# The branch locus for one-dimensional Pisot tiling spaces

Marcy Barge, Beverly Diamond and Richard Swanson

###### Abstract.

If $\varphi$ is a Pisot substitution of degree $d$, then the inflation and
substitution homeomorphism $\Phi$ on the tiling space $\mathcal{T}_{\Phi}$
factors via geometric realization onto a d-dimensional solenoid. Under this
realization, the collection of $\Phi$-periodic asymptotic tilings corresponds
to a finite set that projects onto the  branch locus in a d-torus. We prove
that if two such tiling spaces are homeomorphic, then the resulting branch
loci are the same up to the action of certain affine maps on the torus.

2000 Mathematics Subject Classification: _Primary:_ 37B05; _Secondary:_ 37A30,
37B50, 54H20

##  1\. Introduction

In this paper we introduce the branch locus, a new topological invariant for
one-dimensional Pisot substitution tiling spaces. A substitution on $n$
letters is a map from an alphabet $\mathcal{A}=\\{1,2,\ldots,n\\}$ into
$\mathcal{A}^{*}$, where $\mathcal{A}^{*}$ is the collection of finite and
nonempty words from $\mathcal{A}$. The _abelianization matrix_ of $\varphiup$
is defined as $A=(a_{ij})$, where $a_{ij}=$ number of occurrences of $i$ in
$\varphiup(j)$. The substitution $\varphiup$ is _Pisot_ provided the Perron-
Frobenius eigenvalue $\lambdaup$ of $A$ is a Pisot-Vijayaraghavan number
($\lambdaup>1$ and all algebraic conjugates of $\lambdaup$ are strictly inside
the unit circle). The _degree_ of $\varphiup$ is the degree of the minimal
polynomial of $\lambdaup$.

Associated with any substitution $\varphiup$, there is a tiling space
$\mathcal{T}_{\varphiup}$ consisting of certain tilings of the real line; if
$(\lambdaup_{1},\ldots,\lambdaup_{n})$ is a positive left eigenvector of $A$,
then $\mathcal{T}_{\varphiup}$ consists of all tilings of $\mathbb{R}$ by
translates of the prototiles $P_{i}=[0,\lambdaup_{i}]$, $i=1,\ldots,n$, with
the property that the word spelled out by any finite patch of consecutive
tiles in the tiling occurs as a factor of $\varphiup^{m}(i)$ for some
$i\in\mathcal{A}$ and $m\in\mathbb{N}$. The substitution $\varphiup$ is
_primitive_ if for some $m\in\mathbb{N}$, every entry of $A^{m}$ is strictly
positive, and _aperiodic_ if no element of $\mathcal{T}_{\varphiup}$ is
periodic under translation. For the remainder of the paper, all substitutions
will be assumed primitive and aperiodic.

Define the topology of $\mathcal{T}_{\varphiup}$ by stipulating that two
tilings are close provided a small translate of one is identical to the other
in a large neighborhood of the origin. Under the assumption that $\varphiup$
is primitive and aperiodic, $\mathcal{T}_{\varphiup}$ is a continuum (a
compact, connected metric space). For such a $\varphiup$, _inflation and
substitution_ is the homeomorphism
${\Phi}:\mathcal{T}_{\varphiup}\rightarrow\mathcal{T}_{\varphiup}$ that
replaces each tile $t+P_{i}=[t,\lambdaup_{i}+t]$ of a tiling $T$ in
$\mathcal{T}_{\varphiup}$ by the patch $[\lambdaup
t,\lambdaup\lambdaup_{i}+\lambdaup t]$ tiled by translates of prototiles
following the pattern of the word $\varphiup(i)$. There is also a minimal and
uniquely ergodic $\mathbb{R}$-action on $\mathcal{T}_{\varphiup}$, called the
_translation flow_ , given by $T=\\{T_{i}\\}_{i=-\infty}^{\infty}\mapsto
T-t:=\\{T_{i}-t\\}_{i=-\infty}^{\infty}$, for $t\in\mathbb{R}$.

The topology of a substitution tiling space is of interest for a number of
reasons. Physics provides one source of motivation. Suppose that $T$ is a
tiling in the substitution tiling space $\mathcal{T}_{\varphiup}$. Placing an
atom at the end of each tile creates a one-dimensional material which is
called a _quasi-crystal_ if its diffraction spectrum is pure point (the atoms
must be ‘weighted’ according to the tile types they lie in). Bombieri and
Taylor ([BT]) proved that if $\varphiup$ is Pisot, such a material has a
nontrivial discrete component in its spectrum. Whether such a material has
pure point spectrum when $\varphiup$ is irreducible unimodular Pisot (i.e.,
$\varphiup$ is Pisot with ${\rm degree}(\lambdaup)=d=n=|\mathcal{A}|$, and
$\det(A)=\pm 1$) remains an open question. Lee, Moody and Solomyak ([LMS])
have proved that the diffraction spectrum of the material is pure point if and
only if the dynamical spectrum of the translation flow on
$\mathcal{T}_{\varphiup}$ is pure discrete. It follows from [BSw2] that if
$\mathcal{T}_{\varphiup}$ and $\mathcal{T}_{\psiup}$ are homeomorphic tiling
spaces, then the tiling flow on $\mathcal{T}_{\varphiup}$ is pure discrete if
and only if the tiling flow on $\mathcal{T}_{\psiup}$ is pure discrete. That
is, the question of whether or not a one-dimensional material built from a
substitution has pure point diffraction spectrum is a _topological_ question
about the corresponding tiling space.

Substitution tiling spaces also arise in the study of hyperbolic attractors.
R.F. Williams ([Wi1]) proved that every hyperbolic one-dimensional attractor
is topologically conjugate with the shift map on the inverse limit of an
expanding endomorphism of a branched one-manifold, and, with minor
restrictions on the map of the branched one-manifold, all such inverse limits
can be realized as hyperbolic attractors. More recently, Anderson and Putnam
([AP]) proved that inflation and substitution on a one-dimensional
substitution tiling space is conjugate with the shift on the inverse limit of
an expanding endomorphism of a branched one-manifold. As a consequence, every
orientable hyperbolic one-dimensional attractor is either a substitution
tiling space, for which the underlying manifold is branched, or a classical
solenoid, for which the underlying manifold is the circle. Modeling an
attractor as a tiling space provides a much clearer view of its global
topology than one gets from considering an inverse limit description: moving
along an arc component in the attractor is simply translating a tiling, and
the patterns of consecutive tiles determine the recurrence properties of the
translates.

Although the “inverse limit on branched manifolds” description of tiling
spaces will not play an explicit role in this paper, the intuitive content of
our main result, and the rationale for the terminology “branch locus” that we
introduce, has its origin in that description. There are, of course, no actual
branch points in a tiling space: in the one-dimensional case, every point has
a neighborhood that is homeomorphic with the product of an arc and a Cantor
set. Nevertheless, an inverse limit description of the tiling space gives a
sequence of approximating branched one-manifolds. In the limit, the ghost of
the branches can be observed in the existence of asymptotic composants: two
distinct tilings $T,T^{\prime}$ are _asymptotic_ provided
$d(T-t,T^{\prime}-t)\rightarrow 0$ as $t\rightarrow\infty$ or as
$t\rightarrow-\infty$. The arc components of asymptotic tilings are called
_asymptotic composants_. Partially sewing up such asymptotic composants
results in a space that does have branching: the new, branched, space
corresponds to an inverse limit on a branched manifold with periodic branch
points.

Although this “ghost branching” in $\mathcal{T}_{\varphiup}$ seems to have no
clear location, we will see that, in case the substitution is Pisot, the
branching occurs in well defined relative geometrical patterns. The
appropriate underlying geometry is that of the $d$-dimensional torus, where
$d$ is the degree of $\lambdaup$. Our main result is that if
$\mathcal{T}_{\varphiup}$ and $\mathcal{T}_{\psiup}$ are homeomorphic tiling
spaces, then their branch loci, nonempty finite sets of points in the
$d$-torus that we define in §2, are equal modulo the action of a certain
collection of affine endomorphisms of the torus. Thus the branch locus becomes
a topological invariant. We illustrate this by distinguishing pairs of tiling
spaces that are otherwise difficult to tell apart (see §3).

The idea for considering “branching” in one-dimensional tiling spaces arose in
discussions the first author had with Søren Eilers regarding the topological
content of the $K_{0}$-group of the Matsumoto algebra associated with a
substitution. Eilers, Restorff and Ruiz ([ERR]) have shown that this (ordered)
group is also a complete invariant of the Matsumoto algebra and, consequently,
the Matsumoto algebra is a topological invariant of the tiling space.

How is the Matsumoto algebra (equivalently, its $K_{0}$-group) reflected in
the topology of the tiling space? In § 5 we show that the branch locus
provides a partial answer. We use the branch locus to define the “Pisot part
of the augmented dimension group”, an ordered group that is a flow equivalence
invariant of the substitution and an ordered subgroup of the augmented
cohomology group of the tiling space, which, in turn, is closely related to
the Matsumoto $K_{0}$-group (see [CE1], [CE2] for a description of the
Matsumoto $K_{0}$-group in the substitutive setting and [BSm] for an account
of the relationship between the Matsumoto $K_{0}$-group of a substitutive
system and the augmented cohomology of the associated tiling space).

##  2\. Geometric Realization

For convenience, we will use the “strand space” model for the tiling space
(see [BK] and [BBK]) which we recall now. Let $\varphiup$ be a Pisot
substitution of degree $d$ on $n$ letters with abelianization $A$ and Perron-
Frobenius eigenvalue $\lambdaup$. There is an (unique) $A$-invariant
decomposition $\mathbb{R}^{n}=V\oplus W$ such that $V$ contains a right
Perron-Frobenius eigenvector $\omegaup$ associated with $\lambdaup$ and $\dim
V=d$111$V$ is the kernel of $m(A)$, if $m(x)$ denotes the minimal polynomial
of $\lambdaup$. . We can always choose rational bases for $V$ and $W$. There
is a further $A|_{V}$-invariant splitting
$V=\mathbb{E}^{u}\oplus\mathbb{E}^{s}$ obtained by letting $\mathbb{E}^{s}$ be
the space orthogonal to a left Perron-Frobenius eigenvector of $A|_{V}$ and
$\mathbb{E}^{u}$ be the span of $\omegaup$. Let
$\mathrm{pr}_{V}:\mathbb{R}^{n}\to V,\ \mathrm{pr}_{s}:V\to\mathbb{E}^{s},\
\mathrm{and}\ \mathrm{pr}_{u}:V\to\mathbb{E}^{u}$ denote the projections,
resp., along $W$, $\mathbb{E}^{u}$ and $\mathbb{E}^{s}$, and let $\Gamma$
denote the $A$-invariant lattice $\mathrm{pr}_{V}\mathbb{Z}^{n}$.

If $e_{i},i=1,\ldots,n$, are the standard basis vectors in $\mathbb{R}^{n}$,
define $\varv_{i}:=\mathrm{pr}_{V}(e_{i})$, and let
$\sigmaup_{i}:=\\{t\varv_{i}:0\leq t\leq 1\\}$ denote the oriented segment
representing $\varv_{i}$. Even if $\sigmaup_{i}=\sigmaup_{j}$ for some $i\neq
j$, we wish to distinguish between these segments: we call $\sigmaup_{i}$ a
_(labeled) segment of type $i$_. An oriented broken line
$\gammaup=\\{\sigmaup_{i_{k}}+x_{k}\\}$, $x_{k}\in V$, consisting of a
collection of translated copies of the basic segments meeting tip-to-tail and
with connected union, will be called a _strand_. We denote the space of bi-
infinite strands in $V$ by

$\mathcal{F}:=\\{\gammaup:\gammaup\textrm{ is a bi-infinite strand in }V\\}.$

The substitution $\varphiup$ induces the _inflation and substitution map_
${\Phi}:\mathcal{F}\rightarrow\mathcal{F}$ as follows: for each edge
(translated segment) $\sigmaup_{i_{k}}+x_{k}$ in strand $\gammaup$, replace
that edge by the collection of edges $\sigmaup_{j_{1}}+Ax_{k}$,
$\sigmaup_{j_{2}}+Ax_{k}+\varv_{j_{1}}$, $\ldots$,
$\sigmaup_{j_{\ell}}+Ax_{k}+\varv_{j_{1}}+\cdots+\varv_{j_{\ell-1}}$, where
$\varphiup(i_{k})=j_{1}j_{2}\cdots j_{\ell}$. That is, $\Phi(\gammaup)$ is
obtained by applying the linear map $A$ to each edge of $\gammaup$, then
breaking up the image into translated segments following the pattern
determined by applying $\varphiup$ to the type of the edge.

For $R>0$, let $\mathcal{F}^{R}$ denote the subset of $\mathcal{F}$ consisting
of those strands all of whose edges are within distance $R$ of
$\mathbb{E}^{u}$. There is then an $R_{0}$ so that

$\mathcal{F}_{\varphiup}:=\bigcap_{n\geq 0}{\Phi}^{n}(\mathcal{F}^{R})$

is independent of $R\geq R_{0}$. The set $\mathcal{F}_{\varphiup}$ has a
natural metric topology in which $\gammaup$ and $\gammaup^{\prime}$ are close
if a small translate of $\gammaup$ by a vector in $V$ lines up exactly with
$\gammaup^{\prime}$, segment types being considered, in a large neighborhood
of the origin. With respect to this topology, ${\Phi}$ and the _translation
flow_ ,

$\gammaup=\left\\{\sigmaup_{i_{k}}+x_{k}\right\\}\mapsto\gammaup-t:=\left\\{\sigmaup_{i_{k}}+x_{k}-t\omegaup\right\\},$

are continuous.

In some cases there may be a few translation orbits in
$\mathcal{F}_{\varphiup}$ that correspond to strands with acceptable heads and
tails but that are joined in an unnatural way (this happens, for instance,
when there are letters $a$ and $b$ with $\varphiup(a)=a\cdots$,
$\varphiup(b)=\cdots b$, but the word $ba$ never occurs in $\varphiup^{m}(i)$,
$i\in\mathcal{A}$, $m\in\mathbb{N}$). We eliminate these chimeras by defining
$\mathcal{T}_{\varphiup}$ to be the $\omegaup$-limit set, under translation
flow, of any $\gammaup\in\mathcal{F}_{\varphiup}$. That is, for any
$\gammaup\in\mathcal{F}_{\varphiup}$,

$\mathcal{T}_{\varphiup}:=\bigcap_{T>0}\textrm{cl}\\{\gammaup-t:t\in[T,\infty)\\}.$

Inflation and substitution,
${\Phi}:\mathcal{T}_{\varphiup}\rightarrow\mathcal{T}_{\varphiup}$, is a
homeomorphism from $\mathcal{T}_{\varphiup}$ onto itself, and the translation
flow on $\mathcal{T}_{\varphiup}$ is minimal and uniquely ergodic.

One reason for using the rather elaborate strand space model of the tiling
space is that it permits a simple and concrete definition of geometric
realization. One could factor the tiling dynamics onto those of a solenoid, by
choosing the lattice $\Gamma$ in $V$ and mapping
$\gammaup\in\mathcal{T}_{\varphiup}$ to

$(\min I_{0}+\Gamma,\min I_{1}+\Gamma,\ldots)\in\varprojlim F_{A|_{V}},$

where $F_{A|_{V}}:V/\Gamma\rightarrow V/\Gamma$ is defined by
$F_{A|_{V}}(v+\Gamma)=Av+\Gamma$, $I_{k}$ is any edge in
${\Phi}^{-1}(\gammaup)$ and $\min I_{k}$ is its initial vertex. This does give
a well defined surjection of $\mathcal{T}_{\varphiup}$ onto the
$d$-dimensional solenoid $\varprojlim F_{A|_{V}}$ that semiconjugates
inflation and substitution with the shift, as well as translation flow with a
Kronecker action. However, to maximize the size of the factor, we will define
a coarser, more natural lattice. Toward this end, denote the collection of
_return vectors_ by

$\displaystyle\Theta(i):=$ $\displaystyle\\{v\in\Gamma:\textrm{ there exists
}\gammaup\in\mathcal{T}_{\varphiup}\textrm{ containing edges }I,I^{\prime},$
$\displaystyle\hskip 108.405pt\textrm{ each of type }i,\textrm{ with
}I^{\prime}=I+v\\}.$

It is not difficult to show that the subgroup

$\Sigma_{\infty}:=\left<\bigcup_{k\in\mathbb{Z}}(A|_{V})^{k}\Theta(i)\right>$

of $V$ generated by $\bigcup_{k\in\mathbb{Z}}(A|_{V})^{k}\Theta(i)$ is
independent of $i\in\mathcal{A}$. The _return lattice_

$\Sigma:=\Sigma_{\infty}\cap\Gamma$

is invariant under $A$ and also of rank $d$.

Note: In case $A$ is irreducible ($d=n$), then $\Sigma=\Gamma=\mathbb{Z}^{n}$.
In § 3, we will see an example where $\Sigma$ is strictly coarser than
$\Gamma$.

To define geometric realization onto a solenoid determined by $\Sigma$, we
must determine some appropriate translations. We begin by noticing that for
each $i,j\in\mathcal{A}$, there is a well defined “transition vector”
$\varw_{ij}\in\Gamma/\Sigma$ so that if $\gammaup\in\mathcal{T}_{\varphiup}$
and $I$ and $J$ are edges of $\gammaup$ of types $i$ and $j$ resp., then

$(\min J-\min I)\textrm{ mod }\Sigma=\varw_{ij}.$

Then, since $\varw_{ij}+\varw_{jk}+\varw_{ki}=0$ for all
$i,j,k\in\mathcal{A}$, there are $u_{i}\in\Gamma/\Sigma$, for
$i\in\mathcal{A}$, so that for all $i,j$,

$\varw_{ij}=u_{j}-u_{i}.$

We may then normalize the $u_{i}$ so that

$Au_{i}=u_{i^{\prime}}$

for all $i\in\mathcal{A}$, where $i^{\prime}$ denotes the initial letter of
$\varphiup(i)$. Letting $F_{A|_{V}}$ denote the map induced by $A$ on the
torus $V/\Sigma$, we have a well defined map
$\varg_{\varphiup}:\mathcal{T}_{\varphiup}\rightarrow\varprojlim F_{A|_{V}}$
(we will always use the lattice $\Sigma$ in place of $\Gamma$ in the
definition of $F_{A|_{V}}$) given by

$\varg_{\varphiup}(\gammaup)=(\min I_{0}+u_{i_{0}},\min
I_{1}+u_{i_{1}},\ldots)$

where $I_{k}$ is any edge of ${\Phi}^{-k}(\gammaup)$ of type $i_{k}$. The map
$\varg_{\varphiup}$ is called _geometric realization_.

Note: The map $\varg_{\varphiup}$ depends on the choice of the
$u_{i}\in\Gamma/\Sigma$, which are not uniquely defined even after the
normalization.

From results of [BK] and [BBK], the map $\varg_{\varphiup}$ is boundedly
finite-to-one and almost everywhere $m$-to-$1$, where $m$ is the _coincidence
rank_ of $\varphiup$, and the tiling flow on $\mathcal{T}_{\varphiup}$ has
pure discrete spectrum if and only if $m=1$. Moreover, $\varg_{\varphiup}$ is
optimal in the sense that any other factoring of ${\Phi}$ onto a solenoidal
shift factors through $\varg_{\varphiup}$.222 For the translation flow on
$\mathcal{T}_{\varphiup}$, the Kronecker flow,
$(z_{0},z_{1},\ldots)\mapsto(z_{0}-t\omegaup,z_{1}-t\lambdaup^{-1}\omegaup,\ldots)$
on $\varprojlim F_{A|V}$ is the maximal equicontinuous factor.

Geometric realization expresses the underlying solenoidal nature of Pisot
tiling spaces. But tiling spaces, unlike solenoids, are not homogeneous.
_Asymptotic tilings_ are one kind of inhomogeneity collapsed out by geometric
realization. We say the tilings
$\gammaup,\gammaup^{\prime}\in\mathcal{T}_{\varphiup}$ are _forward (backward,
resp.) asymptotic_ provided $d(\gammaup-t,\gammaup^{\prime}-t)\rightarrow 0$
as $t\rightarrow\infty$ ($-\infty$, resp.).333There is a weakening of
asymptoticity, called regional proximality, with the property that
$\varg_{\varphiup}(\gammaup)=\varg_{\varphiup}(\gammaup^{\prime})$ if and only
if $\gammaup$ and $\gammaup^{\prime}$ are regionally proximal, see [Aus]. If
$\gammaup$ and $\gammaup^{\prime}$ are asymptotic, where
$\gammaup\neq\gammaup^{\prime}$, there is a unique $t$ so that $\gammaup-t$
and $\gammaup^{\prime}-t$ are periodic under inflation and substitution.
Moreover, the set $\mathcal{C}_{\varphiup}$ consisting of those tilings
$\gammaup\in\mathcal{T}_{\varphiup}$ for which there is a tiling
$\gammaup^{\prime}\neq\gammaup$ with $\gammaup$ and $\gammaup^{\prime}$
asymptotic and ${\Phi}$-periodic is finite and nonempty (see [BD1]). We will
call the elements of $\mathcal{C}_{\varphiup}$ _special tilings_.

Let $L:(V,\Sigma)\rightarrow(\mathbb{R}^{d},\mathbb{Z}^{d})$ be a linear
isomorphism, and
$F_{L}:V/\Sigma\rightarrow\mathbb{T}^{d}:=\mathbb{R}^{d}/\mathbb{Z}^{d}$ the
induced isomorphism. Also, let $M=M_{\varphiup}$ denote the integer matrix
representing the linear isomorphism $L\circ A|_{V}\circ L^{-1}$ in the
standard basis, and let $F_{M}:\mathbb{T}^{d}\hookleftarrow$ denote the
corresponding toral endomorphism.

The _branch locus_ of $\mathcal{T}_{\varphiup}$ is the set

$Br(\varphiup):=(F_{L}\circ\pi_{0}\circ\varg_{\varphiup})(\mathcal{C}_{\varphiup})\subset\mathbb{T}^{d},$

where $\pi_{0}:\varprojlim F_{A|V}\rightarrow V/\Sigma$ is projection onto the
$0^{th}$ coordinate. Note that $F_{M}(Br(\varphiup))=Br(\varphiup)$. Also note
that $Br(\varphiup)$ depends not only on $\varphiup$ but also on the choice of
$\\{u_{i}\\}$ in the definitions of $\varg_{\varphiup}$ and the linear
isomorphism $L$ made in the construction of $\varg_{\varphiup}$. Our main
theorem is that this dependence is limited.

######  Theorem 1.

Suppose that $\varphiup$ and $\psiup$ are primitive, aperiodic, Pisot
substitutions whose tiling spaces are homeomorphic. Then there are $d\times d$
integer matrices $S,T$, and $m_{0},\ m_{1}\in\mathbb{N}$ so that
$M_{\varphiup}^{m_{0}}=ST$, $M_{\psiup}^{m_{1}}=TS$, and translations
$\tauup_{0},\ \tauup_{1}$ on $\mathbb{T}^{d}$ so that

$Br(\psiup)=(\tauup_{0}\circ
F_{T})\left(Br(\varphiup)\right)\quad\text{and}\quad
Br(\varphiup)=(\tauup_{1}\circ F_{S})\left(Br(\psiup)\right)$

In case $\lambdaup$ is a _Pisot unit_ , that is, $\det(A|_{V})=\pm 1$;
calculations are simplified.

######  Corollary 2.

Suppose that $\varphiup$ and $\psiup$ are primitive and aperiodic Pisot
substitutions with $\lambdaup_{\varphiup}$ a Pisot unit of degree $d$, and
that $\mathcal{T}_{\varphiup}$ is homeomorphic with $\mathcal{T}_{\psiup}$.
Then there are a $T\in GL(d,\mathbb{Z})$ and a translation $\tauup$ on
$\mathbb{T}^{d}$ so that

$Br(\psiup)=\tauup\circ F_{T}(Br(\varphiup)).$

Before proving the theorem, we look at a few examples.

##  3\. Examples

######  Example 1.

Consider the substitution
$\displaystyle\varphiup:\begin{cases}1\quad\rightarrow&121\\\
2\quad\rightarrow&312\\\ 3\quad\rightarrow&213\end{cases}.\quad$ In this
instance,

$\displaystyle A=\left(\begin{array}[]{ccc}2&1&1\\\ 1&1&1\\\
0&1&1\end{array}\right)$, $\lambdaup=3$ and
$V=\mathbb{E}^{u}=\left\\{t\left(\begin{array}[]{c}3\\\ 2\\\
1\end{array}\right):t\in\mathbb{R}\right\\}$. We have
$\varv_{i}=\mathrm{pr}_{V}e_{i}=\displaystyle\frac{1}{6}\left(\begin{array}[]{c}3\\\
2\\\ 1\end{array}\right)$, which we denote by $v$, so that
$\Gamma=\mathbb{Z}v$. Now consider the strand $\gammaup$, fixed under
${\Phi}$,

$\gammaup=\left\\{\ldots,\sigmaup_{1}-2v,\sigmaup_{2}-v,\sigmaup_{1},\sigmaup_{2}+v,\sigmaup_{1}+2v,\ldots\right\\}.$

That is, $\gammaup$ follows the pattern of the fixed word $\cdots
312.121\cdots$ of $\varphiup$. Clearly, if $I$ and $I^{\prime}$ are two edges
of $\gammaup$ of type 1, then $\min I^{\prime}-\min I$ is of the form $2kv$,
where $k\in\mathbb{Z}$ (and $k=1$ occurs). Thus

$\Sigma=\left\langle\bigcup_{k\in\mathbb{Z}}(A|_{V})^{k}\Theta(1)\right\rangle\cap\Gamma=\left<\bigcup_{k\in\mathbb{Z}}(3^{k})(2v)\right>\cap\Gamma=2\Gamma=2\mathbb{Z}v.$

The transition vectors are (mod $\Sigma$) $\varw_{21}=v$, $\varw_{31}=v$, and
$\varw_{23}=0$. Choosing $u_{1}=0$, $u_{2}=v$, $u_{3}=v$, we have
$\varw_{ij}=u_{j}-u_{i}$ and $Au_{i}=3u_{i}=u_{i^{\prime}}$ (mod $\Sigma$),
where $\varphiup(i)=i^{\prime}\cdots$. Finally, we map $V/\Sigma$ onto
$\mathbb{R}/\mathbb{Z}=\mathbb{T}^{1}$ by $F_{L}$, where
$L:V\rightarrow\mathbb{R}$ is defined by $L(tv)=\frac{t}{2}v$. Then geometric
realization of $\mathcal{T}_{\varphiup}$ onto a 3-adic solenoid is given by
$\varg_{\varphiup}(\gammaup)=(\min I_{0}-u_{i_{0}},\min
I_{1}-u_{i_{1}},\ldots)$, where $I_{k}$ is an edge of ${\Phi}^{-k}(\gammaup)$
of type $i_{k}$ and $\min I_{k}$ is taken mod $\Sigma$. The branch locus of
$\varphiup$ is $(F_{L}\circ\pi_{0}\circ\varg_{\varphiup})(C_{\varphiup})$.

There are procedures for finding asymptotic tilings described in [BD1] and
[BDH], and an applet for this purpose can be found at [E1]. For this
$\varphiup$, there is a single pair of backward asymptotic tilings (of period
2 under ${\Phi}$) and a single pair of forward asymptotic tilings (fixed by
${\Phi}$) in
$\mathcal{C}_{\varphiup}=\\{\alphaup,\alphaup^{\prime},\betaup,\betaup^{\prime}\\}$,
where

$\alphaup:=\\{\ldots,\sigmaup_{2}-v,\sigmaup_{1},\sigmaup_{2}+v,\sigmaup_{1}+2v,\ldots\\}$

follows the pattern $\cdots 312.121\cdots$,

$\alphaup^{\prime}:=\\{\ldots,\sigmaup_{3}-v,\sigmaup_{1},\sigmaup_{2}+v,\sigmaup_{1}+2v,\ldots\\}$

follows the pattern $\cdots 213.121\cdots,$

$\betaup:=\\{\ldots,\sigmaup_{1}-v,\sigmaup_{3},\sigmaup_{1}+v,\sigmaup_{2}+2v,\ldots\\}$

follows the pattern $\cdots 121.312\cdots$ and

$\betaup^{\prime}:=\\{\ldots,\sigmaup_{1}-v,\sigmaup_{2},\sigmaup_{1}+v,\sigmaup_{3}+2v,\ldots\\}$

follows the pattern $\cdots 121.213\cdots$. Thus
$Br(\varphiup)=\\{0+\mathbb{Z},1/2+\mathbb{Z}\\}$. The periodic forward
asymptotic tilings are “halfway around” the tiling space from the periodic
backward asymptotic tilings. ∎

Note: One can show that $\Gamma/\Sigma\simeq\mathbb{Z}/h\mathbb{Z}$, where $h$
is the height of the substitution $\varphiup$ (see [D]), if $\varphiup$ is
constant length (as in Example 1) or, more generally, $\varphiup$ is Pisot of
degree 1. In this sense, the group $\Gamma/\Sigma$ is an extension of the
notion of the height of a constant length substitution to the higher degree
setting.

######  Example 2.

We will show that two particular tiling spaces are not homeomorphic by
considering their branch loci. The Matsumoto $K_{0}$-group ([M]) and the
crossing group ([BSw1] do not distinguish these spaces. Also, we have not been
able to distinguish these spaces by the generalized Bowen-Franks trace
([BSw1]) or the proximality structures described in [BD2]. The (unordered)
augmented cohomology groups of these spaces are isomorphic, but one can show
that the Pisot parts of the augmented dimension groups (see § 5) are not
order isomorphic.

Consider the substitutions

$\varphiup:\begin{cases}a&\to\,aaa^{12}b^{16}aa\\\
b&\to\,aba^{14}b^{14}ba\end{cases}\quad\text{and}\quad\psiup:\begin{cases}a&\to\,aaaa^{11}b^{16}aa\\\
b&\to\,aaba^{13}b^{14}ba\end{cases}.$

Here $A_{\varphiup}=A_{\psiup}=\begin{pmatrix}16&16\\\ 16&16\end{pmatrix}$,
$\varv_{\varphiup}=\varv_{\psiup}=\mathbb{E}^{u}=\left\\{t\left(\begin{array}[]{c}1\\\
1\end{array}\right):t\in\mathbb{R}\right\\}$,
$\varv_{1}=\varv_{2}=v=\begin{pmatrix}\frac{1}{2}\\\ \frac{1}{2}\end{pmatrix}$
for both $\varphiup$ and $\psiup$, and
$\Sigma_{\varphiup}=\Gamma_{\varphiup}=\Gamma_{\psiup}=\Sigma_{\psiup}=\mathbb{Z}v$.
In geometric realization all the $u_{i}$ can be taken to be 0\. Choosing
$L:tv\mapsto t$, one gets
$Br(\varphiup)=\\{1/31+\mathbb{Z},30/31+\mathbb{Z}\\}$,
$Br(\psiup)=\\{2/31+\mathbb{Z},30/31+\mathbb{Z}\\}$, and
$M=M_{\varphiup}=M_{\psiup}=(32)$. So, if $\mathcal{T}_{\varphiup}$ is
homeomorphic with $\mathcal{T}_{\psiup}$, there must be $S=(s)$ and $T=(t)$,
where $s,t\in\mathbb{Z}$, an $m\in\mathbb{N}$ so that $M^{m}=ST$, and a
translation $\tauup$ on $\mathbb{T}^{1}=\mathbb{R}/\mathbb{Z}$ so that
$\tauup\circ F_{T}Br(\varphiup)=Br(\psiup)$. The only possibility for $t$ is
$\pm 2^{k}$, where $k\in\mathbb{N}$. Then $\tauup\left((\pm
2^{k})\\{1/31,30/31\\}\right)=\\{2/31,30/31\\}(\textrm{mod}1)$ implies that
$(\pm 2^{k})(29/31)=(28/31)(\textrm{mod }1)$, hence that $(\pm
2^{k})(29)=28(\textrm{mod }31)$, or $\pm 2^{k+1}=3(\textrm{mod }31)$, which is
not the case for any $k\in\mathbb{N}$. It follows from Theorem 1 that
$\mathcal{T}_{\varphiup}$ and $\mathcal{T}_{\psiup}$ are not homeomorphic. ∎

######  Example 3.

Define $\varphiup$: $\begin{array}[]{cl}1&\rightarrow 1122111\\\ 2&\rightarrow
1211111221\end{array}$ and $\psiup$: $\begin{array}[]{cl}1&\rightarrow
1121211\\\ 2&\rightarrow 1211112211\end{array}.$

The matrix $A=A_{\varphiup}=A_{\psiup}=\begin{pmatrix}5&7\\\
2&3\end{pmatrix}$, so $\lambdaup=\lambdaup_{\varphiup}=\lambdaup_{\psiup}$ is
a Pisot unit and Corollary 2 applies. As $\lambdaup$ has degree 2,
$\Sigma_{\varphiup}=\Sigma_{\psiup}=\mathbb{Z}^{2}$. Taking $L$ to be the
identity map, geometrical realization for both $\mathcal{T}_{\varphiup}$ and
$\mathcal{T}_{\psiup}$ takes the form $\gammaup\mapsto\min I\ (\textrm{mod
}\mathbb{Z}^{2})$, where this last $I$ is an edge of $\gammaup$. Each of
$\varphiup$ and $\psiup$ has a pair of backward and a pair of forward
asymptotic tilings, fixed under inflation and substitution. Consider, for
example, the tilings for $\varphiup$ following the patterns

$\cdots 11221111\dot{1}22111\cdots$

and

$\cdots 11221111\dot{2}11111221\cdots,$

where the dot in each word above indicates the position of the origin. To
locate the corresponding strands (call them $\gammaup_{1}$ and
$\gammaup_{2}$), we seek $x\in\mathbb{R}^{2}$ so that

${\Phi}(\sigmaup_{1}+x)=\\{\sigmaup_{1}+x-e_{1},\sigmaup_{1}+x,\sigmaup_{2}+x+e_{1},\ldots\\}$

or, equivalently,

${\Phi}(\sigmaup_{2}+x)=\\{\sigmaup_{1}+x-e_{1},\sigmaup_{2}+x,\sigmaup_{1}+x+e_{2},\ldots\\}.$

Since

${\Phi}(\sigmaup_{1}+x)=\\{\sigmaup_{1}+Ax,\sigmaup_{1}+Ax+e_{1},\ldots\\}$

we must have $Ax=x-e_{1}$. This yields
$x=\left(\\!\\!\begin{array}[]{r}\frac{1}{3}\\\
-\frac{1}{3}\end{array}\right)$, a vertex of both $\gammaup_{1}$ and
$\gammaup_{2}$. Thus

$\varg_{\varphiup}(\gammaup_{1})=\varg_{\varphiup}(\gammaup_{2})=\begin{pmatrix}\,\,\frac{1}{3}\\\
\\!-\frac{1}{3}\end{pmatrix}=\begin{pmatrix}\frac{1}{3}\\\
\frac{2}{3}\end{pmatrix}(\textrm{mod }\mathbb{Z}^{2})\in Br(\varphiup).$

Similar calculations yield:

$Br(\varphiup)=\left\\{\left(\begin{array}[]{c}\frac{1}{3}\\\
\frac{2}{3}\end{array}\right)+\mathbb{Z}^{2},\begin{pmatrix}\frac{2}{3}\\\
\frac{1}{3}\end{pmatrix}+\mathbb{Z}^{2}\right\\}$

and

$Br(\psiup)=\left\\{\begin{pmatrix}\frac{1}{3}\\\
\frac{2}{3}\end{pmatrix}+\mathbb{Z}^{2},\begin{pmatrix}\frac{1}{2}\\\
0\end{pmatrix}+\mathbb{Z}^{2}\right\\}.$

Were $\mathcal{T}_{\varphiup}$ and $\mathcal{T}_{\psiup}$ homeomorphic, there
would be a $T\in GL(2,\mathbb{Z})$ and a translation $\tauup$ on
$\mathbb{T}^{2}$ so that $(\tauup\circ F_{T})(Br(\varphiup))=Br(\psiup)$. In
other words,

$\displaystyle F_{T}\left[\begin{pmatrix}\frac{1}{3}\\\
\frac{2}{3}\end{pmatrix}-\begin{pmatrix}\frac{2}{3}\\\
\frac{1}{3}\end{pmatrix}+\mathbb{Z}^{2}\right]=\pm\left[\begin{pmatrix}\frac{1}{3}\\\
\frac{2}{3}\end{pmatrix}-\begin{pmatrix}\frac{1}{2}\\\
0\end{pmatrix}+\mathbb{Z}^{2}\right].$

That is, $T\in GL(2,\mathbb{Z})$ would satisfy

$\frac{1}{3}\,T\begin{pmatrix}1\\\
2\end{pmatrix}-\frac{1}{6}\,\begin{pmatrix}1\\\
2\end{pmatrix}\in\mathbb{Z}^{2}.$

Since this is not possible, Corollary 2 implies that $\mathcal{T}_{\varphiup}$
and $\mathcal{T}_{\psiup}$ are not homeomorphic. ∎

######  Example 4.

$\varphiup:=\begin{cases}a\to aaa^{2}b^{4}cac\\\ b\to aba^{7}b^{6}c^{5}bbc\\\
c\to aaba^{6}b^{6}c^{3}cbc\end{cases}\qquad\psiup:=\begin{cases}a\to
aaab^{4}caac\\\ b\to aaba^{5}b^{5}c^{5}bbbac\\\ c\to
aabbaa^{4}b^{4}c^{3}cbbac\end{cases}$

with abelianization $\displaystyle A=\begin{pmatrix}5&8&8\\\ 4&9&8\\\
2&6&5\end{pmatrix}$ . The projection along $\begin{pmatrix}2\\\ 0\\\
\\!\\!-1\end{pmatrix}$ onto the Pisot subspace $V$ is given by
$\displaystyle\mathrm{pr}_{V}=\begin{pmatrix}0&1&0\\\ 0&1&0\\\
\frac{1}{2}&\\!\\!-\\!\frac{1}{2}&1\end{pmatrix}$ . In this case the lattices
$\Gamma$ and $\Sigma$ are equal, since return vectors span $\Gamma$ over
$\mathbb{Z}$ (the repeated letters $aa,\ bb,\ {\rm and}\ cc$ yield return
vectors $\varv_{i}=\mathrm{pr}_{V}(e_{i}),\ i=1,2,3$).

The set
$\displaystyle\\{\varv_{1},\varv_{1}+\varv_{2}\\}=\left\\{\begin{pmatrix}0\\\
0\\\ \frac{1}{2}\end{pmatrix},\linebreak\begin{pmatrix}1\\\ 1\\\
0\end{pmatrix}\right\\}$ is a basis for $\Sigma$ over $\mathbb{Z}$. In these
coordinates, $A|_{V}$ takes the form $\displaystyle M=\begin{pmatrix}5&16\\\
4&13\end{pmatrix}$.

There are one pair of fixed backward asymptotic tilings and two pairs of fixed
forward asymptotic tilings for both $\varphiup$ and $\psiup$. We solve the
equations for the branch locus points in a manner similar to the foregoing
examples:

$\begin{array}[]{c|c}\underline{Br(\varphiup)\ (\textrm{mod}\
\mathbb{Z}^{2})}&\underline{Br(\psiup)\ (\textrm{mod}\
\mathbb{Z}^{2})}\\\\[7.22743pt] (M-I)^{-1}\left(\\!\\!\begin{array}[]{r}-1\\\
0\end{array}\\!\right)=\begin{pmatrix}\frac{3}{4}\\\
\frac{3}{4}\end{pmatrix}&\ (M-I)^{-1}\left(\\!\\!\begin{array}[]{r}-2\\\
0\end{array}\\!\right)=\begin{pmatrix}\frac{1}{2}\\\
\frac{1}{2}\end{pmatrix}\\\\[21.68121pt]
(M-I)^{-1}\left(\\!\\!\begin{array}[]{r}2\\\
0\end{array}\\!\right)=\begin{pmatrix}\frac{1}{2}\\\
\frac{1}{2}\end{pmatrix}&(M-I)^{-1}\left(\\!\\!\begin{array}[]{r}3\\\
0\end{array}\\!\right)=\begin{pmatrix}\frac{3}{4}\\\
\frac{3}{4}\end{pmatrix}\\\\[21.68121pt]
(M-I)^{-1}\left(\\!\\!\begin{array}[]{r}1\\\
1\end{array}\\!\right)=\begin{pmatrix}\frac{1}{4}\\\
0\end{pmatrix}&(M-I)^{-1}\left(\\!\\!\begin{array}[]{r}1\\\
2\end{array}\\!\right)=\begin{pmatrix}\frac{1}{4}\\\
\frac{3}{4}\end{pmatrix}\end{array}$

There are infinitely many different triangles on the 2-torus having a given
set of three points as vertices. In case the vertices are the points of
$Br(\varphiup)$, we write the general edge vectors, up to integer translation
in the lift, as $(\frac{1}{4}+p,\frac{1}{2}+q)$ and
$(\frac{1}{4}+r,\frac{1}{4}+s)$. The area has the form

$\left|\frac{1}{16}-\frac{1}{8}+\frac{1}{4}(p+s+4ps-q-2r-4rq)\right|\text{.}$

Since $p,q,r,s$ are integers, the minimum area is $\frac{1}{16}$ (e.g.
$p=q=r=s=0$). The argument for $Br(\psiup)$ is similar, but the minimum area
is $\frac{1}{8}$. Were $\mathcal{T}_{\varphiup}$ and $\mathcal{T}_{\psiup}$
homeomorphic, the map $\tauup\circ T$ of Corollary 6 would preserve the areas
of these triangles. We conclude they are not homeomorphic. ∎

######  Example 5.

The pair

$\varphiup:=\begin{cases}a\to aabbcac\\\ b\to ababccbcc\\\ c\to
aaa^{4}b^{6}c^{8}\end{cases}\qquad\psiup:=\begin{cases}a\to caabbac\\\ b\to
cababcbcc\\\ c\to caaa^{4}b^{6}c^{7}\end{cases}$

provides an example of substitutions $\varphiup$ and $\psiup$ such that the
Perron-Frobenius eigenvalue $\lambdaup_{\varphiup}=\lambdaup_{\psiup}$ is
Pisot but not a unit. The reader can check that the branch locus $Br(\psiup)$
is colinear, whereas $Br(\varphiup)$ is not. By Theorem 1,
$\mathcal{T}_{\varphiup}$ and $\mathcal{T}_{\psiup}$ are not homeomorphic. ∎

##  4\. Proof of Theorem 1

Suppose that $\varphiup$ is a primitive and aperiodic substitution with
_language_

$\mathcal{L}_{\varphiup}:=\\{w\in\mathcal{A}^{*}:w\textrm{ is a factor of
}\varphiup^{n}(i)\textrm{ for some }i\in\mathcal{A},n\in\mathbb{N}\\}$

and _allowed bi-infinite words_

$X_{\varphiup}:=\\{\cdots x_{-1}x_{0}x_{1}\cdots\,|\
x_{n}\in\mathcal{A},x_{n}\cdots x_{n+k}\in\mathcal{L}_{\varphiup}\textrm{ for
}n\in\mathbb{Z},k\in\mathbb{N}\\}.$

Suppose that the finite subset
$W=\\{\varw_{1},\ldots,\varw_{k}\\}\subset\mathcal{L}_{\varphiup}$ has the
properties that (1) each element of $X_{\varphiup}$ can be uniquely factored
as a product of elements of $W$, and (2) for each $\varw_{i}\in W$,
$\varphiup(\varw_{i})$ can be uniquely factored as a product of elements of
$W$. We may then define a substitution
$\varphiup^{\prime}:\\{1,\ldots,k\\}\rightarrow\\{1,\ldots,k\\}^{*}$ by the
rule $\varphiup^{\prime}(i)=i_{1}\cdots i_{\ell}$ provided
$\varphiup(\varw_{i})=\varw_{i_{1}}\cdots\varw_{i_{\ell}}$. We will call such
a $\varphiup^{\prime}$ a _rewriting of $\varphiup$_, and the morphism
$\rhoup:\\{1,\ldots,k\\}\rightarrow\mathcal{A}^{*}$ given by
$\rhoup(i)=\varw_{i}$ the associated _rewriting morphism_. Note that
$\rhoup\circ\varphiup^{\prime}=\varphiup\circ\rhoup$. It is not hard to see
that $\varphiup^{\prime}$ is also primitive and aperiodic.

Suppose now that $\varphiup^{\prime}$ and $\varphiup$ are substitutions and
$\rhoup$ is a morphism with
$\rhoup\circ\varphiup^{\prime}=\varphiup\circ\rhoup$. Then the abelianizations
$A,A^{\prime}$ and $P$ of $\varphiup,\varphiup^{\prime}$ and $\rhoup$, resp.,
satisfy $PA^{\prime}=AP$. Consequently, if $w$ and $w^{\prime}$ are positive
right Perron-Frobenius eigenvectors of $A$ and $A^{\prime}$, resp., then
$Pw^{\prime}=aw$ for some $a>0$, and $A$ and $A^{\prime}$ have the same
Perron-Frobenius eigenvalue. Thus $\varphiup^{\prime}$ is Pisot if and only if
$\varphiup$ is. In this case, let $V$ and $V^{\prime}$ be the invariant Pisot
subspaces (with rational bases) corresponding to $A$ and $A^{\prime}$, resp.
Since $w\in PV^{\prime}\cap V$, $PV^{\prime}\cap V$ is a nontrivial rational
subspace of $V$. The characteristic polynomial of $A|_{V}$ is irreducible, and
$PV^{\prime}\cap V$ is $A$-invariant, which implies that $PV^{\prime}\cap V=V$
and $V\subset PV^{\prime}$. But $\dim V=d=\dim V^{\prime}$, hence
$P(V^{\prime})=V$ and $P|_{V^{\prime}}:V^{\prime}\rightarrow V$ is invertible.
Furthermore, if $\varphiup$ and $\varphiup^{\prime}$ are Pisot, there is a
natural continuous surjection
$\overline{\rhoup}:\mathcal{T}_{\varphiup^{\prime}}\rightarrow\mathcal{T}_{\varphiup}$
that conjugates ${\Phi}^{\prime}$ with ${\Phi}$, defined as follows: Given an
edge $I=\sigmaup_{1}^{\prime}+x$ in $V^{\prime}$, with $\rhoup(i)=i_{1}\cdots
i_{l}$ and $\varv_{j}:=\pi_{V}e_{j}$, let $\overline{\rhoup}(I)$ denote the
finite strand in $V$ defined by

$\overline{\rhoup}(I):=\\{\sigmaup_{i_{1}}+Px,\sigmaup_{i_{2}}+Px+\varv_{i_{1}},\ldots,\sigmaup_{i_{\ell}}+Px+\varv_{i_{1}}+\cdots+\varv_{i_{\ell-1}}\\}.$

If $\gammaup^{\prime}=\\{\sigmaup^{\prime}_{i_{k}}+x_{k}:k\in\mathbb{Z}\\}$ is
any strand in $V^{\prime}$, let $\overline{\rhoup}(\gammaup^{\prime})$ denote
the strand in $V$ made up of the finite substrands
$\overline{\rhoup}(\sigmaup_{i_{k}}+x_{k})$ given by

$\overline{\rhoup}(\gammaup^{\prime})=\bigcup_{k\in\mathbb{Z}}\overline{\rhoup}(\sigmaup^{\prime}_{i_{k}}+x_{k}).$

Observe that
$\overline{\rhoup}\circ{\Phi}^{\prime}={\Phi}\circ\overline{\rhoup}$. Since
$P\mathbb{E}^{u}_{\varphiup^{\prime}}=\mathbb{E}^{u}_{\varphiup}$, there is an
$R>0$ such that
$\overline{\rhoup}(\mathcal{T}_{\varphiup^{\prime}})\subset\mathcal{F}^{R}_{\varphiup}$.
Thus if $\gammaup^{\prime}\in\mathcal{T}_{\varphiup^{\prime}}$,

$\overline{\rhoup}(\gammaup^{\prime})=\overline{\rhoup}\left(({\Phi}^{\prime})^{n}({\Phi}^{\prime})^{-n}(\gammaup^{\prime})\right)={\Phi}^{n}(\overline{\rhoup}\left(({\Phi}^{\prime})^{-n}(\gammaup^{\prime})\right)\in{\Phi}^{n}\mathcal{F}^{R}_{\varphiup}$

for all $n\in\mathbb{N}$, so

$\overline{\rhoup}(\gammaup^{\prime})\in\bigcap_{n\in\mathbb{N}}{\Phi}^{n}(\mathcal{F}^{R}_{\varphiup})=\mathcal{T}_{\varphiup}.$

Continuity of $\overline{\rhoup}$ is clear, and from the fact that
$\overline{\rhoup}(\gammaup^{\prime}-t)=\overline{\rhoup}(\gammaup^{\prime})-at$
and minimality of the tiling flow, it follows that
$\overline{\rhoup}:\mathcal{T}_{\varphiup^{\prime}}\rightarrow\mathcal{T}_{\varphiup}$
is surjective.

Suppose that $V$ and $V^{\prime}$ are finite dimensional real vector spaces
and $\Sigma\subset V$, $\Sigma^{\prime}\subset V^{\prime}$ are lattices. By a
map $T:(V,\Sigma)\rightarrow(V^{\prime},\Sigma^{\prime})$, we will mean a
linear transformation $T:V\rightarrow V^{\prime}$ with
$T(\Sigma)\subset\Sigma^{\prime}$. Two maps
$A:(V,\Sigma)\rightarrow(V,\Sigma)$ and
$A^{\prime}:(V^{\prime},\Sigma^{\prime})\rightarrow(V^{\prime},\Sigma^{\prime})$
are _shift-equivalent_ , $A\sim_{SE}A^{\prime}$, provided there are maps
$T:(V,\Sigma)\rightarrow(V^{\prime},\Sigma^{\prime})$ and
$S:(V^{\prime},\Sigma^{\prime})\rightarrow(V,\Sigma)$ and natural numbers
$m,n$ so that $A^{m}=ST$, $(A^{\prime})^{n}=TS$, $A^{\prime}T=TA$, and
$SA^{\prime}=AS$; $S$ and $T$ _express the shift equivalence_. The relation
$\sim_{SE}$ is an equivalence relation, and if $A\sim_{SE}A^{\prime}$, then
$A^{k}\sim_{SE}(A^{\prime})^{k}$ for all $k\in\mathbb{N}$. Furthermore, if
$\hat{F}_{T}:\varprojlim F_{A}\rightarrow\varprojlim F_{A^{\prime}}$ is
defined by

$\hat{F}_{T}(x_{1}+\Sigma,x_{2}+\Sigma,\ldots)=(Tx_{1}+\Sigma^{\prime},Tx_{2}+\Sigma^{\prime},\ldots)$

(and $\hat{F}_{S},\hat{F}_{A^{\prime}}$ and $\hat{F}_{A}$ are defined
similarly), and if $\Sigma$ and $\Sigma^{\prime}$ are co-compact, then
$\hat{F}_{T}$ and $\hat{F}_{S}$ are topological isomorphisms of the solenoids
that conjugate the shifts $\hat{F}_{A}$ and $\hat{F}_{A^{\prime}}$.

######  Lemma 3.

Suppose that $\varphiup$ and $\varphiup^{\prime}$ are Pisot substitutions and
that either
(a) $\varphiup^{\prime}$ is a rewriting of $\varphiup$ with rewriting morphism
$\rhoup$, or
(b) there are morphisms $\alphaup,\betaup$ so that
$\varphiup=\alphaup\circ\betaup$ and
$\varphiup^{\prime}=\betaup\circ\alphaup$.
Let $A,V$ and $\Sigma$ ($A^{\prime},V^{\prime}$ and $\Sigma^{\prime}$, resp.)
be the abelianization, invariant Pisot subspace and return lattice for
$\varphiup$ ($\varphiup^{\prime}$, resp.). Then, in the case of (a),
(1a)
$\overline{\rhoup}:\mathcal{T}_{\varphiup^{\prime}}\rightarrow\mathcal{T}_{\varphiup}$
is a homeomorphism,
or, in the case of (b),
(1b)
$\overline{\alphaup}:\mathcal{T}_{\varphiup^{\prime}}\rightarrow\mathcal{T}_{\varphiup}$
and
$\overline{\betaup}:\mathcal{T}_{\varphiup}\rightarrow\mathcal{T}_{\varphiup^{\prime}}$
are homeomorphisms.
Also,
(2) the maps $A|_{V}:(V,\Sigma)\rightarrow(V,\Sigma)$ and
$A^{\prime}|_{V^{\prime}}:(V^{\prime},\Sigma^{\prime})\rightarrow(V^{\prime},\Sigma^{\prime})$
are shift equivalent.
Furthermore, if
$\varg_{\varphiup}:\mathcal{T}_{\varphiup}\rightarrow\varprojlim F_{A|_{V}}$
and
$\varg_{\varphiup^{\prime}}:\mathcal{T}_{\varphiup^{\prime}}\rightarrow\varprojlim
F_{A^{\prime}|_{V^{\prime}}}$ are geometric realizations onto the natural
solenoids, and $T:(V,\Sigma)\rightarrow(V^{\prime},\Sigma^{\prime})$ and
$S:(V^{\prime},\Sigma^{\prime})\rightarrow(V,\Sigma)$ express the shift
equivalence in (2), then there are translations $\tauup$ and $\tauup^{\prime}$
on $\varprojlim F_{A|_{V}}$ and $\varprojlim F_{A^{\prime}|_{V^{\prime}}}$ so
that, in the case of (a),
(3a)
$\varg_{\varphiup}\circ\overline{\rhoup}=\tauup\circ\hat{F}_{S}\circ\varg_{\varphiup^{\prime}}$,
and
$\varg_{\varphiup^{\prime}}\circ\overline{\rhoup}^{\,-1}=\tauup^{\prime}\circ\hat{F}_{T}\circ\hat{F}^{m}_{A|_{V}}\circ\varg_{\varphiup}$
for some $m\in\mathbb{N}$,
or, in the case of (b),
(3b)
$\varg_{\varphiup}\circ\overline{\alphaup}=\tauup\circ\hat{F}_{S}\circ\varg_{\varphiup^{\prime}}$,
and
$\varg_{\varphiup^{\prime}}\circ\overline{\betaup}=\tauup^{\prime}\circ\hat{F}_{T}\circ\varg_{\varphiup}$.

Proof: Suppose that $\varphiup^{\prime}$ is a rewriting of $\varphiup$ with
rewriting morphism $\rhoup$ having abelianization $P$, and suppose that
$\gammaup^{\prime}$,
$\gammaup^{\prime\prime}\in\mathcal{T}_{\varphiup^{\prime}}$ are such that
$\overline{\rhoup}(\gammaup^{\prime})=\overline{\rhoup}(\gammaup^{\prime\prime})$.
Let $w^{\prime}$ and $w^{\prime\prime}$ be the words of
$X_{\varphiup^{\prime}}$ spelled out by $\gammaup^{\prime}$ and
$\gammaup^{\prime\prime}$, where some location is chosen for the decimal point
indicating the location of the $0^{th}$ letter. Then $\rhoup(\varw^{\prime})$
and $\rhoup(\varw^{\prime\prime})$ agree, up to shift, so that $w^{\prime}$
and $w^{\prime\prime}$ must agree, up to shift, by the uniqueness of
factorization in $X_{\varphiup}$ in the definition of rewriting. It follows
that $\gammaup^{\prime}=\gammaup^{\prime\prime}-t$ for some $t$. But then
$\overline{\rhoup}(\gammaup^{\prime\prime})=\overline{\rhoup}(\gammaup^{\prime})=\overline{\rhoup}(\gammaup^{\prime\prime}-t)=\overline{\rhoup}(\gammaup^{\prime\prime})-at$,
hence $t=0$ and $\gammaup^{\prime}=\gammaup^{\prime\prime}$. That is,
$\overline{\rhoup}$ is a homeomorphism.

In case $\varphiup=\alphaup\circ\betaup$ and
$\varphiup^{\prime}=\betaup\circ\alphaup$, we have that
$\overline{\alphaup}\circ\overline{\betaup}={\Phi}$ and
$\overline{\betaup}\circ\overline{\alphaup}={\Phi}^{\prime}$, so both
$\overline{\alphaup}$ and $\overline{\betaup}$ must be homeomorphisms.

Again, suppose that $\varphiup^{\prime}$ is a rewriting of $\varphiup$. If
$v\in\Theta^{\prime}(i)$ is a return vector for $\varphiup^{\prime}$, say
$\gammaup^{\prime}\in\mathcal{T}_{\varphiup^{\prime}}$ has edges $I$ and $I+v$
of type $i$, then
$\overline{\rhoup}\gammaup^{\prime}\in\mathcal{T}_{\varphiup}$ has edges $J$
and $J+Pv$ of type $i_{1}$, where $\rhoup(i)=i_{1}\cdots$. Thus
$Pv\in\Theta(i_{1})$ is a return vector for $\varphiup$. It follows that
$P\Sigma^{\prime}\subset\Sigma$.

While it is not necessarily the case that $P\Sigma^{\prime}=\Sigma$, we shall
see that $A^{m}\Sigma\subset P\Sigma^{\prime}$ for some $m\in\mathbb{N}$. To
this end, suppose that $I$ is an edge in $V$ of type $i$, and $I$, $I+v$ are
edges of $\gammaup\in\mathcal{T}_{\varphiup}$ with $v\neq 0$. Let
$\gammaup^{\prime}:=\overline{\rhoup}^{\,-1}(\gammaup)$, and for each
$n\in\mathbb{N}$, let $\delta^{n}$ and $\eta^{n}$ be maximal substrands of
$({\Phi}^{\prime})^{n}(\gammaup^{\prime})$ with the property that
$\overline{\rhoup}(\delta^{n})\subset{\Phi}^{n}(I)$ and
$\overline{\rhoup}(\eta^{n})\subset{\Phi}^{n}(I+v)$. If
$x^{n},y^{n}\in\mathcal{L}_{\varphiup^{\prime}}$ are the words corresponding
to $\delta^{n}$, $\eta^{n}$, resp., then we have the factorizations
$p^{n}\rhoup(x^{n})s^{n}$ and $q^{n}\rhoup(y^{n})r^{n}$ of $\varphiup^{n}(i)$
in which the words $p^{n}$, $s^{n}$, $r^{n}$,
$q^{n}\in\mathcal{L}_{\varphiup}$ are of bounded length. Putting a decimal
point (to mark the position of the $0^{th}$ coordinate) on the left of
$\varphiup^{n}(i)$, we may choose $n_{k}\rightarrow\infty$ and
$m_{k}\in\mathbb{N}$, with $m_{k}$ on the order of $\frac{n_{k}}{2}$, so that
$\sigmaup^{m_{k}}(\varphiup^{n_{k}}(i))$ converges to a bi-infinite word $w\in
X_{\varphiup}$ (here $\sigmaup$ is the shift that takes $.abc\cdots$ to
$a.bc\cdots$). Let $i^{k}$ be the letter (in
$\mathcal{A}=\mathcal{A}_{\varphiup}$) immediately to the right of the decimal
point in $\sigmaup^{m_{k}}(\varphiup^{n_{k}}(i))$, and let $x_{0}^{n_{k}}$ and
$y_{0}^{n_{k}}$ denote the letters (in
$\mathcal{A}^{\prime}=\mathcal{A}_{\varphiup^{\prime}}$) of $x^{n_{k}}$ and
$y^{n_{k}}$, resp., so that $\rhoup(x_{0}^{n_{k}})$ and
$\rhoup(y_{0}^{n_{k}})$ contain (the identified occurrence of) $i^{k}$.
Putting a decimal point immediately to the left of $x_{0}^{n_{k}}$ and
$y_{0}^{n_{k}}$ in $x^{n_{k}}$ and $y^{n_{k}}$, resp., we may choose a
subsequence $n_{k_{\ell}}$ so that $x^{n_{k_{\ell}}}$ and $y^{n_{k_{\ell}}}$
converge to bi-infinite words, say $x$ and $y$, resp., in
$X_{\varphiup^{\prime}}$. Now $\rhoup(x)$ and $\rhoup(y)$ give factorings of
$\varw\in X_{\varphiup}$ by the words $\varw_{i}=\rhoup(j)$, for
$j\in\mathcal{A}^{\prime}$. By uniqueness, $x=y$. Thus for large $\ell$,
$x_{0}^{n_{k_{\ell}}}=y_{0}^{n_{k_{\ell}}}$. Pick such an $\ell$, let
$m={n_{k_{\ell}}}$, $j=x_{0}^{m}=y_{0}^{m}$, and $J,J+\varv^{\prime}$ be the
edges of the substrands $\delta^{m}$, $\eta^{m}$ of
$({\Phi}^{\prime})^{m}(\gammaup^{\prime})$ corresponding to $x_{0}^{m}$ and
$y_{0}^{m}$, resp. Then $P(\varv^{\prime})=A^{m}\varv$. That is, given
$v\in\Theta(i)$, a return vector for $\varphiup$, there is
$\varv^{\prime}\in\Theta^{\prime}(j)$, a return vector for
$\varphiup^{\prime}$, with
$\varv=(A|_{V})^{-m}P(\varv^{\prime})=P(A^{\prime}|_{V^{\prime}})^{-m}(\varv^{\prime})$.

Now suppose that $\varv_{1},\ldots,\varv_{d}$ is a basis for $\Sigma$, the
return lattice for $\varphiup$. Each $\varv_{j}$ is then an integer
combination of elements of $\bigcup_{k\in\mathbb{Z}}(A|_{V})^{k}\Theta(i)$.
The preceding argument shows that there is an $m\in\mathbb{N}$ so that
$A^{m}\varv_{j}$ is in $P\Sigma^{\prime}$ for all $j=1,\ldots,d$. That is,
$A^{m}\Sigma\subset P\Sigma^{\prime}$. Let $S:=P|_{V^{\prime}}$ and
$T:=(P|_{V^{\prime}})^{-1}(A|_{V})^{m}$ (recall that $P|_{V^{\prime}}$ is
invertible because $\varphiup$ and $\varphiup^{\prime}$ are Pisot). We have
$S:(V^{\prime},\Sigma^{\prime})\rightarrow(V,\Sigma)$,
$T:(V,\Sigma)\rightarrow(V^{\prime},\Sigma^{\prime})$,
$A|_{V}S=SA^{\prime}|_{V^{\prime}}$, $TA|_{V}=A^{\prime}|_{V^{\prime}}T$,
$ST=(A|_{V})^{m}$, and $TS=(A^{\prime}|_{V^{\prime}})^{m}$.

Still in the rewriting case, suppose that
$\varg_{\varphiup}:\mathcal{T}_{\varphiup}\rightarrow\varprojlim F_{A|_{V}}$
and
$\varg_{\varphiup^{\prime}}:\mathcal{T}_{\varphiup^{\prime}}\rightarrow\varprojlim
F_{A^{\prime}|_{V^{\prime}}}$ are geometric realizations onto the natural
solenoids defined by the selection of $\\{u_{i}\\}$ in $\Gamma/\Sigma$ and
$\\{u^{\prime}_{i}\\}$ in $\Gamma^{\prime}/\Sigma^{\prime}$, resp. (see the
definition of geometric realization in §2). Let
$\\{\varw_{ij}\\}\subset\Gamma/\Sigma$ and
$\\{w^{\prime}_{ij}\\}\subset\Gamma^{\prime}/\Sigma^{\prime}$ denote the
corresponding transition vectors (so $\varw_{ij}=u_{j}-u_{i}$ and
$w^{\prime}_{ij}=u^{\prime}_{j}-u^{\prime}_{i}$, for all $i,j$). Let
$\gammaup^{\prime}\in\mathcal{T}_{\varphiup^{\prime}}$ have edge $I$ of type
$i$. Then $\varg_{\varphiup^{\prime}}(\gammaup^{\prime})$ has
$0^{th}$coordinate

$(\varg_{\varphiup^{\prime}}(\gammaup^{\prime}))_{0}=(\min
I+\Sigma^{\prime})-u^{\prime}_{i}$

in $V^{\prime}/\Sigma^{\prime}.$ The strand
$\overline{\rhoup}(\gammaup^{\prime})$ in $\mathcal{T}_{\varphiup}$ has an
edge $J$ of type $i_{1}$, where $\rhoup(i)=i_{1}\cdots$, with $\min J=P\min
I$. Thus

$(\varg_{\varphiup}(\overline{\rhoup}(\gammaup^{\prime})))_{0}=((P\min
I)+\Sigma)-u_{i_{1}}.$

We have

$\displaystyle F_{P}((\varg_{\varphiup^{\prime}}(\gammaup^{\prime}))_{0})$
$\displaystyle=$ $\displaystyle\left((P\min
I\right)+\Sigma)-F_{P}u^{\prime}_{i}$ $\displaystyle=$
$\displaystyle\left(\varg_{\varphiup}(\overline{\rhoup}(\gammaup^{\prime}))\right)_{0}+u_{i_{1}}-F_{P}u^{\prime}_{i}.$

We claim that $u_{i_{1}}-F_{P}u^{\prime}_{i}$ is independent of
$i\in\mathcal{A}^{\prime}$. Indeed,

$\displaystyle(u_{i_{1}}-F_{P}u^{\prime}_{i})-(u_{j_{1}}-F_{P}u^{\prime}_{j})$
$\displaystyle=$
$\displaystyle(u_{i_{1}}-u_{j_{1}})-(F_{P}(u^{\prime}_{i}-u^{\prime}_{j}))$
$\displaystyle=$ $\displaystyle\varw_{j_{1}i_{1}}-F_{P}(w^{\prime}_{ji}).$

But if $\gammaup^{\prime}\in\mathcal{T}_{\varphiup^{\prime}}$ has edge $I$ of
type $i$ and edge $J$ of type $j$, so that $(\min J-\min
I)+\Sigma=w^{\prime}_{ji}$, then $\overline{\rhoup}(\gammaup^{\prime})$ has
corresponding edges of type $i_{1}$ and $j_{1}$ with initial vertices
differing by $P(\min J)-P(\min I)$. That is,
$\varw_{j_{1}i_{1}}=F_{P}(\varw^{\prime}_{ji})$. It follows that if
$\tauup_{1}$ is translation by $F_{P}u^{\prime}_{i}-u_{i_{1}}$, then

$(\varg_{\varphiup}\circ\rhoup)_{0}=\tauup_{1}\circ
F_{P|_{V^{\prime}}}\circ(\varg_{\varphiup^{\prime}})_{0}.$

Similarly, $k^{th}$ coordinates satisfy

$(\varg_{\varphiup}\circ\rhoup)_{k}=\tauup_{1}\circ
F_{P|_{V^{\prime}}}\circ(\varg_{\varphiup^{\prime}})_{k}.$

Note that

$\displaystyle F_{A|_{V}}(u_{i_{1}}-F_{P|_{V^{\prime}}}u^{\prime}_{i})$
$\displaystyle=$ $\displaystyle
F_{P|_{V^{\prime}}}u_{i_{1}}-F_{P|_{V^{\prime}}}F_{A|_{V^{\prime}}}u^{\prime}_{i}$
$\displaystyle=$ $\displaystyle u_{i_{1}}-F_{P|_{V^{\prime}}}u^{\prime}_{i}$

from the normalization requirement on the $\\{u_{i}\\},\\{u^{\prime}_{i}\\}$,
so translation by this same element in each coordinate in $\varprojlim
F_{A|_{V}}$ defines a translation $\tauup$ on $\varprojlim F_{A|_{V}}$. We
have

$\varg_{\varphiup}\circ\overline{\rhoup}=\tauup\circ\hat{F}_{P|_{V^{\prime}}}\circ\varg_{\varphiup^{\prime}}.$

Thus with $S=P|_{V^{\prime}}$ and $T=(P|_{V^{\prime}})^{-1}\circ(A|_{V})^{m}$,
we have conclusion (3a).

The proofs of (2), in case $\varphiup=\alphaup\circ\betaup$ and
$\varphiup^{\prime}=\betaup\circ\alphaup$, and (3b) are similar (although more
straightforward). ∎

With the notation of Lemma 3, and with $C_{\varphiup}$ and
$C_{\varphiup^{\prime}}$ the collections of special tilings for $\varphiup$
and $\varphiup^{\prime}$ resp., note that $\overline{\rhoup}$,
$\overline{\alphaup}$, and $\overline{\betaup}$ determine bijections between
$C_{\varphiup}$ and $C_{\varphiup^{\prime}}$, and $\hat{F}_{A|_{V}}$ maps
$\varg_{\varphiup}(C_{\varphiup})$ bijectively onto itself. This yields the
following consequence of Lemma 3.

######  Corollary 4.

If $\varphiup$ and $\varphiup^{\prime}$ are Pisot substitutions with one a
rewriting of the other, or for which there are morphisms $\alphaup,\betaup$
with $\varphiup=\alphaup\circ\betaup$ and
$\varphiup^{\prime}=\betaup\circ\alphaup$, then
$A|_{V}:(V,\Sigma)\rightarrow(V,\Sigma)$ and
$A^{\prime}|_{V^{\prime}}:(V^{\prime},\Sigma^{\prime})\rightarrow(V^{\prime},\Sigma^{\prime})$
are shift equivalent, expressed by maps
$S:(V^{\prime},\Sigma^{\prime})\rightarrow(V,\Sigma)$ and
$T:(V,\Sigma)\rightarrow(V^{\prime},\Sigma^{\prime})$, with

$\tauup\circ
F_{T}\circ\pi_{0}\circ\varg_{\varphiup}(C_{\varphiup})=\pi_{0}\circ\varg_{\varphiup^{\prime}}(C_{\varphiup^{\prime}})$

and

$\tauup^{\prime}\circ
F_{S}\circ\pi_{0}\circ\varg_{\varphiup^{\prime}}(C_{\varphiup^{\prime}})=\pi_{0}\circ\varg_{\varphiup}(C_{\varphiup}).$

Let us say that the substitutions $\varphiup$ and $\psiup$ are in the same
rewriting class if there are substitutions $\varphiup_{0}=\varphiup,\
\varphiup_{1},\ \dots,\varphiup_{n}=\psiup$ with the property that for each
$i=0,\dots,n-1$, one of $\varphiup_{i}$ and $\varphiup_{i+1}$ is a rewriting
of the other. The proof of the rigidity result in [BSw2] (see Theorem 7 in the
next section) establishes the following:

######  Lemma 5.

Suppose that $\varphiup$ and $\psiup$ are substitutions such that
$\mathcal{T}_{\varphiup}$ is homeomorphic with $\mathcal{T}_{\psiup}$. Then
there are $m,n\in\mathbb{N}$ and substitutions $\varphiup^{\prime}$ and
$\psiup^{\prime}$ such that (1) $\varphiup^{\prime}$ and $\varphiup^{m}$ are
in the same rewriting class, as are $\psiup^{\prime}$ and $\psiup^{n}$, and
(2) there are morphisms $\alphaup$ and $\betaup$ such that
$\varphiup^{\prime}=\alphaup\circ\betaup$ and
$\psiup^{\prime}=\betaup\circ\alphaup$.

Proof: See [BSw2, Theorem 2.1]. ∎

To prove Theorem 1, let $\varphiup$ and $\psiup$ be Pisot of degree $d$, with
geometric realizations $\varg_{\varphiup}$ and $\varg_{\psiup}$ of
$\mathcal{T}_{\varphiup}$ and $\mathcal{T}_{\psiup}$. Fix isomorphisms
$L_{\varphiup}:(V_{\varphiup},\Sigma_{\varphiup})\rightarrow(\mathbb{R}^{d},\mathbb{Z}^{d})$
and
$L_{\psiup}:(V_{\psiup},\Sigma_{\psiup})\rightarrow(\mathbb{R}^{d},\mathbb{Z}^{d})$,
and let
$Br(\varphiup)=F_{L_{\varphiup}}\circ\pi_{0}(\varg_{\varphiup}(C_{\varphiup}))$
and $Br(\psiup)=F_{L_{\psiup}}\circ\pi_{0}(\varg_{\psiup}(C_{\psiup}))$ be the
corresponding branch loci. Let $m,n,\varphiup^{\prime}$ and $\psiup^{\prime}$
be as in Lemma 5. As $\mathcal{T}_{\varphiup}=\mathcal{T}_{\varphiup^{m}}$ and
$\mathcal{T}_{\psiup}=\mathcal{T}_{\psiup^{n}}$, we may take
$\varg_{\varphiup^{m}}=\varg_{\varphiup}$,
$\varg_{\psiup^{n}}=\varg_{\psiup}$, $L_{\varphiup^{m}}=L_{\varphiup}$ and
$L_{\psiup^{n}}=L_{\psiup}$, so that $Br(\varphiup^{m})=Br(\varphiup)$ and
$Br(\psiup^{m})=Br(\psiup)$. Apply Corollary 4 repeatedly to get translations
$\eta$ and $\hat{\eta}$ (this last determined by $\eta$ and $L_{\psiup}$), and
a map
$T_{1}:(V_{\varphiup},\Sigma_{\varphiup})\rightarrow(V_{\psiup},\Sigma_{\psiup})$
for which there are $k,\ell\in\mathbb{N}$ and a map
$S_{1}:(V_{\psiup},\Sigma_{\psiup})\rightarrow(V_{\varphiup},\Sigma_{\varphiup})$
such that

$S_{1}T_{1}=(A_{\varphiup}^{m}|_{V_{\varphiup}})^{k},\quad
T_{1}S_{1}=(A_{\psiup}^{n}|_{V_{\psiup}})^{\ell}$

and

$\displaystyle Br(\psiup)$ $\displaystyle=$ $\displaystyle
F_{L_{\psiup}}\left((\varg_{\psiup^{n}}(C_{\psiup^{n}}))_{0}\right)$
$\displaystyle=$ $\displaystyle F_{L_{\psiup}}(\eta\circ
F_{T_{1}}\left(\varg_{\varphiup^{m}}(C_{\varphiup^{m}}))_{0}\right)$
$\displaystyle=$ $\displaystyle F_{L_{\psiup}}\left(\eta\circ
F_{T_{1}}\left(L^{-1}_{\varphiup}(Br(\varphiup)\right)\right)$
$\displaystyle=$ $\displaystyle\hat{\eta}\circ
F_{L_{\psiup}T_{1}L^{-1}_{\varphiup}}.$

If $M_{\varphiup}=L_{\varphiup}A_{\varphiup}L^{-1}_{\varphiup}$,
$M_{\psiup}=L_{\psiup}A_{\psiup}L^{-1}_{\psiup}$,
$T=L_{\psiup}T_{1}L^{-1}_{\varphiup}$, and
$S=L_{\varphiup}S_{1}L^{-1}_{\psiup}$ (expressed as matrices in the standard
basis on $\mathbb{R}^{d}$), $m_{0}=m^{k}$ and $\tauup_{0}=\hat{\eta}$, then
$Br(\psiup)=\tauup_{0}\circ F_{T}(Br(\varphiup))$ with
$ST=(M_{\varphiup})^{m_{0}}$. Similarly, if $m_{1}=n^{\ell}$, there is a
translation $\tauup_{1}$ with $Br(\varphiup)=\tauup_{1}\circ
F_{S}\left((Br(\psiup)\right)$ and $TS=(M_{\psiup})^{m_{1}}$. ∎

##  5\. The Pisot Part of the Augmented Dimension Group

In this section we give a cohomological interpretation of Theorem 1. For a
Pisot substitution $\varphiup$ of degree $d$, let $L$ and $M$ be as in the
definition of $Br(\varphiup)$. The induced
$F_{M}:H^{1}(\mathbb{T}^{d};\mathbb{R})\hookleftarrow$ is then a vector space
isomorphism with simple eigenvalue $\lambdaup=\lambdaup_{\varphiup}$ and a
$(d-1)$-dimensional invariant subspace complementary to the eigenspace of
$\lambdaup$. This $(d-1)$-dimensional space splits
$H^{1}(\mathbb{T}^{d};\mathbb{R})$ into two closed invariant half-spaces, one
of which contains the cocycle dual to $(F_{L})_{*}([\omegaup])$, where
$[\omegaup]$ is the 1-cycle generated by the positive eigenvector $\omegaup$
of $A=A_{\varphiup}$. This half-space is called the _positive cone_ in
$H^{1}(\mathbb{T}^{d};\mathbb{R})$ and is denoted by
$H^{1}(\mathbb{T}^{d};\mathbb{R})^{+}$. The inclusion
$\imath:\mathbb{Z}\to\mathbb{R}$ induces $\imath^{*}:H^{1}(\mathbb{T}^{d})\to
H^{1}(\mathbb{T}^{d};\mathbb{R})$ (an unspecified coefficient ring is
understood to be $\mathbb{Z}$), and we define the positive cone in
$H^{1}(\mathbb{T}^{d})$ to be
$H^{1}(\mathbb{T}^{d})^{+}:=(\imath^{*})^{-1}(H^{1}(\mathbb{T};\mathbb{R})^{+})$.
Likewise, the positive cone in $H^{1}(\mathbb{T}^{d},Br(\varphiup))$ is

$H^{1}(\mathbb{T}^{d},Br(\varphiup))^{+}:=(\jmath^{*})^{-1}\left(H^{1}(\mathbb{T}^{d})^{+}\right),$

$\jmath^{*}:H^{1}(\mathbb{T}^{d},Br(\varphiup))\to H^{1}(\mathbb{T}^{d})$ the
natural homomorphism.

We have

$F_{M}^{*}:\left(H^{1}(\mathbb{T}^{d},Br(\varphiup)),(H^{1}(\mathbb{T}^{d},Br(\varphiup))^{+}\right)\to\left(H^{1}(\mathbb{T}^{d},Br(\varphiup)),(H^{1}(\mathbb{T}^{d},Br(\varphiup))^{+}\right),$

and we define
$PDG(\varphiup):=\varprojlim\left(F_{B}^{*}:\,H^{1}\left(\mathbb{T}^{d},Br(\varphiup)\right)\to
H^{1}\left(\mathbb{T}^{d},Br(\varphiup)\right)\right)$, with positive cone

$PDG(\varphiup)^{+}:=\left\\{[(k,g)]\in PDG(\varphiup)\,:\ g\in
H^{1}\left(\mathbb{T}^{d},Br(\varphiup)\right)^{+}\right\\}$

The _shift isomorphism_ on $\left(PDG(\varphiup),PDG(\varphiup)^{+}\right)$ is
the ordered isomorphism given by
$[(k,g)]\mapsto\left[\left(k,F_{M}^{*}(g)\right)\right]$.

A homeomorphism of tiling spaces,
$f:\mathcal{T}_{\varphiup}\to\mathcal{T}_{\psiup}$, is _orientation preserving
(reversing)_ provided it takes the positive flow direction in
$\mathcal{T}_{\varphiup}$ to the positive (negative) flow direction in
$\mathcal{T}_{\psiup}$; that is, the function
$s:\,\mathcal{T}_{\varphiup}\times\mathbb{R}^{+}\to\mathbb{R}$ defined by
$f(\gammaup-t)=f(\gammaup)-s(\gammaup,t)$ is positive (negative).

######  Theorem 6.

If $\varphiup$ and $\psiup$ are Pisot substitutions, and the tiling spaces
$\mathcal{T}_{\varphiup}$ and $\mathcal{T}_{\psiup}$ are orientation
preserving (reversing) homeomorphic, then there is an ordered isomorphism
between $\displaystyle\left(PDG(\varphiup),PDG(\varphiup)^{+}\right)$ and
$\displaystyle\left(PDG(\psiup),PDG(\psiup)^{+}\right)\
\left({\mathrm{r}esp.}\
\displaystyle\left(PDG(\psiup),-PDG(\psiup)^{+}\right)\right)$ that conjugates
some positive powers of the shift isomorphisms.

Proof: Let $T,\ S,\ \tauup_{0},\ \tauup_{1},\ m_{0},\ {\rm and}\ m_{1}$ be as
in Theorem 1. The commuting diagram

$\displaystyle H^{1}\left(\mathbb{T}^{d},Br(\varphiup)\right)$$\displaystyle
F_{M_{\varphiup}^{m_{0}}}^{*}$$\displaystyle
H^{1}\left(\mathbb{T}^{d},Br(\varphiup)\right)$$\displaystyle(\tauup_{1}\circ
F_{S})^{*}$$\displaystyle(\tauup_{1}\circ
F_{S})^{*}$$\displaystyle(\tauup_{0}\circ F_{T})^{*}$$\displaystyle
H^{1}\left(\mathbb{T}^{d},Br(\psiup)\right)$$\displaystyle
H^{1}\left(\mathbb{T}^{d},Br(\psiup)\right)$$\displaystyle
F_{M_{\psiup}^{m_{1}}}^{*}$

induces an isomorphism
$\displaystyle\left[(k,g))\right]\to\left[(k,(\tauup_{1}\circ
F_{S})^{*}(g)\right]$ that conjugates the ${m_{1}}^{\rm th}$ power of the
shift on $PDG(\psiup)$ with the ${m_{0}}^{\rm th}$ power of the shift on
$PDG(\varphiup)$. Since the order is dynamically defined, this conjugacy alone
guarantees that order is either preserved or reversed. If
$\mathcal{T}_{\varphiup}$ and $\mathcal{T}_{\psiup}$ are orientation
preserving homeomorphic, then $S$ and $T$ take $L_{\psiup}(\omegaup_{\psiup})$
and $L_{\varphiup}(\omegaup_{\varphiup})$ to positive multiples of
$L_{\varphiup}(\omegaup_{\varphiup})$ and $L_{\psiup}(\omegaup_{\psiup})$,
resp. (here, $L_{\varphiup}$ and $L_{\psiup}$ denote the $L$ in the definition
of $Br(\varphiup)$ and $Br(\psiup)$, and $\omegaup_{\varphiup}$,
$\omegaup_{\psiup}$ are positive eigenvectors of $A_{\varphiup}$,
$A_{\psiup}$, resp.). Thus $(\displaystyle\tauup_{1}\circ
F_{S})^{*}\left(H^{1}(\mathbb{T}^{d},Br(\varphiup))^{+}\right)=H^{1}(\mathbb{T}^{d},Br(\psiup))^{+}$.

If $\mathcal{T}_{\varphiup}$ and $\mathcal{T}_{\psiup}$ are orientation
reversing homeomorphic, then $\mathcal{T}_{\varphiup}$ and
$\mathcal{T}_{\overline{\psiup}}$ are orientation preserving homeomorphic,
$\overline{\psiup}$ being the reverse of $\psiup$ (if $\psiup(i)=i_{1}\cdots
i_{k}$ then $\overline{\psiup}(i)=i_{k}\cdots i_{1}$). The above applied to
$(PDG(\varphiup),PDG(\varphiup)^{+})$ and $(PDG(\psiup),PDG(\psiup)^{+})$
together with the isomorphism
$(PDG(\overline{\psiup}),PDG(\overline{\psiup})^{+})\to(PDG(\psiup),-PDG(\psiup)^{+})$
induced by $F_{-{\rm Id}}^{*}$ yields the conclusion of the theorem in the
orientation reversing case. ∎

######  Example 6.

Consider $\displaystyle\varphiup:\begin{cases}a\quad\rightarrow&ababba\\\
b\quad\rightarrow&aabbba\end{cases}$ and
$\psiup:\begin{cases}a\quad\rightarrow&a^{47}ab^{18}bb^{29}\\\
b\quad\rightarrow&a^{47}bb^{18}ab^{29}\end{cases}$.

One computes:

$\begin{array}[]{ll}\left(PDG(\varphiup),PDG(\varphiup)^{+}\right)\cong&\left(DG\begin{pmatrix}6&0\\\
1&1\end{pmatrix},DG\begin{pmatrix}6&0\\\
1&1\end{pmatrix}^{+}\right),\quad\text{and}\\\\[21.68121pt]
\left(PDG(\psiup),PDG(\psiup)^{+}\right)\cong&\left(DG\begin{pmatrix}96&0\\\
19&1\end{pmatrix},DG\begin{pmatrix}96&0\\\
19&1\end{pmatrix}^{+}\right).\end{array}$

These groups are order isomorphic but, because no power of $96$ is a power of
$6$, the shift isomorphisms do not have conjugate powers. By Theorem 6,
$\mathcal{T}_{\varphiup}$ and $\mathcal{T}_{\psiup}$ are not homeomorphic. ∎

We now relate the pair $(PDG(\varphiup),PDG(\varphiup)^{+})$ to the _ordered
augmented cohomology_ of the tiling space $\mathcal{T}_{\varphiup}$. We begin
with a description of the _augmented tiling space_ (for more detail, and a
precise description as an inverse limit, see [BSm]).

Let $\varphiup$ be any primitive aperiodic substitution with tiling space
$\mathcal{T}_{\varphiup}$. Let
$\\{\gammaup_{1}^{f},\dots,\gammaup_{n_{f}}^{f}\\}$ be a collection of forward
asymptotic special tilings, exactly one chosen from each forward asymptotic
equivalence class, and $\\{\gammaup_{1}^{b},\dots,\gammaup_{n_{b}}^{b}\\}$ a
collection of backward asymptotic special tilings, one from each backward
asymptotic equivalence class. Let $R_{j}^{f}$, $j=1,\dots,n_{f}$, and
$R_{i}^{b}$, $i=1,\dots n_{b}$, be the rays

$R_{j}^{f}:=\left\\{\\{j\\}\times\\{\gammaup_{j}^{f}-t\\}\,:\ t\geq
0\right\\}\quad\text{and}\quad
R_{i}^{b}:=\left\\{\\{i\\}\times\\{\gammaup_{i}^{b}-t\\}\,:\ t\geq
0\right\\}.$

The augmented tiling space $\widetilde{\mathcal{T}}_{\varphiup}$ is defined to
be the union

$\widetilde{\mathcal{T}}_{\varphiup}:=\mathcal{T}_{\varphiup}\ \ \cup\ \
\left.\left(\bigcup_{j=1}^{n_{f}}R_{j}^{f}\,\cup\,\bigcup_{i=1}^{n_{b}}R_{i}^{b}\right)\
\right/\,\left\\{\\{j\\}\times\\{\gammaup_{j}^{f}\\},\
\\{i\\}\times\\{\gammaup_{i}^{b}\\}\right\\},$

in which all of the endpoints of the rays have been identified to a single
branch point. The metric on $\mathcal{T}_{\varphiup}$ is extended to
$\widetilde{\mathcal{T}}_{\varphiup}$ in such a way that

$d\left(\gammaup_{j}^{f}-t,\\{j\\}\times\\{\gammaup_{j}^{f}-t\\}\right)=\frac{1}{1+t}=d\left(\gammaup_{i}^{b}+t,\\{i\\}\times\\{\gammaup_{i}^{b}+t\\}\right),$

for $t\geq 0$, making the ray $R_{j}^{f}$ asymptotic to the forward orbit of
$\gammaup_{j}^{f}$ and the ray $R_{i}^{b}$ asymptotic to the backward orbit of
$\gammaup_{i}^{b}$.

The homeomorphism $\Phi$ extends to
$\widetilde{\Phi}:\widetilde{\mathcal{T}}_{\varphiup}\hookleftarrow$ with
$\displaystyle\widetilde{\Phi}\left(\\{j\\}\times\\{\gammaup_{j}^{f}-t\\}\right)=\\{j^{\prime}\\}\times\\{\gammaup_{j^{\prime}}^{f}-\lambdaup
t\\}$, where $j^{\prime}$ is such that $\Phi(\gammaup_{j}^{f})$ is forward
asymptotic to $\gammaup_{j^{\prime}}^{f}$, and similarly for the rays
$R_{i}^{b}$. We order $H^{1}(\widetilde{\mathcal{T}}_{\varphiup})$ in such a
way that
$\displaystyle\widetilde{\Phi}^{*}:\left(H^{1}(\widetilde{\mathcal{T}}_{\varphiup}),\,H^{1}(\widetilde{\mathcal{T}}_{\varphiup})^{+}\right)\hookleftarrow$
is an order isomorphism. First note that
$\displaystyle\Phi^{*}\,:\,H^{1}(\mathcal{T}_{\varphiup};\mathbb{R})\hookleftarrow$
is a vector space isomorphism with simple eigenvalue
$\lambdaup=\lambdaup_{\varphiup}$ and codimension one invariant subspace
complementary to the eigenspace of $\lambdaup$. This codimension one space
splits $H^{1}(\mathcal{T}_{\varphiup})$ into two closed half-spaces, one of
which corresponds to the positive direction of the flow444Specifically, there
is an orientation preserving map of $\mathcal{T}_{\varphiup}$ onto the circle
$\mathbb{T}^{1}$. Let $H^{1}(\mathcal{T}_{\varphiup};\mathbb{R})^{+}$ contain
the positive half-line in $H^{1}(\mathcal{T}^{1};\mathbb{R})$ pulled back to
$H^{1}(\mathcal{T}_{\varphiup};\mathbb{R})$. ; call that space
$H^{1}(\mathcal{T}_{\varphiup};\mathbb{R})^{+}$. Let
$H^{1}(\widetilde{\mathcal{T}}_{\varphiup})^{+}=(\imath_{*}\jmath^{*})^{-1}\left(H^{1}(\mathcal{T}_{\varphiup};\mathbb{R})^{+}\right)$,
where $\imath_{*}:H^{1}(\mathcal{T}_{\varphiup})\to
H^{1}(\mathcal{T}_{\varphiup};\mathbb{R})$ is induced by
$\imath:\mathbb{Z}\to\mathbb{R}$ and
$\jmath^{*}:H^{1}(\widetilde{\mathcal{T}}_{\varphiup})\to
H^{1}(\mathcal{T}_{\varphiup})$ is induced by the inclusion
$j:\mathcal{T}_{\varphiup}\to\widetilde{\mathcal{T}}_{\varphiup}$. The pair
$\displaystyle\left(H^{1}(\widetilde{\mathcal{T}}_{\varphiup}),\,H^{1}(\widetilde{\mathcal{T}}_{\varphiup})^{+}\right)$
is the _ordered augmented cohomology_ of $\mathcal{T}_{\varphiup}$. Call the
isomorphism
$\displaystyle\widetilde{\Phi}^{*}:\left(H^{1}(\widetilde{\mathcal{T}}_{\varphiup}),\,H^{1}(\widetilde{T}_{\varphiup})^{+}\right)\hookleftarrow$
the shift isomorphism.

The rigidity result of [BSw2] is

######  Theorem 7.

If $f:\mathcal{T}_{\varphiup}\to\mathcal{T}_{\psiup}$ is a homeomorphism of
substitution tiling spaces, then there are $m,n\in{\mathbb{N}}$ and
$h:\mathcal{T}_{\varphiup}\to\mathcal{T}_{\psiup}$ isotopic to $f$ so that
$h\circ\Phi^{m}=\Psi^{n}\circ h$.

######  Corollary 8.

If $\mathcal{T}_{\varphiup}$ and $\mathcal{T}_{\psiup}$ are orientation
preserving (reversing) homeomorphic, then the ordered cohomologies
$\displaystyle\left(H^{1}(\widetilde{\mathcal{T}}_{\varphiup}),H^{1}(\widetilde{\mathcal{T}}_{\varphiup})^{+}\right)$
and
$\displaystyle\left(H^{1}(\widetilde{\mathcal{T}}_{\psiup}),H^{1}(\widetilde{\mathcal{T}}_{\psiup})^{+}\right)$
(resp.,
$\displaystyle\left(H^{1}(\widetilde{\mathcal{T}}_{\psiup}),-H^{1}(\widetilde{\mathcal{T}}_{\psiup})^{+}\right)$)
are isomorphic by an isomorphism that conjugates some positive powers of the
shift isomorphisms.

Proof: The homeomorphism $h$ of Theorem 7 extends to a homeomorphism
$\widetilde{h}:\widetilde{\mathcal{T}}_{\varphiup}\to\widetilde{\mathcal{T}}_{\psiup}$
that conjugates $\widetilde{\Phi}^{m}$ with $\widetilde{\Psi}^{n}$. ∎

The corollary — without the conjugacy in the conclusion — appears in [BSm].
Also, in that paper, a nonnegative integer matrix $\widetilde{A}^{t}$ is
constructed with ordered dimension group
$\displaystyle\left(DG(\widetilde{A}^{t}),DG(\widetilde{A}^{t})^{+}\right)$
isomorphic to
$\displaystyle\left(H^{1}(\widetilde{\mathcal{T}}_{\varphiup}),H^{1}(\widetilde{\mathcal{T}}_{\varphiup})^{+}\right)$.

Viewing $\displaystyle\left(PDG(\varphiup),PDG(\varphiup)^{+}\right)$ as the
Pisot part of the augmented cohomology is justified by our final theorem. A
formula for the Pisot part as the dimension group of an integer matrix will
arise in the proof.

######  Theorem 9.

If $\varphiup$ is a Pisot substitution, then there is an ordered embedding of
$\displaystyle\left(PDG(\varphiup),PDG(\varphiup)^{+}\right)$ into
$\displaystyle\left(H^{1}(\widetilde{\mathcal{T}}_{\varphiup}),H^{1}(\widetilde{\mathcal{T}}_{\varphiup})^{+}\right)$
that commutes with a positive power of the shift isomorphisms.

######  Example 7.

If $\varphiup$ is the Thue-Morse substitution ($1\to 12,\ 2\to 21$), then

$\left(PDG(\varphiup),PDG(\varphiup)^{+}\right)=\left(\mathbb{Z}\left[\small\frac{1}{2}\normalsize\right],\,\left\\{x\in\mathbb{Z}\left[\small\frac{1}{2}\normalsize\right],\
x\geq 0\right\\}\right),$

with shift isomorphism $x\mapsto 2x$, and

$\left(H^{1}(\widetilde{\mathcal{T}}_{\varphiup}),H^{1}(\widetilde{\mathcal{T}}_{\varphiup})^{+}\right)=\left(\mathbb{Z}\left[\small\frac{1}{2}\normalsize\right]\oplus\mathbb{Z}^{4},\
\left\\{(x,y)\in\mathbb{Z}\text{\tiny$\left[\frac{1}{2}\right]$\normalsize}\oplus\mathbb{Z}^{4}:\
x\geq 0\right\\}\right),$

with shift isomorphism $(x,y)\mapsto(2x,y)$. ∎
Proof:(Theorem 9) Let $\varphiup$ be a Pisot substitution. To simplify, we
pass to a _prepared_ substitution (see [BSm]). We may suppose all special
asymptotic tilings in $\mathcal{T}_{\varphiup}$ are fixed by $\varphiup^{n}$,
for some $n\in\mathbb{N}$, and $\varphiup^{n}$ has a fixed bi-infinite word
$\cdots b.a\cdots$. So we have $\varphiup^{n}(a)=a\cdots$,
$\varphiup^{n}(b)=\cdots b$ and $ba$ occurs in $\varphiup^{k}(a)$, for some
$k\in\mathbb{N}$.

We rewrite $\varphiup^{n}$ with stopping rule $b$ and starting rule $a$ as
follows. Let $\mathcal{U}=\\{u_{1},\dots,u_{q}\\}$ be the finite collection of
words with the properties

  * (_i_)

each $u_{i}$ occurs as a factor of the infinite word
$a\varphiup^{n}(a)\varphiup^{2n}(a)\cdots$;

  * (_i_ i)

each $u_{i}$ has the form $a\cdots b$.

  * (_i_ ii)

$ba$ is not a factor of any $u_{i},\ i=1,\dots,q$.

Then each word $\varphiup^{n}(u_{i})$ factors uniquely in the form
$u_{i_{1}}\cdots u_{i_{p}}$. From this, define $\psiup$ by
$\psiup(i)=i_{1}\cdots i_{p}$, $i=1,\dots,q$. The substitution $\psiup$ is
primitive and aperiodic Pisot ($\lambdaup_{\psiup}=\lambdaup_{\varphiup}^{n}$)
and _strictly proper_ : there are $r,s\in\\{1,2,\dots,q\\}$ so that
$\psiup(i)=r\cdots s$ for $i=1,\dots,q$. Also, the special asymptotic tilings
in $\mathcal{T}_{\psiup}$ are all fixed by $\Psi$. Then
$\mathcal{T}_{\varphiup^{n}}=\mathcal{T}_{\varphiup}$ and
$\mathcal{T}_{\psiup}$ are orientation preserving homeomorphic (this is a
special case of Lemma 3) by a homeomorphism that conjugates $\Phi^{n}$ with
$\Psi$. Hence,
$\displaystyle\left(H^{1}(\widetilde{\mathcal{T}}_{\varphiup}),H^{1}(\widetilde{\mathcal{T}}_{\varphiup})^{+}\right)$
and
$\displaystyle\left(H^{1}(\widetilde{\mathcal{T}}_{\psiup}),H^{1}(\widetilde{\mathcal{T}}_{\psiup})^{+}\right)$
are isomorphic by an isomorphism that conjugates $(\Phi^{n})^{*}$ with
$\Psi^{*}$ (Corollary 8) and
$\displaystyle\left(PDG(\varphiup),PDG(\varphiup)^{+}\right)=\left(PDG(\varphiup^{n}),PDG(\varphiup^{n})^{+}\right)$
is isomorphic to $\displaystyle\left(PDG(\psiup),PDG(\psiup)^{+}\right)$
(Theorem 6). The isomorphism conjugates the $(\ell n)^{\rm th}$ power of the
shift on $\left(PDG(\varphiup),PDG(\varphiup)^{+}\right)$ to the $\ell^{\rm
th}$ power of the shift on $\left(PDG(\psiup),PDG(\psiup)^{+}\right)$ for some
$\ell\in\mathbb{N}$. Thus once we prove the theorem for such substitutions
$\psiup$, we have proved it for all Pisot substitutions.

Let $n_{f}$ denote the number of equivalence classes of forward asymptotic
special tilings in $\mathcal{T}_{\psiup}$, and $n_{b}$ the number of special
backward classes. As in [BSm], we may select special asymptotic tilings
$\gammaup_{j}^{f}$, $j=1,\dots,n_{f}$, one from each forward class, and
$\gammaup_{i}^{b}$, $i=1,\dots,n_{b}$, one from each backward class, with the
properties

  * (i)

if $J$ is an edge of $\gammaup_{j}^{f}$ (resp., $I$ an edge of
$\gammaup_{i}^{b}$) that meets $\mathbb{E}^{s}$, then $J$ (resp., $I$) meets
$\mathbb{E}^{s}$ in its interior;

  * (ii)

if $k$ is the type of $J$ (resp., of $I$), then
$\psiup(k)=p_{j}^{f}ks_{j}^{f}$ (resp., $\psiup(k)=p_{i}^{b}ks_{i}^{b}$) with
$p_{j}^{f}$ and $s_{j}^{f}$ (resp., $p_{i}^{b}$ and $s_{i}^{b}$) nonempty and

$\varw_{j}^{f}:=\cdots\psiup^{2}(p_{j}^{f})\psiup(p_{j}^{f})p_{j}^{f}\psiup(s_{j}^{f})\psiup^{2}(s_{j}^{f})\cdots\
\text{(resp.,}\
\varw_{i}^{b}:=\cdots\psiup(p_{i}^{b})p_{i}^{b}\psiup(s_{i}^{b})\cdots)$

is the bi-infinite word corresponding to $\gammaup_{j}^{f}$ (resp.,
$\gammaup_{i}^{b}$).

In particular, and this will be important later,

$A(\min I)+\mathrm{pr}_{V}\llbracket p^{b}_{i}\rrbracket=\min
I\quad\text{and}\quad A(\max J)-\mathrm{pr}_{V}\llbracket
p_{j}^{f}\rrbracket=\max J,$

where $\llbracket u\rrbracket$ denotes the abelianization of the word $u$.

Let $E$ denote the $n\times(n_{f}+n_{b}-1)$ matrix with $ij^{\rm th}$ entry

$E_{ij}=\begin{cases}\text{number of occurrences of $i$ in
$p_{1}^{b}s_{j}^{f},\ \text{if}\ 1\leq j\leq n_{f}$}\\\\[7.22743pt]
\text{number of occurrences of $i$ in $p^{b}_{j-n_{f}+1}s^{f}_{1},\ \text{if}\
n_{f}<j\leq n_{f}+n_{b}-1$.}\end{cases}$

Let $A=A_{\psiup}$ be the abelianization of $\psiup$, and let $I$ be the
$(n_{f}+n_{b}-1)\times(n_{f}+n_{b}-1)$ identity matrix. The _augmented matrix_
for $\psiup$ is

$\widetilde{A}=\begin{pmatrix}A&E\\\ O&I\end{pmatrix},$

of size ${\widetilde{n}}=n+n_{f}+n_{b}-1$. The _augmented dimension group_ for
$\psiup$ is the pair

$\left(DG(\widetilde{A}^{t}),\,DG(\widetilde{A}^{t})^{+}\right),\quad\text{where}\quad
DG(\widetilde{A}^{t}):=\varinjlim\widetilde{A}^{t}:\mathbb{Z}^{{\widetilde{n}}}\hookleftarrow$

with $DG(\widetilde{A}^{t})^{+}$ determined (dynamicallly) as before:

Since $\lambdaup_{\psiup}$ is simple, $\widetilde{A}^{t}$ has a codimension
one invariant subspace $W$ (in $\mathbb{R}^{{\widetilde{n}}}$) complementary
to the eigenspace of $\lambdaup_{\psiup}$. Let

$\begin{array}[]{r}(\mathbb{Z}^{{\widetilde{n}}})^{+}:=\\{x\in\mathbb{Z}^{{\widetilde{n}}}\,:\
x\ \text{is in the half-space determined by $W$ containing}\\\ \text{ the
nonnegative eigenvector corresponding to}\ \lambdaup_{\psiup}\\}.\end{array}$

Then $DG(\widetilde{A}^{t})^{+}:=\left\\{[k,g]\in DG(\widetilde{A}^{t})\,:\
g\in(\mathbb{Z}^{{\widetilde{n}}})^{+}\right\\}$.

We know from [BSm] that $\left(DG(\widetilde{A}^{t}),\
DG(\widetilde{A}^{t})^{+}\right)$ is isomorphic to
$\left(H^{1}(\tilde{T}_{\psiup}),\ H^{1}(\tilde{T}_{\psiup})^{+}\right)$ via
an isomorphism that conjugates the shifts. Our remaining task, then, is to
embed $\left(PDG(\psiup),PDG(\psiup)^{+}\right)$ into
$\left(DG(\widetilde{A}^{t}),\ DG(\widetilde{A}^{t})^{+}\right)$.

Besides enabling us to describe its augmented cohomology as a dimension group,
another advantage of a prepared substitution (like $\psiup$) is that its
return lattice $\Sigma$ is the same as
$\Gamma=\mathrm{pr}_{V}(\mathbb{Z}^{n})$, making formulas for geometric
realization simpler. To show these lattices agree, let
$\gammaup\in\mathcal{T}_{\psiup}$, and let $J$ be an edge of type $j$ in
$\gammaup$ followed by the edge $I$ of type $i$. Since $\psiup$ is strictly
proper, $\Psi(J)$ begins with an edge of the same type, say $r$, as does
$\Psi(I)$. Thus the vector

$\begin{array}[]{r}\min\Psi(I)-\min\Psi(J)=\max\Psi(J)-\min\Psi(J)=\mathrm{pr}_{V}(Ae_{j})\\\
=(A|_{V})\mathrm{pr}_{V}e_{j}=(A|_{V})\varv_{j}\in\Theta(j).\end{array}$

So, $\varv_{j}\in\Sigma$ for all $j$, and $\Sigma=\Gamma$.

We may suppose, without loss of generality, that
$\\{\varv_{1},\dots,\varv_{d}\\}$ is a basis for $\Sigma=\Gamma$ , taken from
the basis $\\{\varv_{1},\dots,\varv_{n}\\}$ for $V$. Let
$\overline{B}r(\psiup)=\pi_{0}\varg_{\psiup}(\mathcal{C}_{\psiup})$ be the
branch locus in $V/\Sigma$ (so
$F_{L_{\psiup}}\left(\overline{B}r(\psiup)\right)=Br(\psiup)$). If $\gammaup$
and $\gammaup^{\prime}$ are asymptotic in $\mathcal{T}_{\psiup}$, then
$\pi_{0}\varg_{\psiup}(\gammaup)$ and
$\pi_{0}\varg_{\psiup}(\gammaup^{\prime})$ are asymptotic in $V/\Sigma$ under
the Kronecker flow; that is,
$\pi_{0}\varg_{\psiup}(\gammaup)=\pi_{0}\varg_{\psiup}(\gammaup^{\prime})$. It
follows that

$\overline{B}r(\psiup)=\left\\{\pi_{0}\varg_{\psiup}(\gammaup_{1}^{f}),\dots,\pi_{0}\varg_{\psiup}(\gammaup_{n_{f}}^{f})\right\\}\bigcup\left\\{\pi_{0}\varg_{\psiup}(\gammaup_{1}^{b}),\dots,\pi_{0}\varg_{\psiup}(\gammaup_{n_{b}}^{b})\right\\}.$

If $I$ is the unique edge of $\gammaup_{i}^{b}$ that meets $\mathbb{E}^{s}$,
let $x_{i}^{b}:=\min I$; and if $J$ is the unique edge of $\gammaup_{j}^{f}$
that meets $\mathbb{E}^{s}$, let $x_{j}^{f}:=\max J$.

Since $\Sigma=\Gamma$, we may take the $u_{i}=0$ in the definition of
geometric realization $\varg_{\psiup}$. Thus
$\pi_{0}\varg_{\psiup}(\gammaup_{j}^{f})=x^{f}_{j}+\Sigma=:\overline{x}_{j}^{f}$
and
$\pi_{0}\varg_{\psiup}(\gammaup_{i}^{b})=x^{b}_{i}+\Sigma=:\overline{x}_{i}^{b}$.

If $\\#(\overline{B}r(\psiup))=1$, then

$\left(PDG(\psiup),\ PDG(\psiup)^{+}\right)\cong\left(\varinjlim A|_{V},\
(\varinjlim A|_{V})^{+}\right),$

which is easily embedded in $\left(DG(\psiup),\ (DG(\psiup)^{+}\right)$.

Thus we can assume $\\#Br(\psiup)=m_{f}+m_{b}$ with $m_{f}>0$ and $m_{b}>0$,
and after reindexing,

$\overline{B}r(\psiup)=\left\\{\overline{x}_{i}^{b}\,:\
i=1,\dots,m_{b}\right\\}\,\bigcup\,\left\\{\overline{x}_{j}^{f}\,:\
j=1,\dots,m_{f}\right\\}.$

Recall that $\sigmaup_{i}:=\\{t\varv_{i}\,:\ t\in[0,1]\\}$ is the oriented
segment. The homology classes $[\overline{\sigmaup}_{i}]$ of the oriented
cycles $\overline{\sigmaup}_{i}:=\sigmaup_{i}+\Sigma,\ i=1,\dots,d$,
constitute a basis for $H_{1}(V/\Sigma)$.

For each $i=1,\dots,n_{b}$, let $\alphaup_{i}$ denote the oriented line
segment in $V$ from $x_{i}^{b}$ to $0$; for each $j=1,\dots,n_{f}$, let
$\betaup_{j}$ be the directed line segment from $0$ to $x_{j}^{f}$; and, let
$\overline{\alphaup}_{i}:=\alphaup_{i}+\Sigma,\
\overline{\betaup}_{j}:=\betaup_{j}+\Sigma$. Then

$\left\\{[\overline{\sigmaup}_{1}],\dots,[\overline{\sigmaup}_{\alphaup}],[\overline{\alphaup}_{1}\overline{\betaup}_{1}],\,[\overline{\alphaup}_{1}\overline{\betaup}_{2}],\dots,[\overline{\alphaup}_{1}\overline{\betaup}_{m_{f}}],\,[\overline{\alphaup}_{2}\overline{\betaup}_{1}],\,[\overline{\alphaup}_{3}\overline{\betaup}_{1}],\dots,[\overline{\alphaup}_{m_{b}}\overline{\betaup}_{1}]\right\\}$

is a basis for $\displaystyle
H_{1}\left(V/\Sigma,\,\overline{B}r(\psiup)\right)$.

Let $\mathcal{L}:\mathbb{Z}^{\widetilde{n}}\to\mathbb{Z}^{\widetilde{n}}$
(recall ${\widetilde{n}}=n+n_{f}+n_{b}-1$) denote the homomorphism represented
by $\widetilde{A}$ in the standard basis
$\mathcal{B}=\\{e_{1},\dots,e_{\widetilde{n}}\\}$. For convenience, relabel
the basis elements as follows:

$e_{ij}=\begin{cases}e_{i}&\text{for $j=0$ and $i=1,\dots,n$}\\\
e_{n+j}&\text{for $i=1$ and $j=1,\dots,n_{f}$}\\\ e_{n+n_{f}+i-1}&\text{for
$j=2$ and $i=2,\dots,n_{b}$.}\end{cases}$

We define a homomorphism $\displaystyle P:\mathbb{Z}^{\widetilde{n}}\to
H_{1}\left(V/\Sigma,\overline{B}r(\psiup)\right)$ on the basis $\mathcal{B}$
as follows:

$\begin{array}[]{l}P(e_{i0}):=\sum_{j=1}^{d}r_{ij}[\overline{\sigmaup}_{j}],\
\text{provided}\ \varv_{i}=\sum_{j=1}^{d}r_{ij}\varv_{j},\ i=1,\dots
n\text{;}\\\\[7.22743pt]
P(e_{ij}):=[\overline{\alphaup}_{i}\overline{\betaup}_{j}],\ i=1,\
j=2,\dots,n_{f}\ \text{and}\ j=1,\ i=2,\dots,n_{b}.\end{array}$

Clearly, $P$ is surjective. The relation $P\mathcal{L}=(F_{A|_{V}})_{*}P$
evidently holds on the basis elements $e_{10},\dots,e_{n0}$, so consider
$e_{ij}\in\mathcal{B}$ with $j>0$. Let $p_{i}:=\llbracket
p^{b}_{i}\rrbracket\in\mathbb{Z}^{n},\ s_{j}=\llbracket
s_{j}^{f}\rrbracket\in\mathbb{Z}^{n}$, and let
$\imath:\mathbb{Z}^{n}\to\mathbb{Z}^{\widetilde{n}}$ be given by
$\imath(x_{1},\dots,x_{n})=\sum_{i=1}^{n}x_{i}\,e_{i0}$. Then,
$\mathcal{L}(e_{ij})=\imath(p_{i})+\imath(s_{j})+e_{ij}$ and

$P\mathcal{L}(e_{ij})=P\mathcal{L}(p_{i})+P\mathcal{L}(s_{j})+P(e_{ij})=\sum_{k=1}^{d}a_{k}[\overline{\sigmaup}_{k}]+\sum_{k=1}^{d}b_{k}[\overline{\sigmaup}_{k}]+[\overline{\alphaup}_{i}\overline{\betaup})j],$

where $\mathrm{pr}_{V}(p_{i})=\sum_{k=1}^{d}a_{k}\varv_{k}$ and
$\mathrm{pr}_{V}(s_{j})=\sum_{k=1}^{d}b_{k}\varv_{k}$.

On the other hand,
$(F_{A|_{V}})_{*}P(e_{ij})=(F_{A|_{V}})_{*}([\overline{\alphaup}_{i}\overline{\betaup}_{j}])=[\overline{A(\alphaup_{i}\betaup_{j})}]$,
if $A(\alphaup_{i}\betaup_{j})$ denotes the image of the directed curve
$\alphaup_{i}\betaup_{j}$ under the linear map $A:V\to V$. We claim that
$\overline{A(\alphaup_{i}\betaup_{j})}$ is homologous to
$\overline{\rhoup}_{i}+\overline{\alphaup}_{i}\overline{\betaup}_{j}+\overline{\eta}_{j}$,
where $\rhoup_{i}:=\\{t\mathrm{pr}_{V}(p_{i})\,:\ 0\leq t\leq 1\\}$ and
$\eta_{j}:=\\{t\mathrm{pr}_{V}(s_{j})\,:\ 0\leq t\leq 1\\}$ are directed
segments.
Indeed, this follows from $A(-x_{i}^{b})=\mathrm{pr}_{V}(p_{i})+(-x^{b}_{i})$
and $Ax_{j}^{f}=x_{j}^{f}+\mathrm{pr}_{V}(s_{j})$ in the cover $V$ of
$V/\Sigma$ (See Figure 1). Clearly $\overline{\rhoup}_{i}$ is homologous to
$\sum_{k=1}^{d}a_{k}\overline{\sigmaup}_{k}$ and $\overline{\eta}_{j}$ is
homologous to $\sigmaup_{k=1}^{d}b_{k}\overline{\sigmaup}_{k}$. Thus
$P\mathcal{L}(e_{ij})=(F_{A|_{V}})_{*}P(e_{ij})$.

Since $P$ is surjective, the dual

$P^{t}:{\rm
Hom}\left(H_{1}\left(V/\Sigma,\overline{B}r(\psiup)\right),\mathbb{Z}\right)\cong
H^{1}\left(V/\Sigma,\overline{B}r(\psiup)\right)\to{\rm
Hom}(\mathbb{Z}^{\widetilde{n}},\mathbb{Z})\cong\mathbb{Z}^{\widetilde{n}}$

is injective. As $\mathcal{L}^{t}P^{t}=P^{t}(F_{A|_{V}})^{*}$, we know $P^{t}$
induces an injection

$\widehat{P}^{t}:\varinjlim
F_{A|V}^{*}=PDG(\psiup)\to\varprojlim\mathcal{L}^{t}=DG(\psiup)$

that commutes with the shifts. Since the positive cones are determined
dynamically, $\widehat{P}^{t}$ either preserves or reverses order. As
$[\sigmaup_{1}]^{*}\in H^{1}(V/\Sigma,\,\overline{B}r(\psiup))^{+}$ and
$\widehat{P}^{t}[\sigmaup_{1}]^{*}=[e_{1}]^{*}\in(DG(\psiup))^{+},\
\widehat{P}^{t}$ preserves order.

Finally,

$\widehat{F}_{L_{\psiup}}^{*}:\varinjlim\left(F_{M_{\psiup}}^{*}:H^{1}(\mathbb{T}^{d},Br(\psiup))\hookleftarrow\right)=PDG(\psiup)\to\varinjlim\left(F_{A|_{V}}^{*}:H^{1}\left(V/\Sigma,\overline{B}r(\psiup)\right)\hookleftarrow\right)$

by
$\widehat{F}_{L_{\psiup}}^{*}([(k,g)])=\left[\left(k,\,F_{L_{\psiup}}^{*}(g)\right)\right]$
is an ordered isomorphism (that commutes with the shifts), which embeds
$\left(PDG(\psiup),PDG(\psiup)^{+}\right)$ into
$\left(DG(\psiup),DG(\psiup)^{+}\right)$.

Figure 1. A Homology
$\psmatrix[mnode=R,rowsep=1pt,colsep=6pt]&\large{\mathbb{E}^{s}}\small{*}\\\
\\\ \\\ \large{\gammaup_{i}^{b}}\\\ \\\ \small{*}\large{\ \gammaup_{j}^{f}}\\\
\small{*}\\\ \\\ x_{i}^{b}\small{*}\small{*}\small{*}\\\ \small{*}\\\
\,\cdot\large{Ax_{j}^{f}}\\\ \\\ \\\ \\\ \small{*}\\\ \small{*}\circ\\\ \\\
\\\ \large{Ax_{i}^{b}}\small{*}\small{*}\\\ \small{*}\large{\,\,x_{j}^{f}}\\\
\small{*}\\\ \\\ \small{*}\\\ \small{*}\\\ \\\ \small{*}\\\ \\\ \small{*}\\\
\\\ \\\ \small{*}\psset{linestyle=solid,arrowsize=5.69055pt
0.0,linewidth=1.0pt}{\psset{linestyle=dotted}\ncline{6,24}{4,27}}{\ncline{7,21}{6,24}}{\ncline{->}{16,21}{20,25}}^{\betaup_{j}}{\ncline{->}{20,25}{11,33}}^{\eta_{j}}{\ncline{->}{16,21}{11,33}}^{A\betaup_{j}}{\ncline{11,33}{9,35}}{\psset{linestyle=dotted}\ncline{9,35}{6,38}}{\ncline{->}{19,5}{16,21}}_{A\alphaup_{i}}{\ncline{->}{9,15}{16,21}}_{\alphaup_{i}}{\ncline{19,25}{18,28}}{\psset{linestyle=dotted}\ncline{19,28}{15,32}}{\ncline{15,32}{11,33}}{\ncline{11,33}{9,35}}{\psset{linestyle=dotted}\ncline{9,35}{6,38}}{\ncline{9,19}{7,21}}{\ncline{10,12}{9,15}}{\ncline{9,15}{9,19}}^{I}{\ncline{->}{19,5}{9,15}}_{\rhoup_{i}}{\psset{linestyle=dotted}\ncline{1,12}{31,28}}{\psset{linestyle=dotted}\ncline{28,18}{26,20}}{\ncline{26,20}{24,23}}{\ncline{21,3}{19,5}}{\ncline{19,5}{16,6}}{\psset{linestyle=dotted}\ncline{16,6}{10,12}}{\psset{linestyle=dotted}\ncline{23,1}{21,3}}{\ncline{24,23}{20,25}}^{J}{\ncline{20,25}{19,28}}$

∎

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Department of Mathematics, Montana State University, Bozeman, MT 59717

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Department of Mathematics, College of Charleston, Charleston, SC 29424

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