{\mathbb{R}}^{d})$ is uniquely ergodic.

###### Proof.

Linear repetitivity of primitive self-similar tiling spaces is well-known,
see, e.g. [35], and the last lemma implies that this is the case for
$X^{\mathfrak{f}}_{\omega}$. Linear repetitivity implies the existence of
uniform patch frequencies [25] and hence unique ergodicity. ∎

The unique invariant probability measure for
$(X^{\mathfrak{f}}_{\omega},{\mathbb{R}}^{d})$ will be denoted by
$\mu_{\mathfrak{f}}$.

######  Remark 3.13.

The deformed tiling space $X_{\omega}^{\mathfrak{f}}$ falls into the very
general framework of fusion, developed by Sadun and Frank [17]. In fact, our
situation satisfies the conditions of transition-regular, primitive, and
recognizable fusion rules, for which unique ergodicity was proved in [17,
Corollary 3.10]. Moreover, we have a constant number of prototiles (equal to
$m$) at each level, and a constant transition matrix, hence the tiling
dynamical system $(X^{\mathfrak{f}}_{\omega},{\mathbb{R}}^{d})$, with the
unique invariant probability measure $\mu_{\mathfrak{f}}$, is not strongly
mixing by [17, 4.13].

We will need a formula for the unique invariant measure $\mu_{\mathfrak{f}}$
on $X^{\mathfrak{f}}_{\omega}$. Recall the notion of transversal, defined by

$\Upsilon({\mathcal{P}}^{\mathfrak{f}})=\\{{\mathcal{T}}^{\mathfrak{f}}\in
X^{\mathfrak{f}}_{\omega}:\
{\mathcal{P}}^{\mathfrak{f}}\subset{\mathcal{T}}^{\mathfrak{f}}\\}$

for a deformed tiling space $X^{\mathfrak{f}}_{\omega}$ and a deformed patch
${\mathcal{P}}^{\mathfrak{f}}$.

######  Lemma 3.14.

Let $X_{\omega}$ be an FLC primitive aperiodic L-PSS tiling space, with
expansion $\varphi=\theta{\mathcal{O}}$. Fix an elementary admissible shape
deformation ${\mathfrak{f}}$ and consider the corresponding deformed tiling
space $X_{\omega}^{\mathfrak{f}}$. Then there exists
$c_{\omega,{\mathfrak{f}}}>0$, such that for any Borel set $U$ and all
$n\in{\mathbb{N}}$,

(3.16) ${\rm diam}(U)\leq
c_{\omega,{\mathfrak{f}}}\theta^{n}\implies\mu_{\mathfrak{f}}\bigl{(}\Upsilon(T_{j}^{{\mathfrak{f}},n})+U\bigr{)}\geq{\rm
const}\cdot{\mathcal{L}}^{d}(U)\cdot\theta^{-nd},$

with the constants depending only on $\omega$ and on ${\mathfrak{f}}$.

###### Proof.

It is proved in [35, Lemma 2.4] that for an aperiodic primitive self-similar
tiling ${\mathscr{T}}$, there exists $0<c<1$ such that if
${\mathcal{P}}\subset{\mathscr{T}}$ is a patch containing a ball of radius
$R>0$ in its support, then ${\mathcal{P}}+x\not\subset{\mathscr{T}}$ for all
$x$ with $\|x\|\leq cR$. Since the Delone set of the deformed tiling
${\mathscr{S}}^{\mathfrak{f}}=({\mathcal{T}}-v)^{\mathfrak{f}}$ is quasi-
isometric to $\Lambda({\mathcal{T}})=\Lambda({\mathscr{T}})$ by Lemma 3.10,
this property persists, with an appropriate constant, for
${\mathscr{S}}^{\mathfrak{f}}$. By (3.12), the patch
$T_{j}^{{\mathfrak{f}},n}$ contains a ball of radius $\asymp\theta^{n}$ in its
support. We are going to use Proposition 2.2. Represent the set $U$ as a union
of disjoint Borel sets $U_{k}$, each of diameter less than
$\eta({\mathscr{S}}^{\mathfrak{f}})=2r_{\mathfrak{f}}$, the maximal diameter
of a ball contained in the support of every deformed tile. It follows that if
${\rm diam}(U)\leq c_{\omega,{\mathfrak{f}}}\theta^{n}$, with a sufficiently
small constant, independent of $n$, then the sets
$\Upsilon(T_{j}^{{\mathfrak{f}},n})+U_{k}$ are mutually disjoint, and hence by
(2.3),

(3.17)
$\mu_{\mathfrak{f}}\Bigl{(}\Upsilon(T_{j}^{{\mathfrak{f}},n})+U\Bigr{)}=\sum_{k}\mu_{\mathfrak{f}}\Bigl{(}\Upsilon(T_{j}^{{\mathfrak{f}},n})+U_{k}\Bigr{)}=\sum_{k}{\mathcal{L}}^{d}(U_{k})\cdot{\rm
freq}(T_{j}^{{\mathfrak{f}},n})={\mathcal{L}}^{d}(U)\cdot{\rm
freq}(T_{j}^{{\mathfrak{f}},n}).$

By primitivity of the substitution and the Perron-Frobenius Theorem, the
frequency of $\omega^{n}(T_{j})$ in the self-similar tiling space is at least
${\rm const}\cdot\theta^{-nd}$ (in fact, an upper bound holds as well, using
[35, Lemma 2.4] again, but we do not need it), just by counting the number of
$n$-level super-tiles inside $(n+k)$-level super-tiles. By quasi-isometry, the
same asymptotic bound holds for the deformed tiling space, and (3.16) follows.
∎

##  4\. The spectral cocycle; statement of result on local dimension

Let $X_{\omega}$ be an FLC primitive aperiodic L-PSS tiling space. The Fourier
matrix is an $m\times m$ complex matrix-function on ${\mathbb{R}}^{d}$ whose
$(j,k)$-entry is

(4.1)
$[{\mathscr{B}}(\boldsymbol{\lambda})]_{(j,k)}:=\Bigl{[}\sum_{{\mathbf{x}}\in{\mathcal{D}}_{jk}}\exp\bigl{(}-2\pi
i\langle\boldsymbol{\lambda},{\mathbf{x}}\rangle\bigr{)}\Bigr{]},\ \
\boldsymbol{\lambda}\in{\mathbb{R}}^{d},\ \ 1\leq j,k\leq m,$

see [5, (29)]. First, we need to define a “deformed” Fourier matrix. Recall
that the set of integer linear combinations of elementary recurrence classes
is a subgroup $\Gamma<H_{1}(AP_{0}(X_{\omega}),{\mathbb{Z}})$, for which we
fixed a set of free generators $\\{a_{1},\ldots,a_{s}\\}$. The decomposition
with respect to this basis is given by $\alpha:\Gamma\to{\mathbb{Z}}^{s}$, the
address map. Recall (3.9) saying that the expansion induces a linear mapping
on ${\mathbb{Z}}^{s}$ given by a matrix $M$, so that

$\alpha\varphi=M\alpha.$

For a shape parameter ${\mathfrak{f}}$ (a representative of a class in
$H^{1}(AP_{0}(X_{\omega});{\mathbb{R}}^{d})$), we considered the row vector
${\bf L}_{\mathfrak{f}}=({\mathfrak{f}}(a_{1}),\ldots,{\mathfrak{f}}(a_{s})),$
see (3.8). We also discussed that elements of ${\mathcal{D}}_{jk}$ can be
identified with elementary recurrences, so
$\alpha({\mathbf{x}})\in{\mathbb{Z}}^{s}$ is well-defined for
${\mathbf{x}}\in{\mathcal{D}}_{jk}$. Now we define the deformed Fourier matrix
by

(4.2) $\Bigl{[}\sum_{{\mathbf{x}}\in{\mathcal{D}}_{jk}}\exp\bigl{(}-2\pi
i\langle\boldsymbol{\lambda},{\bf
L}_{\mathfrak{f}}\alpha({\mathbf{x}})\rangle\bigr{)}\Bigr{]}_{j,k\leq
m}=\Bigl{[}\sum_{{\mathbf{x}}\in{\mathcal{D}}_{jk}}\exp\bigl{(}-2\pi
i\langle{\bf L}^{\sf
T}_{\mathfrak{f}}\boldsymbol{\lambda},\alpha({\mathbf{x}})\rangle_{{}_{{\mathbb{R}}^{s}}}\bigr{)}\Bigr{]}_{j,k\leq
m},\ \ \boldsymbol{\lambda}\in{\mathbb{R}}^{d}.$

The spectral cocycle will be over the toral endomorphism

${\mathbf{z}}\mapsto M^{\sf T}{\mathbf{z}}\ \mbox{mod}\ {\mathbb{Z}}^{s},$

where the superscript ${\sf T}$ indicates the transpose. It will be an
endomorphism of the $s$-torus ${\mathbb{T}}^{s}$ if $\det(M)\neq 0$;
otherwise, we can restrict it to a lower-dimensional invariant sub-torus. Let

(4.3)
${\mathscr{M}}({\mathbf{z}}):=\Bigl{[}\sum_{{\mathbf{x}}\in{\mathcal{D}}_{jk}}\exp\bigl{(}-2\pi
i\langle{\mathbf{z}},\alpha({\mathbf{x}})\rangle_{{}_{{\mathbb{R}}^{s}}}\bigr{)}\Bigr{]}_{j,k\leq
m},\ \ {\mathbf{z}}\in{\mathbb{R}}^{s};$

this is closely related to the deformed Fourier matrix, which was defined as
${\mathscr{M}}({\bf L}_{\mathfrak{f}}^{\sf T}\boldsymbol{\lambda})$, for
$\boldsymbol{\lambda}\in{\mathbb{R}}^{d}$, in (4.2). Because of periodicity,
${\mathscr{M}}$ is well-defined on the torus
${\mathbb{T}}^{s}={\mathbb{R}}^{s}/{\mathbb{Z}}^{s}$.

######  Definition 4.1.

The matrix product

${\mathscr{M}}({\mathbf{z}},n):={\mathscr{M}}\bigl{(}{(M^{\sf
T})}^{n-1}{\mathbf{z}}\bigr{)}\cdot\ldots\cdot{\mathscr{M}}({\mathbf{z}}),\ \
{\mathbf{z}}\in{\mathbb{T}}^{s},\ n\in{\mathbb{N}},$

will be called the spectral cocycle associated with the L-PSS tiling space
$X_{\omega}$.

We will see below that for ${\mathbf{z}}={\bf L}_{\mathfrak{f}}^{\sf
T}\boldsymbol{\lambda}$, the growth behavior of
${\mathscr{M}}({\mathbf{z}},n)$ in some sense “controls” the local behavior of
spectral measures at $\boldsymbol{\lambda}$.

As a consistency check, consider the case of unperturbed self-similar tiling,
when ${\bf L}_{\mathfrak{f}}=[a_{1},\ldots,a_{s}]$. Then
${\mathbf{z}}=[a_{1},\ldots,a_{s}]^{\sf T}\boldsymbol{\lambda}$ and

$\langle{\mathbf{z}},\alpha({\mathbf{x}})\rangle=\langle\boldsymbol{\lambda},[a_{1},\ldots,a_{s}]\alpha({\mathbf{x}})\rangle=\langle\boldsymbol{\lambda},{\mathbf{x}}\rangle,$

by the definition of the address map $\alpha$. So we obtain
${\mathscr{M}}({\mathbf{z}})={\mathscr{B}}(\boldsymbol{\lambda})$ (the Fourier
matrix) from (4.3), (4.1). Further,

(4.4) $M^{\sf T}{\mathbf{z}}=M^{\sf T}[a_{1},\ldots,a_{s}]^{\sf
T}\boldsymbol{\lambda}=[a_{1},\ldots,a_{s}]^{\sf T}\varphi^{\sf
T}\boldsymbol{\lambda},$

by the definition of $M$, hence

${\mathscr{M}}({\mathbf{z}},n)=\prod_{j=0}^{n-1}{\mathscr{B}}\bigl{(}(\varphi^{\sf
T})^{j}\boldsymbol{\lambda}\bigr{)},$

which agrees with the Fourier matrix cocycle of Baake et al., see [5, 6].

Define the pointwise upper Lyapunov exponent of the cocycle
${\mathscr{M}}({\mathbf{z}},n)$ at the point ${\mathbf{z}}\in{\mathbb{R}}^{s}$
by

(4.5)
$\chi^{+}({\mathbf{z}}):=\limsup_{n\to\infty}\frac{1}{n}\log\|{\mathscr{M}}({\mathbf{z}},n)\|;$

we omit the superscript if the limit exists. In the “unperturbed,” self-
similar case, it becomes

$\chi^{+}(\boldsymbol{\lambda}):=\limsup_{n\to\infty}\frac{1}{n}\log\Bigl{\|}\prod_{j=0}^{n-1}{\mathscr{B}}\bigl{(}(\varphi^{\sf
T})^{j}\boldsymbol{\lambda}\bigr{)}\Bigr{\|}.$

We will also need a more refined version of the Lyapunov exponent: for
$\vec{\zeta}\in{\mathbb{C}}^{m}$, let

(4.6)
$\chi^{+}({\mathbf{z}},\vec{\zeta}):=\limsup_{n\to\infty}\frac{1}{n}\log\|{\mathscr{M}}({\mathbf{z}},n)\vec{\zeta}\|;$

Obviously, $\chi^{+}({\mathbf{z}},\vec{\zeta})\leq\chi^{+}({\mathbf{z}})$ for
all $\vec{\zeta}$.

Observe that ${\mathscr{M}}({\bf 0})={\sf S}_{\omega}^{\sf T}$, the transpose
substitution matrix of $\omega$, see (2.9). Its PF eigenvalue is
$\vartheta_{1}=|\det\varphi|=\theta^{d}$, where $\theta$ is the expansion
constant, by self-similarity. Thus $\chi({\bf 0})=d\log\theta$. Since every
entry of ${\mathscr{M}}({\mathbf{z}})$ in absolute value is less than or equal
to the corresponding entry of ${\mathscr{M}}({\bf 0})$, a non-negative
primitive matrix, we obtain that

$\chi^{+}({\mathbf{z}})\leq d\log\theta\ \ \mbox{for all}\
{\mathbf{z}}\in{\mathbb{R}}^{s}.$

The lower local dimension of a finite positive Borel measure $\nu$ at a point
$x$ is defined by

(4.7) ${d}^{-}(\nu,x)=\liminf_{r\to 0}\frac{\log\nu(B_{r}(x))}{\log r}.$

Equivalently,

(4.8) ${d}^{-}(\nu,x)=\sup\bigl{\\{}\gamma\geq 0:\ \nu(B_{r}(x))\leq
Cr^{\gamma},\ \mbox{for all}\ r>0,\ \mbox{for some}\ C>0\bigr{\\}}.$

Note that the definition is not changed if we only require the upper bound for
the measure of balls of sufficiently small radius, since the measure is
finite. We will say that a measure $\nu$ is Hölder regular on a set $F$ if

(4.9) $\exists\,\alpha>0\ \ \mbox{such that}\ \ {d}^{-}(\nu,x)\geq\alpha\ \
\mbox{for all}\ x\in F.$

Let ${\mathfrak{f}}$ be an elementary admissible deformation of $X_{\omega}$.
Recall that the deformation was well-defined only on a transversal of the
tiling space, namely, on the tilings having a vertex at the origin, whereas
the deformed tiling space $X_{\omega}^{\mathfrak{f}}$ is simply the
translation orbit closure of one representative deformed tiling. For
concreteness, we fixed a vertex $v\in{\mathcal{T}}$, an L-PSS tiling in
$X_{\omega}$, and define the deformed prototiles and super-prototiles for
$j\leq m$ by

$T_{j}^{\mathfrak{f}}:=(T_{j}-v)^{\mathfrak{f}}\in({\mathcal{T}}-v)^{\mathfrak{f}};\
\ \ \
T_{j}^{{\mathfrak{f}},n}=(\omega^{n}(T_{j})-v)^{\mathfrak{f}}\subset({\mathcal{T}}-v)^{\mathfrak{f}},\
\ j\leq m.$

see (3.5), (3.6).

For a tiling ${\mathscr{S}}^{\mathfrak{f}}\in X_{\omega}^{\mathfrak{f}}$ and
$j\leq m$, let ${\mathscr{L}}_{j}({\mathscr{S}}^{\mathfrak{f}})$ be the Delone
set of translation vectors between the prototiles $T_{j}^{\mathfrak{f}}$ and
the tiles in ${\mathscr{S}}^{\mathfrak{f}}$ equivalent to it, that is,

(4.10) ${\mathscr{S}}^{\mathfrak{f}}=\bigcup_{j\leq
m}\bigl{(}T_{j}^{\mathfrak{f}}+{\mathscr{L}}_{j}({\mathscr{S}}^{\mathfrak{f}})\bigr{)}.$

Now we are ready to state our main result. Assume for simplicity that the test
function $\phi$ is a TLC function of level 0 on $X^{\mathfrak{f}}_{\omega}$,
represented as

(4.11)
$\phi({\mathscr{S}}^{\mathfrak{f}})=\sum_{j=1}^{m}\sum_{x\in{\mathscr{L}}_{j}({\mathscr{S}}^{\mathfrak{f}})}\delta_{x}*\psi_{j}(0),$

where $\psi_{j}$ is an integrable function, with ${\rm supp}(\psi_{j})\subset
T_{j}^{\mathfrak{f}}$. General TLC functions may be represented in a similar
way, using higher-level supertiles.

######  Theorem 4.2.

Let ${\mathcal{T}}$ be an FLC primitive aperiodic L-PSS tiling and
$X_{\omega}$ the corresponding tiling space. Let $\Gamma$ be the
${\mathbb{Z}}$-module generated by elementary recurrences for ${\mathcal{T}}$
and $\\{a_{1},\ldots,a_{s}\\}$ a set of free generators for $\Gamma$. Let
${\mathfrak{f}}$ be a shape parameter defining an admissible elementary
deformation, ${\bf
L}_{\mathfrak{f}}=[{\mathfrak{f}}(a_{1}),\ldots,{\mathfrak{f}}(a_{s})]$, and
$X^{\mathfrak{f}}_{\omega}$ the deformed substitution tiling space. Consider
the measure-preserving system
$(X^{\mathfrak{f}}_{\omega},{\mathbb{R}}^{d},\mu_{\mathfrak{f}})$. For a TLC
function $\phi$ on $X^{\mathfrak{f}}_{\omega}$ of the form (4.11), let
$\sigma_{\phi}$ be the corresponding spectral measure on ${\mathbb{R}}^{d}$.
Then

(4.12) ${d}^{-}(\sigma_{\phi},\boldsymbol{\lambda})\geq
2\min\left\\{d-\frac{\chi^{+}({\mathbf{z}},\vec{\zeta})}{\log\theta},1\right\\},\
\ \boldsymbol{\lambda}\in{\mathbb{R}}^{d}\setminus\\{0\\},$

where ${\mathbf{z}}={\bf L}_{\mathfrak{f}}^{\sf T}\boldsymbol{\lambda}$ and
$\vec{\zeta}=[\widehat{\psi}_{1}(\boldsymbol{\lambda}),\ldots,\widehat{\psi}_{m}(\boldsymbol{\lambda})]^{\sf
T}$.

For $\boldsymbol{\lambda}=0$, suppose that $\phi$ is orthogonal to constants,
i.e., $\int_{X_{\omega}^{\mathfrak{f}}}\phi\,d\mu_{\mathfrak{f}}=0$. Then

(4.13) ${d}^{-}(\sigma_{\phi},0)\geq
2\min\left\\{d-\frac{\log|\vartheta_{2}|}{\log\theta},1\right\\},$

where $\vartheta_{2}$ is the second eigenvalue of ${\sf S}_{\omega}$ (the PF
eigenvalue being $\vartheta_{1}=\theta^{d}$).

######  Remark 4.3.

(a) This is an extension to higher dimensions of the lower bound in [10,
Theorem 4.6] (in the case of a single substitution); it is essentially
contained in [39], although it is not stated there in terms of the spectral
cocycle.

(b) The reason for the “switch” in the estimate at
$\chi^{+}({\mathbf{z}})=(d-1)\log\theta$ is due to “boundary effects,” as in
[7, 33, 39].

(c) For unperturbed tiling self-similar spaces, there are more precise, two-
sided bounds for the local dimension of the spectral measure at zero and even
asymptotic expansions; see [7] (for one-dimensional tilings) and [14] (for
$d>1$).

(d) Juan Marshall Maldonaldo [30, Theorem 3.10] proved that for a self-similar
tiling in ${\mathbb{R}}^{d}$, with an expansion diagonalizable over
${\mathbb{R}}$ that is strongly non-Pisot (i.e., has an eigenvalue outside the
unit circle), spectral measures are log-Hölder regular. This means that they
satisfy an estimate of the form $\sigma_{f}(B_{r}(\boldsymbol{\lambda}))\leq
C_{\boldsymbol{\lambda}}\log(1/r)^{\gamma}$, $r>0$, with $\gamma>0$
independent of $\boldsymbol{\lambda}\neq 0$. This is an extension of [8,
Theorem 5.1] which obtained the result for $d=1$.

(e) Using the Fourier matrix cocycle (4.4), Baake, Grimm, and Mañibo [5,
Theorem 5.7] proved that for an unperturbed self-similar tiling (even for
self-affine and even without the FLC assumption), if there exists
${\varepsilon}>0$ such that
$\chi^{+}(\boldsymbol{\lambda})\leq\frac{d}{2}\log\theta-{\varepsilon}$ for
Lebesgue-a.e. $\boldsymbol{\lambda}$, then the diffraction spectrum (which is
essentially a “part” of the dynamical spectrum) is purely singular. For $d=1$
this follows from [10, Cor. 4.7], but for $d\geq 2$ we cannot make such a
conclusion by our methods, essentially because
$(d-1)\log\theta\geq\frac{d}{2}\log\theta$.

(f) It would be interesting to also obtain upper bounds for the local
dimension of spectral measures, by analogy with [10]. We have some partial
results in this direction and hope to return to this question in the future.

##  5\. Eigenvalues, quantitative Host-Veech criterion, and Hölder regularity
of spectral measures

The material in this section is not particularly new, but included for
completeness, and we also emphasize the connections.

###  5.1. Eigenvalues

Recall that $\boldsymbol{\lambda}\in{\mathbb{R}}^{d}$ is a topological
eigenvalue for the tiling dynamical system
$(X^{\mathfrak{f}}_{\omega},{\mathbb{R}}^{d})$ if there exists a continuous
function $\phi:X^{\mathfrak{f}}_{\omega}\to{\mathbb{C}}$ such that

(5.1) $\phi({\mathscr{S}}^{\mathfrak{f}}-{\mathbf{z}})=\exp\bigl{(}-2\pi
i\langle\boldsymbol{\lambda},{\mathbf{z}}\rangle\bigr{)}\cdot\phi({\mathscr{S}}^{\mathfrak{f}})\
\ \mbox{for all}\ \ {\mathscr{S}}^{\mathfrak{f}}\in
X^{\mathfrak{f}}_{\omega},\ {\mathbf{z}}\in{\mathbb{R}}^{d}.$

######  Theorem 5.1 (variant of [13, Theorem 4.1]).

Let ${\mathcal{T}}$ be an FLC primitive aperiodic L-PSS tiling and
$X_{\omega}$ the corresponding tiling space. Let ${\mathfrak{f}}$ be a shape
parameter defining an elementary admissible deformation,
$X^{\mathfrak{f}}_{\omega}$ the deformed substitution tiling space, and ${\bf
L}_{\mathfrak{f}}$ given by (3.8). A vector
$\boldsymbol{\lambda}\in{\mathbb{R}}^{d}$ is in the topological point spectrum
of $(X^{\mathfrak{f}}_{\omega},{\mathbb{R}}^{d})$ if and only if for every
elementary recurrence vector ${\bf v}$ of ${\mathcal{T}}$,

(5.2) $\langle\boldsymbol{\lambda},{\bf L}_{\mathfrak{f}}M^{n}{\bf
v}\rangle\to 0\ \mbox{(mod 1)},\ \ n\to\infty,$

where the convergence is exponentially fast and uniform in ${\bf v}$. Here $M$
is the matrix from (3.9).

In fact, our setting is more general, since [13] assumed a pure dilation
expansion map, whereas we allow an expansion which is a general similitude.
However, the proof transfers almost verbatim. Furthermore, in [13, Theorem
4.1] only uniform in ${\bf v}$ convergence is claimed, but the exponential
rate follows from the proof immediately.

Next we show that the same condition characterizes measurable eigenvalues for
the uniquely ergodic system
$(X^{\mathfrak{f}}_{\omega},{\mathbb{R}}^{d},\mu)$. Thus, for admissible
deformations of pseudo-self-similar tiling spaces weak mixing is equivalent to
topological weak mixing. These results follow a long line of earlier work,
starting with Host [20], who obtained a similar criterion for the eigenvalues
of (symbolic) substitution ${\mathbb{Z}}$-actions, in terms of return words,
and also proved that every measure-theoretic eigenvalue is topological. For
interval exchange transformations and translation flows an analogous criterion
was obtained by Veech [40].

######  Proposition 5.2.

Under the assumptions of Theorem 5.1, a vector
$\boldsymbol{\lambda}\in{\mathbb{R}}^{d}$ is in the topological point spectrum
of $(X^{\mathfrak{f}}_{\omega},{\mathbb{R}}^{d})$ if and only if it is in the
point spectrum of the uniquely ergodic measure-preserving system
$(X^{\mathfrak{f}}_{\omega},{\mathbb{R}}^{d},\mu_{\mathfrak{f}})$.

The proof is a minor variation of the argument in [34]; we sketch it for
completeness in Section 7. Note that

(5.3) $\langle\boldsymbol{\lambda},{\bf L}_{\mathfrak{f}}M^{n}{\bf
v}\rangle=\langle{\bf L}^{\sf T}_{\mathfrak{f}}\boldsymbol{\lambda},M^{n}{\bf
v}\rangle_{{}_{{\mathbb{R}}^{s}}}=\bigl{\langle}{(M^{\sf T})}^{n}({\bf L}^{\sf
T}_{\mathfrak{f}}\boldsymbol{\lambda}),{\bf
v}\bigr{\rangle}_{{}_{{\mathbb{R}}^{s}}},$

where $\langle\cdot,\cdot\rangle$ is the inner product in ${\mathbb{R}}^{d}$,
as opposed to the inner product in ${\mathbb{R}}^{s}$ in the right-hand side.

######  Lemma 5.3.

There exists a uniform $c_{1}>0$ with the following property. Let ${\bf v}$ be
a recurrence vector for ${\mathcal{T}}$, such that for all $j\leq m$ there
exists $k$ satisfying

(5.4) $\exists\,z_{1},z_{2}\in{\mathcal{D}}_{jk}\ \ \mbox{such that}\ \ {\bf
v}=\alpha(z_{1},z_{2}).$

Then

(5.5) $\|{\mathscr{M}}({\bf L}^{\sf
T}_{\mathfrak{f}}\boldsymbol{\lambda},n)\|\leq{\rm
const}\cdot\theta^{nd}\prod_{i=0}^{n-1}\left(1-c_{1}\bigl{\|}\bigl{\langle}{(M^{\sf
T})}^{i}({\bf L}^{\sf T}_{\mathfrak{f}}\boldsymbol{\lambda}),{\bf
v}\bigr{\rangle}_{{}_{{\mathbb{R}}^{s}}}\bigr{\|}_{{}_{{\mathbb{R}}/{\mathbb{Z}}}}^{2}\right),\
\ n\geq 1.$

For the proof, see [39]; it is a generalization of [8] to higher rank actions.
∎

In view of primitivity and repetitivity, for any given recurrence vector ${\bf
v}$, the property (5.4) holds if we replace $\omega$ by a sufficiently hight
power. Passing from $\omega$ to $\omega^{k}$ for some $k\in{\mathbb{N}}$, we
can assume without loss of generality that (5.4) holds for a set of recurrence
vectors ${\bf v}$ generating ${\mathbb{Z}}^{s}$.

The following lemma is elementary, see e.g., [11, Lemma 5.1].

######  Lemma 5.4.

Let ${\mathbb{Z}}^{s}={\mathbb{Z}}[{\bf v}_{1},\ldots,{\bf v}_{\ell}]$ for
some $\ell\geq s$. Then we have for ${\mathbf{x}}\in{\mathbb{R}}^{s}$:

$\max_{j\leq\ell}\|\langle{\mathbf{x}},{\bf
v}_{j}\rangle\|_{{\mathbb{R}}/{\mathbb{Z}}}\asymp\|{\mathbf{x}}\|_{{\mathbb{R}}^{s}/{\mathbb{Z}}^{s}},$

with implied constants depending only on the generating set ${\bf
v}_{1},\ldots,{\bf v}_{\ell}$.

######  Corollary 5.5.

Under the assumptions of Theorem 5.1, there exists
$k=k(\omega)\in{\mathbb{N}}$ such that

(i) $\boldsymbol{\lambda}\in{\mathbb{R}}^{d}$ is in the discrete spectrum of
the system $(X^{\mathfrak{f}}_{\omega},{\mathbb{R}}^{d})$ (in the topological
or measurable category) if and only if

$\lim_{i\to\infty}{\bigl{\|}(M^{\sf T})^{ki}({\bf L}^{\sf
T}_{\mathfrak{f}}\boldsymbol{\lambda})\bigr{\|}}_{{\mathbb{R}}^{s}/Z^{s}}=0,$

and the convergence is exponential.

(ii)

$\|{\mathscr{M}}({\bf L}^{\sf
T}_{\mathfrak{f}}\boldsymbol{\lambda},n)\|\leq{\rm
const}\cdot\theta^{nd}\prod_{i=0}^{n-1}\left(1-\widetilde{c}_{1}{\bigl{\|}(M^{\sf
T})^{ki}({\bf L}^{\sf
T}_{\mathfrak{f}}\boldsymbol{\lambda})\bigr{\|}}^{2}_{{\mathbb{R}}^{s}/{\mathbb{Z}}^{s}}\right),\
\ n\geq 1.$

###### Proof.

(i) follows from Theorem 5.1 and Lemma 5.4.

(ii) is immediate from (5.4) and Lemmas 5.3 and 5.4. ∎

Thus, weak mixing of the tiling dynamical system is equivalent to

$\forall\ \boldsymbol{\lambda}\in{\mathbb{R}}^{d}\setminus\\{0\\},\ \|(M^{\sf
T})^{ki}({\bf L}^{\sf
T}_{\mathfrak{f}}\boldsymbol{\lambda})\|_{{\mathbb{R}}^{s}/{{\mathbb{Z}}^{s}}}\not\to
0,\ \ i\to\infty,\ \ \mbox{for}\ k=k(\omega).$

(this is analogous to the Veech criterion [40]). On the other hand, if
$\|(M^{\sf T})^{ki}({\bf L}^{\sf
T}_{\mathfrak{f}}\boldsymbol{\lambda})\|_{{\mathbb{R}}^{s}/{{\mathbb{Z}}^{s}}}\not\to
0$ in some “quantitative way” (say, with a positive frequency the distance is
greater than $\delta$) for all
$\boldsymbol{\lambda}\in{\mathbb{R}}^{d}\setminus\\{0\\}$, then we get
quantitative weak mixing.

###  5.2. Hölder regularity

######  Theorem 5.6.

Let ${\mathcal{T}}$ be an FLC primitive aperiodic L-PSS tiling and
$X_{\omega}$ the corresponding tiling space. Let $\Gamma$ be the
${\mathbb{Z}}$-module generated by elementary recurrences for ${\mathcal{T}}$
and $M$ be the integer matrix of the induced action of the expansion map on
$\Gamma$. Let ${\mathcal{M}}$ be an open set of elementary admissible
deformations. If the dimension of the (strictly) expanding subspace of $M$ is
at least $d+1$, then for Lebesgue-a.e. $[{\mathfrak{f}}]\in{\mathcal{M}}$,
there exist a representative ${\mathfrak{f}}$ of $[{\mathfrak{f}}]$ such that
the deformed ${\mathbb{R}}^{d}$ action on $X_{\omega}^{\mathfrak{f}}$ has
uniformly Hölder-regular spectral measures. More precisely, there exists
$\alpha>0$, depending only on the tiling space and the substitution, such that
for a.e. $[{\mathfrak{f}}]\in{\mathcal{M}}$ with admissible representative
${\mathfrak{f}}$ and every transversally locally constant function $\phi$ on
$X_{\omega}^{\mathfrak{f}}$, the spectral measure $\sigma_{\phi}$, associated
with the uniquely ergodic dynamical system
$(X_{\omega}^{\mathfrak{f}},{\mathbb{R}}^{d},\mu_{\mathfrak{f}})$, satisfies
$d^{-}(\sigma_{\phi},\boldsymbol{\lambda})\geq\alpha$ for all
$\boldsymbol{\lambda}\in{\mathbb{R}}^{d}\setminus\\{0\\}$.

This is an extension of [39, Cor. 1] to the case of pseudo self-similar
tilings with a general (not necessarily pure dilation) expansion map.
Analogously to [39, Th. 2], one can also deduce uniform rates of weak mixing
for functions with sufficient regularity in the leaf direction, as well as
bounds on integrals of correlations.

In many cases Theorem 5.6 may be applied based only on the knowledge of the
algebraic properties of the expansion $\varphi$. It is well-known that all the
eigenvalues of $\varphi$ must be (real or complex) algebraic integers, see
e.g., [28, Cor. 4.2]. A family of algebraic integers
$\Theta=\\{\theta_{1},\ldots,\theta_{d}\\}$, all of absolute value greater
than one, is called a Pisot family if every Galois conjugate of every
$\theta_{j}\in\Theta$ is either another element of $\Theta$ or lies inside the
unit circle. For $d=1$ this is the definition of a (real) Pisot number, and
for $d=2$, with $\theta_{1},\theta_{2}$ complex conjugates, this is the
definition of a complex Pisot number. Under very general conditions, if the
set of eigenvalues of $\varphi$ is a Pisot family, then the tiling dynamical
system is not weakly mixing, see [29], and sometimes even pure discrete. We
will say that $\Theta$ is strongly non-Pisot if there exists
$\theta_{j}\in\Theta$ such that at least one of its Galois conjugates has
absolute value strictly greater than one.

######  Corollary 5.7.

Under the assumptions of Theorem 5.6, suppose that the set of eigenvalues of
$\varphi$ is strongly non-Pisot. Then for Lebesgue-a.e. admissible deformation
$[{\mathfrak{f}}]\in{\mathcal{M}}$ the deformed ${\mathbb{R}}^{d}$ action on
$X^{\mathfrak{f}}$ has uniformly Hölder-regular spectral measures.

###### Derivation of Corollary 5.7 assuming Theorem 5.6.

Let $\Gamma$ be the ${\mathbb{Z}}$-module generated by elementary recurrences
for ${\mathcal{T}}$. The expansion $\varphi$ induces an endomorphism of
$\Gamma$ given by the integer matrix $M$. It follows that all the eigenvalues
of $\varphi$ are algebraic integers, and moreover, every eigenvalue of
$\varphi$ is also an eigenvalue of $M$, see e.g., [37, Lemma 1.4.5]. But $M$
is an integer matrix, hence every Galois conjugate of its every eigenvalue is
also an eigenvalue of $M$. By the strongly non-Pisot assumption, the dimension
of the expanding subpace for $M$ is at least $d+1$, and the claim follows from
Theorem 5.6. ∎

To conclude this section, we want to show that the above results can be
extended to general PSS tilings, not necessarily the “special” L-PSS ones, and
we summarize now how this is done.

Let $\mathcal{T}$ be a PSS tiling and $X$ its tiling space. By Remark 2.8 (c),
there is a tiling space $X^{\prime}$ and a tiling $\mathcal{T}^{\prime}\in
X^{\prime}$ which is L-PSS and MLD equivalent to $\mathcal{T}$. Denote by
$\Phi:X\rightarrow X^{\prime}$ the homeomorphism of tiling spaces defined by
the MLD equivalence. Let $\mathcal{M}^{\prime}:=\mathcal{M}(X^{\prime})\subset
H^{1}(X^{\prime};\mathbb{R}^{d})$ be the set of non-singular classes of
deformations, as defined in §3.3, and
$\mathcal{M}:=\Phi^{*}\mathcal{M}^{\prime}\subset H^{1}(X;\mathbb{R}^{d})$ its
pullback under the MLD map.

Consider a class $[\mathfrak{f}]^{\prime}\in\mathcal{M}^{\prime}\subset
H^{1}(X^{\prime};\mathbb{R}^{d})$ with representative $\mathfrak{f}$, a
$\mathbb{R}^{d}$-valued, $\mathcal{T}^{\prime}$-equivariant, admissible smooth
1-form. First, we claim that $\mathfrak{f}$ is also $\mathcal{T}$-equivariant.
Indeed, the value of $\mathfrak{f}(x)$ depends on
$\mathcal{O}^{+}_{\mathcal{T}^{\prime}}(B_{R^{\prime}}(x))$ for some
$R^{\prime}>0$. Moreover, by MLD equivalence, the patch
$\mathcal{O}^{+}_{\mathcal{T}^{\prime}}(B_{R^{\prime}}(x))$ is determined by
the patch $\mathcal{O}^{+}_{\mathcal{T}}(B_{R}(x))$ for some $R>0$. Thus
whenever
$\mathcal{O}^{+}_{\mathcal{T}}(B_{R}(x))=\mathcal{O}^{+}_{\mathcal{T}}(B_{R}(y))+x-y$
we have that $\mathfrak{f}(x)=\mathfrak{f}(y)$, that is, $\mathfrak{f}$ is
$\mathcal{T}$-equivariant. Thus $\mathfrak{f}$ has a class in
$H^{1}(X;\mathbb{R}^{d})$, which is denoted by $[\mathfrak{f}]$, and it is
indeed the image of $[\mathfrak{f}]^{\prime}$ under the pullback map:
$[\mathfrak{f}]=\Phi^{*}[\mathfrak{f}]^{\prime}$.

Since $[\mathfrak{f}]^{\prime}\in\mathcal{M}^{\prime}$, by the results of
§3.3, the representative $\mathfrak{f}$ defines a homeomorphism
$H_{\mathfrak{f}}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ through
$H_{\mathfrak{f}}(x)=\int_{0}^{x}\mathfrak{f}$, which satisfies
$H_{\mathfrak{f}}(\mathcal{T}-t)=\mathcal{T}-H_{\mathfrak{f}}(t)$. Let
$\mathcal{T}^{\mathfrak{f}^{\prime}}:=H_{\mathfrak{f}}(\mathcal{T}^{\prime})$
and $\mathcal{T}^{\mathfrak{f}}:=H_{\mathfrak{f}}(\mathcal{T})$ be the
deformed tilings, which are both FLC and repetitive. As such, they have well-
defined tiling spaces $X^{\mathfrak{f}^{\prime}}$ and $X^{\mathfrak{f}}$,
respectively. Consider the map
$\Phi_{\mathfrak{f}}:=H_{\mathfrak{f}}\circ\Phi\circ H_{\mathfrak{f}}^{-1}$
sending $\mathcal{T}^{\mathfrak{f}}$ to $\mathcal{T}^{\mathfrak{f}^{\prime}}$.
This map extends by minimality to all of $X^{\mathfrak{f}}$, and gives a map
$\Phi_{\mathfrak{f}}:X^{\mathfrak{f}}\rightarrow X^{\mathfrak{f}^{\prime}}$.
This map is an MLD equivalence: indeed, since $\mathfrak{f}$ is $C^{\infty}$
and bounded, the distorsion of the map $H_{\mathfrak{f}}$ (and that of its
inverse) is bounded over sets with diameter less than some finite fixed
diameter. As such, the tile(s) covering $x\in\mathbb{R}^{d}$ in
$\mathcal{T}^{\mathfrak{f}^{\prime}}$ is determined by the patch
$\mathcal{O}^{-}_{\mathcal{T}^{\mathfrak{f}}}(B_{R}(x))$ for some $R$ large
enough.

Finally, note that a function $\phi:X^{\mathfrak{f}}\rightarrow\mathbb{R}$ is
TLC if and only if $\phi=\Phi_{\mathfrak{f}}^{*}\phi^{\prime}$ for some TLC
function $\phi^{\prime}:X^{\mathfrak{f}^{\prime}}\rightarrow\mathbb{R}$. As
such, we have that
$S_{R}^{y}(\phi,\boldsymbol{\lambda})=S_{R}^{\Phi_{\mathfrak{f}}(y)}(\phi^{\prime},\boldsymbol{\lambda})$
for any $\boldsymbol{\lambda}\in\mathbb{R}^{d}$, where
$S_{R}^{y}(\phi,\boldsymbol{\lambda})$ is the twisted ergodic integral of
$\phi$, introduced in §7.1, and used in all proofs of bounds of lower local
dimension. As such, bounds on $S_{R}^{y}(\phi,\boldsymbol{\lambda})$ are
equivalent to bounds on
$S_{R}^{\Phi_{\mathfrak{f}}(y)}(\phi^{\prime},\boldsymbol{\lambda})$ (see
Lemma 7.1). Thus we have proved the following.

######  Corollary 5.8.

Theorem 5.6 holds with the weaker assumption of $\mathcal{T}$ being pseudo
self-similar, not necessarily L-PSS. Moreover, if the set of eigenvalues of
the inflation of a PSS tiling is strongly non-Pisot, then for Lebesgue-a.e.
class $[{\mathfrak{f}}]\in{\mathcal{M}}$, there exists a representative
${\mathfrak{f}}$ such that the deformed ${\mathbb{R}}^{d}$ action on
$X^{\mathfrak{f}}$ has uniformly Hölder-regular spectral measures.

###  5.3. Quantitative Veech Criterion

The proof of Theorem 5.6 is based on the following proposition, where we
assume that $k=1$ without loss of generality. This type of result goes by the
name of Quantitative Veech Criterion.

######  Proposition 5.9.

Let $[\mathfrak{f}]\in\mathcal{M}$ and $\mathfrak{f}$ an admissible
representative. If there exist $\rho,\delta\in(0,\frac{1}{2})$ such that

(5.6)
$\limsup_{N\rightarrow\infty}\frac{1}{n}\left|\left\\{n\in\mathbb{N}\cap(0,N):{\bigl{\|}(M^{\sf
T})^{n}({\bf L}^{\sf
T}_{\mathfrak{f}}\boldsymbol{\lambda})\bigr{\|}}_{{\mathbb{R}}^{s}/{\mathbb{Z}}^{s}}<\rho\right\\}\right|<1-\delta$

for $\boldsymbol{\lambda}\in{\mathbb{R}}^{d}\setminus\\{0\\}$, then there
exists $\alpha>0$, depending only on $\rho$ and $\delta$, such that for any
transversally locally constant function
$\phi:X^{\mathfrak{f}}\rightarrow\mathbb{R}$ holds
$d^{-}(\sigma_{\phi},\boldsymbol{\lambda})\geq\alpha$.

###### Proof.

This follows from Lemma 5.3 and (4.12). For details, see [11, §5] and [39,
Prop. 4]. ∎

Recall that when we deform tilings and their spaces we do so through
admissible representatives ${\mathfrak{f}}$ of classes in ${\mathcal{M}}$. If
we pick a basis ${\bf a}=\\{a_{1},\ldots,a_{s}\\}$ of the
${\mathbb{Z}}$-module $\Gamma$, then we obtain the shape vector ${\bf
L}_{\mathfrak{f}}$ be evaluating ${\mathfrak{f}}$ on each element $a_{i}$ and
obtaining the vector ${\mathfrak{f}}(a_{i})\in\mathbb{R}^{d}$. The action of
${\mathfrak{f}}$ on this basis is by definition independent of representative
${\mathfrak{f}}$ of its class $[{\mathfrak{f}}]$ used and is given by the
$d\times s$ matrix ${\bf L}_{\mathfrak{f}}$. Therefore the shape vector is
assosciated to a class and not representative. Below we write
${\mathcal{M}}_{\bf a}$ for an open set of $d\times s$ matrices which are
sufficiently small perturbations of
$[{\mathfrak{f}}(a_{1}),\ldots,{\mathfrak{f}}(a_{s})]$. In particular, all
matrices in ${\mathcal{M}}_{\bf a}$ are uniformly bounded and have maximal
rank $d$; more precisely, there are $d$ columns such that the determinant of
the corresponding $d\times d$ matrix is bounded away from zero in modulus by a
uniform $c>0$.

###### Proof sketch of Theorem 5.6.

This is a brief sketch; for details the reader should consult [11] and [39].
Denote

${\varepsilon}_{n}({\mathbf{z}}):={\|(M^{\sf
T})^{n}{\mathbf{z}}\|}_{{\mathbb{R}}^{s}/{\mathbb{Z}}^{s}},\ \
{\mathbf{z}}\in{\mathbb{R}}^{s}.$

The exceptional set is defined as follows: for $B>1$ and $N\in{\mathbb{N}}$
let

(5.7)
$\begin{split}E_{N}(\rho,\delta,B)&=\left\\{{\mathbf{z}}\in{\mathbb{R}}^{s}:\
B^{-1}\leq\|{\mathbf{z}}\|\leq B,\ \ \bigl{|}\\{n\leq
N:{\varepsilon}_{n}({\mathbf{z}})\geq\rho\\}\bigr{|}<\delta N\right\\},\\\
{\mathcal{E}}_{N}(\rho,\delta,B)&=\left\\{{\mathfrak{f}}\in{{\mathcal{M}}_{\bf
a}}:\ \exists\,\boldsymbol{\lambda}\in{\mathbb{R}}^{d},\ \
L_{\mathfrak{f}}^{\sf T}\boldsymbol{\lambda}\in
E_{N}(\rho,\delta,B)\right\\},\end{split}$

and

${\mathfrak{E}}(\rho,\delta,B):=\bigcap_{N_{0}=1}^{\infty}\bigcup_{N=N_{0}}^{\infty}{\mathcal{E}}_{N}(\rho,\delta,B).$

The theorem easily follows from Proposition 5.9, combined with the next lemma,
in view of the fact that ${\mathcal{M}}_{\bf a}$ is an open subset of
${\mathbb{R}}^{sd}$. ∎

######  Lemma 5.10.

Let $E^{+}$ be the (strictly) expanding subspace for the linear map $M^{\sf
T}$ on ${\mathbb{R}}^{s}$. There is $\rho>0$ such that given any
${\varepsilon}_{1}>0$, for every $\delta>0$ sufficiently small, for all $B>1$,

(5.8) $\dim_{H}({\mathfrak{E}}(\rho,\delta,B))\leq\gamma:=sd-\dim
E^{+}+d+{\varepsilon}_{1}.$

###### Proof.

The proof is a version of the “Erdős-Kahane argument” (or a kind of
quantitative “linear exclusion” in the spirit of [1, Section 7]). We provide a
somewhat detailed sketch, since the proofs in [11, §5] and [39, Prop. 4] are
rather technical, and our situation here is simpler: we apply a fixed
transformation $M^{\sf T}$ instead of a random sequence, so there is no need
for Oseledets Theorem. Let

(5.9) $(M^{\sf
T})^{n}{\mathbf{z}}={\mathbf{K}}_{n}({\mathbf{z}})+{\boldsymbol{\epsilon}}_{n}({\mathbf{z}}),\
\ n\geq 0,$

where ${\mathbf{K}}_{n}({\mathbf{z}})\in{\mathbb{Z}}^{s}$ is (a) nearest
lattice point to $(M^{\sf T})^{n}{\mathbf{z}}$. It will be convenient to use
the $\ell^{\infty}$ norm, so that
$\|{\boldsymbol{\epsilon}}_{n}({\mathbf{z}})\|_{\infty}={\varepsilon}_{n}({\mathbf{z}})\leq
1/2$. It follows from (5.9) that

${\mathbf{K}}_{n+1}({\mathbf{z}})+{\boldsymbol{\epsilon}}_{n+1}({\mathbf{z}})=M^{\sf
T}{\mathbf{K}}_{n}({\mathbf{z}})+M^{\sf
T}{\boldsymbol{\epsilon}}_{n}({\mathbf{z}}),\ \ n\geq 0,$

hence

(5.10) $\|{\mathbf{K}}_{n+1}({\mathbf{z}})-M^{\sf
T}{\mathbf{K}}_{n}({\mathbf{z}})\|_{\infty}=\|M^{\sf
T}{\boldsymbol{\epsilon}}_{n}({\mathbf{z}})-{\boldsymbol{\epsilon}}_{n+1}({\mathbf{z}})\|_{\infty},\
\ n\geq 0.$

Using that $M^{\sf T}{\mathbf{K}}_{n}({\mathbf{z}})\in{\mathbb{Z}}^{s}$, we
immediately obtain the following:

######  Lemma 5.11.

For all $n\geq 0$, independent of ${\mathbf{z}}\in{\mathbb{R}}^{s}$, holds:

(i) Given ${\mathbf{K}}_{n}({\mathbf{z}})$, there are at most $(\|M^{\sf
T}\|_{\infty}+2)^{d}$ possibilities for ${\mathbf{K}}_{n+1}({\mathbf{z}})$;

(ii) Let

(5.11) $\rho:=2(\|M^{\sf T}\|_{\infty}+1)^{-1}.$

If
$\max\\{{\varepsilon}_{n}({\mathbf{z}}),{\varepsilon}_{n+1}({\mathbf{z}})\\}<\rho$,
then ${\mathbf{K}}_{n+1}({\mathbf{z}})=M^{\sf
T}{\mathbf{K}}_{n}({\mathbf{z}})$.

We will use $\rho$ from (5.11) in Lemma 5.10. Lemma 5.11 implies that if
$\delta>0$ is very small, then there is a good upper bound on the set of
possible finite sequences
$\\{{\mathbf{K}}_{0}({\mathbf{z}}),\ldots,{\mathbf{K}}_{N}({\mathbf{z}})\\}$
for ${\mathbf{z}}\in E_{N}(\rho,\delta,B)$. We will next show that this yields
an efficient covering of $E_{N}(\rho,\delta,B)$.

Denote by $E^{-}$ the contracting subspace for $M^{\sf T}$ on
${\mathbb{R}}^{s}$, complementary to $E^{+}$. Let $M^{\sf T}_{+}$ be the
restriction of $M^{\sf T}$ to $E^{+}$. Let $P_{+}$ and $P_{-}$ be the
projections onto $E^{+},E^{-}$ respectively, commuting with $M^{\sf T}$. From
(5.9) we obtain, using that $P_{+}$ commutes with $M^{\sf T}$:

$P_{+}{\mathbf{z}}=(M_{+}^{\sf
T})^{-n}P_{+}({\mathbf{K}}_{n}({\mathbf{z}})+{\boldsymbol{\epsilon}}_{n}({\mathbf{z}})).$

Let $\kappa=\dim E^{+}$ and let $\theta_{\kappa}$ be the minimal eigenvalue of
$M$ greater than 1. Fix any $r\in(1,\theta_{\kappa})$. Then

$\|(M_{+}^{\sf T})^{-n}P_{+}\|\leq Cr^{-n},\ \ n\geq 0,$

and hence

$\|P_{+}{\mathbf{z}}-(M_{+}^{\sf
T})^{-n}P_{+}{\mathbf{K}}_{n}({\mathbf{z}})\|\leq Cr^{-n},\ \ n\geq 0,$

where $C$ depends only on $M$. Thus, the knowledge of
${\mathbf{K}}_{n}({\mathbf{z}})$ yields an approximation of order $\sim
r^{-n}$ for ${\mathbb{P}}\,_{\mathbf{z}}$. Now, using the combinatorial
counting argument from [11, §5], we can conclude that the set
$\\{P_{+}{\mathbf{z}}:{\mathbf{z}}\in E_{N}(\rho,\delta,B)\\}$ may be covered
by $O_{B,M}(1)\cdot\exp[L(\frac{1}{\delta}\log\delta)N]$ balls of radius
$r^{-N}$, where $L$ is a uniform constant. Choosing $\delta>0$ sufficiently
small guarantees that this number is at most $O_{B,M}(1)\cdot
r^{N{\varepsilon}_{1}}$. This yields a covering of $E_{N}(\rho,\delta,B)$ by
$O_{B,M}(1)\cdot r^{N(s-\kappa+{\varepsilon}_{1})}$ balls of radius $Cr^{-N}$,
since $s-\kappa=\dim E^{-}$ and there are a priori bounds $\|z\|\leq B$.

Finally, we need to pass from the covering of
$E_{N}(\rho,\delta,B)\subset{\mathbb{R}}^{s}$ to a covering of
${\mathcal{E}}_{N}(\rho,\delta,B)\subset{\mathcal{M}}_{\bf
a}\subset{\mathbb{R}}^{ds}$. The latter is the set of matrices ${\bf
L}_{\mathfrak{f}}$ such that ${\bf L}_{\mathfrak{f}}^{\sf
T}\boldsymbol{\lambda}={\mathbf{z}}\in E_{N}(\rho,\delta,B)$. The assumptions
on ${\mathcal{M}}_{\bf a}$ guarantees that $\boldsymbol{\lambda}$ is bounded
away from 0 and $\infty$ in norm.

For ${\mathbf{z}}\in{\mathbb{R}}^{s}$ the set of ${\bf L}_{\mathfrak{f}}$ such
that ${\bf L}_{\mathfrak{f}}^{\sf T}\boldsymbol{\lambda}={\mathbf{z}}$, with
${\mathbf{z}}$ fixed, can be parameterized (at least, locally) as follows.
Choose $d$ rows of ${\bf L}_{\mathfrak{f}}$ with a determinant bounded away
from zero in modulus. Solving the $d\times d$ system yields a unique solution
$\boldsymbol{\lambda}$. Then the remaining $s-d$ rows in the linear system
define a co-dimension 1 hyperplane each as the set of possibilities for the
rows of ${\bf L}_{\mathfrak{f}}$, resulting in a set of total dimension
$d\times d+(s-d)\times(d-1)=sd-s+d$. Hence the set
${\mathcal{E}}_{N}(\rho,\delta,B)$ may be covered by

$O_{B,M}(1)\cdot r^{N(sd+d-\kappa+{\varepsilon}_{1})}=O_{B,M}(1)\cdot
r^{\gamma}$

balls of radius $Cr^{-N}$. Now (5.8) follows by a standard estimate of the
Hausdorff dimension of limsup sets. ∎

######  Remark 5.12.

The last proof highlights the fact that the main “play” in the proof of Hölder
regularity occurs in the strictly expanding subspace $E^{+}$ for $M^{\sf T}$.
One may ask what is the role of the contracting subspace. The following result
of Clark and Sadun [13] shows that by perturbing the deformation in the
contracting direction we obtain a topologically conjugate system, hence all
the spectral properties remain unchanged.

######  Proposition 5.13 (corollary of [13, Theorems 2.2 and 3.1]).

Let ${\mathcal{T}}$ be an FLC primitive aperiodic L-PSS tiling and
$X_{\omega}$ the corresponding tiling space. For two classes
$[{\mathfrak{f}}],[{\mathfrak{g}}]\in\mathcal{M}\subset
H^{1}(X_{\omega},{\mathbb{R}}^{d})$ consider the deformed tiling spaces
$X_{\omega}^{\mathfrak{f}}$ and $X_{\omega}^{\mathfrak{g}}$, where
${\mathfrak{f}},{\mathfrak{g}}$ are admissible representatives. Let $E^{-}$ be
the contracting subspace for $M^{\sf T}$, i.e., the span of (generalized)
eigenvectors corresponding to eigenvalues less than one in modulus. If there
exists $k\in{\mathbb{N}}$ such that $(M^{\sf T})^{k}({\bf
L}_{\mathfrak{f}})-{\bf L}_{\mathfrak{g}}\in E^{-}$ or $(M^{\sf T})^{k}({\bf
L}_{\mathfrak{g}})-{\bf L}_{\mathfrak{f}}\in E^{-}$, then
$(X^{\mathfrak{f}}_{\omega},{\mathbb{R}}^{d})$ and
$(X^{\mathfrak{g}}_{\omega},{\mathbb{R}}^{d})$ are topologically conjugate.

Although [13] assumed a pure dilation expansion map, their proof extends
verbatim.

###  5.4. One-dimensional substitution tilings revisited

For one-dimensional substitution tilings, there is a space of “natural”
deformations, obtained simply by changing the tile sizes. Similarly to [13,
Section 5], we can extend [8, Theorem 4.1]. First, we need to introduce some
notation. Let $\zeta$ be a primitive aperiodic substitution on $d$ symbols,
$(X_{\zeta},T,\mu)$ the (2-sided) uniquely ergodic tiling
${\mathbb{Z}}$-action, and
$({\mathfrak{X}}_{\zeta}^{\vec{s}},h_{t},\widetilde{\mu})$ the suspension flow
corresponding to a “roof vector” $\vec{s}\in{\mathbb{R}}^{d}_{+}$. A word $v$
is called a return vector for $\zeta$ if $vc$ occurs in $x\in X_{\zeta}$ where
$c$ is the 1st letter of $v$, that is, $v$ separates the two consecutive
occurrences of $c$. Denote by $\vec{\ell}(v)$ the “population vector” of a
word $v$. Let $\Gamma_{\zeta}<{\mathbb{Z}}^{d}$ be the ${\mathbb{Z}}$-module
generated by $\\{\vec{\ell}(v):\ v$ is a return vector for $\zeta\\}$, and the
“essential subspace” $V_{\zeta}:=\Gamma_{\zeta}\otimes{\mathbb{R}}$ the real
linear span of $\Gamma_{\zeta}$. First we recall the earlier result:

######  Theorem 5.14 ([8, Th. 4.1]).

Let $\zeta$ be a primitive aperiodic substitution on $d$ symbols with a
substitution matrix ${\sf S}_{\zeta}$. Suppose that the characteristic
polynomial of ${\sf S}_{\zeta}$ is irreducible over ${\mathbb{Q}}$ and the
second eigenvalue satisfies $|\theta_{2}|>1$. Then for Lebesgue-a.e.
$\vec{s}\in{\mathbb{R}}^{d}_{+}$ the spectral measures of TLC functions for
the system $({\mathfrak{X}}_{\zeta}^{\vec{s}},h_{t},\widetilde{\mu})$ are
Hölder regular away from zero, with a uniform Hölder exponent.

For comparison, now we have the following result, which is essentially a
special case of Theorem 5.6, with $d=1$.

######  Theorem 5.15.

Let $\zeta$ be a primitive aperiodic substitution on $d$ symbols with a
substitution matrix ${\sf S}_{\zeta}$ and the essential subspace $V_{\zeta}$.
Suppose that the second eigenvalue of ${\sf S}_{\zeta}|_{V_{\zeta}}$ satisfies
$|\theta_{2}|>1$. Then for Lebesgue-a.e. $\vec{s}\in V_{\zeta}$ the spectral
measures of TLC functions for the system
$({\mathfrak{X}}_{\zeta}^{\vec{s}},h_{t},\widetilde{\mu})$ are Hölder regular
away from zero, with a uniform Hölder exponent.

######  Remark 5.16.

(a) It is easy to see that if ${\sf S}_{\zeta}$ is irreducible, then
$V_{\zeta}={\mathbb{R}}^{d}$, however, the latter often happens even when
${\sf S}_{\zeta}$ is reducible.

(b) Suppose that $V_{\zeta}\neq{\mathbb{R}}^{d}$. Then it follows from [12]
that for $\vec{s}\in{\mathbb{R}}^{d}_{+}$ that is orthogonal to $V_{\zeta}$,
the suspension flow $({\mathfrak{X}}_{\zeta}^{\vec{s}},h_{t},\widetilde{\mu})$
has point spectrum containing ${\mathbb{Z}}$. If
$\vec{s},\vec{s}^{\prime}\in{\mathbb{R}}^{d}_{+}$ are such that
$\vec{s}-\vec{s}^{\prime}$ is orthogonal to $V_{\zeta}$, then the tiling
spaces ${\mathfrak{X}}_{\zeta}^{\vec{s}}$ and
${\mathfrak{X}}_{\zeta}^{\vec{s}^{\prime}}$ are MLD, hence the flows are
topologically conjugate.

(c) As it was pointed out in [13, Section 5], suspension flows
$({\mathfrak{X}}_{\zeta}^{\vec{s}},h_{t},\widetilde{\mu})$ do not necessarily
supply the entire space of admissible deformations; sometimes it happens that
collaring will result in a group (the analog of $\Gamma_{\zeta}$) of higher
rank than $d$.

##  6\. Examples

###  6.1. Kenyon’s (pseudo) self-similar tilings

In [24], given integers $p,q\geq 0$ and $r\in\mathbb{N}$, Kenyon introduced an
algebraic construction of pseudo self-similar tilings using parallelograms,
from which one can obtain a true self-similar tiling of $\mathbb{R}^{2}$. The
nature of the construction is such that there is an obvious subspace
$\mathcal{M}_{p,q,r}$ of deformation parameters which are accessible without
having to compute the cohomology of the associated tiling spaces. In this
section we give sufficient conditions under which a typical deformation of
Kenyon’s tilings, in the natural deformation space $\mathcal{M}_{p,q,r}$, we
obtain tiling spaces which are quantitatively weak mixing.

Let us review the construction: let $a,b,c$ be three vectors in
$\mathbb{R}^{2}$ pointing in different directions, and let $F$ be the set of
polygonal paths starting at the origin, each of which is a translate of $\pm
a,\pm b,$ or $\pm c$ without backtracking (i.e. $x$ is not followed by $-x$).
A product can be defined on $F$ by concatenating paths and errasing any
backtrack. As such any element in $F$ defines an element of the free group
$F(a,b,c)$ on three generators and a natural isomorphism
$h:F(a,b,c)\rightarrow F$.

Pick $p,q$ be non-negative integers, $r\in\mathbb{N}$, and let
$\phi:F(a,b,c)\rightarrow F(a,b,c)$ be the endomorphism defined by

(6.1) $\begin{split}\phi(a)&=b,\\\ \phi(b)&=c,\\\
\phi(c)&=c^{p}a^{-r}b^{-q}.\end{split}$

The three commutators $A=[a,b]$, $B=[b,c]$ and $C=[a,c]$ define three closed
paths which enclose parallelograms which we label $A,B,C$. The action of the
endomorphism on the commutators is thus

(6.2) $\begin{split}\phi(A)&=B,\\\
\phi(B)&=c^{p}a^{-r}[a^{r},c](b^{-q}[b^{q},c]b^{q})a^{r}c^{-q},\\\
\phi(C)&=[b,c^{p}]c^{p}a^{-r}[a^{r},b]a^{r}c^{-p}.\end{split}$

Consider the polynomial

(6.3) $f(z)=z^{3}-pz^{2}+qz+r.$

We have to impose the assumptions that $f$ is irreducible over ${\mathbb{Q}}$
and has a complex root $\lambda$ of absolute value greater than 1: a complex
Perron number, that is, a non-real algebraic integer strictly greater in
absolute value than its Galois conjugates other than $\overline{\lambda}$.
This does not always happen: for example, $f$ has a root $-1$, hence
reducible, if $r=p+q+1$, and $z^{3}-4z^{2}+z+1$ has three real zeros. Later we
will also need the condition for all three roots to be outside of the unit
circle, that is, for $\lambda$ not to be a complex Pisot number.

######  Lemma 6.1.

(i) The polynomial $f$ has a complex root if and only if $p^{2}<3q$, or
$p^{2}\geq 3q$ and

(6.4) $27r>2(p^{2}-3q)(p+\sqrt{p^{2}-3q})-3pq.$

(ii) If $f$ does have a complex root $\lambda$, then it is complex Perron,
unless $p=q=0$.

(iii) If the above conditions are satisfied, all the roots are outside of the
unit circle if and only if $r>p+q+1$. If $r<p+q+1$, then $\lambda$ is a
complex Pisot number.

(iv) The polynomial $f$ is reducible over ${\mathbb{Q}}$ if and only if it has
an integer root, which is necessarily a divisor of $r$.

###### Proof.

(i) Note first that $df/dz=3z^{2}-2pz+q$ has zeros $(p\pm\sqrt{p^{2}-3q})/3$,
hence if $p^{2}<3q$, we know that there is only one real root of $f$, hence
there is a complex root. If $p^{2}\geq 3q$, then both (or the unique) extremal
points $z_{1}\leq z_{2}$ are non-negative, and the condition for having a
complex root is that $f(z_{2})>0$ (keeping in mind that $f(0)=r>0$). A
computation (left to the reader) shows that this is equivalent to (6.4).

(ii) Suppose that the conditions from part (i) hold, and let $\lambda$ be the
complex root of $f$ with a positive imaginary part. If $p=q=0$, then
$\lambda^{3}=-r$, hence it is not a complex Perron number. Suppose that
$\max\\{p,q\\}\geq 1$. There is one negative zero $-\alpha$ and two complex
zeros $\lambda$ and $\overline{\lambda}$, such that
$|\lambda|^{2}\cdot\alpha=r$. Observe that

$f(-r^{1/3})=-r-pr^{2/3}-qr^{1/3}+r<0,$

hence $\alpha<r^{1/3}\implies|\lambda|>\alpha$, and so $\lambda$ is complex
Perron.

(iii) Assuming $\lambda$ is complex Perron, all zeros are greater than one in
absolute value if and only if $-\alpha<-1$, and this is equivalent to
$f(-1)=-1-p-q+r>0$, implying the first claim. If $f(-1)<0$, then
$-\alpha\in(-1,0)$ and $\lambda$ is complex Pisot.

(iv) The (ir)reducibility claim is immediate, since $f$ is a monic polynomial.
∎

For the rest of the section we assume that $p,q,r$ are such that $f$ is
irreducible over ${\mathbb{Q}}$ and $\lambda$ is a complex Perron number. In
the above construction let $a,b,c$ be $1,\lambda,\lambda^{2}\in\mathbb{C}$. We
can now express the operation in terms of parallelograms obtained from (6.2).
With slight abuse of notation,

(6.5) $\begin{split}\omega(A)&=B,\\\
\omega(B)&=\left[\bigcup_{j=0}^{q-1}B-r-j\lambda+p\lambda^{2}\right]\cup\left[\bigcup_{j=1}^{r}C-j+p\lambda^{2}\right],\\\
\omega(C)&=\left[\bigcup_{j=1}^{r}A-j+p\lambda^{2}\right]\cup\left[\bigcup_{j=0}^{p-1}B-j\lambda^{2}+p\lambda^{2}\right].\end{split}$

This is not exactly a substitution rule, but a “substitution with
amalgamation”. We will show how this gives both pseudo self-similar tilings
and self-similar tilings as a limit of the pseudo self-similar ones.

Using the rules $\omega(K+x)=\omega(K)+\lambda x$ and $\omega(K_{1}\cup
K_{2})=\omega(K_{1})\cup\omega(K_{2})$ for $K\in\\{A,B,C\\}$, we can iterate
the rules given in (6.5) to obtain larger and larger patches
$K_{n}=\omega^{n}(K)$ for every $n\in\mathbb{N}$. The patches $K_{n}$ grow in
area exponentially with $n$ while containing the origin, and so in the limit
they define tilings $\mathcal{T}_{K}$ of $\mathbb{R}^{2}$ belonging to the
same tiling space $X_{p,q,r}$.

######  Proposition 6.2.

(i) The space $X_{p,q,r}$ is the tiling space of a pseudo self-similar tiling
given by the “substitution with amalgamation” rule (6.5). Moreover, it is an
L-PSS tiling.

(ii) This tiling space is FLC, repetitive, and aperiodic.

###### Proof.

(i) For $K\in\\{A,B,C\\}$ and
$\mathscr{K}\in\\{\mathscr{A},\mathscr{B},\mathscr{C}\\}$ define the rescaled
tiles

$\mathscr{K}_{n}=\lambda^{-n}K_{n}=\lambda^{-n}\omega^{n}(K).$

These satisfy

$\omega(\mathscr{K}_{n})=\omega(\lambda^{-n}\omega^{n}(K_{n}))=\lambda^{-n}\omega^{n+1}(K)=\lambda^{-n}K_{n+1}=\lambda\mathscr{K}_{k+1}.$

Taking the limit as $n\rightarrow\infty$, there is a convergence of tiles
$\mathscr{K}_{n}\rightarrow\mathscr{K}$ (see, e.g., [34, Lemma 7.7]) and we
obtain an actual self-similar tiling [24] defined by

(6.6) $\begin{split}\lambda\mathscr{A}&=\mathscr{B},\\\
\lambda\mathscr{B}&=\left[\bigcup_{j=0}^{q-1}\mathscr{B}-r-j\lambda+p\lambda^{2}\right]\cup\left[\bigcup_{j=1}^{r}\mathscr{C}-j+p\lambda^{2}\right],\\\
\lambda\mathscr{C}&=\left[\bigcup_{j=1}^{r}\mathscr{A}-j+p\lambda^{2}\right]\cup\left[\bigcup_{j=0}^{p-1}\mathscr{B}-j\lambda^{2}+p\lambda^{2}\right].\end{split}$

Let $\widetilde{X}_{p,q,r}$ the tiling space associated to the primitive
substitution (6.6). It follows by construction that $\mathcal{T}\in X_{p,q,r}$
if and only if there exist countable sets
$\Lambda_{A},\Lambda_{B},\Lambda_{C}\subset\mathbb{R}^{2}$ such that

(6.7) $\mathcal{T}=(A+\Lambda_{A})\cup(B+\Lambda_{B})\cup(C+\Lambda_{C})\hskip
21.68121pt\mbox{ and }\hskip
21.68121pt\mathscr{T}=(\mathscr{A}+\Lambda_{A})\cup(\mathscr{B}+\Lambda_{B})\cup(\mathscr{C}+\Lambda_{C}).$

As such, $\mathcal{T}$ and $\mathscr{T}$ are MLD and therefore so are
$\lambda\mathcal{T}$ and $\lambda\mathscr{T}$. Since $\mathscr{T}$ is locally
derivable from $\lambda\mathscr{T}$, $\mathcal{T}$ is locally derivable from
$\lambda\mathcal{T}$, so $\mathcal{T}$ is pseudo self-similar with expanding
map $\lambda$.

(ii) The substitution matrix from Kenyon’s construction from the polynomial
$x^{3}-px^{2}+qx+r=0$, where $p,q\geq 0$ and $r\in\mathbb{N}$, is obtained
from (6.6) and is

(6.8) ${\sf S}_{p,q,r}=\left(\begin{array}[]{ccc}0&0&r\\\ 1&q&p\\\
0&r&0\end{array}\right),$

which has characteristic polynomial $x^{3}-qx^{2}-prx-r^{2}$, with the Perron-
Frobenius eigenvalue equal to $|\lambda|^{2}$. The FLC property is immediate
by construction, and repetitivity follows from the fact that the substitution
matrix ${\sf S}_{p,q,r}$ is primitive. To show aperiodicity, it is convenient
to work with the self-similar tiling space $\widetilde{X}_{p,q,r}$. Note that
if $0\neq x\in{\mathbb{R}}^{2}$ is a period, then $\lambda\cdot x$ is a period
as well, whence there is a lattice of periods and all tile frequencies must be
rational. However, the frequencies are given by the components of the Perron
eigenvector of ${\sf S}_{p,q,r}$, which are irrational, since we assumed
irreducibility of $f(z)$. This is a contradiction, and the proof is complete.

Finally, observe that all the conditions of an L-PSS tiling are satisfied, see
Definition 2.6. ∎

######  Proposition 6.3.

Suppose that $f(z)=z^{3}-pz^{2}+qz+r$ is irreducible over ${\mathbb{Q}}$ and
has a complex zero $\lambda$. Then the dynamics on the tiling spaces
$X_{p,q,r}$ and $\widetilde{X}_{p,q,r}$ are weakly mixing if and only if
$r>p+q+1$.

###### Proof.

By [34, Theorem 5.1] and [35], the dynamics on the tiling space
$\widetilde{X}_{p,q,r}$ are weakly mixing if and only if $\lambda$ is not a
complex Pisot number. Lemma 6.1 says that this is equivalent (provided the
other assumptions hold) to $r>p+q+1$. The result for $X_{p,q,r}$ then follows
by MLD equivalence to $\widetilde{X}_{p,q,r}$. ∎

In Figures 1-3 below we show several examples of Kenyon’s tilings. The colors
(in the electronic version) correspond to distinct collared tiles.

Figure 1. $(p,q,r)=(1,1,1)$, not weak mixing, 13 collared tiles, level 13
super-tile  Figure 2. $(p,q,r)=(1,1,4)$, weak mixing, 43 collared tiles, level
6 super-tile Figure 3. $(p,q,r)=(1,2,5)$, weak mixing, 36 collared tiles,
level 7 super-tile

####  6.1.1. Deformations

Although _all_ deformations of the tiling space $X_{p,q,r}$ are given by an
open subset of $H^{1}(X_{p,q,r};\mathbb{R}^{2})$ (see Section 3), here we
focus on elementary deformations, corresponding to perturbations of the
vectors $1,\lambda,\lambda^{2}\in\mathbb{C}$ which define the parallelograms.
This saves us the the work of having to compute the entire space of
deformations $H^{1}(X_{p,q,r};\mathbb{R}^{2})$, which may or may not have
higher dimension than 6. This means that the role of the subgroup
$\Gamma<H_{1}(AP_{0}(X),{\mathbb{Z}})$ in Section 3.4 will be played by the
subgroup
${\mathbb{Z}}[1,\lambda,\lambda^{2}]<{\mathbb{C}}\cong{\mathbb{R}}^{2}$.

From (6.5) and (6.6) it follows that the tiling spaces $X_{p,q,r}$ and
$\widetilde{X}_{p,q,r}$ have the same group generated by the return vectors
$\Gamma_{p,q,r}$, and this group is necessarily a subgroup of
${\mathbb{Z}}[1,\lambda,\lambda^{2}]$. In fact, we have

######  Lemma 6.4.

$\Gamma_{p,q,r}=\mathbb{Z}[1,\lambda,\lambda^{2}]$.

###### Proof.

By construction, $\Gamma_{p,q,r}$ is a subgroup of
$\mathbb{Z}[1,\lambda,\lambda^{2}]$, invariant under multiplication by
$\lambda$. If $r\geq 2$, we immediately obtain that $1\in\Gamma_{p,q,r}$ from
(6.6), since the substitution of ${\mathscr{B}}$ contains two translates of
${\mathscr{C}}$ differing by 1, and the claim follows. If $r=1$ and $q\geq 2$,
then $\lambda\in\Gamma_{p,q,r}$ by the formula for $\lambda{\mathscr{B}}$, and
then $\\{\lambda,\lambda^{2},\lambda^{3}\\}\subset\Gamma_{p,q,r}$. If $r=1$,
then the algebraic number $\lambda$ is a unit, and we conclude that
$1\in\Gamma_{p,q,r}$ by (6.3). A similar argument works for $r=1$ and $p\geq
2$, since then $\lambda^{2}\in\Gamma_{p,q,r}$. The remaining cases are when
$(p,q,r)\in\\{(1,0,1),(0,1,1),(1,1,1)\\}$, which are treated separately. For
instance, if $(p,q,r)=(1,0,1)$ we obtain by iterating (6.6) that
$\lambda^{3}{\mathscr{C}}$ contains ${\mathscr{C}}-1+\lambda^{3}$ and
${\mathscr{C}}-1+\lambda^{2}+\lambda^{3}$, hence
$\lambda^{2}\in\Gamma_{p,q,r}$ and we conclude as above. The remaining two
cases are similar and are left to the reader. ∎

To proceed, we need the sets ${\mathcal{D}}_{ij}$ from the definition of self-
similar tiling (2.4) to be subsets of $\Gamma_{p,q,r}$, and this holds in our
case. In fact, identifying the prototile labels by
$({\mathscr{A}},{\mathscr{B}},{\mathscr{C}})\equiv(1,2,3)$, we obtain from
(6.6):

$\begin{split}{\mathcal{D}}_{12}=\\{0\\},\ \ \ \ \
&{\mathcal{D}}_{22}=\\{-r-j\lambda+p\lambda^{2}\\}_{j=0}^{q-1},\ \ \ \ \
{\mathcal{D}}_{23}=\\{-j+p\lambda^{2}\\}_{j=1}^{r},\\\
&{\mathcal{D}}_{31}=\\{-j+p\lambda^{2}\\}_{j=1}^{r},\ \ \ \ \ \ \ \ \
{\mathcal{D}}_{32}=\\{-j\lambda^{2}+p\lambda^{2}\\}_{j=0}^{p-1},\end{split}$

with all the remaining ${\mathcal{D}}_{ij}$ being empty.

Let
$\alpha_{p,q,r}:{\mathbb{Z}}[1,\lambda,\lambda^{2}]=\Gamma_{p,q,r}\rightarrow\mathbb{Z}^{3}$
be the address map defined by
$\alpha_{p,q,r}(n_{1}+n_{2}\lambda+n_{3}\lambda^{2})=(n_{1},n_{2},n_{3})\in\mathbb{Z}^{3}$.
The inverse map is explicitly given by

$\alpha^{-1}_{p,q,r}(n_{1},n_{2},n_{3})=\left(\begin{array}[]{ccc}1&\Re(\lambda)&\Re(\lambda^{2})\\\
0&\Im(\lambda)&\Im(\lambda^{2})\end{array}\right)\left(\begin{array}[]{c}n_{1}\\\
n_{2}\\\ n_{3}\end{array}\right)=V_{p,q,r}\left(\begin{array}[]{c}n_{1}\\\
n_{2}\\\ n_{3}\end{array}\right)=n_{1}+n_{2}\lambda+n_{3}\lambda^{2},$

where $V_{p,q,r}$ is the matrix with column vectors
$1,\lambda,\lambda^{2}\in\mathbb{C}$.

There is a neighborhood $\mathcal{M}_{p,q,r}$ of
$(1,\lambda,\lambda^{2})\in\mathbb{C}^{3}\cong\mathbb{R}^{6}$ which
parametrizes non-degenerate deformations of parallelograms $A,B,C$. In fact,
one can check that the deformed tiling is well-defined if we let the vector
$a$ point in the positive direction of the $x$-axis, and vectors $b,c$ point
into the 1st and 2nd quadrant respectively. (To this configuration we can of
course apply a $GL(2,{\mathbb{R}})$ map.)

We denote the deformed parallelograms by
$\\{A^{\mathfrak{f}},B^{\mathfrak{f}},C^{\mathfrak{f}}\\}$, for
$\mathfrak{f}\in\mathcal{M}_{p,q,r}$ close enough to
$(1,\lambda,\lambda^{2})\in\mathbb{C}^{3}$. This in turn defines deformed
patches $\\{A_{n}^{\mathfrak{f}},B_{n}^{\mathfrak{f}},C_{n}^{\mathfrak{f}}\\}$
obtained by deforming the individual parallelograms in the patches
$\\{A_{n},B_{n},C_{n}\\}$ for all $n\geq 0$. Given that

$\mathcal{T}_{K}=\bigcup_{n\geq 0}K_{n},$

for $K_{n}\in\\{A_{n},B_{n},C_{n}\\}$, the deformed patches define a
deformation of $\mathcal{T}_{K}$ by

$\mathcal{T}_{K}^{\mathfrak{f}}:=\bigcup_{n\geq 0}K_{n}^{\mathfrak{f}},$

for
$K_{n}^{\mathfrak{f}}\in\\{A_{n}^{\mathfrak{f}},B_{n}^{\mathfrak{f}},C_{n}^{\mathfrak{f}}\\}$.
We denote by $X^{\mathfrak{f}}_{p,q,r}$ the tiling spaces for
$\mathcal{T}^{\mathfrak{f}}_{K}$ which are deformations of the tiling space
$X_{p,q,r}$.

To deform $X_{p,q,r}$ we deform $V_{p,q,r}$ as a natural subset of
$\mathbb{C}^{3}\cong\mathbb{R}^{6}$. Let $\mathfrak{f}\in\mathcal{M}_{p,q,r}$
be close to $(1,\lambda,\lambda^{2})$ and denote by $V_{p,q,r}^{\mathfrak{f}}$
the matrix associated with this deformation. (This is the matrix ${\bf
L}_{\mathfrak{f}}$ in our case.) In other words, since $V_{p,q,r}$ has columns
$1,\lambda,\lambda^{2}$, the matrix $V_{p,q,r}^{\mathfrak{f}}$ has columns
$v_{1}^{\mathfrak{f}},v_{2}^{\mathfrak{f}}$ and $v_{3}^{\mathfrak{f}}$ (these
are the vectors $a,b,c$ mentioned above), which are respectively close to
$1,\lambda$ and $\lambda^{2}$. The new group generated by the set of return
vectors is therefore

$\Gamma^{\mathfrak{f}}_{p,q,r}=V^{\mathfrak{f}}_{p,q,r}\cdot\alpha_{p,q,r}(\Gamma_{p,q,r}).$

The expansion map $\varphi$ for the PSS tiling is multiplication by $\lambda$
on ${\mathbb{C}}$, which induces a linear map on
${\mathbb{Z}}[1,\lambda,\lambda^{2}]\cong{\mathbb{Z}}^{3}$, given by the
matrix

(6.9) $M=\mathcal{G}_{p,q,r}:=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\
-r&-q&p\end{array}\right)$

which has characteristic polynomial $f(z)=z^{3}-pz^{2}+qz+r$.

In order to write down the spectral cocycle, we recall (4.3) and Definition
4.1 to obtain, denoting $e(t):=\exp(-2\pi it)$:

${\mathscr{M}}({\mathbf{z}})=\Bigl{[}\sum_{{\mathbf{x}}\in{\mathcal{D}}_{jk}}e\bigl{(}\langle{\mathbf{z}},\alpha({\mathbf{x}})\rangle\bigr{)}\Bigr{]}_{j,k\leq
3};\ \ {\mathscr{M}}({\mathbf{z}},n)={\mathscr{M}}\bigl{(}{(M^{\sf
T})}^{n-1}{\mathbf{z}}\bigr{)}\cdot\ldots\cdot{\mathscr{M}}({\mathbf{z}}),\ \
{\mathbf{z}}\in{\mathbb{T}}^{3},\ n\in{\mathbb{N}}.$

For example, if we take $(p,q,r)=(1,1,1)$ for simplicity, then
${\mathcal{D}}_{12}=\\{0\\}$,
${\mathcal{D}}_{22}={\mathcal{D}}_{23}={\mathcal{D}}_{31}=\\{-1+\lambda^{2}\\}$,
${\mathcal{D}}_{32}=\\{\lambda^{2}\\}$, which yields

${\mathscr{M}}({\mathbf{z}})=\left[\begin{array}[]{ccc}0&1&0\\\
0&e(-z_{1}+z_{3})&e(-z_{1}+z_{3})\\\
e(-z_{1}+z_{3})&e(z_{3})&0\end{array}\right].$

We now wish to extend Proposition 6.3 in terms of both deformations and the
Hölder property for the corresponding spectral measures.

######  Proposition 6.5.

Suppose that $f(z)=z^{3}-pz^{2}+qz+r$ is irreducible over ${\mathbb{Q}}$ and
has a complex zero $\lambda$. If $r>p+q+1$, then

(i) for every admissible deformation parameter
${\mathfrak{f}}\in{\mathcal{M}}_{p,q,r}$ outside of a set of codimension 1,
the uniquely ergodic dynamics on $X_{p,q,r}^{\mathfrak{f}}$ is weakly mixing;

(ii) for Lebesgue almost every deformation parameter
${\mathfrak{f}}\in{\mathcal{M}}_{p,q,r}$, the spectral measures for TLC
functions associated to the uniquely ergodic dynamics on
$X_{p,q,r}^{\mathfrak{f}}$ have positive local dimension.

###### Proof sketch.

(i) We can apply Theorem 5.1 and Proposition 5.2. For recurrence vector ${\bf
v}$ in (5.2) we can take ${\bf e}_{1}=(1,0,0)^{\sf T}$, representing
$1\in\Gamma_{p,q,r}$ in the basis $\\{1,\lambda,\lambda^{2}\\}$. These results
say that if $\boldsymbol{\lambda}\in{\mathbb{R}}^{2}$ is an eigenvalue
(topological or measure-theoretic), then $\langle{\bf L}^{\sf
T}_{\mathfrak{f}}\boldsymbol{\lambda},M^{n}{\bf e}_{1}\rangle\to 0$ (mod 1) as
$n\to\infty$, where ${\bf L}^{\sf T}=V_{p,q,r}^{\mathfrak{f}}$ in our case.
Since all the eigenvalues of $M$ are greater than one in modulus, a standard
argument (see, e.g., [13]) implies that for $\boldsymbol{\lambda}\neq 0$ this
can happen only when $\langle{\bf L}^{\sf
T}_{\mathfrak{f}}\boldsymbol{\lambda},M^{n}{\bf e}_{1}\rangle=0$ for all $n$
sufficiently large. But then the columns of $V_{p,q,r}^{\mathfrak{f}}$ must be
rationally dependent, and such ${\mathfrak{f}}$ form a countable union of
subspaces of codimension 1.

(ii) This is a special case of Corollary 5.7, in view of Lemma 6.1. ∎

######  Remark 6.6.

If $f(z)=z^{3}-pz^{2}+qz+r$ is irreducible over ${\mathbb{Q}}$, has a complex
zero $\lambda$, and $r<p+q+1$, then $\lambda$ is a complex Pisot number. Then
it follows from [13], see Proposition 5.13, that all admissible deformations
of the tiling space result in a topologically conjugate system.

###  6.2. Square tilings

This class of examples is obtained when all tiles are unit cubes (located at
the vertices of ${\mathbb{Z}}^{d}$), but with different labels, and the
expansion map is a diagonal map $qI$, where $q\in{\mathbb{N}}$, $q\geq 2$.
This is essentially a $d$-dimensional generalization of symbolic substitutions
of constant length.

######  Example 6.7.

All the tiles have the unit square as its support and are distinguished only
by the labels. Let ${\mathcal{A}}=\left\\{\begin{tabular}[]{|c|}\hline\cr 0\\\
\hline\cr\end{tabular}\,,\begin{tabular}[]{|c|}\hline\cr 1\\\
\hline\cr\end{tabular}\,,\begin{tabular}[]{|c|}\hline\cr 2\\\
\hline\cr\end{tabular}\right\\}$; the expansion is pure dilation by a factor
of 6:

$\begin{tabular}[]{|c|}\hline\cr 0\\\ \hline\cr\end{tabular}\ \to\
\begin{tabular}[]{|c|c|c|c|c|c|}\hline\cr 2&2&2&2&2&2\\\ \hline\cr
2&2&2&2&2&2\\\ \hline\cr 1&0&1&0&0&1\\\ \hline\cr 1&0&0&1&0&1\\\ \hline\cr
2&2&2&2&2&2\\\ \hline\cr 2&2&2&2&2&2\\\ \hline\cr\end{tabular}\,,\ \ \ \ \ \
\begin{tabular}[]{|c|}\hline\cr 1\\\ \hline\cr\end{tabular}\ \to\
\begin{tabular}[]{|c|c|c|c|c|c|}\hline\cr 2&2&2&2&2&2\\\ \hline\cr
2&2&2&2&2&2\\\ \hline\cr 1&0&1&0&0&1\\\ \hline\cr 1&0&1&1&0&1\\\ \hline\cr
2&2&2&2&2&2\\\ \hline\cr 2&2&2&2&2&2\\\ \hline\cr\end{tabular}\,,\ \ \ \ \ \
\begin{tabular}[]{|c|}\hline\cr 2\\\ \hline\cr\end{tabular}\ \to\
\begin{tabular}[]{|c|c|c|c|c|c|}\hline\cr 0&1&0&0&0&1\\\ \hline\cr
0&0&1&1&0&0\\\ \hline\cr 2&2&2&2&2&2\\\ \hline\cr 2&2&2&2&2&2\\\ \hline\cr
1&0&0&1&0&0\\\ \hline\cr 1&0&0&1&0&1\\\ \hline\cr\end{tabular}$

Figure 4. Square tiling and its deformation

The substitution $\omega$ was chosen in such a way that every tiling in the
tiling space has two periodic rows, consisting entirely of 2’s, followed by
two rows consisting entirely of 0’s and 1’s, again followed by two periodic
rows, consisting entirely of 2’s, etc., ad infinitum. The substitution is
clearly primitive. Although it has periodic rows, there are no global
translational periods. This can be seen, e.g., by observing that the
substitution is recognizable: the pattern  1 0 1 1 can only occur in
$\omega\left(\,\begin{tabular}[]{|c|}\hline\cr 1\\\
\hline\cr\end{tabular}\,\right)$ at the center of a super-tile. Moreover, it
is easy to see that the substitution “forces the border” [2].

We consider elementary deformations. Namely, we deform the vectors-edges in a
way consistent with the Anderson-Putnam complex ${AP}_{0}(X)$, so that the sum
of the vectors around each tile is zero. It is easy to see that all horizontal
edges are identified in the AP-complex, but there are two kinds of vertical
edges: those of the tiles labeled 2, and those of the tiles labeled 0 or 1.
For each tile edge $e$, the deformation of the edge opposite to $e$ must be
the opposite of the deformation of $e$; this will ensure consistency and all
the deformed tiles will be parallelograms.

On the other hand, working with recurrence will lead to the same conclusion.
There are recurrences with return vectors $(1,0)$ and $(0,1)$. All
“horizontal” recurrences project to a multiple of the same cycle , whereas the
“vertical ones” are of two types, similarly to the above. These recurrences
form a basis for the lattice of all recurrences. It is not hard to compute the
linear map induced by the expansion — it is a block matrix, with the blocks
corresponding to “side substitutions”:

$\left[\begin{array}[]{ccc}2&4&0\\\ 4&2&0\\\ 0&0&6\end{array}\right].$

The eigenvalues are $6,6,2$, so now $\dim E^{+}=3$, and we conclude that a.e.
deformation has quantitative weak mixing.

##  7\. Proofs

###  7.1. Spectral estimate

The bounds for local dimension for spectral measures are based on growth
estimates of twisted ergodic integrals. The following lemma, which essentially
goes back to Hof [19], will be used in the proof of Theorem 4.2.

######  Lemma 7.1 ([39, Lem. 1]).

Let $(Y,\mu,h_{t})_{t\in{\mathbb{R}}^{d}}$ be an ergodic probability-
preserving ${\mathbb{R}}^{d}$ action and $\phi\in L^{2}(Y,\mu)$. For
$\boldsymbol{\lambda}\in{\mathbb{R}}^{d}$ and $y\in Y$ the twisted ergodic
integral is

$S_{R}^{y}(\phi,\boldsymbol{\lambda}):=\int_{Q_{R}}e^{-2\pi
i\langle\boldsymbol{\lambda},\tau\rangle}\phi\circ h_{t}(y)\,d\tau,$

where $Q_{R}=[-R,R]^{d}$. Suppose that for some
$\boldsymbol{\lambda}\in{\mathbb{R}}^{d},\ R_{0}>0,\phi\in L^{2}(Y,\mu)$ and
$\alpha\in(0,d)$,

$\|S_{R}^{y}(\phi,\boldsymbol{\lambda})\|_{L^{2}}\leq C_{1}R^{d-\alpha}\ \
\mbox{for all}\ R\geq R_{0}.$

Then

$\sigma_{\phi}(B_{r}(\boldsymbol{\lambda}))\leq Cr^{2\alpha}\ \ \mbox{for
all}\ r<1/(2R_{0}),$

for some $C>0$. In particular, the lower local dimension of the spectral
measure satisfies

$d^{-}(\sigma_{\phi},\boldsymbol{\lambda})\geq 2\alpha.$

###  7.2. Twisted ergodic integrals over deformed super-tiles

Recall that we are working with an L-PSS tiling space $X_{\omega}$ with
expansion map $\varphi$, where $\omega$ is a combinatorial substitution, or
“substitution-with-amalgamation,” for which a pseudo-self-similar tiling
${\mathcal{T}}$ is a fixed point. The prototiles $\\{T_{j}\\}_{j\leq m}$ are
actual specific tiles of ${\mathcal{T}}$. Let ${\mathfrak{f}}$ be an
admissible deformation. In order to consider deformed tiles and super-tiles,
we fixed a vertex $v$ of ${\mathcal{T}}$ and defined
$T_{j}^{\mathfrak{f}}:=(T_{j}-v)^{\mathfrak{f}}$ and
$T_{j}^{{\mathfrak{f}},n}=(\omega^{n}(T_{j})-v)^{\mathfrak{f}}$ for $j\leq m$,
see (3.5) and (3.6).

Below, when writing $\int_{T_{j}^{{\mathfrak{f}},n}}$ and the like, we mean
integration over the support of the corresponding patch.

Let $\phi$ is a TLC function of level 0 on $X^{\mathfrak{f}}_{\omega}$ of the
form (4.11). Let $j\leq m$ and ${\mathcal{T}}^{\mathfrak{f}}\in
X_{\omega}^{\mathfrak{f}}$ be such that
$(T_{j}^{{\mathfrak{f}},n})\subset{\mathcal{T}}^{\mathfrak{f}}$ for all $n\geq
0$. Then, in view of (3.10),

$\displaystyle\int_{T_{j}^{{\mathfrak{f}},1}}e^{-2\pi
i\langle\boldsymbol{\lambda},{\bf
t}\rangle}\phi({\mathcal{T}}^{\mathfrak{f}}-{\bf t})\,d{\bf t}$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{m}\sum_{{\mathbf{x}}\in{\mathfrak{f}}({\mathcal{D}}_{jk})}e^{-2\pi
i\langle\boldsymbol{\lambda},{\mathbf{x}}\rangle}\cdot\widehat{\psi}_{k}(\boldsymbol{\lambda})$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{m}\sum_{{\mathbf{x}}\in{\mathcal{D}}_{jk}}e^{-2\pi
i\langle{\bf L}_{\mathfrak{f}}^{\sf
T}\boldsymbol{\lambda},\alpha({\mathbf{x}})\rangle_{{}_{{\mathbb{R}}^{s}}}}\cdot\widehat{\psi}_{k}(\boldsymbol{\lambda})$
$\displaystyle=$ $\displaystyle\bigl{[}{\mathscr{M}}({\bf
L}_{\mathfrak{f}}^{\sf
T}\boldsymbol{\lambda})\cdot\widehat{\boldsymbol{\Psi}}(\boldsymbol{\lambda})\bigr{]}(j)=\bigl{[}{\mathscr{M}}({\mathbf{z}})\cdot\widehat{\boldsymbol{\Psi}}(\boldsymbol{\lambda})\bigr{]}(j),$

where

$\widehat{\boldsymbol{\Psi}}(\boldsymbol{\lambda})=\left[\begin{array}[]{c}\widehat{\psi}_{1}\\\
\vdots\\\ \widehat{\psi}_{m}\end{array}\right](\boldsymbol{\lambda}).$

Iterating this, we obtain, in view of (3.11),

$\displaystyle\int_{T_{j}^{{\mathfrak{f}},2}}e^{-2\pi
i\langle\boldsymbol{\lambda},{\bf
t}\rangle}\phi({\mathcal{T}}^{\mathfrak{f}}-{\bf t})\,d{\bf t}$
$\displaystyle=$
$\displaystyle\sum_{s=1}^{m}\sum_{k=1}^{m}\left(\sum_{{\mathbf{x}}\in{\bf
L_{\mathfrak{f}}}M\alpha({\mathcal{D}}_{jk})+{\bf
L_{\mathfrak{f}}}\alpha({\mathcal{D}}_{ks})}e^{-2\pi
i\langle\boldsymbol{\lambda},{\mathbf{x}}\rangle}\right)\widehat{\psi}_{s}(\boldsymbol{\lambda})$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{m}\sum_{s=1}^{m}\left(\sum_{{\mathbf{x}}\in{\mathcal{D}}_{jk}}e^{-2\pi
i\langle M^{\sf T}{\bf L}_{\mathfrak{f}}^{\sf
T}\boldsymbol{\lambda},\alpha({\mathbf{x}})\rangle_{{}_{{\mathbb{R}}^{s}}}}\cdot\sum_{{\mathbf{x}}\in{\mathcal{D}}_{ks}}e^{-2\pi
i\langle{\bf L}_{\mathfrak{f}}^{\sf
T}\boldsymbol{\lambda},\alpha({\mathbf{x}})\rangle_{{}_{{\mathbb{R}}^{s}}}}\right)\widehat{\psi}_{s}(\boldsymbol{\lambda}),$

hence

$\int_{T_{j}^{{\mathfrak{f}},2}}e^{-2\pi i\langle\boldsymbol{\lambda},{\bf
t}\rangle}\phi({\mathcal{T}}^{\mathfrak{f}}-{\bf t})\,d{\bf
t}=\bigl{[}{\mathscr{M}}(M^{\sf
T}{\mathbf{z}}){\mathscr{M}}({\mathbf{z}})\cdot\widehat{\boldsymbol{\Psi}}(\boldsymbol{\lambda})\bigr{]}(j)=\bigl{[}{\mathscr{M}}({\mathbf{z}},2)\cdot\widehat{\boldsymbol{\Psi}}(\boldsymbol{\lambda})\bigr{]}(j),$

and similarly, by induction, for all $n\geq 1$,

(7.1) $\int_{T_{j}^{{\mathfrak{f}},n}}e^{-2\pi
i\langle\boldsymbol{\lambda},{\bf
t}\rangle}\phi({\mathcal{T}}^{\mathfrak{f}}-{\bf t})\,d{\bf
t}=\bigl{[}{\mathscr{M}}({\mathbf{z}},n)\cdot\widehat{\boldsymbol{\Psi}}(\boldsymbol{\lambda})\bigr{]}(j).$

This immediately implies the following.

######  Lemma 7.2.

Suppose that ${\mathcal{S}}^{\mathfrak{f}}\in X^{\mathfrak{f}}_{\omega}$ and
$T_{j}^{{\mathfrak{f}},n}+{\bf y}$ is a super-tile of
${\mathcal{S}}^{\mathfrak{f}}$. Then