#  Measure transfer and $S$-adic developments for subshifts

Nicolas Bédaride  ,  Arnaud Hilion  and  Martin Lustig  Aix Marseille
Université, CNRS, I2M UMR 7373, 13453 Marseille, France Nicolas.Bedaride@univ-
amu.fr Institut de Mathématiques de Toulouse, UMR 5219, Université de
Toulouse, UPS F-31062 Toulouse Cedex 9, France<EMAIL_ADDRESS>toulouse.fr Aix Marseille Université, CNRS, I2M UMR 7373, 13453 Marseille,
France<EMAIL_ADDRESS>
###### Abstract.

Based on previous work of the authors, to any $S$-adic development of a
subshift $X$ a “directive sequence” of commutative diagrams is associated,
which consists at every level $n\geq 0$ of the measure cone and the letter
frequency cone of the level subshift $X_{n}$ associated canonically to the
given $S$-adic development.

The issuing rich picture enables one to deduce results about $X$ with
unexpected directness. For instance, we exhibit a large class of minimal
subshifts with entropy zero that all have infinitely many ergodic probability
measures.

As a side result we also exhibit, for any integer $d\geq 2$, an $S$-adic
development of a minimal, aperiodic, uniquely ergodic subshift $X$, where all
level alphabets $\cal A_{n}$ have cardinality $d\,$, while none of the $d-2$
bottom level morphisms is recognizable in its level subshift
$X_{n}\subseteq\cal A_{n}^{\mathbb{Z}}$.

###### Key words and phrases:

$S$-adic development, measure transfer, ergodic measures, letter frequencies,
eventually recognizable

###### 2010 Mathematics Subject Classification:

Primary 37B10, Secondary 37A25, 37E25

The first author was partially supported by ANR Project IZES ANR-22-CE40-0011

##  1\. Introduction

A subshift over a finite alphabet $\cal A$ is a non-empty, closed and shift-
invariant subset $X\subseteq\cal A^{\mathbb{Z}}$. A very efficient tool to
investigate such a subshift $X$ is given by an $S$-adic development of $X$:
the latter is obtained by a directive sequence $\overleftarrow{\sigma}$ of
monoid morphisms $\sigma_{n}:\cal A^{*}_{n+1}\to\cal A_{n}^{*}$ for all
integers $n\geq 0$, where each $\cal A_{n}$ is again a finite alphabet, and
$\cal A_{n}^{*}$ denotes the free monoid over $\cal A_{n}$. The morphisms
$\sigma_{n}$ here are all assumed to be non-erasing, i.e. none of the letters
of $\cal A_{n+1}$ is mapped to the empty word. The directive sequence
$\overleftarrow{\sigma}$ generates the given subshift $X$ if for some
identification $\cal A=\cal A_{0}$ any finite factor $x_{k}\ldots x_{\ell}$ of
any biinfinite word ${\bf x}=\ldots x_{-1}x_{0}x_{1}\ldots\in X$ is also a
factor of some $\sigma_{0}\circ\ldots\circ\sigma_{n-1}(a_{i})$ with
$a_{i}\in\cal A_{n}$, and conversely: any such ${\bf x}$ belongs to $X$. One
usually also assumes that $\overleftarrow{\sigma}$ is everywhere growing,
which means that
$\underset{n\to\infty}{\liminf}(\min\\{|\sigma_{0}\circ\ldots\circ\sigma_{n-1}(a_{i})|\mid
a_{i}\in\cal A_{n}\\})=\infty$. It is well known that any subshift
$X\subseteq\cal A^{\mathbb{Z}}$ is generated by some everywhere growing
directive sequence $\overleftarrow{\sigma}$.

A directive sequence $\overleftarrow{\sigma}$ as above determines at every
level $n\geq 0$ a level subshift $X_{n}\subseteq\cal A_{n}^{\mathbb{Z}}$,
which is the subshift generated by the truncated sequence
$\overleftarrow{\sigma}\dagger_{n}$, obtained from $\overleftarrow{\sigma}$
through forgetting all levels $k<n$ and the corresponding level morphisms. It
is a straight forward observation that every level morphism $\sigma_{n}$
induces a map $X_{n+1}\to X_{n}$ which is surjective on shift-orbits.

More generally, any non-erasing morphism $\sigma:\cal A^{*}\to\cal B^{*}$
between free monoids over finite alphabets $\cal A$ and $\cal B$ respectively,
defines for any subshift $X\subseteq\cal A^{\mathbb{Z}}$ an image subshift
$\sigma(X)$, and it is natural to ask which properties of $X$ are inherited
(under suitable hypotheses) by the image subshift $\sigma(X)$. In our cousin
paper [5] we have formally introduced and studied, for any such morphism
$\sigma$, a measure transfer map $\sigma_{X}M:\cal M(X)\to\cal M(\sigma(X))$,
where $\cal M(X)$ denotes the measure cone on $X$, i.e. the set of all shift-
invariant Borel measures on the subshift $X$. The map $\sigma_{X}M$ is the
restriction/co-restriction of a map $\sigma M:\cal M(\cal
A^{\mathbb{Z}})\to\cal M(\cal B^{\mathbb{Z}})$ which is linear, functorial and
commutes with the support map on subshifts (see section 3.1).

We thus obtain canonically, for any everywhere growing directive sequence
$\overleftarrow{\sigma}=(\sigma_{n}:\cal A^{*}_{n+1}\to\cal A^{*}_{n})_{n\geq
0}$ as above, an induced sequence $\cal M(\overleftarrow{\sigma})=(\sigma
M_{n}:\cal M(X_{n+1})\to\cal M(X_{n}))_{n\geq 0}$ of linear maps $\sigma
M_{n}:=\sigma M_{X_{n+1}}$ on the measure cones $\cal M(X_{n+1})$.

Furthermore, any invariant measure $\mu$ on a subshift $X\subseteq\cal
A^{\mathbb{Z}}$ defines canonically a letter frequency vector $\vec{v}(\mu)$
in the non-negative cone $\mathbb{R}_{\geq 0}A$ of the vector space
$\mathbb{R}A$, where for each letter $a_{i}\in\cal A$ the coordinate of
$\vec{v}(\mu)$ is given by the measure $\mu([a_{i}])$ of the cylinder
$[a_{i}]$. The latter consists of all biinfinite words ${\bf x}\in\cal
A^{\mathbb{Z}}$ as above for which the letter with index 1 satisfies
$x_{1}=a_{i}$. The cone of all such letter frequency vectors is denoted by
$\cal C(X)\subseteq\mathbb{R}_{\geq 0}A\,$; it gives rise to a canonical
linear evaluation map $\zeta_{X}:\cal M(X)\to\cal C(X)$ which by definition is
surjective.

It has been shown in [5] that the linear map $\mathbb{R}A\to\mathbb{R}B$,
defined by the incidence matrix $M(\sigma)$ of any non-erasing free monoid
morphism $\sigma:\cal A^{*}\to\cal B^{*}$, commutes via the evaluation maps
$\zeta_{\cal A^{\mathbb{Z}}}$ and $\zeta_{\cal B^{\mathbb{Z}}}$ with the
measure transfer map $\sigma M$. We thus obtain, for any directive sequence
$\overleftarrow{\sigma}$ as above, a rather useful commutative diagram:

$\,\,\,\,\ldots\,\,\overset{\sigma_{n+1}M}{\longrightarrow}\,\,\,\cal
M(X_{n+1})\overset{\sigma_{n}M}{\longrightarrow}\,\,\cal
M(X_{n})\,\,\overset{\sigma_{n-1}M}{\longrightarrow}\,\,\,\ldots\,\,\,\,\,\,\overset{\sigma_{2}M}{\longrightarrow}\,\,\cal
M(X_{1})\overset{\sigma_{1}M}{\longrightarrow}\cal M(X)$
$\qquad\qquad\qquad\downarrow\zeta_{X_{n+1}}\qquad\,\,\,\,\downarrow\zeta_{X_{n}}\qquad\qquad\qquad\qquad\,\,\,\,\downarrow\zeta_{X_{1}}\qquad\qquad\downarrow\zeta_{X}$
$\ldots\,\,\overset{M(\sigma_{n+1})}{\longrightarrow}\cal
C(X_{n+1})\overset{M(\sigma_{n})}{\longrightarrow}\cal
C(X_{n})\overset{M(\sigma_{n-1})}{\longrightarrow}\,\,\ldots\,\,\overset{M(\sigma_{2})}{\longrightarrow}\cal
C(X_{1})\,\,\overset{M(\sigma_{1})}{\longrightarrow}\,\,\cal C(X)$

A measure tower $\overleftarrow{\mu}=(\mu_{n})_{n\geq 0}$ on a directive
sequence $\overleftarrow{\sigma}$ as above, defined by postulating
$\mu_{n}\in\cal M(X_{n})$ and $\sigma_{n}M(\mu_{n+1})=\mu_{n}\,$, defines a
tower of letter frequency vectors
$\vec{v}(\overleftarrow{\mu})=(\vec{v}(\mu_{n}))_{n\geq 0}$ which satisfy
$M(\sigma_{n+1})\cdot\vec{v}(\mu_{n+1})=\vec{v}(\mu_{n})$. This last equality
had been used in [3] as defining equality for what was called there a vector
tower over the directive sequence $\overleftarrow{\sigma}$. A linear
evaluation map $\mathfrak{m}:\cal V(\overleftarrow{\sigma})\to\cal M(X)$, from
the set $\cal V(\overleftarrow{\sigma})$ of all such vector towers, to the
measure cone $\cal M(X)$ of the subshift $X$ generated by
$\overleftarrow{\sigma}$, has been established in [3], and the map
$\mathfrak{m}$ is shown in [3] to be always surjective, as long as
$\overleftarrow{\sigma}$ is everywhere growing (but no other hypotheses are
needed). We obtain:

######  Proposition 1.1.

For any everywhere growing directive sequence $\overleftarrow{\sigma}$ there
is a canonical linear bijection between the cone $\cal
V(\overleftarrow{\sigma})$ of vector towers and the cone $\cal
M(\overleftarrow{\sigma})$ of measure towers on $\overleftarrow{\sigma}$,
given by the letter frequency map

$\overleftarrow{\mu}=(\mu_{n})_{n\geq
0}\,\,\mapsto\,\,\overleftarrow{v}=(\vec{v}_{n})_{n\geq 0}\,,$

with $\vec{v}_{n}=\vec{v}(\mu_{n})=(\mu_{n}([a_{k}]))_{a_{k}\in\cal A_{n}}$
for all levels $n\geq 0$.

Based on the main result of our previous paper [3] (quoted below as Theorem
2.7) we derive from this set-up the following consequence (see Proposition
4.4):

######  Theorem 1.2.

For any non-erasing monoid morphism $\sigma:\cal A^{*}\to\cal B^{*}$ and any
subshift $X\subseteq\cal A^{\mathbb{Z}}$ the induced measure transfer map
$\sigma M$ maps the measure cone $\cal M(X)$ of $X$ surjectively to the
measure cone $\cal M(\sigma(X))$ of the image subshift $\sigma(X)$:

$\sigma M(\cal M(X))=\cal M(\sigma(X))$

This general surjectivity result for the measure transfer map $\sigma M$ is
mirrored in the special case where $\sigma$ is recognizable in $X$ (see
Definition 3.5) by the the following fact, proved below in Corollary 3.9:

######  Proposition 1.3.

If a non-erasing morphism $\sigma:\cal A^{*}\to\cal B^{*}$ is recognizable in
a subshift $X\subseteq\cal A^{\mathbb{Z}}$, then the measure transfer map
$\sigma M_{X}:\cal M(X)\to\cal M(\sigma(X))$ is injective.

We apply this injectivity result to any directive sequence
$\overleftarrow{\sigma}=(\sigma_{n})_{n\geq 0}\,$, where each level map
$\sigma_{n}$ is assumed to be recognizable in the corresponding level subshift
$X_{n+1}\,$. Such totally recognizable directive sequences (or slight
variations of it) have recently received a lot of attention (see for instance
[1], [9], [13], [17]), and they are shown to play a central role in the
$S$-adic approach to symbolic dynamics. We obtain (see Theorem 5.7):

######  Theorem 1.4.

For any totally recognizable everywhere growing directive sequence
$\overleftarrow{\sigma}$, with generated subshift
$X=X_{\tiny\overleftarrow{\sigma}}$, the linear surjective map of cones

$\mathfrak{m}:\cal V(\overleftarrow{\sigma})\to\cal M(X)$

is a bijection.

We combine this result with a construction from our earlier paper [4], where
for any integer $d\geq 2$ a subshift $X$ with $d$ distinct invariant ergodic
probability measures has been shown to exist, while $X$ is defined by an
everywhere growing directive sequence with level alphabets $\cal A_{n}$ that
all have cardinality $\mbox{card}(\cal A_{n})=d$. This construction is used in
section 7 below to define a large “diagonal” family $\mathfrak{X}$ of
directive sequences $\overleftarrow{\sigma}$ and to give a quick proof (see
Theorem 7.4) that they all generate subshifts
$X_{\tiny\overleftarrow{\sigma}}$ which have a remarkable property, exhibited
first by a quite different and more elaborate method for very particular
subshifts in a recent paper by Cyr-Kra (see [11]):

######  Corollary 1.5.

For any directive sequence $\overleftarrow{\sigma}\in\mathfrak{X}$ the
subshift $X_{\tiny\overleftarrow{\sigma}}$ is minimal, has topological entropy
$h_{X_{\tiny\overleftarrow{\sigma}}}=0$ and admits infinitely many distinct
ergodic probability measures in $\cal M(X_{\tiny\overleftarrow{\sigma}})$.

The directive sequences considered in the last corollary are all totally
recognizable, and they are “large”, in that their alphabet rank, i.e. the
limit inferior of the cardinality of the level alphabets, is infinite. For
finite alphabet rank, on the other hand, the condition “totally recognizable”
can be replaced by a distinctly weaker condition: in this case, the linear map
defined by the incidence matrix $M(\sigma_{n})$ is for any sufficiently high
level $n\geq 0$ a forteriori (from the surjectivity result in Theorem 1.2)
injective on the subspace spanned by the cone $\cal M(X_{n})$. In the special
– but rather frequent – case that this injectivity property of the
$M(\sigma_{n})$ is also true for all low levels, the bijectivity of the map
$\mathfrak{m}$ as in Theorem 1.4 above is a direct consequence of our set-up.
We thus obtain (see Corollary 6.3):

######  Corollary 1.6.

Let $X\subseteq\cal A^{\mathbb{Z}}$ be a subshift generated by an everywhere
growing directive sequence $\overleftarrow{\sigma}=(\sigma_{n})_{n\geq 0}$ of
finite alphabet rank. Assume that for any $n\geq 0$ the incidence matrix
$M(\sigma_{n})$ is invertible over $\mathbb{R}$. Then any invariant measure
$\mu$ on the subshift $X$ is determined by the letter frequency vector
associated to $\mu$, i.e. by the values $\mu([a_{k}])$ for all $a_{k}\in\cal
A$.

This generalizes a result of [6], obtained under additional hypotheses by very
different methods.

A slightly more general situation than considered in Theorem 1.4, which
deserves some particular interest, occurs if the given directive sequence is
only eventually recognizable, i.e. only for sufficiently high levels one
assumes that the level morphisms are recognizable in the corresponding level
subshift. In section 8 we investigate non-recognizable morphisms, and in
particular we show in Corollary 8.5 the following result, which is somewhat
surprising, in view of the claims in [13] and [17] (see Remark 8.7).

######  Proposition 1.7.

For any integer $n_{0}\geq 0$ there exists an everywhere growing directive
sequence $\overleftarrow{\sigma}=(\sigma_{n})_{n\geq 0}$ with the following
properties:

  1. (1)

For any $n\geq n_{0}$ the level alphabets satisfy $\cal A_{n}=\cal A_{n_{0}}$
and the level morphisms are stationary: $\sigma_{n}=\sigma_{n_{0}}$.
Furthermore, each level morphisms $\sigma_{n}$ is recognizable in the level
subshift $X_{n+1}$.

  2. (2)

For any level $n$ with $0\leq n\leq n_{0}-1$ we have $\mbox{card}(\cal
A_{n})=n+2=\mbox{card}(\cal A_{n+1})-1$, and none of the level morphisms
$\sigma_{n}$ is recognizable in the level subshift $X_{n+1}$.

  3. (3)

All level subshifts $X_{n}$ are minimal, uniquely ergodic, and aperiodic.

(In fact, each level subshift $X_{n}$ is actually an interval exchange
subshift, obtained from the stable lamination of a pseudo-Anosov homeomorphism
on a suitably punctured surface.)

Acknowledgements: The authors would like to thank Fabien Durand and Samuel
Petite for encouraging remarks and interesting comments.

##  2\. Terminology, notation, conventions and some quotes

In this section we first recall some standard terminology from symbolic
dynamics (see subsection 2.1), then summarize the notation introduced in [5]
and some of its results (see subsection 3.1), and in subsection 2.2 we recall
some classical $S$-adic terminology and quote the main result from [3], which
plays a key role later in this paper.

###  2.1. Standard terminology from symbolic dynamics

Throughout this paper we denote by $\cal A,\cal B$ or $\cal C$ non-empty
finite sets, called alphabets, and by $\cal A^{*},\cal B^{*}$ or $\cal C^{*}$
the free monoid over those alphabets. Every element $w\in\cal A^{*}$ is a word
in the letters $a_{1},a_{2},\ldots,a_{d}$ of $\cal A$, i.e.

$w=x_{1}x_{2}\ldots x_{n}\qquad\text{with}\qquad
x_{i}\in\\{a_{1},a_{2},\ldots,a_{d}\\}=\cal A$

for any $i=1,\ldots,n\,$, and the empty word is denoted by $\varepsilon$. Here
$n$ is the length of $w$, denoted by $|w|$, and one sets $|\varepsilon|=0$. We
immediately verify the formula $|w|=\underset{a_{j}\in\cal
A}{\sum}|w|_{a_{j}}\,$, where $|w|_{a_{j}}$ denotes the number of occurrences
of the letter $a_{j}$ in $w$. More generally, for any second word $u\in\cal
A^{*}$ we denote by $|w|_{u}$ the number of (possibly overlapping) occurrences
of $u$ as subword $x_{k}\ldots x_{\ell}$ (also called a factor) of $w$.

Any monoid morphism $\sigma:\cal A^{*}\to\cal B^{*}$ is determined by the
family of letter images $\sigma(a_{i})\in\cal B^{*}$ for all $a_{i}\in\cal A$,
and this family can be chosen freely. Such a morphism $\sigma$ is non-erasing
if $|\sigma(a_{i})|\geq 1$ for all $a_{i}\in\cal A$. All morphisms considered
in this paper will be non-erasing. Note that any composition of non-erasing
morphisms is non-erasing.

Every monoid morphism $\sigma:\cal A^{*}\to\cal B^{*}$ induces canonically a
linear map $\mathbb{R}_{\geq 0}A\to\mathbb{R}_{\geq 0}B\,$, given by the
incidence matrix

(2.1) $M(\sigma)=(|\sigma(a_{j})|_{b_{i}})_{b_{i}\in\cal B,\,a_{j}\in\cal
A}\,.$

To any alphabet $\cal A$ there is also associated the full shift $\cal
A^{\mathbb{Z}}$; its elements are written as biinfinite words

(2.2) ${\bf x}=\ldots x_{i-1}x_{i}x_{i+1}\ldots$

with $x_{i}\in\cal A$ for any index $i\in\mathbb{Z}$. The set $\cal
A^{\mathbb{Z}}$ is naturally equipped with the product topology (with respect
to the discrete topology on $\cal A$), and $\cal A^{\mathbb{Z}}$ is a Cantor
set unless $\mbox{card}(\cal A)=1$. Furthermore, the space $\cal
A^{\mathbb{Z}}$ comes naturally with a shift-operator $T$, defined for any
${\bf x}$ as in (2.2) by $T({\bf x})=\ldots y_{i-1}y_{i}y_{i+1}\ldots$ with
$y_{i}=x_{i+1}$ for any $i\in\mathbb{Z}\,$. The shift-operator acts as
homeomorphism on the space $\cal A^{\mathbb{Z}}\,$; for convenience it will
always be denoted by the symbol $T$, independently of the choice of the given
alphabet $\cal A$.

For any integers $k\leq l$ we denote by ${\bf x}_{[k,\ell]}$ the subword
(again also called factor) $x_{k}\ldots x_{\ell}$ of the biinfinite word ${\bf
x}$ as in (2.2). We also consider the one-sided infinite positive half-word
${\bf x}_{[1,\infty)}=x_{1}x_{2}\ldots$ of $\bf x$.

To any word $w\in\cal A^{*}$ there is associated the cylinder
$[w]\subseteq\cal A^{\mathbb{Z}}\,$, which consists of all words ${\bf
x}\in\cal A^{\mathbb{Z}}$ which satisfy ${\bf x}_{[1,|w|]}=w$. If $w$ is the
empty word, then $[w]=\cal A^{\mathbb{Z}}$. The set of all cylinders $[w]$
together with their shift translates $T^{m}([w])$ for any $m\in\mathbb{Z}$
constitute a basis for the above specified topology of the space $\cal
A^{\mathbb{Z}}$.

A non-empty subset $X\subseteq\cal A^{\mathbb{Z}}$ is a subshift if $X$ is
closed and if $T(X)=X$. A subshift $X$ is minimal if none of its subsets is a
subshift except $X$ itself. This is equivalent to the statement that for any
${\bf x}\in X$ the shift-orbit $\cal O({\bf x})=\\{T^{m}({\bf x})\mid
m\in\mathbb{Z}\\}$ is dense in $X$. A minimal subshift $X$ is either
uncountably infinite or else it is finite: in this case $X$ consists of the
single shift-orbit $X=\cal O(w^{\pm\infty})$ of some periodic word
$w^{\pm\infty}=\ldots www\ldots\,$, which is well defined for any non-empty
$w\in\cal A^{*}$ by the convention $w^{\pm\infty}_{[1,\infty)}=www\ldots\,\,$.
It follows that any infinite minimal subshift is in particular aperiodic,
which means that $X$ doesn’t contain any periodic word $w^{\pm\infty}$.

Any subshift $X\subseteq\cal A^{\mathbb{Z}}$ defines a language $\cal L(X)$
which consists of all words $w\in\cal A^{*}$ that occur as factor in some
${\bf x}\in X$. Conversely, every infinite subset $\cal L\subseteq\cal A^{*}$
generates a subshift $X(\cal L)\subseteq\cal A^{\mathbb{Z}}$, defined by the
property that any word from $\cal L(X)$ must occur as factor in some
$w^{\prime}\in\cal L$.

For any subshift $X\subseteq\cal A^{\mathbb{Z}}$ and any $n\in\mathbb{N}$ one
denotes by $p_{X}(n)$ the number of words in $\cal L(X)$ of length $n$. The
following limit is well defined and is known as topological entropy $h_{X}$ of
the subshift $X$:

(2.3) $h_{X}=\lim_{n\to\infty}\frac{\log p(n)}{n}$

Any non-erasing monoid morphism $\sigma:\cal A^{*}\to\cal B^{*}$ defines
canonically a map

(2.4) $\sigma^{\mathbb{Z}}:\cal A^{\mathbb{Z}}\to\cal B^{\mathbb{Z}}$

where for any ${\bf x}\in\cal A^{\mathbb{Z}}$ the image ${\bf
y}=\sigma^{\mathbb{Z}}({\bf x})\in\cal B^{\mathbb{Z}}$ is defined by extending
$\sigma$ first to the positive half-word ${\bf x}_{[1,\infty)}$ to define
${\bf y}_{[1,\infty)}$, and subsequently extending $\sigma$ to all of $\bf x$.

For almost all subshifts $X\subseteq\cal A^{Z}$ the image set
$\sigma^{\mathbb{Z}}(X)$ will not be shift-invariant and hence not be a
subshift. However, there is a canonical image subshift $\sigma(X)$ of $X$,
which admits several naturally equivalent definitions:

######  Remark 2.1.

The following three definitions of the image subshift $Y:=\sigma(X)$ are
equivalent, for any non-erasing monoid morphism $\sigma:\cal A^{*}\to\cal
B^{*}$:

  1. (1)

$Y$ is the intersection of all subshifts that contain the set
$\sigma^{\mathbb{Z}}(X)$.

  2. (2)

$Y$ is the the union of all shift-orbits $\cal O(\sigma({\bf x}))$, for any
${\bf x}\in X$. [Note here (see Lemma 2.4 of [5]) that this union is always
closed, a fact that a priori can not be taken for granted.]

  3. (3)

$Y$ is the subshift generated by the language $\sigma(\cal L(X))$. Thus $Y$
consists of all biinfinite words ${\bf y}\in\cal B^{\mathbb{Z}}$ with the
property that every factor of $\bf y$ is also a factor of some word in
$\sigma(\cal L(X))$.

We observe directly the following consequence:

######  Lemma 2.2.

Let $\sigma:\cal A^{*}\to\cal B^{*}$ be a non-erasing monoid morphism, and let
$X\subseteq\cal A^{\mathbb{Z}}$ be any subshift. If $\cal L\subseteq\cal
A^{*}$ is a language that generates $X$, then $\sigma(\cal L)$ generates
$\sigma(X)$. $\sqcup$$\sqcap$

An invariant measure on $\cal A^{\mathbb{Z}}$ is a finite Borel measure $\mu$
on $\cal A^{\mathbb{Z}}$ which is invariant under the homeomorphism $T$ (= the
shift operator). The set of all such invariant measures is denoted by $\cal
M(\cal A^{\mathbb{Z}})$. For any subshift $X\subseteq\cal A^{\mathbb{Z}}$ we
denote by $\cal M(X)\subseteq\cal M(\cal A^{\mathbb{Z}})$ the set of those
invariant measures $\mu$ for which their support satisfies
$\mbox{Supp}(\mu)\subseteq X$. For notational convenience we identify any such
$\mu$ with its restriction to $X$.

Any invariant measure $\mu\in\cal M(\cal A^{\mathbb{Z}})$ defines a function

$\cal A^{*}\,\to\,\mathbb{R}_{\geq 0}\,\,,\,\,\,\,w\mapsto\mu([w])$

which for convenience is also denoted by $\mu$, yielding $\mu(w)=\mu([w])$ for
any $w\in\cal A^{*}$. This function is a weight function in that it satisfies
the Kirchhoff equalities

(2.5) $\mu(w)=\sum_{a_{i}\in\cal A}\mu(a_{i}w)=\sum_{a_{i}\in\cal
A}\mu(wa_{i})$

for any $w\in\cal A^{*}$. Conversely, it is well known that any weight
function $\mu:\cal A^{*}\to\mathbb{R}_{\geq 0}$ defines an invariant measure
$\mu\in\cal M(\cal A^{\mathbb{Z}})$ which satisfies $\mu([w])=\mu(w)$. The set
$\cal M(\cal A^{\mathbb{Z}})$ can hence be understood as subset of the
infinite dimensional non-negative cone $\mathbb{R}_{\geq 0}^{\cal
A^{*}}=\\{\underset{w\in\cal A^{*}}{\sum}x_{w}\vec{e}_{w}\mid x_{w}\geq 0\\}$,
from which it inherits the product topology; the latter coincides with the
more generally known weak∗-topology on the measure cone $\cal M(\cal
A^{\mathbb{Z}})$.

A measure $\mu\in\cal M(\cal A^{\mathbb{Z}})$ is a probability measure if its
total mass satisfies $\mu(\cal A^{\mathbb{Z}})=1$. A measure $\mu\in\cal
M(\cal A^{\mathbb{Z}})$ is ergodic if $\mu$ can not be written as linear
combination with positive coefficients of two distinct probability measures.
For any subshift $X\subseteq\cal A^{\mathbb{Z}}$ the number $e(X)$ of ergodic
probability measures in $\cal M(X)$ can be finite or infinite; it is equal to
the dimension of the linear convex cone $\cal M(X)\subseteq\mathbb{R}_{\geq
0}^{\cal A^{*}}$. For any subshift $X\subseteq\cal A^{\mathbb{Z}}$ we have
$e(X)\geq 1$; if $e(X)=1$ the subshift $X$ is called uniquely ergodic.

The support $\mbox{Supp}(\mu)$ of any $\mu\in\cal A^{\mathbb{Z}}$ is always a
subshift $X\subseteq\cal A^{\mathbb{Z}}$; if $\mu$ is ergodic, then
$X=\mbox{Supp}(\mu)$ is a minimal subshift. The converse conclusion doesn’t
hold (see section 7 below).

Any non-trivial word $w\in\cal A^{*}$ defines a characteristic measure
$\mu_{w}\in\cal M(\cal A^{\mathbb{Z}})$: if $w$ is not a proper power, then
$\mu_{w}$ is given by

(2.6) $\mu_{w}(B)\,\,:=\,\,\mbox{card}(B\cap\cal O(w^{\pm\infty}))$

for any measurable set $B\subseteq\cal A^{\mathbb{Z}}$. If on the other hand
$w={u}^{m}$ for some $u\in\cal A^{*}$ and some integer $m\geq 2$, where $u$ is
assumed not to be a proper power, then one has

$\mu_{w}\,\,:=\,\,m\cdot\mu_{u}$

In either case, it follows that $\frac{1}{|w|}\mu_{w}$ is a probability
measure. The set of weighted characteristic measures $\lambda\,\mu_{w}$ (for
any $\lambda>0$) is known to be dense in $\cal M(\cal A^{\mathbb{Z}})$. The
support of any characteristic measure is given by

(2.7) $\mbox{Supp}(\mu_{w})=\cal O(w^{\pm\infty})\,.$

To any alphabet $\cal A$ one associates canonically the non-negative alphabet
cone $\mathbb{R}_{\geq 0}A=\\{\underset{a_{k}\in\cal
A}{\sum}x_{k}\vec{e}_{a_{k}}\mid x_{k}\geq 0\\}$. For any invariant measure
$\mu$ on $\cal A^{\mathbb{Z}}$ the evaluation on the letter cylinders
$[a_{k}]$ for all $a_{k}\in\cal A$ defines a letter frequency vector

(2.8) $\vec{v}(\mu):=\sum_{a_{k}\in\cal A}\mu([a_{k}])\,\vec{e}_{a_{k}}\,,$

so that one has a canonical $\mathbb{R}_{\geq 0}$-linear map of cones, denoted
by

(2.9) $\zeta A:\cal M(\cal A^{\mathbb{Z}})\to\mathbb{R}_{\geq
0}A\,,\,\,\mu\mapsto\vec{v}(\mu)\,.$

For any subshift $X\subseteq\cal A^{\mathbb{Z}}$ the restriction of this map
to $\cal M(X)$ will be denoted by $\zeta_{X}$. The image of this map is a
cone, denoted by

(2.10) $\cal C(X):=\zeta_{X}(\cal M(X))\subseteq\mathbb{R}_{\geq 0}A\,,$

and called the letter frequency cone of the subshift $X$. For simplicity we
will below, for any subshift $X\subseteq\cal A^{\mathbb{Z}}$ and any morphism
$\sigma:\cal A^{*}\to\cal B^{*}\,$, use the symbol $M(\sigma)$ to denote all
three linear maps

(2.11) $\mathbb{R}A\to\mathbb{R}B\,,\,\,\mathbb{R}_{\geq
0}A\to\mathbb{R}_{\geq 0}B\,\,$ and $\,\,\cal C(X)\to\cal C(\sigma(X))$

defined by the incidence matrix of the morphism $\sigma$.

More details about these basic facts and some references can be found in
section 2 of [5].

###  2.2. Measures on subshifts via vector towers on directive sequences

In order to state Theorem 2.7 below, which is the main purpose of this
subsection, we first recall some standard notation that is also used later.

A directive sequence $\overleftarrow{\sigma}=(\sigma_{n})_{n\geq 0}$ consists
of level morphisms

(2.12) $\sigma_{n}:\cal A_{n+1}^{*}\to\cal A_{n}^{*}$

for any level $n\geq 0$, where each $\cal A_{n}$ is a finite non-empty set,
called the level $n$ alphabet. We sometimes use the less formal but more
suggestive notation

$\overleftarrow{\sigma}=\sigma_{0}\circ\sigma_{1}\circ\sigma_{2}\circ\ldots$

to denote a directive sequence.

For any integers $m>n\geq 0$ we define the telescoped level morphism

$\sigma_{[n,m)}:=\sigma_{n}\circ\sigma_{n+1}\circ\ldots\circ\sigma_{m-1}$

as well as the level $n$ truncated directive sequence

(2.13) $\overleftarrow{\sigma}\dagger_{\\!n}=(\sigma_{k})_{k\geq n}\,.$

Any directive sequence $\overleftarrow{\sigma}$ as in (2.12) above generates a
subshift $X=X_{\tiny\overleftarrow{\sigma}}$ over the base alphabet $\cal
A_{0}\,$, defined by the convention that ${\bf x}\in\cal A_{0}^{\mathbb{Z}}$
belongs to $X$ if and only if for any finite factor $w$ of $\bf x$ there
exists some level $n\geq 1$ and some letter $a_{j}\in\cal A_{n}$ such that $w$
is also a factor of $\sigma_{[0,n-1)}(a_{j})$.

For any level $n\geq 0$ a directive sequence $\overleftarrow{\sigma}$ as above
defines an intermediate level subshift $X_{n}\subseteq\cal A_{n}^{\mathbb{Z}}$
which is generated by the truncated sequence
$\overleftarrow{\sigma}\dagger_{\\!n}\,$:

(2.14) $X_{n}:=X_{\tiny\overleftarrow{\sigma}\dagger_{\\!n}}$

The subshift $X_{n}$ is the image subshift of the analogously defined level
$n+1$ intermediate subshift $X_{n+1}$ under the morphism $\sigma_{n}\,$, i.e.:

(2.15) $X_{n}=\sigma_{n}(X_{n+1})\qquad\text{for any level}\qquad n\geq 0$

In this paper we will almost exclusively consider directive sequences which
are everywhere growing, by which we mean that the sequence of minimal level
letter image lengths

(2.16)
$\beta_{-}(n):=\min{\big{\\{}}\,|\sigma_{[0,n-1)}(a_{j})|\,\,{\big{|}}\,\,a_{j}\in\cal
A_{n}{\big{\\}}}$

tends to $\infty$ for $n\to\infty$. We have (see for instance Proposition 5.10
of [3]):

######  Fact 2.3.

For every subshift $X\subseteq\cal A^{\mathbb{Z}}$ there exists an everywhere
growing directive sequence $\overleftarrow{\sigma}$ that generates $X$. More
precisely, using the notation from (2.12), one has

(2.17) $\cal A_{0}=\cal A\qquad\text{and}\qquad
X_{\tiny\overleftarrow{\sigma}}=X.$

######  Remark 2.4.

If a directive sequence $\overleftarrow{\sigma}=(\sigma_{n})_{n\geq 0}$ is
everywhere growing, it could well be that some of the telescoped level maps
$\sigma_{[n,m)}:\cal A_{m+1}^{*}\to\cal A_{n}^{*}$ maps a generator
$a_{i}\in\cal A_{m+1}$ to the empty word in $\cal A_{n}^{*}$. In this case,
since the equivalences from Remark 2.1 could fail to hold (see Example 5.1 of
[2]), it is advisable to admit as “generated subshift”
$X_{\tiny\overleftarrow{\sigma}}$ only those ${\bf x}\in\cal
A_{0}^{\mathbb{Z}}$ which can be lifted to any level of
$\overleftarrow{\sigma}$.

With this alteration a simple direct argument shows that quotienting out any
such “eventually erased” letter yields new alphabets $\cal
A^{\prime}_{n}\subseteq\cal A_{n}$ and new level maps
$\sigma^{\prime}_{n}:{\cal A^{\prime}}_{n+1}^{*}\to{\cal A^{\prime}}_{n}^{*}$
such that the issuing directive sequence
$\overleftarrow{\sigma}^{\prime}=(\sigma^{\prime}_{n})_{n\geq 0}$ is
everywhere growing and generates the same subshift as the original sequence
$\overleftarrow{\sigma}$. In addition, any of the morphisms
$\sigma^{\prime}_{n}$ or $\sigma^{\prime}_{[n,m)}$ is non-erasing.

We can hence (and will from now on quietly) assume that any everywhere growing
directive sequence consists of non-erasing level maps only.

######  Remark 2.5.

(1) A directive sequence $\overleftarrow{\sigma}$ that generates a subshift
$X$ is also called an $S$-adic development (or $S$-adic expansion) of $X$,
where $S$ stands sometimes for an (often assumed to be finite) set of
substitutions which contains all level morphisms. This concept and in
particular the terminology “$S$-adic” has been introduced by S. Ferenczi in
[19]. In this context one often assumes that the sequence
$\overleftarrow{\sigma}$ has finite alphabet rank. By this we mean that there
is a uniform upper bound to the cardinality of any level alphabet, so that we
can identify all level alphabets with a single finite alphabet $\cal A$.

(2) If the set $S$ consists of a single endomorphism $\sigma:\cal A^{*}\to\cal
A^{*}$, then the $S$-adic subshift $X$, which is generated by the stationary
directive sequence $\overleftarrow{\sigma}=(\sigma_{n})_{n\geq 0}$ with
$\sigma_{n}=\sigma$ for all $n\geq 0$, is called substitutive. It is important
to note that we require here the substitution $\sigma$ (or rather: the above
stationary directive sequence $\overleftarrow{\sigma}$) to be everywhere
growing. The term “substitution” itself is often used synonymous to
“endomorphism of a free monoid”, but sometimes (varying) additional conditions
are imposed (see for instance [16]).

(3) A very convenient criterion to ensure the condition “everywhere growing”
is given as follows:

  1. ($\mathfrak{Y}$)

Let $\overleftarrow{\sigma}=(\sigma_{n})_{n\geq 0}$ be a directive sequence,
and assume that for every level $n\geq 0$ there exists a level $m>n$ such that
the telescoped incidence matrix $M(\sigma_{[n,m)})$ is positive (i.e. it has
all coefficients $>0$).

One verifies easily that any directive sequence $\overleftarrow{\sigma}$ which
satisfies the criterion ($\mathfrak{Y}$) is indeed everywhere growing.

(4) The criterion ($\mathfrak{Y}$) has another important consequence, namely
that the subshift $X$ generated by $\overleftarrow{\sigma}$ is minimal. For
this conclusion we cite Theorem 5.3 of [7], proved originally in [14].

In [3] to any directive sequence $\overleftarrow{\sigma}$ as in (2.12) there
has been associated the set $\cal V(\overleftarrow{\sigma})$ of vector towers
$\overleftarrow{v}=(\vec{v}_{n})_{n\geq 0}$ over $\overleftarrow{\sigma}$.
Such a vector tower 111 The terminological specification
$\overleftarrow{\sigma}$-compatible vector tower used in [4] has been dropped
here, as all “vector towers” occurring in the present paper satisfy the
compatibility condition (2.19) for any $n\geq 0$. consists of non-negative
vectors

(2.18) $\vec{v}_{n}=\underset{a_{j}\in\cal
A}{\sum}\vec{v}_{n}(a_{j})\,\vec{e}_{a_{j}}\in\mathbb{R}_{\geq 0}^{\cal
A_{n}}$

that are subject to the compatibility condition

(2.19) $\vec{v}_{n}=M(\sigma_{n})\cdot\vec{v}_{n+1}$

for all $n\geq 0$. It has been shown (see [3], Remark 9.5) that for any word
$w\in\cal A_{0}^{*}$ and any such vector tower $\overleftarrow{v}$ the
sequence of sums

$\sum_{a_{j}\,\in\cal A_{n}}\vec{v}_{n}(a_{j})\,|\sigma_{[0,n)}(a_{j})|_{w}\,$

is bounded above and increasing, as long as $\overleftarrow{\sigma}$ is
everywhere growing (but no other condition is needed). This gives:

######  Remark 2.6.

(1) The value

(2.20) $\mu^{\tiny\overleftarrow{v}}(w):=\lim_{n\to\infty}\sum_{a_{j}\,\in\cal
A_{n}}\vec{v}_{n}(a_{j})\,|\sigma_{[0,n)}(a_{j})|_{w}$

is well defined for any $w\in\cal A_{0}^{*}$. It is shown in [3], Propositions
7.4 and 9.4, that the issuing function $\mu^{\tiny\overleftarrow{v}}:\cal
A_{0}^{*}\to\mathbb{R}_{\geq 0}$ satisfies the Kirchhoff equalities (2.5), so
that we can summarize:

(2) Any vector tower $\overleftarrow{v}$ on an everywhere growing directive
sequence $\overleftarrow{\sigma}$ defines via equality (2.20) an invariant
measure on the subshift $X$ generated by $\overleftarrow{\sigma}$, denoted by
$\mu^{\tiny\overleftarrow{v}}\in\cal M(X)$.

In terms of $S$-adic language, the main result of [3] translates directly into
the following (see also section 3 of [4]):

######  Theorem 2.7 ([3]).

Let $\overleftarrow{\sigma}=(\sigma_{n})_{n\geq 0}$ be an everywhere growing
directive sequence which generates the subshift
$X:=X_{\tiny\overleftarrow{\sigma}}$. Then the map

${\mathfrak{m}}_{\tiny\overleftarrow{\sigma}}:\cal
V(\overleftarrow{\sigma})\to\cal
M(X)\,,\,\,\overleftarrow{v}\mapsto\mu^{\tiny\overleftarrow{v}}$

is linear and surjective. $\sqcup$$\sqcap$

For any of the level alphabets $\cal A_{n}$ of a directive sequence
$\overleftarrow{\sigma}$ as above we consider the projection map 222 The map
$pr_{n}$ was denoted in [3] and [4] by $\mathfrak{m}_{n}$, but we decided to
reserve this notation here for the more telling maps introduced below in
section 5. of the set of vector towers to the corresponding non-negative
alphabet cone:

$pr_{n}:\cal V(\overleftarrow{\sigma})\to\mathbb{R}_{\geq 0}^{\cal
A_{n}}\,,\,\,\overleftarrow{v}=(\vec{v}_{n})_{n\geq 0}\mapsto\vec{v}_{n}$

On the base level $n=0$ this projection splits over the evaluation map
$\zeta_{\cal A_{0}}$ from (2.9) via the map
$\mathfrak{m}_{\tiny\overleftarrow{\sigma}}$ from the last theorem. More
precisely, this gives (see [3], Proposition 10.2 (1) and (2)):

######  Proposition 2.8.

For any subshift $X\subseteq\cal A^{\mathbb{Z}}$, generated by an everywhere
growing directive sequence $\overleftarrow{\sigma}$ as in (2.12) and (2.17),
one has:

  1. (1)

The map $\zeta_{X}:\cal M(X)\to\mathbb{R}_{\geq 0}^{\cal
A}\,,\,\,\mu\mapsto(\mu([a_{k}])_{a_{k}\in\cal A}$ satisfies:

$pr_{0}=\zeta_{X}\circ{\mathfrak{m}}_{\tiny\overleftarrow{\sigma}}$

  2. (2)

In particular, for the letter frequency cone $\cal C(X)=\rm{im}(\zeta_{X})$
(see (2.10)) this gives:

$\cal C(X)=\zeta_{X}({\mathfrak{m}_{\tiny\overleftarrow{\sigma}}}(\cal
V(\overleftarrow{\sigma})))$

  3. (3)

Alternatively, the letter frequency cone is obtained as nested intersection as
follows:

$\cal C(X):=\bigcap_{n\geq 1}\,M(\sigma_{[0,n)})(\mathbb{R}_{\geq 0}^{\cal
A_{n}})$

  4. (4)

In particular, $\dim\cal C(X)$ is a lower bound to the number $e(X)$ of
distinct ergodic probability measures on $X$. $\sqcup$$\sqcap$

The following statement is the translation of Remark 9.2 (3) of [3] into the
terminology used here.

######  Lemma 2.9.

For any vector tower $\overleftarrow{v}=(\vec{v}_{n})_{n\geq 0}$ over an
everywhere growing directive sequence $\overleftarrow{\sigma}$ as in (2.12)
one has

$\lim_{n\to\infty}\sum_{a_{j}\in\cal A_{n}}\vec{v}_{n}(a_{j})=0\,,$

where the coefficient $\vec{v}_{n}(a_{j})\in\mathbb{R}_{\geq 0}$ is defined in
equality (2.18). $\sqcup$$\sqcap$

##  3\. The measure transfer and its injectivity for recognizable morphisms

In this section we will first recall the definition of the measure transfer
map and quote some basic properties derived in [5] (see subsection 3.1 below),
then recall the definition and some related properties of recognizable
morphisms (see subsection 3.2 below), and in subsection 3.3 we will derive the
injectivity result from the title of this section.

###  3.1. The measure transfer and some results from [5]

For any non-erasing monoid morphism $\sigma:\cal A^{*}\to\cal B^{*}$ we define
the subdivision alphabet $\cal A_{\sigma}=\\{a_{i}(k)\mid a_{i}\in\cal A\text{
and }1\leq k\leq|\sigma(a_{i})|\\}$. The morphism $\sigma$ now defines a
subdivision morphism $\pi_{\sigma}:\cal A^{*}\to\cal A_{\sigma}^{*}$ and a
letter-to-letter morphism $\alpha_{\sigma}:\cal A_{\sigma}^{*}\to\cal B^{*}$,
given for any $a_{i}\in\cal A$ and any $a_{i}(k)\in\cal A_{\sigma}$ by:

$\pi_{\sigma}(a_{i})=a_{i}(1)\,a_{i}(2)\ldots
a_{i}(|\sigma(a_{i})|)\qquad\text{and}\qquad\alpha_{\sigma}(a_{i}(k))=[\sigma(a_{i})]_{k}$

Here by $[\sigma(a_{i})]_{k}$ we mean the $k$-th letter of the word
$\sigma(a_{i})\in\cal B^{*}$. We obtain directly:

######  Fact 3.1.

For any non-erasing monoid morphism $\sigma:\cal A^{*}\to\cal B^{*}$ one has:

$\sigma\,\,=\,\,\alpha_{\sigma}\circ\pi_{\sigma}$

For any word $w\in\cal A_{\sigma}^{*}$ we denote by $\widehat{w}\in\cal A^{*}$
the shortest word such that $\pi_{\sigma}(\widehat{w})$ contains $w$ as
factor. If such $\widehat{w}$ exists, it is unique; otherwise we treat
$\widehat{w}$ as formal symbol and we set

(3.1) $\mu(\widehat{w})=0$

for any $\mu\in\cal M(\cal A^{\mathbb{Z}})$.

For any measure $\mu\in\cal M(\cal A^{\mathbb{Z}})$ a measure
$\mu^{\pi_{\sigma}}\in\cal M(\cal A_{\sigma}^{\mathbb{Z}})$ is defined in
section 3.1 of [5] by setting $\mu^{\pi_{\sigma}}([w]):=\mu([\widehat{w}])$,
where $[\widehat{w}]$ is the cylinder associated to the word $\widehat{w}$
(see subsection 2.1). On the other hand, for any measure $\mu^{\prime}\in\cal
M(\cal A_{\sigma}^{\mathbb{Z}})$ the classical push-forward measure
$(\alpha_{\sigma})_{*}(\mu^{\prime})$ is an invariant measure on $\cal
B^{\mathbb{Z}}$, since $\alpha_{\sigma}$ is letter-to-letter. We thus obtain
(see [5], section 3):

######  Theorem 3.2.

Let $\sigma:\cal A^{*}\to\cal B^{*}$ be a non-erasing morphism of free
monoids.

(1) For any invariant measure $\mu$ on $\cal A^{\mathbb{Z}}$ an invariant
measure $\mu^{\sigma}$ on $\cal B^{\mathbb{Z}}$ is given by

$\mu^{\sigma}=(\alpha_{\sigma})_{*}(\mu^{\pi_{\sigma}})\,.$

(2) For any word $w^{\prime}\in\cal B^{*}$ the “transferred measure”
$\mu^{\sigma}$ takes on the cylinder $[w^{\prime}]$ the value

$\mu^{\sigma}([w^{\prime}])=\sum_{w_{i}\in\alpha_{\sigma}^{-1}(w^{\prime})}\mu([\widehat{w}_{i}])\,.$

(3) The issuing measure transfer map

$\sigma M:\cal M(\cal A^{\mathbb{Z}})\to\cal M(\cal
B^{\mathbb{Z}})\,,\,\,\mu\mapsto\mu^{\sigma}$

induced by the morphism $\sigma$ has the following properties:

  1. (3a)

The map $\sigma M$ is linear (over $\mathbb{R}$) and continuous (with respect
to the weak∗-topology).

  2. (3b)

The map $\sigma M$ is functorial.

  3. (3c)

If $X$ is the support of $\mu$, then $\sigma(X)$ is the support of
$\mu^{\sigma}$. Hence $\sigma M$ induces in particular on any subshift
$X\subseteq\cal A^{\mathbb{Z}}$ a restriction/co-restriction map

$\sigma M_{X}:\cal M(X)\to\cal M(\sigma(X))\,.$

$\sqcup$$\sqcap$

We also list the following more technical properties derived in [5]:

######  Proposition 3.3.

Let $\sigma:\cal A^{*}\to\cal B^{*}$ be a non-erasing free monoid morphism,
and let $\sigma M$ be the induced transfer map on the measure cones. Let
$\mu\in\cal M(\cal A^{\mathbb{Z}})$ be an invariant measure on the full shift
$\cal A^{\mathbb{Z}}$, and denote as before by $\mu^{\sigma}=\sigma M(\mu)$
the transferred measure on $\cal B^{\mathbb{Z}}$. Then one has:

  1. (a)

The total mass of the transferred measure $\mu^{\sigma}$ is given by the
formula

$\mu^{\sigma}(\cal B^{\mathbb{Z}})=\sum_{a_{k}\in\cal A}\sum_{b_{j}\in\cal
B}|\sigma(a_{k})|_{b_{j}}\cdot\mu(a_{k})\,.$

In particular, if $\mu$ is a probability measure, then in general
$\mu^{\sigma}$ will not be probability.

  2. (b)

For any generator $b_{j}\in\cal B$ we have:

$\mu^{\sigma}([b_{j}])=\sum_{a_{k}\in\cal
A}|\sigma(a_{k})|_{b_{j}}\cdot\mu(a_{k})$

In particular, for the letter frequency vectors from (2.8) we obtain:

(3.2) $\vec{v}(\mu^{\sigma})\,\,=\,\,M(\sigma)\cdot\vec{v}(\mu)$

In other words (see Proposition 4.5 of [5]), the measure transfer map $\sigma
M$ commutes via the evaluation maps $\zeta A$ and $\zeta B$ from (2.9) with
the linear map induced by $\sigma$ on the non-negative cone $\mathbb{R}_{\geq
0}A\,$:

$\zeta B\circ\sigma M\,\,=\,\,M(\sigma)\circ\zeta A$

  3. (c)

For any $w\in\cal A^{*}$ the cylinder measures satisfy:

$\mu^{\sigma}([\sigma(w)])\,\,\geq\,\,\mu([w])$

  4. (d)

For any word $w\in\cal A^{*}$ the characteristic measure $\mu_{w}$ satisfies:

$\sigma M(\mu_{w})=\mu_{\sigma(w)}$

$\sqcup$$\sqcap$

It remains to quote a useful evaluation technique for the transferred measure,
derived in section 4 of [5] from what is stated above as part (2) of Theorem
3.2. For this purpose we define for any non-erasing morphism $\sigma:\cal
A^{*}\to\cal B^{*}$ and any $w\in\cal A^{*},\,u\in\cal B^{*}$ the number
$\lfloor\sigma(w)\rfloor_{u}$ of essential occurrences of $u$ in $\sigma(w)$,
by which we mean that the first letter of $u$ occurs in the $\sigma$-image of
first letter of $w$, and the last letter of $u$ occurs in the $\sigma$-image
of last letter of $w$. By $\langle\sigma\rangle$ we denote the smallest length
of any of the letter images $\sigma(a_{i})$.

######  Proposition 3.4.

([5], Proposition 4.2) Let $\sigma:\cal A^{*}\to\cal B^{*}$ be any non-erasing
monoid morphism, and let $\mu\in\cal M(\cal A^{\mathbb{Z}})$. Then for any
$w^{\prime}\in\cal B^{*}$ with $|w^{\prime}|\geq 2$ the transferred measure
$\mu^{\sigma}=\sigma M(\mu)$ takes on the cylinder $[w^{\prime}]$ the value

$\mu^{\sigma}([w^{\prime}])=\sum_{{\big{\\{}}w_{j}\in\cal
A^{*}\,{\big{|}}\,|w_{j}|\leq\frac{|w^{\prime}|-2}{\langle\sigma\rangle}+2{\big{\\}}}}{\lfloor\sigma(w_{j})\rfloor}_{w^{\prime}}\cdot\mu([w_{j}])\,.$

$\sqcup$$\sqcap$

###  3.2. Recognizable morphisms and some related properties

The following notion has become more and more central to symbolic dynamics
(see for instance [9], [13], [15] or [16]):

######  Definition 3.5.

Let $\sigma:\cal A^{*}\to\cal B^{*}$ be a non-erasing morphism, and let
$X\subseteq\cal A^{\mathbb{Z}}$ be a subshift over $\cal A$. Then $\sigma$ is
said to be recognizable in $X$ if the following conclusion is true:

Consider biinfinite words ${\bf x},{\bf x^{\prime}}\in X\subseteq\cal
A^{\mathbb{Z}}$ and ${\bf y}\in\cal B^{\mathbb{Z}}$ which satisfy:

  1. (*)

${\bf y}=T^{k}(\sigma^{\mathbb{Z}}({\bf x}))$ and ${\bf
y}=T^{\ell}(\sigma^{\mathbb{Z}}({\bf x^{\prime}}))$ for some integers $k,\ell$
which satisfy $0\leq k\leq|\sigma(x_{1})|-1$ and
$0\leq\ell\leq|\sigma(x^{\prime}_{1})|-1$, where $x_{1}$ and $x^{\prime}_{1}$
are the first letters of the positive half-words ${\bf
x_{[1,\infty)}}=x_{1}x_{2}\ldots$ of ${\bf x}$ and ${\bf
x^{\prime}_{[1,\infty)}}=x^{\prime}_{1}x^{\prime}_{2}\ldots$ of ${\bf
x^{\prime}}$ respectively.

Then one has ${\bf x}=\bf x^{\prime}$ and $k=\ell$.

As we will see in the next subsection, recognizability in a subshift is much
related to the following:

######  Definition 3.6 ([5], Section 5).

For any non-erasing monoid morphism $\sigma:\cal A^{*}\to\cal B^{*}$ and any
subshift $X\subseteq\cal A^{\mathbb{Z}}$ we define the following two
properties:

  1. (1)

$\sigma$ is shift-orbit injective in $X$: Any $\bf x$ and $\bf y$ in $X$ have
images $\sigma({\bf x})$ and $\sigma({\bf y})$ in the same shift-orbit if and
only $\bf x$ and $\bf y$ lie in a common shift-orbit.

  2. (2)

$\sigma$ is shift-period preserving in $X$: For any periodic biinfinite word
$w^{\pm\infty}=\ldots www\ldots\in X$ the word $w$ can be written as proper
power if and only if $\sigma(w)$ can be written as proper power.

Here $w\in\cal A^{*}\smallsetminus\\{\varepsilon\\}$ is a proper power 333
Elements in $\cal A^{*}$ which are not a proper power are sometimes called
“primitive”. However, since $\cal A^{*}$ is canonically embedded into the free
group $F(\cal A)$, where the notion of “primitive elements” is classical, but
has a different meaning, we believe it is better not to use this terminology
for a different purpose. if $w=u^{m}$ for some $u\in\cal A^{*}$ and some
integer $m\geq 2$.

The following useful property is a direct consequence of the previous
definition (see Lemma 5.2 of [5]).

######  Lemma 3.7.

Let $\sigma_{1}:\cal A^{*}\to\cal B^{*}$ and $\sigma_{2}:\cal B^{*}\to\cal
C^{*}$ be two non-erasing morphisms, and consider a subshift $X\subseteq\cal
A^{\mathbb{Z}}$ as well as its image subshift $Y=\sigma_{1}(X)\subseteq\cal
B^{\mathbb{Z}}$. Then we have:

  1. (1)

The composed morphism $\sigma_{2}\circ\sigma_{1}:\cal A^{*}\to\cal C^{*}$ is
shift-orbit injective in $X$ if and only if $\sigma_{1}$ is shift-orbit
injective in $X$ and $\sigma_{2}$ is shift-orbit injective in $Y$.

  2. (2)

The composed morphism $\sigma_{2}\circ\sigma_{1}:\cal A^{*}\to\cal C^{*}$ is
shift-period preserving in $X$ if and only if $\sigma_{1}$ is shift-period
preserving in $X$ and $\sigma_{2}$ is shift-period preserving in $Y$.
$\sqcup$$\sqcap$

###  3.3. Injectivity of the measure transfer for recognizable morphisms

Let $\sigma:\cal A^{*}\to\cal B^{*}$ be a non-erasing morphism of free
monoids, and let $\pi_{\sigma}:\cal A^{*}\to\cal A_{\sigma}^{*}$ and
$\alpha_{\sigma}:\cal A_{\sigma}^{*}\to\cal B^{*}$ be the canonical
subdivision morphism and the induced letter-to-letter morphism associated to
$\sigma$ which satisfy $\sigma=\alpha_{\sigma}\circ\pi_{\sigma}$ (see Fact
3.1). For any subshift $X\subseteq\cal A^{\mathbb{Z}}$ we consider the image
subshift $\pi_{\sigma}(X)\subseteq\cal A_{\sigma}^{\mathbb{Z}}$ and the
induced restriction/co-restriction

$\alpha_{\sigma}^{X}:\pi_{\sigma}(X)\to\sigma(X)$

of the map $\alpha_{\sigma}^{\mathbb{Z}}:\cal A_{\sigma}^{\mathbb{Z}}\to\cal
B^{\mathbb{Z}}$ to $\pi_{\sigma}(X)$ and $\sigma(X)$ respectively.

######  Proposition 3.8.

For any non-erasing morphism $\sigma:\cal A^{*}\to\cal B^{*}$ and any subshift
$X\subseteq\cal A^{\mathbb{Z}}$ the following statements are equivalent:

  1. (1)

$\sigma$ is recognizable in $X$.

  2. (2)

$\alpha_{\sigma}^{X}$ is an isomorphism of subshifts.

  3. (3)

$\alpha_{\sigma}$ is shift-orbit injective and shift-period preserving in
$\pi_{\sigma}(X)$.

  4. (4)

$\sigma$ is shift-orbit injective and shift-period preserving in $X$.

###### Proof.

We first note that by definition $\alpha_{\sigma}^{X}$ is continuous and
surjective, so that claim (2) is equivalent to stating that
$\alpha_{\sigma}^{X}$ is injective.

Next we observe that claim (1) is equivalent to stating that
$\alpha_{\sigma}^{X}$ is recognizable in $\pi_{\sigma}(X)$. This is a direct
consequence of the product decomposition
$\sigma=\alpha_{\sigma}\circ\pi_{\sigma}$ from Fact 3.1 and of Lemma 3.5 of
[9], since every subdivision morphism $\pi_{\sigma}$ is recognizable in the
full shift, as follows directly from the definition of $\pi_{\sigma}$.

In order to show the equivalence (1) $\Longleftrightarrow$ (2) we apply
Definition 3.5 to the morphism $\alpha_{\sigma}$ and the subshift
$\pi_{\sigma}(X)\,$: we observe that, since $|\alpha_{\sigma}(x)|=1$ for any
letter $x\in\cal A_{\sigma}\,$, in the hypothesis (*) of Definition 3.5 the
integers $k$ $\ell$ are necessarily equal to 0. But in this case the
conclusion ${\bf x}={\bf x}^{\prime}$ stated there amounts precisely to
assuring that the map $\alpha_{\sigma}^{\mathbb{Z}}$ is injective on
$\pi_{\sigma}(X)$, or in other words, that $\alpha_{\sigma}^{X}$ is injective.

The equivalence (2) $\Longleftrightarrow$ (3) is immediate, since any
subshift-isomorphism preserves orbits and shift-periods, while conversely, any
shift-orbit injective letter-to-letter morphism could only fail to be
injective if on some periodic orbit the shift-period is not preserved.

Finally, the equivalence (3) $\Longleftrightarrow$ (4) is a direct consequence
of Lemma 3.7, since every subdivision morphism $\pi_{\sigma}$ is shift-orbit
injective and shift-period preserving in the full shift (see Lemma 5.3 of
[5]). $\sqcup$$\sqcap$

Note that the equivalence of the statements (1) and (2) from Proposition 3.8
has already been observed in [16], Proposition 2.4.24. Indeed, Fabien Durand
has suggested to us to use this equivalence in order to derive the following
corollary. In the mean time we have obtained a result which is actually a bit
stronger: it turns out (see Theorem 5.5 of [5]) that the hypothesis “shift-
orbit injective” suffices to obtain the same conclusion as stated in Corollary
3.9 below, but the proof is much less direct.

######  Corollary 3.9.

For any non-erasing morphism $\sigma:\cal A^{*}\to\cal B^{*}$ and any subshift
$X\subseteq\cal A^{\mathbb{Z}}$ the measure transfer map
$\sigma_{X}M:\mu\to\mu^{\sigma}$ is injective if $\sigma$ is recognizable in
$X$.

###### Proof.

We decompose $\sigma=\alpha_{\sigma}\circ\pi_{\sigma}$ as in Fact 3.1, so that
from the functoriality of the measure transfer (see property (3b) of Theorem
3.2) we have $\sigma_{X}M=(\alpha_{\sigma}^{X})M\circ\pi_{\sigma}M$. The
injectivity of $\pi_{\sigma}M$ is immediate from the definition of a
subdivision morphism (see Lemma 5.4 of [5]), and the injectivity of
$(\alpha_{\sigma}^{X})M$ is a direct consequence of Proposition 3.8 (2).
$\sqcup$$\sqcap$

######  Remark 3.10.

Consider any non-erasing morphism $\sigma:\cal A^{*}\to\cal B^{*}$ and any
subshift $X\subseteq\cal A^{\mathbb{Z}}$ with image subshift
$Y=\sigma(X)\subseteq\cal B^{\mathbb{Z}}$.

(1) Assume that the subshift $Y$ contains a periodic word $w^{\pm\infty}$ for
some $w\in\cal B^{*}\smallsetminus\\{\varepsilon\\}$, and that the morphism
$\sigma$ is shift–orbit injective. Then, in order for $\sigma$ to be shift-
period preserving in $X$, a necessary condition is that at least one of the
letters $a_{i}\in\cal A$ satisfies $|\sigma(a_{i})|\leq|w|$.

As a consequence, unless a given subshift $Y$ is aperiodic, in any everywhere
growing $S$-adic development of $Y$ there will always be infinitely many level
morphisms which are not recognizable in their corresponding level subshift.

(2) This has sparked the following weakening of the notion of
“recognizability” which has become recently very popular (see for instance
[2]).

The morphism $\sigma$ is said to be recognizable for aperiodic points in $X$
if the conclusion in Definition 3.5 holds under the strengthened assumption
that ${\bf y}$ is not a periodic word.

(3) From the above proof of Proposition 3.8 we observe that the property
“shift-orbit injective in $X$” implies the property “recognizable for
aperiodic points in $X$”.

Indeed, since the subdivision morphism $\pi_{\sigma}$ is always shift-orbit
injective and shift-periodic preserving (and thus recognizable) in the full
shift, the property “$\sigma$ is recognizable for aperiodic points in $X$” is
equivalent to “$\alpha_{\sigma}^{X}$ is recognizable for aperiodic points in
$\pi_{\sigma}(X)$”. This in turn is equivalent to stating that every non-
periodic word in $\sigma(X)$ has precisely one preimage under the letter-to-
letter map $\alpha_{\sigma}$. But since we assume that $\sigma$ and hence
$\alpha_{\sigma}^{X}$ is shift-orbit injective, two distinct such preimages
must lie in the same shift-orbit, which implies that their image in
$\sigma(X)$ must be periodic.

##  4\. The measure transfer via vector towers

In this section we will consider a subshift $X$ given by means of a directive
sequence, an invariant measures $\mu$ on $X$ given by means of a vector tower
on this directive sequence, and a morphism $\tau:X\to Y=\tau(X)$ which we use
to build a new directive sequence for $Y$ by simply adding $\tau$ at the
bottom to the given sequence. Then the given vector tower is naturally
transferred to a new vector tower on the new directive sequence, and, as do
all vector towers, it defines an invariant measure $\mu^{\prime}$ on the
subshift $Y$ generated by this new sequence. The main goal of this section is
to show that the new measure $\mu^{\prime}$ is precisely the image of given
measure $\mu$ under the transfer map $\tau M$ induced by the morphism $\tau$
(see Theorem 3.2 (1)).

For convenience we summarize the running hypotheses for this section as
follows:

######  Assumption 4.1.

Let $\tau:\cal A^{*}\to\cal B^{*}$ be a non-erasing morphism of free monoids
over finite alphabets $\cal A$ and $\cal B$, and let
$\overleftarrow{\sigma}=(\sigma_{n})_{n\geq 0}$ be an everywhere growing
directive sequence with base level alphabet $\cal A_{0}=\cal A$. Let
$X:=X_{\tiny\overleftarrow{\sigma}}\subseteq\cal A^{\mathbb{Z}}$ be the
subshift generated by $\overleftarrow{\sigma}$, and denote by $Y:=\tau(X)$ the
image subshift of $X$ given by the morphism $\tau$ (see Remark 2.1).

######  Definition-Remark 4.2.

Let $\tau$ and $\overleftarrow{\sigma}=(\sigma_{n})_{n\geq 0}$ as well as the
subshifts $X$ and $Y$ be as in Assumption 4.1.

(1) We define a second “prolonged” directive sequence
$\overleftarrow{\sigma}^{\tau}=(\sigma^{\prime}_{n})_{n\geq 0}$ by setting
$\sigma^{\prime}_{n}:=\sigma_{n-1}$ for any level $n\geq 1$ and
$\sigma^{\prime}_{0}:=\tau$, and observe from Lemma 2.2 that the subshift
$X_{\tiny\overleftarrow{\sigma}^{\tau}}$ generated by
$\overleftarrow{\sigma}^{\tau}$ agrees precisely with the $\tau$-image
subshift $Y=\tau(X)\in\cal B^{\mathbb{Z}}$.

(2) Consider now any vector tower $\overleftarrow{v}=(\vec{v}_{n})_{n\geq 0}$
over $\overleftarrow{\sigma}$, and let
$\mu=\mathfrak{m}_{\tiny\overleftarrow{\sigma}}(\overleftarrow{v})$ be the
invariant measure on $X$ associated to $\overleftarrow{v}$ via Theorem 2.7. We
obtain a prolonged vector tower
$\overleftarrow{v}^{\tau}=(\vec{v}\,^{\prime}_{\\!n})_{n\geq 0}$ over
$\overleftarrow{\sigma}^{\tau}$ by setting
$\vec{v}\,^{\prime}_{\\!n}=\vec{v}_{n-1}$ for any level $n\geq 1$, and by
setting $\vec{v}\,^{\prime}_{\\!0}:=M(\tau)\cdot\vec{v}_{0}\,$. Let
$\mu^{\prime}$ be the associated measure on $Y$, i.e.

(4.1)
$\mu^{\prime}=\mathfrak{m}_{\tiny\overleftarrow{\sigma}^{\tau}}(\overleftarrow{v}^{\tau})\,.$

We can now link up the measure transfer map defined and studied in [5] with
the technology of vector towers from our previous papers [3],[4]. The
following will be the basis for all results presented in this paper:

######  Proposition 4.3.

Let $\tau,\overleftarrow{\sigma}$ and $X$ be as in Assumption 4.1, and let
$\overleftarrow{v}=(\vec{v}_{n})_{n\geq 0}$ be a vector tower over
$\overleftarrow{\sigma}$, with associated invariant measure
$\mu=\mathfrak{m}_{\tiny\overleftarrow{\sigma}}(\overleftarrow{v})$ on $X$.
Let $\overleftarrow{\sigma}^{\tau}=(\sigma^{\prime}_{n})_{n\geq 0}\,$,
$\overleftarrow{v}^{\tau}=(\vec{v}\,^{\prime}_{\\!n})_{n\geq 0}$ and
$\mu^{\prime}=\mathfrak{m}_{\tiny\overleftarrow{\sigma}^{\tau}}(\overleftarrow{v}^{\tau})$
be as in Definition-Remark 4.2.

Then the measure transfer map $\tau M:\cal M(\cal A^{\mathbb{Z}})\to\cal
M(\cal B^{\mathbb{Z}})$ induced by the morphism $\tau$ satisfies:

$\mu^{\prime}=\tau M(\mu)\,\,\,[=\mu^{\tau}]$

###### Proof.

In this proof we will freely use the terminology from [5] as recalled in
section 3.1.

For any word $w^{\prime}\in\cal B^{*}$ we consider in the subdivision monoid
$\cal A_{\tau}^{*}$ the subset $W(w^{\prime})$ of preimages $w_{i}$ of
$w^{\prime}$ under the induced letter-to-letter morphism $\alpha_{\tau}:\cal
A_{\tau}^{*}\to\cal B^{*}$. For each $w_{i}\in W(w^{\prime})$ consider (as in
the paragraph subsequent to Fact 3.1) the word $\widehat{w}_{i}\in\cal A^{*}$
defined by the conditions that (a) its canonically subdivided image
$\pi_{\tau}(\widehat{w}_{i})$ contains $w_{i}\,$, and that (b) the word
$\widehat{w}_{i}$ is shortest among all words in $\cal A^{*}$ which satisfy
(a). Recall from (3.1) that either $\widehat{w}_{i}$ exists and is unique, or
else we formally set $\mu(\widehat{w}_{i})=0$ for any $\mu\in\cal M(\cal
A^{\mathbb{Z}})$. From Theorem 3.2 (2) we know that
$\mu^{\tau}(w^{\prime})=\underset{w_{i}\in
W(w^{\prime})}{\sum}\mu(\widehat{w}_{i})\,$ (with $\mu^{\tau}=\tau M(\mu)$ as
before).

For the purpose of using formula (2.20) we consider now the value of the
approximating sum on its right hand side of this formula for any (large) level
$n-1$, for each of the words $\widehat{w}_{i}$ and the given vector tower
$\overleftarrow{v}$, i.e. the term (see (2.18) for the notation)

(4.2) $\sum_{a\,\in\cal
A_{n-1}}\vec{v}_{n-1}(a)\,|\sigma_{[0,n-1)}(a)|_{\widehat{w}_{i}}\,.$

We sum up the results of (4.2) over all $w_{i}\in W(w^{\prime})$ to get

(4.3) $\sum_{w_{i}\in W(w^{\prime})}\sum_{a\,\in\cal
A_{n-1}}\vec{v}_{n-1}(a)\,|\sigma_{[0,n-1)}(a)|_{\widehat{w}_{i}}\,,$

and compare the obtained sum to the limit on the right hand side of (2.20),
when this formula is applied to $w^{\prime}$ and to the vector tower
$\overleftarrow{v}^{\tau}$ over the prolonged directive sequence
$\overleftarrow{\sigma}^{\tau}$. The $n$-th term of this limit gives the sum

(4.4) $\sum_{a\,\in\cal
A_{n-1}}\vec{v}\,^{\prime}_{\\!n}(a)\,|\sigma^{\prime}_{[0,n)}(a)|_{w^{\prime}}\,.$

When comparing the two sums (4.3) and (4.4) we keep in mind that according to
the set-up from Definition-Remark 4.2 for any $n\geq 1$ we have
$\vec{v}\,^{\prime}_{\\!n}(a)=\vec{v}_{n-1}(a)$ for any $a\in\cal A_{n-1}\,$,
as well as $\sigma^{\prime}_{[0,n)}=\tau\circ\sigma_{[0,n-1)}\,$.

We now notice that each occurrence of any of the $\widehat{w}_{i}$ in any of
the image words $\sigma_{[0,n-1)}(a)$ with $a\in\cal A_{n-1}$ defines
precisely an occurrence of $w_{i}$ in $\pi_{\tau}(\sigma_{[0,n-1)}(a))$, and
thus an occurrence of $w^{\prime}$ in
$\alpha_{\tau}(\pi_{\tau}(\sigma_{[0,n-1)}(a)))=\tau(\sigma_{[0,n-1)}(a))=\sigma^{\prime}_{[0,n)}(a)$.
Furthermore, two distinct occurrences of $\widehat{w}_{i}$ in some
$\sigma_{[0,n-1)}(a)$ define distinct occurrences of $w_{i}$ in
$\pi_{\tau}(\sigma_{[0,n-1)}(a))$, and thus distinct occurrences of
$w^{\prime}$ in $\sigma^{\prime}_{[0,n)}(a)$. The same is true for occurrences
of distinct $\widehat{w}_{i}$ in $\sigma_{[0,n-1)}(a)$. It follows (using the
above recalled equality $\vec{v}\,^{\prime}_{\\!n}=\vec{v}_{n-1}$) that

(4.5) $\sum_{w_{i}\in W(w^{\prime})}\,\sum_{a\,\in\cal
A_{n-1}}\vec{v}_{n-1}(a)\,|\sigma_{[0,n-1)}(a)|_{\widehat{w}_{i}}\leq\sum_{a\,\in\cal
A_{n-1}}\vec{v}\,^{\prime}_{n}(a)\,|\sigma^{\prime}_{[0,n)}(a)|_{w^{\prime}}$

But the opposite inequality is also true, up to a constant $K_{n}$ which only
depends on $\overleftarrow{\sigma}$ and not on $\overleftarrow{v}\,$:

(4.6) $\sum_{a\,\in\cal
A_{n-1}}\vec{v}\,^{\prime}_{n}(a)\,|\sigma^{\prime}_{[0,n)}(a)|_{w^{\prime}}\leq\sum_{w_{i}\in
W(w^{\prime})}\,\sum_{a\,\in\cal
A_{n-1}}\vec{v}_{n-1}(a)\,|\sigma_{[0,n-1)}(a)|_{\widehat{w}_{i}}+K_{n}$

Indeed, any occurrence of $w^{\prime}$ in $\sigma^{\prime}_{[0,n)}(a)$ defines
in a unique manner an occurrence of some $w_{i}$ in
$\pi_{\tau}(\sigma_{[0,n-1)}(a))$. The latter defines (again uniquely) an
occurrence of $\widehat{w}_{i}$ in $\sigma_{[0,n-1)}(a)$, unless the
corresponding occurrence of $w_{i}$ in $\pi_{\tau}(\sigma_{[0,n-1)}(a))$ takes
place in a suffix or prefix of length bounded by the maximum
$m(w^{\prime})\geq 0$ of all $|\widehat{w}_{i}|$. We hence deduce:

$K_{n}\leq 2m(w^{\prime})\sum_{a\,\in\cal A_{n-1}}\vec{v}_{n-1}(a)$

It follows now from Lemma 2.9 that the right hand side of the last inequality
tends to 0 for $n\to\infty$, so that we obtain from (4.5) and (4.6) through
the above definitions
$\mu=\mathfrak{m}_{\tiny\overleftarrow{\sigma}}(\overleftarrow{v})=\mu^{\tiny\overleftarrow{v}}$
and
$\mu^{\prime}=\mathfrak{m}_{\tiny\overleftarrow{\sigma}^{\tau}}(\overleftarrow{v}^{\tau})=\mu^{\tiny\overleftarrow{v}^{\tau}}$
the desired result

$\mu^{\tau}(w^{\prime})=\sum_{w_{i}\in
W(w^{\prime})}\left(\lim_{n\to\infty}\sum_{a\,\in\cal
A_{n-1}}\vec{v}_{n-1}(a)\,|\sigma_{[0,n-1)}(a)|_{\widehat{w}_{i}}\right)$
$=\lim_{n\to\infty}\sum_{a\,\in\cal
A_{n-1}}\vec{v}\,^{\prime}_{n}(a)\,|\sigma^{\prime}_{[0,n)}(a)|_{w^{\prime}}=\mu^{\prime}(w^{\prime})$

for any $w^{\prime}\in\cal B^{*}$. $\sqcup$$\sqcap$

As a first application of the above shown “basic” Proposition 4.3 we derive:

######  Proposition 4.4.

For any non-erasing morphism $\sigma:\cal A^{*}\to\cal B^{*}$ and any subshift
$X\subseteq\cal A^{\mathbb{Z}}$ with image subshift $\sigma(X)$ the induced
measure transfer map

$\sigma M:\cal M(\cal A^{\mathbb{Z}})\to\cal M(\cal
B^{\mathbb{Z}})\,,\,\,\mu\mapsto\mu^{\sigma}$

maps the measure cone $\cal M(X)$ surjectively to the measure cone $\cal
M(\sigma(X))$.

###### Proof.

We consider any everywhere growing directive sequence $\overleftarrow{\sigma}$
which generates $X$; from Fact 2.3 we know that such $\overleftarrow{\sigma}$
exists for any subshift $X$. By prolonging $\overleftarrow{\sigma}$ through
the morphism $\sigma$ as explained above in Definition-Remark 4.2 we obtain
any everywhere growing directive sequence
$\overleftarrow{\sigma}^{\prime}:=\overleftarrow{\sigma}^{\sigma}$ which
generates $\sigma(X)$. We then apply Theorem 2.7 to obtain for any measure
$\mu^{\prime}\in\cal M(\sigma(X))$ a vector tower $\overleftarrow{v}^{\prime}$
on $\overleftarrow{\sigma}^{\prime}$ with
$\mathfrak{m}_{\tiny\overleftarrow{\sigma}^{\prime}}(\overleftarrow{v}^{\prime})=\mu$.
Truncating now the last term of $\overleftarrow{v}^{\prime}$ gives a vector
tower $\overleftarrow{v}$ on $\overleftarrow{\sigma}$, which by Remark 2.6 (2)
defines a measure
$\mu:=\mathfrak{m}_{\tiny\overleftarrow{\sigma}}(\overleftarrow{v})$ on $X$.
We can now apply Proposition 4.3 to obtain
$\mu^{\prime}=\mu^{\sigma}\,[=\sigma M(\mu)]$. $\sqcup$$\sqcap$

######  Remark 4.5.

We’d like to note that as a result of the material presented in this section
we have now derived an alternative way how to understand the transferred
measure $\mu^{\sigma}=\sigma M(\mu)\in\cal M(\cal B^{\mathbb{Z}})$, for any
non-erasing morphism $\sigma:\cal A^{*}\to\cal B^{*}$ and any invariant
measure $\mu\in\cal M(\cal A^{\mathbb{Z}})$.

It turns out that in many circumstances the use of vector towers as presented
here is more convenient when dealing with $\mu^{\sigma}$ in practice, compared
to the definition as studied in sections 3 and 4 of [5], and also compared to
the approximation method via weighted characteristic measures indicated in
Remark 3.9 of [5].

##  5\. Measure towers and vector towers

Throughout this section we will assume that

$\overleftarrow{\sigma}=(\sigma_{n}:\cal A_{n+1}^{*}\to\cal A_{n}^{*})_{n\geq
0}$

is an everywhere growing directive sequence which generates a subshift
$X=X_{0}\subseteq\cal A_{0}^{\mathbb{Z}}$ (and where all level maps
$\sigma_{n}$ are non-erasing, see Remark 2.4). As in (2.14) we denote for any
level $k\geq 0$ by $X_{k}\subseteq\cal A_{k}^{\mathbb{Z}}$ the intermediate
subshift of level $k$, which is generated by the truncated sequence
$\overleftarrow{\sigma}\dagger_{\\!k}=(\sigma_{n}:\cal A_{n+1}^{*}\to\cal
A_{n}^{*})_{n\geq k}\,$ from (2.13).

######  Definition 5.1.

A measure tower on $\overleftarrow{\sigma}$, denoted by
$\overleftarrow{\mu}=(\mu_{n})_{n\geq 0}$, is given by a sequence of measures
$\mu_{n}\in\cal M(\cal A_{n}^{\mathbb{Z}})$ which satisfy:

$\mu_{n}=\sigma_{n}M(\mu_{n+1})$

The set of measure towers on $\overleftarrow{\sigma}$ will be denoted by $\cal
M(\overleftarrow{\sigma})$.

We will now construct a particular type of measure towers on a given directive
sequence $\overleftarrow{\sigma}$ as above, starting from a vector tower
$\overleftarrow{v}=(\vec{v}_{n})_{n\geq 0}$ on $\overleftarrow{\sigma}$. We
first observe that for any intermediate level $k\geq 0$ we obtain from
$\overleftarrow{\sigma}$ via the truncated directive sequence
$\overleftarrow{\sigma}\dagger_{\\!k}$ a “truncated evaluation map”
$\mathfrak{m}_{k}:=\mathfrak{m}_{\tiny\overleftarrow{\sigma}\dagger_{\\!k}}:\cal
V(\overleftarrow{\sigma}\dagger_{\\!k})\to\cal M(X_{k})$. From the vector
tower $\overleftarrow{v}$ we obtain similarly a “truncated” vector tower
$\overleftarrow{v}\dagger_{\\!k}=(\vec{v}_{n})_{n\geq k}$ on
$\overleftarrow{\sigma}\dagger_{\\!k}\,$, which defines the corresponding
shift-invariant “level $k$ measure”

(5.1) $\mu_{k}:=\mathfrak{m}_{k}(\overleftarrow{v}\dagger_{\\!k})$

on the subshift $X_{k}\subseteq\cal A_{k}^{\mathbb{Z}}$. From Proposition 4.3
we obtain directly that $\sigma_{k}M(\mu_{k+1})=\mu_{k}$ for all $k\geq 0$, so
that we have:

######  Lemma 5.2.

For any vector tower $\overleftarrow{v}$ on an everywhere growing directive
sequence $\overleftarrow{\sigma}$ the family of level $k$ measures $\mu_{k}$
as in (5.1), for all $k\geq 0$, defines a measure tower
$\overleftarrow{\mathfrak{m}}(\overleftarrow{v}):=(\mu_{k})_{k\geq 0}$ on
$\overleftarrow{\sigma}$. $\sqcup$$\sqcap$

######  Definition-Remark 5.3.

Conversely, every measure tower $\overleftarrow{\mu}=(\mu_{n})_{n\geq 0}$ on a
directive sequence $\overleftarrow{\sigma}$ as above determines a vector tower
$\overleftarrow{\zeta}(\overleftarrow{\mu})=(\vec{v}_{n})_{n\geq 0}$ on
$\overleftarrow{\sigma}$, given by the letter frequency vectors
$\vec{v}_{n}:=\vec{v}(\mu_{n})=\zeta_{X_{n}}(\mu_{n})$ from (2.8) and (2.9).
The fact that $\overleftarrow{\zeta}(\overleftarrow{\mu})$ is indeed a vector
tower, i.e. that compatibility conditions (2.19) are satisfied, is a direct
application of Proposition 4.5 of [5], stated above as equality (3.2).

The above set-up of measure towers and vector towers over a given directive
sequence is very natural, and indeed, it turns out that the two are
essentially equivalent. More precisely, we obtain:

######  Proposition 5.4.

For any everywhere growing directive sequence $\overleftarrow{\sigma}$ there
is a canonical $\mathbb{R}_{\geq 0}$-linear bijection

$\overleftarrow{\zeta}:\cal M(\overleftarrow{\sigma})\to\cal
V(\overleftarrow{\sigma})$

between the cone of measure towers on one hand and the cone of vector towers
on the other, given by the map

$\overleftarrow{\mu}\mapsto\overleftarrow{\zeta}(\overleftarrow{\mu})\qquad\text{and
its
inverse}\qquad\overleftarrow{v}\mapsto\overleftarrow{\mathfrak{m}}(\overleftarrow{v})\,.$

###### Proof.

The fact, that the composition
$\overleftarrow{\zeta}\circ\overleftarrow{\mathfrak{m}}$ gives the identity on
$\cal V(\overleftarrow{\sigma})$, follows directly from Proposition 2.8 (1),
when applied to all truncated sequences $\overleftarrow{\sigma}\dagger_{\\!k}$
with $k\geq 0$. We obtain in particular that the map
$\overleftarrow{\mathfrak{m}}$ is injective.

On the other hand, we can apply Theorem 2.7 to each of the truncated sequences
$\overleftarrow{\sigma}\dagger_{\\!k}$ to obtain the surjectivity of the map
$\mathfrak{m}_{k}:\cal V(\overleftarrow{\sigma}\dagger_{\\!k})\to\cal
M(X_{k})$ for any level $k\geq 0$. It follows then directly from the
definition set up in Lemma 5.2 above that the map
$\overleftarrow{\mathfrak{m}}:\cal V(\overleftarrow{\sigma})\mapsto\cal
M(\overleftarrow{\sigma})$ must be surjective.

Hence $\overleftarrow{\mathfrak{m}}$ is a bijective map, which implies that
$\overleftarrow{\zeta}$ must also be bijective, and that
$\overleftarrow{\mathfrak{m}}\circ\overleftarrow{\zeta}$ is the identity on
$\cal M(\overleftarrow{\sigma})$.

The linearity of the maps $\overleftarrow{\zeta}$ and
$\overleftarrow{\mathfrak{m}}$ is a direct consequence of the linearity (see
section 2.2) of the maps $\mathfrak{m}_{k}$ and $\zeta_{X_{n}}$ used in the
above definitions of the measure or vector towers
$\overleftarrow{\mathfrak{m}}(\overleftarrow{v})=(\mathfrak{m}_{k}(\overleftarrow{v}\dagger_{\\!k}))_{k\geq
0}$ and
$\overleftarrow{\zeta}(\overleftarrow{\mu})=(\zeta_{X_{n}}(\mu_{n}))_{n\geq
0}$ respectively. $\sqcup$$\sqcap$

Although a bit similar in notation, the two cones $\cal
M(\overleftarrow{\sigma})$ and $\cal M(X_{\tiny\overleftarrow{\sigma}})$
should not be confused. Indeed, without further assumptions on the given set-
up, the structure of the cone $\cal M(\overleftarrow{\sigma})$ of measure
towers will not only depend on the given subshift
$X=X_{\tiny\overleftarrow{\sigma}}$ but can vary quite a bit depending on the
choice of the $S$-adic development $\overleftarrow{\sigma}$ of $X$. More
precisely, we have:

######  Remark 5.5.

For any everywhere growing directive sequence $\overleftarrow{\sigma}$ which
generates a subshift $X=X_{\tiny\overleftarrow{\sigma}}$ the composition

(5.2)
$\mathfrak{m}_{\tiny\overleftarrow{\sigma}}\circ\overleftarrow{\zeta}:\cal
M(\overleftarrow{\sigma})\to\cal M(X)$

is linear and surjective since $\overleftarrow{\zeta}$ is linear and bijective
by Proposition 5.4, and $\mathfrak{m}_{\tiny\overleftarrow{\sigma}}$ is linear
and surjective by Theorem 2.7. However, in general the map
$\mathfrak{m}_{\tiny\overleftarrow{\sigma}}\circ\overleftarrow{\zeta}$ will be
far from being injective.

We thus consider the following strengthening on the hypotheses of the given
directive sequence, which has been considered already by several other authors
in a related context (compare Definition 4.1 of [9] or subsection 3.3 of
[13]):

######  Definition 5.6.

A directive sequence (or an $S$-adic development)
$\overleftarrow{\sigma}=(\sigma_{n})_{n\geq 0}$ is called totally recognizable
if every level map $\sigma_{n}$ is recognizable in the corresponding subshift
$X_{n+1}$ (see Definition 3.5 and Proposition 3.6). If all but finitely many
of the level maps $\sigma_{n}$ are recognizable in $X_{n+1}$, we call
$\overleftarrow{\sigma}$ eventually recognizable.

######  Theorem 5.7.

For any everywhere growing totally recognizable $S$-adic development
$\overleftarrow{\sigma}$ of a subshift $X$ and its associated cone $\cal
V(\overleftarrow{\sigma})$ of vector towers the canonical linear map

$\mathfrak{m}_{\tiny\overleftarrow{\sigma}}:\cal
V(\overleftarrow{\sigma})\to\cal M(X)$

is a bijection.

In particular, for any level $n\geq 0$ the map $\sigma_{[0,n)}M:\cal
M(X_{n})\to\cal M(X)$ is a linear bijection of cones. Similarly, the same
conclusion follows for the map
$\mathfrak{m}_{\tiny\overleftarrow{\sigma}}\circ\overleftarrow{\zeta}$ from
(5.2).

###### Proof.

From the assumption that $\overleftarrow{\sigma}$ is totally recognizable it
follows (using statement (3d) of Theorem 3.2) that the induced linear map

$(\sigma_{n})M_{X_{n+1}}:\cal M(X_{n+1})\to\cal M(X_{n})$

is bijective for any level $n\geq 0$. It follows that the composed map
$m_{\tiny\overleftarrow{\sigma}}\circ\overleftarrow{\zeta}:\cal
M(\overleftarrow{\sigma})\to\cal V(\overleftarrow{\sigma})\to\cal M(X)$ from
(5.2) is bijective. Since we know from Proposition 5.4 that the map
$\overleftarrow{\zeta}$ is a bijection, we deduce that
$m_{\tiny\overleftarrow{\sigma}}$ must be bijective. $\sqcup$$\sqcap$

##  6\. Directive sequences with “small” intermediate letter frequency cones

In this section we will give a first application of the machinery set up in
the previous two sections. But before doing so we want to summarize, for the
convenience of the reader, the various ingredients that the rich picture
issuing from this set-up offers, and to list some basic facts in order to
avoid potential misunderstandings. As an illustration, we give at the end of
this section a detailed example, where all the data listed now can be seen in
practice.

We use the same terminology as previously, i.e. $X\in\cal A^{\mathbb{Z}}$ is a
subshift over the finite alphabet $\cal A=\cal A_{0}$, and
$\overleftarrow{\sigma}=(\sigma_{n}:\cal A_{n+1}^{*}\to\cal A_{n}^{*})_{n\geq
0}$ is an everywhere growing directive sequence which generates $X$. We are
particularly interested in the intermediate letter frequency cones $\cal
C_{n}=\cal C_{n}(\overleftarrow{\sigma})\subseteq\mathbb{R}_{\geq 0}^{\cal
A_{n}}$ and in particular in their dimension

$c_{n}:=\dim\cal C_{n}\leq\mbox{card}(\cal A_{n})\,.$

The cone $\cal C_{n}$ is given as the image of the set $\cal
V(\overleftarrow{\sigma})$ of vector towers under the level $n$ projection map
$pr_{n}\,$, which amounts to stating that $\cal C_{n}$ is the intersection of
the nested images of the non-negative alphabet cones of level $m\geq n$ under
the telescoped level maps, i.e.

$\cal C_{n}=\bigcap\,\\{\mathbb{R}_{\geq 0}^{\cal
A_{n}}\supseteq\ldots\supseteq M(\sigma_{[n,m)})(\mathbb{R}_{\geq 0}^{\cal
A_{m+1}})\supseteq\ldots\\}$

In particular one always has

(6.1) $\cal C_{n}=M(\sigma_{n})(\cal C_{n+1})$

and thus

$c_{n}\leq c_{n+1}$

for all $n\geq 0$.

An alternative interpretation of the intermediate level frequency cones is
given through the level subshifts $X_{n}$ (defined by the truncated directive
sequences $\overleftarrow{\sigma}\dagger_{n}$) and their measure cones $\cal
M(X_{n})$, which, when evaluated via the associated letter frequency vectors,
results into

$\cal C_{n}=\zeta_{X_{n}}(\cal M(X_{n}))\,,$

where $\zeta_{X_{n}}:\cal M(X_{n})\to\mathbb{R}_{\geq 0}^{\cal A_{n}}$ is
given for $\cal A_{n}=\\{a_{n,1},\ldots,a_{n,d(n)}\\}$ by
$\mu\mapsto([\mu(a_{n,1})],\ldots,[\mu(a_{n,d(n)})])$ for any $\mu\in\cal
M(X_{n})$.

Our main focus here is to explain how this set-up and in particular the value
of the $c_{n}$ can be used in order to find out information about the number
$e(X)\in\mathbb{N}\cup\\{\infty\\}$ of invariant ergodic probability measures
on $X$.

######  Remark 6.1.

Under the above stated conditions the following conclusions are immediate:

  1. (1)

It is quite possible that $e(X)>c_{n}$ for some “low” level $n\geq 0$, even if
$\overleftarrow{\sigma}$ is totally recognizable.

  2. (2)

The converse inequality, $e(X)<c_{n}$, is also possible, but in this case the
directive sequence $\overleftarrow{\sigma}$ is not totally recognizable. More
precisely, in this case the telescoped morphism $\sigma_{[0,n)}$ is not
recognizable.

  3. (3)

In any case, we always have

$e(X)\,\,\leq\,\,\lim c_{n}\,\,\leq\,\,\liminf(\mbox{card}\,\cal A_{n})\,,$

but in general both inequalities may well be strict.

  4. (4)

However, if $\overleftarrow{\sigma}$ is totally recognizable, then we have

$e(X)=\lim c_{n}$

In particular, we recover the well known upper bound
$e(X)\leq\liminf(\mbox{card}\,\cal A_{n})$, as well as the lower bounds
$c_{n}\leq e(X)$ for all $n\geq 0$.

From Remark 6.1 (3) we observe directly that for any directive sequence
$\overleftarrow{\sigma}$ with finite alphabet rank (i.e. $\liminf(\mbox{card}\
\cal A_{n})<\infty$) there is a critical level $n_{0}\geq 0$ such that one has

(6.2) $c_{n}=c_{n_{0}}\,\,$ for all $\,\,n\geq n_{0}\,\,$ and
$\,\,c_{n}<c_{n_{0}}\,\,$ for all $\,\,n<n_{0}$ .

More generally, any everywhere growing directive sequence
$\overleftarrow{\sigma}$ (possibly with infinite alphabet rank) which
possesses such a critical level has been termed in [3] thinning, and in the
particular case where the critical level agrees with the base level $n_{0}=0$,
the sequence $\overleftarrow{\sigma}$ has been called thin. Of course, any
thinning sequence can be made thin by simply truncating it at its critical
level (or any level higher up); furthermore, we can telescope all levels below
the critical level into a single “thinning” morphism. Subshifts that are
“thin” in that they are generated by a thin (and in particular everywhere
growing) directive sequence have the following useful property:

######  Proposition 6.2 ([3]).

Let $X\subseteq\cal A^{\mathbb{Z}}$ be a subshift generated by a thin
directive sequence $\overleftarrow{\sigma}$. Then the letter frequency map
$\zeta_{X}:\cal M(X)\to\mathbb{R}_{\geq 0}A$ co-restricts to a linear
bijection

$\cal M(X)\to C(X)\,,\,\,\mu\mapsto(\mu(a_{k}))_{a_{k}\in\cal A}\,.$

In particular, any two invariant measures $\mu_{1}$ and $\mu_{2}$ on $X$ are
equal if and only if one has $\mu_{1}([a_{k}])=\mu_{2}([a_{k}])$ for the
finitely many cylinders $[a_{k}]$ given by all letters $a_{k}\in\cal A$.

This statement can be derived directly from Proposition 10.2 (1) and Corollary
10.4 of [3]. For convenience of the reader we give here a proof in the
terminology introduced above.

###### Proof of Proposition 6.2.

For any two measure $\mu,\mu^{\prime}\in\cal M(X)$ there exist, by Theorem
2.7, vector towers $\overleftarrow{v}=(\vec{v}_{n})_{n\geq 0}$ and
$\overleftarrow{v}^{\prime}=(\vec{v}\,^{\prime}_{\\!n})_{n\geq 0}$ on
$\overleftarrow{\sigma}$ with
$\mathfrak{m}_{\tiny\overleftarrow{\sigma}}(\overleftarrow{v})=\mu$ and
$\mathfrak{m}_{\tiny\overleftarrow{\sigma}}(\overleftarrow{v}^{\prime})=\mu^{\prime}$.
Thus $\mu\neq\mu^{\prime}$ implies
$\overleftarrow{v}\neq\overleftarrow{v}^{\prime}$ and hence
$\vec{v}_{n}\neq\vec{v}\,^{\prime}_{\\!n}$ for some $n\geq 0$. But then we
deduce from (6.1) and the hypothesis that $\dim\cal
C(X_{n})=c_{n}=c_{0}=\dim\cal C(X)$ that
$\vec{v}_{0}=M(\sigma_{[0,n)})(\vec{v}_{n})\neq
M(\sigma_{[0,n)})(\vec{v}\,^{\prime}_{\\!n})=\vec{v}\,^{\prime}_{\\!0}$. From
Proposition 2.8 (1) we know that
$\vec{v}_{0}=pr_{0}(\overleftarrow{v})=\zeta_{X}(\mu)$ and
$\vec{v}\,^{\prime}_{\\!0}=pr_{0}(\overleftarrow{v}^{\prime})=\zeta_{X}(\mu^{\prime})$,
which shows that the map $\zeta_{X}$ is injective. For the linearity of
$\zeta_{X}$ and the equality $\zeta_{X}(\cal M(X))=\cal C(X)$ see (2.9) and
(2.10). $\sqcup$$\sqcap$

Directive sequences of finite alphabet rank occur naturally in many important
contexts in the symbolic dynamics literature (e.g. substitutive subshifts,
IETs, etc). Furthermore, the extra invertibility condition from the following
proposition is rather frequently satisfied.

######  Corollary 6.3.

Let $X\subseteq\cal A^{\mathbb{Z}}$ be a subshift generated by an everywhere
growing directive sequence $\overleftarrow{\sigma}=(\sigma_{n})_{n\geq 0}$ of
finite alphabet rank. Assume that for every $n\geq 0$ the incidence matrix
$M(\sigma_{n})$ is invertible over $\mathbb{R}$. Then any invariant measure
$\mu$ on the subshift $X$ is determined by the evaluation of $\mu$ on the
letter cylinders, i.e. by the values $\mu([a_{k}])$ for all $a_{k}\in\cal A$.

###### Proof.

From (6.1) and the hypothesis that $M(\sigma_{n})$ is invertible it follows
directly that $c_{n+1}=c_{n}$ for all $n\geq 0$, so that the directive
sequence $\overleftarrow{\sigma}$ is thin. Hence the hypotheses of Proposition
6.2 are satisfied, which gives directly the claimed statement.
$\sqcup$$\sqcap$

Note that the conclusion of Corollary 6.3 has recently been proved by Berthé
et al. under somewhat more restrictive hypotheses (see Corollary 4.2 of [6]);
in particular it is required there that every $M(\sigma_{n})$ has determinant
equal to $1$ or to $-1$, and that $X$ is minimal.

######  Remark 6.4.

(1) If in Proposition 6.2 the hypothesis “thin” is replaced by “thinning”,
with critical level $n_{0}\geq 1$, then the conclusion, that any two distinct
measures $\mu\neq\mu^{\prime}\in\cal M(X)$ can be distinguished by the
evaluation on the letter cylinders $[a_{k}]$ for all $a_{k}\in\cal A$, may in
some cases still hold, despite the fact that from the definition of the
critical level we have

$\dim\cal C_{0}=c_{0}<c_{n_{0}}=\dim\cal C_{n_{0}}=\dim\cal M(X_{n_{0}})\,.$

Here the last equality follows from Proposition 6.2, applied to the directive
sequence truncated at the critical level $n_{0}$. The reason, why the above
strict inequality doesn’t contradict the presumed equality $c_{0}=\dim\cal
C_{0}=\dim\cal M(X)$, is that the measure transfer map
$\sigma_{[0,n_{0})}M:\cal M(X_{n_{0}})\to\cal M(X)$ may well not be injective,
in the case that the telescoped level map $\sigma_{[0,n_{0})}$ is not
recognizable in the level subshift $X_{n_{0}}\,$.

However, if $\overleftarrow{\sigma}$ is totally recognizable, or if at least
$\sigma_{[0,n_{0})}$ is recognizable in $X_{n_{0}}$, and if furthermore
$\overleftarrow{\sigma}$ is thinning but not thin, then the conclusion of
Proposition 6.2 necessarily fails: this case is treated in Example 6.5 below.

(2) In view of the fact that the measure transfer map $\sigma M$ induced by a
non-recognizable monoid morphisms $\sigma$ is in general far from being
injective, it seems noteworthy that in Proposition 6.2 and Corollary 6.3 no
recognizability condition on the level maps $\sigma_{n}$ is imposed. One
should recall in this context that in [9], Theorem 5.2, it has been proved
that directive sequences of bounded alphabet rank, with aperiodic level
subshifts, are “eventually recognizable”, i.e. all level maps above some
“other critical level” must be recognizable in their level subshift. But this
“other critical level” may well be a lot bigger than the above critical level
$n_{0}$, and indeed we give in Corollary 8.5 (2) examples of thin directive
sequences where this “other critical level” can be chosen to be arbitrarily
high up, while none of the level morphisms below it is recognizable in its
corresponding level subshift (which is aperiodic for any level).

We now present the promised “detailed example with all above data made
visible”:

######  Example 6.5.

The subshift $X$ in this example consists of two periodic words and is hence
all by itself not so interesting. We chose it in order to give a transparent
presentation of a simple subshift via some not so obvious everywhere growing
directive sequence, which we now describe in detail. We first describe the
level $n=1$, then pass to the base level $n=0$, and finally built the higher
levels $n\geq 2$ on top of the two lowest levels. We also include for each
level $n$ a description of the measure cone $\cal M(X_{n})$ and of the
associated letter frequency cone $\cal C_{n}$.

Set $\cal A_{1}=\\{a,b\\}$, and let $X_{1}\subseteq\cal A_{1}^{\mathbb{Z}}$ be
the union of the two periodic subshifts $\cal O(w^{\pm\infty})$ and $\cal
O({w^{\prime}}^{\pm\infty})$, defined by the words $w=a^{2}b$ and
$w^{\prime}=b^{2}a$. We consider the two characteristic measures
$\mu:=\mu_{w}$ and $\mu^{\prime}:=\mu_{w^{\prime}}\,$, and observe that $\cal
M(X_{1})$ consists of all non-negative linear combinations of these two
measures. The letter frequency map $\zeta_{X_{1}}:\cal M(X_{1})\to\cal
C(X_{1})\subseteq\mathbb{R}_{\geq 0}^{\\{a,b\\}}$ is injective, in that
$\zeta_{X_{1}}(\mu)=2\vec{e}_{a}+\vec{e}_{b}$ and
$\zeta_{X_{1}}(\mu^{\prime})=\vec{e}_{a}+2\vec{e}_{b}$. This results into
$c_{1}=\dim(\cal C_{1})=2$.

For $\cal A_{0}=\\{c,d\\}$ consider now the “Thue-Morse” morphism
$\sigma_{0}:\cal A_{1}^{\mathbb{Z}}\to\cal A_{0}^{\mathbb{Z}}\,,\,\,a\mapsto
cd\,,\,\,b\mapsto dc$, and recall (see Proposition 3.3 (d)) that
$\sigma_{0}M(\mu)=\mu_{\sigma_{0}(w)}$ and
$\sigma_{0}M(\mu^{\prime})=\mu_{\sigma_{0}(w^{\prime})}\,$, with
$\sigma_{0}(w)=cdcddc$ and $\sigma_{0}(w^{\prime})=dcdccd$. Since $cdcddc$ and
$dcdccd$ can not be obtained from each other by a cyclic permutation, we have
$\cal O((cdcddc)^{\pm\infty})\neq\cal O((dcdccd)^{\pm\infty})$, so that from
(2.7) it follows that
$\mbox{Supp}(\mu_{cdcddc})\neq\mbox{Supp}(\mu_{dcdccd})$. We thus deduce for
the image subshift $X_{0}=\sigma_{0}(X_{1})$ that the measure cone $\cal
M(X_{0})$, which is spanned by $\mu_{cdcddc}$ and $\mu_{dcdccd}\,$, is of
dimension $2$.

On the other hand, using Proposition 3.4 (or more directly, equality (2.6)) we
readily compute $\mu_{cdcddc}([cd])=\mu_{cdcddc}([dc])=2$ as well as
$\mu_{dcdccd}([cd])=\mu_{dcdccd}([dc])=2$. It follows that the frequency map
$\zeta=\zeta_{X_{0}}$ is not injective, and that $\cal C_{0}$ has dimension
$c_{0}=1$.

We now define the higher up levels of the directive sequence by setting $\cal
A_{n}=\\{x,y\\}$ for any $n\geq 2$, and by defining all level morphisms
$\sigma_{n}:\cal A_{n+1}\to\cal A_{n}$ for $n\geq 2$ to be equal to the
substitution defined by $x\mapsto x^{2}$, $y\mapsto y^{2}$. It follows that
for $n\geq 2$ all level subshifts $X_{n}$ consist of the two biinfinite
periodic words $x^{\pm\infty}$ and $y^{\pm\infty}$. Moreover, we easily see
that the incidence matrix of $\sigma_{n}$ is equal to 2 times the 2-by-2
identity matrix, $M(\sigma_{n})=2\cdot I_{2}$, so that we have $\cal
M(X_{n})=\cal C_{n}=\mathbb{R}_{\geq 0}^{\\{x,y\\}}$.

It remains now to define $\sigma_{1}:\cal A_{2}\to\cal A_{1}$ via $x\mapsto
w$, $y\mapsto w^{\prime}$, which ensures $\sigma_{1}(X_{2})=X_{1}$, in order
to obtain a directive sequence $\overleftarrow{\sigma}=(\sigma_{n})_{n\geq 0}$
over alphabets that all have cardinality 2. We have shown above that the
critical level of this directive sequence is $n_{0}=1$, while the evaluation
on the cylinders $[\sigma_{0}(a)]=[cd]$ and $[\sigma_{0}(b)]=[dc]$ does not
suffice to distinguish the two measures $\mu_{cdcddc}\neq\mu_{dcdccd}$ that
span $\cal M(X_{0})$.

##  7\. Minimal subshifts with zero entropy and infinitely many ergodic
probability measures

A subshift $X$, which is “small” in that it has topological entropy $h_{X}=0$
(see (2.3)), and simultaneously “large” in that the number $e(X)$ of ergodic
probability measures carried by $X$ is infinite, is a bit of a contradiction
in itself (if one restricts to non-atomic measures). However, such subshifts
are known to exist, but they are not easy to come by. One of the first such
subshift known to us is the Pascal-adic subshift, treated in [22]; more recent
such examples (with additional strong properties, in particular minimality)
are exhibited in [11]. Not surprisingly, there is always a certain amount of
labor involved in order to simultaneously achieve the above two opposite
properties.

In this section we will present an alternative way to construct minimal
subshifts $X$ which satisfy both, $h_{X}=0$ and $e(X)=\infty$. The main
purpose of this section is to underline how directly such examples can be
exhibited by means of the technology established in the previous sections.

We first recall two known results. The first appears as Theorem 4.3 in [7] and
is attributed there to Thierry Monteil; alternatively it can be found in [8]
as Lemma 6.7.1 of Chapter 6, written by Fabien Durand, who told us that the
result can actually be traced back to the paper [10] by Boyle-Handelman.

######  Proposition 7.1.

Let $X$ be a subshift which is generated by a directive sequence
$\overleftarrow{\sigma}=(\sigma_{n})_{n\geq 0}$ with level alphabets $\cal
A_{n}$. Then, for the minimal level letter image length $\beta_{-}(n)$ from
(2.16), one has:

$h_{X}\leq\inf_{n\geq 0}\frac{\log(\text{\rm\mbox{card}}\,\cal
A_{n})}{\beta_{-}(n)}$

######  Proposition 7.2 (Section 4.1 of [4]).

For any integer $d\geq 2$ let $X$ be a subshift which is generated by a
directive sequence ${\overleftarrow{\sigma}}=(\sigma_{n,d})_{n\geq 0}$ with
level alphabets that are all of uniform cardinality $d$ (and are thus
identified with $\cal A_{(d)}=\\{a_{1},\ldots,a_{d}\\}$). Assume that for any
level $n\geq 0$ the incidence matrix of the level map $\sigma_{n,d}$ is given
by

(7.1) $M(\sigma_{n,d})=M_{\ell(n),d}:=\ell(n)I_{d}+1_{d\times d}\,,$

where $I_{d}$ is the identity matrix of size $d\times d$, $1_{d\times d}$ is
the $d\times d$-matrix with all entries equal to 1, and $\ell(n)$ is a
positive integer depending on $n$.

Then $X$ is minimal, and for any sufficiently fast growing sequence
$(\ell(n))_{n\in\mathbb{N}}$ the subshift $X$ admits precisely $d$ distinct
invariant ergodic probability measures.

The use of Proposition 7.1 will be an ingredient in our proof below.
Proposition 7.2, on the other hand, will not be formally used in the sequel,
but it may pay anyway for the reader to look it up: We use below the very same
basic idea as in this earlier result, but do not carry out all calculations as
had been done in section 4 of [4] (where in particular precise lower bounds
for the integers $\ell(n)$ are computed which guarantee the “sufficiently fast
growing” in the above statement).

For the proof below we first need to define for any integer $d\geq 2$ and
alphabet $\cal A_{(d)}=\\{a_{1},\ldots,a_{d}\\}$ the morphism $\tau_{d}:\cal
A_{({d+1})}^{*}\to\cal A_{(d)}^{*}\,$, given by $a_{i}\mapsto a_{i}^{2}$ for
any $a_{i}$ with $1\leq i\leq d$ and $a_{d+1}\mapsto a_{1}a_{2}\ldots a_{d}$.

######  Remark 7.3.

(1) For the morphism $\tau_{d}$ as given above it is easy to see that any
biinfinite word ${\bf y}\in\cal
A_{(d)}^{\mathbb{Z}}\smallsetminus\\{a_{1}^{\pm\infty},a_{2}^{\pm\infty},\ldots,a_{d}^{\pm\infty}\\}$
can be “desubstituted” in at most one way (compare Remark 6.2 (2) of [5]) to
give a biinfinite word ${\bf x}\in\cal A_{({d+1})}^{\mathbb{Z}}$ with
$\tau_{d}(\cal O({\bf x}))=\cal O({\bf y})$. Since for any $i=1,\ldots,n$ the
periodic word $a_{i}^{\pm\infty}$ is the only element ${\bf x}\in\cal
A_{(d+1)}^{\mathbb{Z}}$ with $\tau_{d}(\cal O({\bf x}))=\cal
O(a_{i}^{\pm\infty})$, it follows that $\tau_{d}$ is recognizable in every
subshift which doesn’t contain any of the periodic words $a_{i}^{\pm\infty}$.

(2) Again by elementary desubstitution arguments one verifies quickly that any
morphism $\sigma_{n,d}$ with incidence matrix given by equality (7.1), with
$\ell(n)\geq 2$, is recognizable in the full shift $\cal
A_{(d)}^{\mathbb{Z}}$.

[Indeed, it suffices to check in any biinfinite word ${\bf
y}\in\sigma_{n,d}(\cal A_{(d)}^{\mathbb{Z}})$ for a factor $w\in\cal
A_{(d)}^{*}$ which is “distinguished” in that some letter $a_{i}\in\cal
A_{(d)}$ occurs precisely 3 times in $w$, while all other letters
$a_{j}\in\cal A_{(d)}$ occur at most twice. Such a distinguished word $w$
occurs in $\sigma_{n,d}(a_{i})$, and any such occurrence is contained in the
image of some word from $\cal A_{(d)}^{*}$ of length at most 3. In either case
one verifies quickly that the middle occurrence of $a_{i}$ in $w$ must belong
to $\sigma_{n,d}(a_{i})$. For this middle occurrence $y_{s}$ in the factor
$w=y_{r}\ldots y_{t}$ of ${\bf y}=\ldots y_{n-1}y_{n}y_{n+1}\ldots$ one
considers the factors $w_{+}=y_{s}\ldots y_{t^{\prime}}$ and
$w_{-}=y_{r^{\prime}}\ldots y_{s}$ of ${\bf y}$, with
$y_{s}=y_{s+1}=\ldots=y_{t^{\prime}-1}=a_{i}$ and $y_{t^{\prime}}\neq a_{i}$,
and similarly $y_{s}=y_{s-1}=\ldots=y_{r^{\prime}+1}=a_{i}$ and
$y_{r^{\prime}}\neq a_{i}$. From the fact that $\sigma_{n,d}(a_{i})$ contains
each letter $a_{j}\neq a_{i}$ precisely once one deduces directly that the
words $w_{+}$ and $w_{-}$ determine which occurrence of $a_{i}$ in
$\sigma_{n,d}(a_{i})$ is given by the letter $y_{s}$. It follows that,
starting from $y_{s}\,$, the biininite word ${\bf y}$ can be desubstituted in
precisely one way.]

(3) From the conditions on the incidence matrix $M(\sigma_{n})$ in (7.1) it
follows directly that every word in $\sigma_{n,d}(\cal A_{(d)}^{*})$ must
contain each of the letters of $\cal A_{(d)}$. Hence we observe that
$\sigma_{n,d}(\cal A_{(d)}^{\mathbb{Z}})$ can not contain any of the periodic
words $a_{i}^{\pm\infty}$.

(4) As a consequence of the above observations (1) - (3) we deduce for the
alternating directive sequence

(7.2)
$\overleftarrow{\sigma}=\sigma_{2}\circ\tau_{2}\circ\sigma_{3}\circ\tau_{3}\circ\ldots\,,$

where we set $\sigma_{d}:=\sigma_{d,d}\,$, that each level map is recognizable
in its corresponding level subshift, so that the sequence
$\overleftarrow{\sigma}$ is fully recognizable.

######  Theorem 7.4.

For any integer $d\geq 2$ let $\cal A_{(d)}=\\{a_{1},\ldots,a_{d}\\}$ and let
$\sigma_{d}:\cal A_{(d)}^{*}\to\cal A_{(d)}^{*}$ be a morphism with incidence
matrix $M(\sigma_{d})=M_{\ell(d),d}$ from (7.1), for some integer $\ell(d)\geq
2$ depending on $d$. Let $X$ be the subshift generated by the alternating
directive sequence $\overleftarrow{\sigma}$ given in (7.2).

If the exponent sequence $(\ell(n))_{n\in\mathbb{N}}$ is sufficiently fast
growing, then the subshift $X$ is minimal, has entropy $h_{X}=0$ and admits
infinitely many distinct invariant ergodic probability measures.

(We denote by $\mathfrak{X}$ the class of all subshifts $X\subseteq\cal
A_{(d)}^{\mathbb{Z}}$ which satisfy all of the above conditions.)

###### Proof.

For each integer $d\geq 2$ we identify the finite alphabet $\cal
A_{(d)}=\\{a_{1},a_{2},\ldots,a_{d}\\}$ with the corresponding subset of an
infinite alphabet, via $\cal A_{(2)}\subseteq\cal
A_{(3)}\subseteq\ldots\subseteq\cal A_{(\infty)}=\\{a_{1},a_{2},\ldots\\}$.
For the issuing infinite non-negative cone $\mathbb{R}_{\geq 0}^{\cal
A_{(\infty)}}$ we abbreviate for notational convenience the base unit vectors
to $\vec{e}_{i}:=\vec{e}_{a_{i}}$.

For any level $n=2d-2$ or $n=2d-1$ we consider the subcone $\cal
C^{n}:=\mathbb{R}_{\geq 0}^{\cal A_{(d)}}\subseteq\mathbb{R}_{\geq 0}^{\cal
A_{(\infty)}}$, and in particular the “center vector”
$\vec{c}_{n}=\sum\vec{e}_{i}$ of $\cal C^{n}$. We observe that both families,
the morphisms $\sigma_{d}$ as well as the morphisms $\tau_{d}\,$, induce maps
$M(\sigma_{d}):\cal C^{n}\to\cal C^{n}$ and $M(\tau_{d}):\cal C^{n+1}\to\cal
C^{n}$ respectively which each maps the center vector $\vec{c}_{n}$ (for
$\sigma_{n}$) or $\vec{c}_{n+1}$ (for $\tau_{n}$) to a scalar multiple of the
center vector $\vec{c}_{n}\,$. Furthermore, any unit vector $\vec{e}_{i}$ with
$1\leq i\leq d$ is mapped by both, $M(\sigma_{d})$ and $M(\tau_{d})$, to a
non-negative linear combination
$\lambda_{1}\vec{e}_{i}+\lambda_{2}\vec{c}_{d}$. Note here that (again for
both, $\sigma_{d}$ and $\tau_{d}$,)

(7.3) the quotient $\frac{\lambda_{2}}{\lambda_{1}}$ can be made arbitrarily
small

by choosing $\ell(d)$ sufficiently large.

We now fix some level $n_{0}=2d-2\geq 0$, and for any index $i$ with $1\leq
i\leq d$ we look for a vector tower
$\overleftarrow{v}_{\\!i}=(\vec{v}^{\,i}_{n})_{n\geq n_{0}}$ on the truncated
directed sequence
$\overleftarrow{\sigma}\dagger_{\\!n_{0}}=\sigma_{n_{0}}\circ\tau_{n_{0}}\circ\sigma_{n_{0}+1}\circ\tau_{n_{0}+1}\circ\ldots$
with the property that $\overleftarrow{v}_{\\!i}$ has for any level $n\geq
n_{0}$ a level vector
$\vec{v}^{\,i}_{n}=\lambda_{1,n}\vec{e}_{i}+\lambda_{2,n}\vec{c}_{n}$, with
coefficients

(7.4) $\lambda_{1,n}>0\qquad\text{and} \qquad\lambda_{2,n}>0$

(which must both tend to 0 for $n\to\infty$). From (7.3) we deduce that a
sufficiently large choice of the exponents $\ell(d)$ effects indeed that there
exist families of such coefficients where both of the inequalities in (7.4)
are satisfied, while the compatibility condition (2.19) is maintained, for any
$n\geq n_{0}$. It follows that on the lowest level $n=n_{0}$ (and thus
similarly also on all levels $n\geq n_{0}$) the level vectors
$\vec{v}^{\,1}_{n_{0}},\vec{v}^{\,2}_{n_{0}},\ldots,\vec{v}^{\,d}_{n_{0}}$ are
linearly independent.

For the level subshift $X_{n_{0}}\subseteq\cal A_{(d)}^{\mathbb{Z}}\,$,
generated by the truncated sequence
$\overleftarrow{\sigma}\dagger_{\\!n_{0}}\,$, the truncated evaluation map
$\mathfrak{m}_{n_{0}}:=\mathfrak{m}_{\tiny\overleftarrow{\sigma}\dagger_{\\!n_{0}}}\\!\\!\\!\\!:\overleftarrow{\cal
V}(\overleftarrow{\sigma}\dagger_{\\!n_{0}})\to\cal M(X_{n_{0}})$ from (5.1)
defines $d$ invariant measures $\mu_{1},\ldots,\mu_{d}$ on the level subshift
$X_{n_{0}}$ as images of the $d$ vector towers
$\overleftarrow{v}_{\\!1},\ldots,\overleftarrow{v}_{\\!d}$ respectively:

$\mu_{i}\,\,=\,\,\mathfrak{m}_{n_{0}}(\overleftarrow{v}_{\\!i})$

It follows from Proposition 2.8 (1) that the subcone

$\cal M_{n_{0}}:={\mathbb{R}_{\geq
0}}\,\langle\mu_{1},\ldots,\mu_{d}\rangle\subseteq\cal M(X_{n_{0}})$

spanned by the $\mu_{i}$ has dimension $d$. Since we verified in Remark 7.3
(4) above that each of the maps $\sigma_{j}$ and $\tau_{j}$ is recognizable in
its corresponding level subshift, it follows from Theorem 3.2 (3d) that $\cal
M_{n_{0}}$ is mapped by
$\sigma_{2}M\circ\tau_{2}M\circ\ldots\circ\sigma_{n_{0}-1}M\circ\tau_{n_{0}-1}M$
to a subcone of $\cal M(X)$ that also has dimension $d$.

We have thus proved that $\cal M(X)$ contains subcones of arbitrary large
dimension, and hence must be infinite dimensional, i.e. $e(X)=\infty$. The
desired equality $h_{X}=0$ is immediate from Proposition 7.1 for large
$\ell(d)$, and the minimality of $X$ follows directly from the positivity of
the matrices $M(\sigma_{d})$, see Remark 2.5 (4). $\sqcup$$\sqcap$

##  8\. Non-recognizable directive sequences

The purpose of this section is to show how non-recognizable morphisms appear
naturally in a well known context (IETs and pseudo-Anosov surface
homeomorphisms), and how this phenomenon can be exploited to construct
interesting directive sequences that are not totally recognizable or even not
eventually recognizable.

Our construction will be presented in 4 steps, organized below as follows: In
subsection 8.1 we present our basic quotient construction in geometric
language. In subsection 8.2 we show how the canonical “inverse quotient
construction” is obtained in a natural geometric context, to define a non-
recognizable monoid morphism. In subsection 8.3 the results from the previous
subsections are properly “pasted together” to give a directive sequence where
every level morphism is non-recognizable (and in addition it is a particularly
nice letter-to-letter factor map). Finally, in subsection 8.4 we modify this
sequence slightly to obtain the desired everywhere growing but not
(eventually) recognizable directive sequences. Note that all intermediate
level subshifts which occur in our constructions turn out to be minimal; they
are furthermore both, substitutive and IET.

Before starting the detailed description, we will highlight its essential
features in a special case, in a language that may be more easily accessible
to those of us who are less familiar with Thurston’s work on surface
homeomorphisms:

######  Remark 8.1.

(1) Let us consider the tiling of the real plane $\mathbb{R}^{2}$ by squares
of side length 1 that have their vertices on the points with integer
coordinates. We now pick a slope $s$, say $0<s<1$, and we foliate the plane by
lines that have slope $s$. By choosing the slope $s$ to be irrational, we make
sure that on any line of the foliation there is at most one vertex of our
square tiling. To every line $\ell$ that avoids any such vertex one can
associate canonically a biinfinite word $w(\ell)$ in the letters $h$ and $v$,
which records the sequence of intersections of $\ell$ with a horizontal
(“$h$”) or vertical (“$v$”) line of our square grid. In order to fix an
indexing of the letters of $w(\ell)$ we pick a distinguished “base square” $Q$
and require that $\ell$ passes through the interior of $Q$. We quickly observe
that the orbits in our family of lines $\ell$, with respect to the canonical
$\mathbb{Z}\oplus\mathbb{Z}$-action on $\mathbb{R}^{2}$, are in 1-1 relation
with the shift-orbits of the resulting set of words $w(\ell)$. Indeed, for
this 1-1 relation it suffices to consider the positive half-words of any
$w(\ell)$, so that it extends naturally to the lines $\ell$ that pass over any
of the vertices.

We consider now more closely any of the “troublesome” lines $\ell_{P}$ that
cross over a vertex $P$ of the square grid. To $\ell_{P}$ we associate two
words $w_{\rm{above}}(\ell_{P})$ and $w_{\rm{below}}(\ell_{P})$ in
$\\{h,v\\}^{\mathbb{Z}}$, which are read off from $\ell_{P}$ after isotopying
it slightly in the neighborhood of $P$ so that it passes either above or below
$P$. From the above observed 1-1 relation between the
$\mathbb{Z}\oplus\mathbb{Z}$-orbits of the lines $\ell$ and the shift-orbits
of the corresponding words $w(\ell)$ we deduce that the words
$w_{\rm{above}}(\ell_{P})$ and $w_{\rm{below}}(\ell_{P})$ do not belong to the
same shift-orbit.

The set $X_{s}\subseteq\\{h,v\\}^{\mathbb{Z}}$ of all biinfinite words
$w(\ell)$, including the above defined $w_{\rm{above}}(\ell_{P})$ and
$w_{\rm{below}}(\ell_{P})$, for any line $\ell$ that passes through our
distinguished base square $Q$, is a subshift - indeed, a well known Sturmian
subshift.

(2) We now proceed by subdividing the top and bottom side of each square into
segments of equal length through introducing a new vertex at the midpoint of
any horizontal segment of the square grid. Any transition of a line $\ell$
through the left half of the subdivided horizontal square side will now be
recorded by the letter $h_{\rm{left}}$, and any transition through the right
half by $h_{\rm{right}}$, to give a new biinfinite word
$w^{\prime}(\ell)\in\\{h_{\rm{left}},h_{\rm{right}},v\\}^{\mathbb{Z}}$. The
morphism
$\sigma:\\{h_{\rm{left}},h_{\rm{right}},v\\}^{\mathbb{Z}}\to\\{h,v\\}^{\mathbb{Z}}$
defined by $h_{\rm{left}}\mapsto h,h_{\rm{right}}\mapsto h$ and $v\mapsto v$
maps any $w^{\prime}(\ell)$ to $w(\ell)$, and it will be 1-1, except for the
new “troublesome” lines $\ell_{R}$ that pass through any of the new vertices
$R$ in the middle of our original horizontal square grid intervals. For such
lines we have as before 2 words $w^{\prime}_{\rm{above}}(\ell_{R})$ and
$w^{\prime}_{\rm{below}}(\ell_{R})$, and both have the same image word
$w(\ell_{R})$. Since $w^{\prime}_{\rm{above}}(\ell_{R})$ and
$w^{\prime}_{\rm{below}}(\ell_{R})$ belong as above to distinct shift-orbits,
the morphism $\sigma$ is not shift-orbit injective, and hence not a
recognizable (see Proposition 3.6 (1)).

Clearly, this process can be iterated arbitrarily often, and every time the
obtained morphism is shift-orbit injective except for two particular shift-
orbits, which have the same image orbit.

(3) The above set-up of lines in a square grid of $\mathbb{R}^{2}$ admits a
particularly convincing translation into an IET setting, since for any of the
squares we can use the left hand and the bottom sides together as “bottom
intervals”, and the top side together with the right hand side as “top
intervals”, and the line segments of our foliation that are contained in the
chosen square give canonically a classical IET system. If the chosen square
agrees with the above picked base square $Q$, then the interval coding
associated traditionally to the IET defines a subshift that agrees precisely
with the one given by the set of biinfinite words $w(\ell)$ (or similarly for
$w^{\prime}(\ell)$), which have been read off above from the intersections of
the lines $\ell$ with the given square grid.

After this “appetizer” we now give a detailed description of our construction
in the subsequent 4 subsections. We assume a minimal familiarity with the
basic terminology of Thurston’s work on surfaces, such as “pseudo-Anosov
homeomorphism”, “stable lamination” or “invariant train track”.

###  8.1. The basic geometric quotient construction

We will start by describing our basic geometric construction, using a pseudo-
Anosov homeomorphism $h$ of a compact orientable surface $\Sigma$, and its
expanding invariant lamination $\Lambda^{s}$, which consists of uncountably
many biinfinite geodesics (called “leaves”) with respect to a fixed hyperbolic
structure on $\Sigma$. [The family $\Lambda^{s}$ was called “the stable
lamination” by Thurston, as he was looking at its behavior when lifted to the
universal covering of $\Sigma$, identified with the hyperbolic plane
$\mathbb{H}^{2}$, in the neighborhood of a $\partial\widetilde{h}$-fixed point
on $\partial\mathbb{H}^{2}$ (where $\widetilde{h}$ is a lift of $h$ to
$\mathbb{H}^{2}$ and $\partial\widetilde{h}$ is the canonical extension of
$\widetilde{h}$ to $\partial\mathbb{H}^{2}$).]

It is a standard procedure to translate such laminations (for instance by
using an $h$-invariant train track neighborhood of $\Lambda^{s}$) into a
classical interval exchange setting, which in turn (assuming that
$\Lambda^{s}$ is orientable and $\Sigma$ has at least one boundary component)
allows a direct translation of $\Lambda^{s}$ into a subshift $X\subseteq\cal
A^{\mathbb{Z}}$, where $\cal A$ is given by the intervals in the IET. Since
both of these translations are well known (see for instance [12], [20], [21]),
we will restrict ourselves here only to a description of the geometry of $h$
and $\Lambda^{s}$.

For our purposes it is convenient to impose the following extra conditions:

  1. (H1)

Assume that $\Sigma$ has $r\geq 2$ boundary components, which are all fixed by
$h$.

  2. (H2)

Each complementary component of $\Lambda^{s}$ contains precisely one boundary
component.

(Note that this assumption effects that there is a natural identification of
$\pi_{1}\Sigma$ with the free group $F(\cal A)$.)

  3. (H3)

Each complementary component has at least 2 cusps, and each cusp is fixed by
$h$.

We now pick a particular complementary component $\Sigma_{i}\subseteq\Sigma$
of $\Lambda^{s}$, and assume that $\Sigma_{i}$ has precisely two cusps, and
thus also precisely two boundary leaves $\ell_{1}$ and $\ell_{2}$, which (as
do all boundary leaves of complementary components) will then both belong to
$\Lambda^{s}$. We now pass to a quotient surface $\Sigma^{\prime}$ by “filling
in” the boundary component of $\Sigma$ that is contained in $\Sigma_{i}$,
through identifying all points of the boundary curve in $\Sigma_{i}$ into a
single point $P$ of $\Sigma^{\prime}$. Then $h$ induces a pseudo-Anosov
homeomorphism $h^{\prime}:\Sigma^{\prime}\to\Sigma^{\prime}$ with stable
lamination $\Lambda^{\prime s}$, and there is a canonical quotient map
$q:\Lambda^{s}\to\Lambda^{\prime s}$ that commutes with $h$ and $h^{\prime}$
respectively. The map $q$ is 1-1 everywhere, except at points on the leaves
$\ell_{1}$ and $\ell_{2}$, which are identified by $q$ to a single leaf
$\ell^{\prime}\in\Lambda^{\prime s}$. The leaf $\ell^{\prime}$ is fixed and
expanded by $h^{\prime}$, and the sole $h^{\prime}$-fixed point on
$\ell^{\prime}$ is precisely the above point $P$. This can be seen for example
by the canonical passage from the stable lamination $\Lambda^{s}$ to the
associated stable foliation $\cal F^{s}$ for $h$.

######  Remark 8.2.

(1) There is a remarkable feature here in that both, $\Lambda^{s}$ and
$\Lambda^{\prime s}$ are minimal laminations (i.e. each leaf is dense), while
the map $q$ induces on the leaf spaces of $\Lambda^{s}$ and $\Lambda^{\prime
s}$ a map that is surjective, but not injective.

(2) This is translated (via the associated IETs as indicated above) into a
subshift $X\subseteq\cal A^{\mathbb{Z}}$ that is mapped by a morphism
$\sigma:\cal A^{*}\to\cal A^{\prime*}$ to a subshift
$\sigma(X)=:X^{\prime}\subseteq\cal A^{\prime\mathbb{Z}}$ (for $\cal
A^{\prime*}\subseteq F(\cal A^{\prime})=\pi_{1}\Sigma^{\prime}$, in complete
analogy to $\cal A$ and $\Sigma$ in the above set-up). Here both, $X$ and
$X^{\prime}$, are minimal, while the map induced by $\sigma$ on $X$ is not
shift-orbit injective, so that $\sigma$ is not recognizable in $X$.

(3) More precisely, since there is a natural 1-1 correspondence between the
shift-orbits of $X$ and the leaves of $\Lambda^{s}$ (and similarly for
$X^{\prime}$ and $\Lambda^{\prime s}$), we observe that $\sigma$ maps
precisely two shift-orbits of $X$ to a common image shift-orbit of
$X^{\prime}$, while everywhere else the induced map on shift-orbits is 1-1.

###  8.2. The “inverse” geometric quotient construction

After having presented our basic geometric quotient construction, we will now
describe the precise converse procedure: For this purpose we assume in this
subsection that $\sigma_{0},h_{0},\Lambda_{0}^{s},\cal A_{0}$ and $X_{0}$ are
as $\Sigma,h,\Lambda^{s},\cal A$ and $X$ in subsection 8.1 above, and that in
particular the conditions (H1) - (H3) are satisfied, except that in (H1) we
lower the assumption on the number $r$ of boundary components of $\Sigma_{0}$
to $r\geq 1$. We now select any non-boundary leaf $\ell_{0}$ of
$\Lambda_{0}^{s}$ which is fixed by $h_{0}$:

(8.1) $h_{0}(\ell_{0})=\ell_{0}$

Since $\Lambda_{0}^{s}$ is expanded by $h_{0}$, it follows that there is
precisely one fixed point $P=h_{0}(P)\in\ell$. We derive the surface
$\Sigma_{1}$ from $\Sigma_{0}$ by puncturing a hole in $\Sigma_{0}$ at the
point $P$, and observe from (8.1) that $h_{0}$ induces a homeomorphism
$h_{1}:\Sigma_{1}\to\Sigma_{1}$. Again from considering the stable foliation
$\cal F_{0}^{s}$ associated to $\Lambda_{0}^{s}$, we obtain the stable
lamination $\Lambda_{1}^{s}\subseteq\Sigma_{1}$ for $h_{1}$ from
$\Lambda_{0}^{s}$ by doubling the leaf $\ell_{0}$ into two leaves
$\widehat{\ell}_{0}$ and $\widehat{\ell}^{\prime}_{0}$, which are boundary
leaves of a new complementary component
$\Sigma^{\prime}_{1}\subseteq\Sigma_{1}$ that has no further boundary leaf.
The component $\Sigma^{\prime}_{1}$ contains a new boundary component of
$\Sigma_{1}$ that runs around the puncture where formerly the point
$P\in\Sigma_{0}$ was located.

From this construction we obtain a quotient map
$q_{0}:\Lambda_{1}^{s}\to\Lambda_{0}^{s}$ that satisfies

(8.2) $h_{0}\circ q_{0}\,\,=\,\,q_{0}\circ h_{1}\,,$

and $q_{0}$ is 1-1 everywhere except on the leaves $\widehat{\ell}_{0}$ and
$\widehat{\ell}^{\prime}_{0}\,$, which are identified by $q_{0}$ to the single
leaf $\ell_{0}\in\Lambda_{0}^{s}\,$. We thus observe that the “quotient
procedure” from $\Sigma_{1},h_{1}$ and $\Lambda_{1}^{s}$ to $\Sigma_{0},h_{0}$
and $\Lambda_{0}^{s}$ is precisely the same as described in subsection 8.1
when passing from $\Sigma,h$ and $\Lambda^{s}$ to $\Sigma^{\prime},h^{\prime}$
and $\Lambda^{\prime s}$.

######  Remark 8.3.

In the passage from $\Lambda_{0}^{s}$ to $\Lambda_{1}^{s}$, when translated
into the IET language as in Remark 8.2, we observe that the IET for
$\Lambda_{1}^{s}$ derives from the IET for $\Lambda_{0}^{s}$ by subdividing
one of the intervals (namely the one onto which we choose to isotope $P$ along
the leaf $\ell_{0}$). Hence the alphabet $\cal A_{1}$ for $\Lambda_{1}^{s}$
derives from $\cal A_{0}$ by doubling one of its letters, namely the one
corresponding to the subdivided interval.

For the minimal subshift $X_{1}\subseteq\cal A_{1}^{\mathbb{Z}}$ associated to
$\Lambda_{1}$ and the morphism $\sigma_{0}:\cal A_{1}^{*}\to\cal A_{0}^{*}$
determined by the map $q_{0}$, which maps $X_{1}$ to $X_{0}$ and is non-
recognizable in $X_{1}$, it follows that $\sigma_{0}$ is letter-to-letter, so
that $X_{0}$ is actually a factor of $X_{1}\,$.

###  8.3. Iteration of the inverse quotient construction

We now look for a leaf $\ell_{1}\in\Lambda_{1}^{s}$ with
$h_{1}(\ell_{1})=\ell_{1}\,$. As shown in the previous subsection, this is the
only ingredient needed in order to repeat the above procedure to obtain a
surface $\Sigma_{2}$, a pseudo-Anosov homeomorphism
$h_{2}:\Sigma_{2}\to\Sigma_{2}$ with stable lamination $\Lambda_{2}^{s}$, a
map $q_{1}:\Lambda_{2}^{s}\to\Lambda_{1}^{s}$ and a morphism $\sigma_{1}:\cal
A_{2}^{*}\to\cal A_{1}^{*}$ that is non-recognizable on the minimal subshift
$X_{2}$ which satisfies $\sigma_{1}(X_{2})=X_{1}\,$.

Hence, in order to be able to repeat this procedure infinitely often, with the
purpose to get for any $n\geq 0$ a morphism $\sigma_{n}:\cal
A_{n+1}^{*}\to\cal A_{n}^{*}$ that is non-recognizable on a minimal subshift
$X_{n+1}$ with $\sigma_{n}(X_{n+1})=X_{n}\,$, we just need for any
$\Lambda_{n}^{s}$ a leaf $\ell_{n}\in\Lambda_{n}^{s}$ with
$h_{n}(\ell_{n})=\ell_{n}\,$. However, up to replacing $h_{n}$ by a power
$h_{n}^{t(n)}$ for some suitable integer $t(n)\geq 1$, this is no problem: It
is well known that any pseudo-Anosov map $h$ has infinitely many $h$-periodic
leaves in its stable lamination. We obtain the following result, which is
however only an intermediate step in our construction: In particular, the
subshifts $X_{n}$ are not the intermediate level subshifts of the given
directive sequence $\overleftarrow{\sigma}$.

######  Proposition 8.4.

There exists a directive sequence $\overleftarrow{\sigma}=(\sigma_{n}:\cal
A_{n+1}^{*}\to\cal A_{n}^{*})_{n\geq 0}$ and subshifts $X_{n}\subseteq\cal
A_{n}^{\mathbb{Z}}\,$, such that for any $n\geq 0$ the following hold:

  1. (1)

$\sigma_{n}(X_{n+1})=X_{n}\,$, and $\sigma_{n}$ is not recognizable in
$X_{n+1}\,$.

  2. (2)

$\mbox{card}(\cal A_{n+1})=\mbox{card}(\cal A_{n})+1$

  3. (3)

$\sigma_{n}$ is letter-to-letter. In particular, $\sigma_{n}$ commutes with
the shift operator, and $X_{n}$ is a factor of $X_{n+1}\,$.

  4. (4)

$X_{n}$ is minimal, aperiodic and uniquely ergodic.

  5. (5)

$X_{n}$ is substitutive (see Remark 2.5 (2)) for some primitive substitution
$\tau_{n}:\cal A_{n}^{*}\to\cal A_{n}^{*}\,$.

  6. (6)

$\tau_{n}^{t(n)}\circ\sigma_{n}=\sigma_{n}\circ\tau_{n+1}$ for some integer
$t(n)\geq 1$.

###### Proof.

Properties (1), (2) and (3) have been derived in the construction described
above. The substitution $\tau_{n}$ from (5) is the translation of the
homeomorphism $h_{n}$ into the monoid setting through the canonical embedding
$\cal A_{n}^{*}\subseteq F(\cal A_{n})=\pi_{1}\Sigma_{n}$. The primitivity of
$\tau_{n}$ is a direct consequence of the assumption “pseudo-Anosov” for $h$
and thus for all $h_{n}$. Property (4) is a direct consequence of (5), and (6)
is the translation into the monoid setting of the commutativity relation
$h_{n}^{t(n)}\circ q_{n}=q_{n}\circ h_{n+1}\,$, which is a consequence of the
equality (8.2) together with the above replacement of $h_{n}$ by
$h_{n}^{t(n)}$. $\sqcup$$\sqcap$

###  8.4. Everywhere growing directive sequences that are not (eventually)
recognizable

The sequence $\overleftarrow{\sigma}$ from Proposition 8.4 is not everywhere
growing; in fact, for any integers $m>n\geq 0$ the telescoped level map
$\sigma_{[n,m)}$ is letter-to-letter. However, by choosing suitable “diagonal”
or “eventually horizontal” paths through the infinite commutative diagram
built from the above morphisms $\sigma_{n}$ (“vertical”) and $\tau_{n}$
(“horizontal”) we will derive below everywhere growing directive sequences
with interesting properties.

Using the terminology from Proposition 8.4, we first define for each $n\geq 0$
the morphism

$\sigma^{\prime}_{n}:=\tau_{n}^{t^{\prime}(n)}\circ\sigma_{n}\,\,\,(\,=\sigma_{n}\circ\tau_{n+1}^{s(n)}\,),$

where we set $t^{\prime}(n):=s(n)\,t(n)$ for some suitably chosen integer
$s(n)\geq 1$ which ensures that the incidence matrix
$M(\tau_{n}^{t^{\prime}(n)})$ is positive. Such $s(n)$ exists because of
property (5) of Proposition 8.4, and since $M(\sigma_{n})$ has no zero-
columns, it follows furthermore that

(8.3) the incidence matrix $M(\sigma^{\prime}_{n})$ is positive, for any index
$n\geq 0$.

We now define a directive sequence
$\overleftarrow{\sigma}^{\prime}=(\sigma^{\prime}_{n}:\cal A_{n+1}^{*}\to\cal
A_{n}^{*})_{n\geq 0}$ with intermediate level subshifts called
$X^{\prime}_{n}$. Since $\tau_{n}(X_{n})=X_{n}$ and
$\sigma_{n}(X_{n+1})=X_{n}$, we have

(8.4) $\sigma^{\prime}_{n}(X_{n+1})=X_{n}$

for any $n\geq 0$, so that from the minimality of $X_{n}$ we can deduce
$X_{n}\subseteq X^{\prime}_{n}$. In particular, we obtain from statement (1)
of Proposition 8.4 together with Remark 3.7 that $\sigma^{\prime}_{n}$ is not
recognizable in $X_{n+1}$ and thus neither in $X^{\prime}_{n+1}$. From (8.3)
we obtain directly (see Remark 2.5 (3)) that the sequence
$\overleftarrow{\sigma}^{\prime}$ is everywhere growing.

Furthermore, we define for any integer $k\geq 0$ a directive sequence
$\overleftarrow{\tau}_{\\!k}=(\tau^{\prime}_{n})_{n\geq 0}$ through setting
$\tau^{\prime}_{n}:=\tau_{k}$ for all $n\geq k$ and
$\tau^{\prime}_{n}=\sigma^{\prime}_{n}$ if $0\leq n\leq k-1$. We also specify
the starting surface $\Sigma_{0}$ to be a punctured torus, so that one has
$|\cal A_{0}|=2$, and $X_{0}$ is Sturmian. It follows that for any level
$n\geq k$ the intermediate level $n$ subshift of $\overleftarrow{\tau}_{\\!k}$
is equal to the substitutive subshift $X_{k}$ defined by the substitution
$\tau_{k}$ from statement (5) of Proposition 8.4, so that for every $0\leq
n\leq k-1$ we deduce from (8.4) that the level $n$ subshift is equal to
$X_{n}$. The primitivity of $\tau_{k}$ implies in particular that the
directive sequence $\overleftarrow{\tau}_{\\!k}$ is everywhere growing. Recall
also that (as is true for all stationary sequences, see [1] and the references
given there) the truncated stationary sequence
$\overleftarrow{\tau}_{\\!k}=(\tau^{\prime}_{n})_{n\geq k}$ is totally
recognizable.

We obtain hence as immediate consequence of Proposition 8.4 the following
result; we observe that its parts (2) and (3) give directly the statements
that have been rephrased in the Introduction and stated there as Proposition
1.7:

######  Corollary 8.5.

(1) The directive sequence $\overleftarrow{\sigma}^{\prime}$ is everywhere
growing and satisfies the properties (1), (2), (4), (5) and (6) from
Proposition 8.4, with $\sigma_{n}$ replaced by $\sigma^{\prime}_{n}$.

(2) For any integer $k\geq 0$ there exists a directive sequence
$\overleftarrow{\tau}_{\\!k}$, with level alphabets $\cal A_{n}$ of size
$\mbox{card}(\cal A_{n})=k+2$ for any level $n\geq k$, and $\mbox{card}(\cal
A_{n})=n+2$ if $n\leq k$.

The sequence $\overleftarrow{\tau}_{\\!k}$ is everywhere growing and
eventually recognizable: each of the first $k$ level morphisms on the bottom
of $\overleftarrow{\tau}_{\\!k}$ is not recognizable in its corresponding
level subshift, while all level morphisms of level $n\geq k$ are recognizable
in their corresponding level subshift. Indeed, the sequence
$\overleftarrow{\tau}_{\\!k}$ is stationary above level $k$.

(3) All intermediate level subshifts of the above directive sequences
$\overleftarrow{\tau}_{\\!k}$ are minimal, uniquely ergodic and aperiodic. In
particular, the properties “recognizable”, “shift-orbit injective” (see
Definition 3.6) and “recognizable for aperiodic points” (see Remark 3.10 (2))
are equivalent, for each level morphism in its corresponding intermediate
level subshift. $\sqcup$$\sqcap$

######  Remark 8.6.

It turns out that property (3) of Corollary 8.5 is also true for the directive
sequence $\overleftarrow{\sigma}^{\prime}$. Indeed, from property (6) of
Proposition 8.4 and the well known North-South dynamics induced by any pseudo-
Anosov homeomorphism of $\Sigma$ on the projectivized space of all measured
laminations (= the boundary of Teichmüller space for $\Sigma$) one can deduce
that the inclusion $X_{n}\subseteq X^{\prime}_{n}$ derived after (8.4) is
actually an equality. However, laying out the details of these arguments would
go beyond our self-imposed limits on the amount of Nielsen-Thurston theory
imported into this section.

######  Remark 8.7.

Given any eventually recognizable everywhere growing directive sequence
$\overleftarrow{\sigma}=(\sigma_{n})_{n\geq 0}$ of finite alphabet rank, one
may ask whether there is an upper bound to the number level morphisms
$\sigma_{n}$ which are not recognizable in their corresponding intermediate
level subshift. This question has sparked some interest, see [9] and [13]. It
seems, however, that the examples given in part (2) of Corollary 8.5 above
contradict the bound claimed in Theorem 3.7 of [13]. This could also effect
the upper bound given in Corollary 1.5 of [17] on the number of successive
factor maps, for a large class of subshifts.

In this context we also want to point to Example 7.5 of the very recent paper
[2], where a family of directive sequences is presented that has the same
properties as exhibited in Corollary 8.5 (2) above for the sequences
$\overleftarrow{\tau}_{k}$. The examples from [2] are easier to describe, but
fail to have the extra properties listed in part (3) of Corollary 8.5.

Another construction of a similar kind (but closer our Corollary 8.5 above)
has been communicated to us by Bastiàn Espinoza [18] in the final stages of
the revision of this paper.

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