Measure transfer and S-adic developments for subshifts

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 $\mathcal A_n$ have cardinality $d,$ while none of the $d-2$ bottom level morphisms is recognizable in its level subshift $X_n \subseteq \mathcal A_n^{\mathbb {Z}}$.

1. Introduction

A subshift over a finite alphabet $\mathcal A$ is a non-empty, closed, and shift-invariant subset $X \subseteq \mathcal A^{\mathbb {Z}}$ . 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 $\sigma _n: \mathcal A^*_{n+1} \to \mathcal A_n^*$ for all integers $n \geq 0$ , where each $\mathcal A_n$ is again a finite alphabet, and $\mathcal A_n^*$ denotes the free monoid over $\mathcal A_n$ . The morphisms $\sigma _n$ here are all assumed to be non-erasing, that is, none of the letters of $\mathcal A_{n+1}$ is mapped to the empty word. The directive sequence $\overleftarrow \sigma $ generates the given subshift X if for some identification $\mathcal A = \mathcal A_0$ , any finite factor $x_k \cdots x_\ell $ of any biinfinite word $\mathbf {x} = \cdots x_{-1} x_0 x_1 \cdots \in X$ is also a factor of some $\sigma _0 \circ \cdots \circ \sigma _{n-1}(a_i)$ with $a_i \in \mathcal A_{n}$ , and conversely, any such $\mathbf {x}$ belongs to X. One usually also assumes that $\overleftarrow \sigma $ is everywhere growing, which means that ${\liminf _{n\to \infty }} (\min \{|\sigma _0 \circ \cdots \circ \sigma _{n-1}(a_i)| \mid a_i \in \mathcal A_{n}\}) = \infty $ . It is well known that any subshift $X \subseteq \mathcal A^{\mathbb {Z}}$ 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 ${X_n \subseteq \mathcal A_n^{\mathbb {Z}}}$ , 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 $\sigma : \mathcal A^* \to \mathcal B^*$ between free monoids over finite alphabets $\mathcal A$ and $\mathcal B$ , defines for any subshift $X \subseteq \mathcal A^{\mathbb {Z}}$ 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 $\sigma _X^{\mathcal {M}}: \mathcal M(X) \to \mathcal M(\sigma (X))$ , where $\mathcal M(X)$ denotes the measure cone on X, that is, the set of all shift-invariant Borel measures on the subshift X. The map $\sigma _X^{\mathcal {M}}$ is the restriction/co-restriction of a map $\sigma ^{\mathcal {M}}: \mathcal M(\mathcal A^{\mathbb {Z}}) \to \mathcal M(\mathcal B^{\mathbb {Z}})$ which is $\mathbb R_{\geq 0}$ -linear, functorial, and commutes with the support map on subshifts (see §3.1).

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

Furthermore, any invariant measure $\mu $ on a subshift $X \subseteq \mathcal A^{\mathbb {Z}}$ defines canonically a letter frequency vector $\vec v(\mu )$ in the non-negative cone $\mathbb R_{\geq 0}^{\mathcal {A}}$ of the vector space $\mathbb R^{\mathcal {A}}$ , where for each letter $a_i \in \mathcal A$ , 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 $\mathbf {x} \in \mathcal A^{\mathbb {Z}}$ 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 ${\mathcal C(X) \subseteq \mathbb R_{\geq 0}^{\mathcal {A}}}$ ; it gives rise to a canonical $\mathbb R_{\geq 0}$ -linear evaluation map $\zeta _X: \mathcal M(X) \to \mathcal C(X)$ which by definition is surjective.

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

$$ \begin{align*}\!\! \cdots \,\,\overset{\sigma_{n+1}^{\mathcal{M}}}{\longrightarrow}\,\,\, \mathcal M(X_{n+1}) \overset{\sigma_{n}^{\mathcal{M}}}{\longrightarrow} \,\,\mathcal M(X_n)\,\,\overset{\sigma_{n-1}^{\mathcal{M}}}{\longrightarrow} \,\,\,\cdots \,\,\overset{\sigma_{2}^{\mathcal{M}}}{\longrightarrow} \,\,\mathcal M(X_1) \overset{\sigma_{1}^{\mathcal{M}}}{\longrightarrow} \mathcal M(X)\end{align*} $$

$$ \begin{align*}\qquad \qquad \qquad\downarrow \zeta_{X_{n+1}} \qquad\,\,\,\,\downarrow \zeta_{X_{n}} \qquad \qquad \qquad \qquad\,\,\,\,\downarrow \zeta_{X_{1}} \qquad \qquad\downarrow \zeta_{X}\end{align*} $$

$$ \begin{align*} \cdots \,\,\overset{M(\sigma_{n+1})}{\longrightarrow}\,\, \mathcal C(X_{n+1}) \overset{M(\sigma_{n})}{\longrightarrow} \,\,\mathcal C(X_n)\overset{M(\sigma_{n-1})}{\longrightarrow} \,\,\cdots \,\,\overset{M(\sigma_{2})}{\longrightarrow} \,\,\mathcal C(X_1) \,\,\overset{M(\sigma_{1})}{\longrightarrow} \,\, \,\mathcal C(X)\ \ \end{align*} $$

A measure tower $\overleftarrow \mu = (\mu _n)_{n \geq 0}$ on a directive sequence $\overleftarrow \sigma $ as above, defined by postulating $\mu _n \in \mathcal M(X_n)$ and $\sigma _n^{\mathcal {M}}(\mu _{n+1}) = \mu _n,$ 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 $\mathbb R_{\geq 0}$ -linear evaluation map $\frak m: \mathcal V(\overleftarrow \sigma ) \to \mathcal M(X)$ , from the set $\mathcal V(\overleftarrow \sigma )$ of all such vector towers to the measure cone $\mathcal M(X)$ of the subshift X generated by $\overleftarrow \sigma $ , has been established in [3], and the map $\frak m$ is shown in [3] to be always surjective, as long as $\overleftarrow \sigma $ is everywhere growing (but no other hypotheses are needed). We obtain the following propositon (see Proposition 5.3).

Proposition 1.1. For any everywhere growing directive sequence $\overleftarrow \sigma $ , there is a canonical $\mathbb R_{\geq 0}$ -linear bijection between the cone $\mathcal V(\overleftarrow \sigma )$ of vector towers and the cone $\mathcal M(\overleftarrow \sigma )$ of measure towers on $\overleftarrow \sigma $ , given by the letter frequency map

$$ \begin{align*}\overleftarrow \mu = (\mu_n)_{n \geq 0} \mapsto \overleftarrow v = (\vec v_n)_{n \geq 0},\end{align*} $$

with $\vec v_n = \vec v(\mu _n) =(\mu _n([a_k]))_{a_k \in \mathcal A_n}$ for all levels $n \geq 0$ .

From this set-up, we derive (see Proposition 4.4) the following result. A crucial ingredient in its proof is the main result of our previous paper [3], quoted below as Theorem 2.10.

Theorem 1.2. For any non-erasing monoid morphism $\sigma : \mathcal A^* \to \mathcal B^*$ and any subshift $X \subseteq \mathcal A^{\mathbb {Z}}$ , the induced measure transfer map $\sigma ^{\mathcal {M}}$ maps the measure cone $\mathcal M(X)$ of X surjectively to the measure cone $\mathcal M(\sigma (X))$ of the image subshift $\sigma (X)$ :

$$ \begin{align*}\sigma^{\mathcal{M}}(\mathcal M(X)) = \mathcal M(\sigma(X)).\end{align*} $$

This general surjectivity result for the measure transfer map $\sigma ^{\mathcal {M}}$ 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 $\sigma : \mathcal A^* \to \mathcal B^*$ is recognizable in a subshift $X \subseteq \mathcal A^{\mathbb {Z}}$ , then the measure transfer map $\sigma ^{\mathcal {M}}_X: \mathcal M(X) \to \mathcal M(\sigma (X))$ 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 the following theorem (see Theorem 5.6).

Theorem 1.4. For any totally recognizable everywhere growing directive sequence $\overleftarrow \sigma $ , with generated subshift $X = X_{\tiny \overleftarrow \sigma }$ , the $\mathbb R_{\geq 0}$ -linear surjective map of cones

$$ \begin{align*}\frak m: \mathcal V(\overleftarrow \sigma) \to \mathcal M(X)\end{align*} $$

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 $\mathcal A_n$ that all have cardinality $\mathrm{card}(\mathcal A_n) = d$ . This construction is used in §7 below to define a large ‘diagonal’ family $\frak X$ 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 and Kra (see [11]).

Corollary 1.5. For any directive sequence $\overleftarrow \sigma \in \frak X$ , 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 $\mathcal M(X_{\tiny \overleftarrow \sigma })$ .

The directive sequences considered in the last corollary are all totally recognizable and they are ‘large’, in that their alphabet rank, that is, the limit inferior of the cardinality of the level alphabets, is infinite. For finite alphabet rank, however, 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 $\mathcal M(X_n)$ . 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 $\frak m$ as in Theorem 1.4 above is a direct consequence of our set-up. We thus obtain the following corollary (see Corollary 6.5).

Corollary 1.6. Let $X \subseteq \mathcal A^{\mathbb {Z}}$ 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 $ , that is, by the values $\mu ([a_k])$ for all $a_k \in \mathcal A$ .

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, that is, only for sufficiently high levels, one assumes that the level morphisms are recognizable in the corresponding level subshift. In §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, 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 $\mathcal A_n = \mathcal A_{n_0}$ 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 $\mathrm{card}(\mathcal A_n) = n+2 = \mathrm{card}(\mathcal A_{n+1})-1$ , 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.)

2. Terminology, notation, conventions, and some quotes

In this section, we first recall some standard terminology from symbolic dynamics (see §2.1), then summarize the notation introduced in [5] and some of its results (see §3.1), and in §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 $\mathcal A, \mathcal B$ , or $\mathcal C$ non-empty finite sets, called alphabets, and by $\mathcal A^*, \mathcal B^*$ , or $\mathcal C^*$ the free monoid over those alphabets. Every element $w \in \mathcal A^*$ is a word in the letters $a_1, a_2, \ldots , a_d$ of $\mathcal A$ , that is,

$$ \begin{align*}w = x_1 x_2 \ldots x_n \quad \text{with } x_i \in \{a_1, a_2, \ldots, a_d\} = \mathcal A\end{align*} $$

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 ${|w| = {\sum _{a_j \in \mathcal A}} |w|_{a_j},}$ where $|w|_{a_j}$ denotes the number of occurrences of the letter $a_j$ in w. More generally, for any second word $u \in \mathcal A^*$ , 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 $\sigma : \mathcal A^* \to \mathcal B^*$ is determined by the family of letter images $\sigma (a_i) \in \mathcal B^*$ for all $a_i \in \mathcal A$ , and this family can be chosen freely. Such a morphism $\sigma $ is non-erasing if $|\sigma (a_i)| \geq 1$ for all $a_i \in \mathcal A$ . Note that any composition of non-erasing morphisms is non-erasing. Morphisms which are ‘erasing’ (by which we mean ‘not non-erasing’) can occasionally create unexpected and undesired phenomena (see [2, §5]), which is why we decided to exclude them. It turns out (see Remark 2.5) that in the context considered here, this assumption is almost immaterial.

Assumption 2.1. All morphisms considered in this paper are assumed to be non-erasing.

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

(2.1) $$ \begin{align} M(\sigma) = (|\sigma(a_j)|_{b_i})_{b_i \in \mathcal B, \, a_j \in \mathcal A}. \end{align} $$

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

(2.2) $$ \begin{align} \mathbf{x} = \cdots x_{i-1} x_i x_{i+1} \cdots \end{align} $$

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

For any integers $k \leq l$ , we denote by $\mathbf {x}_{[k, \ell ]}$ the subword (again also called factor) $x_k \cdots x_\ell $ of the biinfinite word $\mathbf {x}$ as in equation (2.2). We also consider the one-sided infinite positive half-word $\mathbf {x}_{[1, \infty )} = x_1 x_2 \cdots $ of $\mathbf {x}$ .

To any word $w \in \mathcal A^*$ , there is associated the cylinder $[w] \subseteq \mathcal A^{\mathbb {Z}},$ which consists of all words $\mathbf {x} \in \mathcal A^{\mathbb {Z}}$ which satisfy $\mathbf {x}_{[1, |w|]} = w$ . If w is the empty word, then $[w] = \mathcal A^{\mathbb {Z}}$ . 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 $\mathcal A^{\mathbb {Z}}$ .

A non-empty subset $X \subseteq \mathcal A^{\mathbb {Z}}$ 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 $\mathbf {x} \in X$ , the shift-orbit $\mathcal O(\mathbf {x}) = \{T^m(\mathbf {x}) \mid m \in \mathbb Z\}$ 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 $X = \mathcal O(w^{\pm \infty })$ of some periodic word $w^{\pm \infty } = \cdots w w w \cdots ,$ which is well defined for any non-empty $w \in \mathcal A^*$ by the convention $w^{\pm \infty }_{[1, \infty )} = w w w \ldots $ (that is, the letter with index 1 in $w^{\pm \infty }$ is the first letter of w). It follows that any infinite minimal subshift is in particular aperiodic, which means that X does not contain any periodic word $w^{\pm \infty }$ .

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

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

(2.3) $$ \begin{align} h_X = \lim_{n \to \infty} \frac{\log p_X(n)}{n}. \end{align} $$

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

(2.4) $$ \begin{align} \sigma^{\mathbb{Z}}: \mathcal A^{\mathbb{Z}} \to \mathcal B^{\mathbb{Z}} \end{align} $$

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

For almost all subshifts $X \subseteq \mathcal A^Z$ , 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 2.2. The following three definitions of the image subshift $Y := \sigma (X)$ are equivalent for any non-erasing monoid morphism $\sigma : \mathcal A^* \to \mathcal B^*$ .

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 $\mathcal O(\sigma (\mathbf {x}))$ , for any $\mathbf {x} \in X$ . (Note here (see [5, Lemma 2.4]) 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 $\sigma (\mathcal L(X))$ . Thus, Y consists of all biinfinite words $\mathbf {y} \in \mathcal B^{\mathbb {Z}}$ with the property that every factor of $\mathbf {y}$ is also a factor of some word in $\sigma (\mathcal L(X))$ .

We observe directly the following consequence which will be used later (see Remark 4.2).

Lemma 2.3. Let $\sigma : \mathcal A^* \to \mathcal B^*$ be a non-erasing monoid morphism and let $X \subseteq \mathcal A^{\mathbb {Z}}$ be any subshift. If $\mathcal L \subseteq \mathcal A^*$ is a language that generates X, then $\sigma (\mathcal L)$ generates $\sigma (X)$ .

An invariant measure on $\mathcal A^{\mathbb {Z}}$ is a finite Borel measure $\mu $ on $\mathcal A^{\mathbb {Z}}$ which is invariant under the homeomorphism T (= the shift operator). The set of all such invariant measures is denoted by $\mathcal M(\mathcal A^{\mathbb {Z}})$ . For any subshift $X \subseteq \mathcal A^{\mathbb {Z}}$ , we denote by $\mathcal M(X) \subseteq \mathcal M(\mathcal A^{\mathbb {Z}})$ 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 $\mu \in \mathcal M(\mathcal A^{\mathbb {Z}})$ defines a function

$$ \begin{align*}\mathcal A^* \, \to \, \mathbb R_{\geq 0},\quad w \mapsto \mu([w])\end{align*} $$

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

(2.5) $$ \begin{align} \mu(w) = \sum_{a_i \in \mathcal A} \mu(a_i w) = \sum_{a_i \in \mathcal A} \mu(w a_i) \end{align} $$

for any $w \in \mathcal A^*$ . Conversely, it is well known that any weight function ${\mu : \mathcal A^* \to \mathbb R_{\geq 0}}$ defines an invariant measure $\mu \in \mathcal M(\mathcal A^{\mathbb {Z}})$ which satisfies $\mu ([w]) = \mu (w)$ . The set $\mathcal M(\mathcal A^{\mathbb {Z}})$ can hence be understood as a subset of the infinite dimensional non-negative cone $\mathbb R_{\geq 0}^{\mathcal A^*} = \{{\sum _{w \in \mathcal A^*}} x_w \vec e_{w} \mid x_w \geq 0\}$ , where $\vec e_{w}$ denotes the unit vector of $\mathbb R^{\mathcal A^*}$ in the direction defined by $w \in \mathcal A^*$ . From this embedding, the set $\mathcal M(\mathcal A^{\mathbb {Z}})$ inherits the product topology; the latter coincides with the more generally known weak $^*$ -topology on the measure cone $\mathcal M(\mathcal A^{\mathbb {Z}})$ .

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

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

A word $w \in \mathcal A^* \smallsetminus \{\varepsilon \}$ is called a proper power if

(2.6) $$ \begin{align} w = u^m \quad\text{for some } u \in \mathcal A^* \text{ and some integer }m \geq 2. \end{align} $$

(Elements in $\mathcal A^*$ which are not a proper power are sometimes called ‘primitive’. However, since $\mathcal A^*$ is canonically embedded into the free group $F(\mathcal A)$ , 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.)

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

(2.7) $$ \begin{align} \mu_w(B) := \mathrm{card}(B \cap \mathcal O(w^{\pm\infty})) \end{align} $$

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

$$ \begin{align*}\mu_w := m \cdot \mu_{u}.\end{align*} $$

In either case, it follows that $({1}/{|w|}) \mu _w$ is a probability measure. The set of weighted characteristic measures $\unicode{x3bb} \, \mu _w$ (for any $\unicode{x3bb}> 0$ ) is known to be dense in $\mathcal M(\mathcal A^{\mathbb {Z}})$ . The support of any characteristic measure is given by

(2.8) $$ \begin{align} \mbox{Supp}(\mu_w) = \mathcal O(w^{\pm \infty}). \end{align} $$

To any alphabet $\mathcal A$ , one associates canonically the non-negative alphabet cone ${\mathbb R_{\geq 0}^{\mathcal {A}} = \{{\sum _{a_k \in \mathcal A}} x_k \vec e_{a_k} \mid x_k \in \mathbb R, x_k \geq 0\}}$ . For any invariant measure $\mu $ on $\mathcal A^{\mathbb {Z}}$ , the evaluation on the letter cylinders $[a_k]$ for all $a_k \in \mathcal A$ defines a letter frequency vector

(2.9) $$ \begin{align} \vec v(\mu) :=\sum_{a_k \in \mathcal A} \mu([a_k]) \, \vec e_{a_k}, \end{align} $$

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

(2.10) $$ \begin{align} \zeta_{\mathcal A^{\mathbb{Z}}}: \mathcal M(\mathcal A^{\mathbb{Z}}) \to \mathbb R_{\geq 0}^{\mathcal{A}}, \quad \mu \mapsto \vec v(\mu). \end{align} $$

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

(2.11) $$ \begin{align} \mathcal C(X) := \zeta_X(\mathcal M(X)) \subseteq \mathbb R_{\geq 0}^{\mathcal{A}}, \end{align} $$

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

(2.12) $$ \begin{align} \mathbb R^{\mathcal{A}} \to \mathbb R^{\mathcal{B}}, \mathbb R_{\geq 0}^{\mathcal{A}} \to \mathbb R_{\geq 0}^{\mathcal{B}}\quad\ \text{and}\ \quad \mathcal C(X) \to \mathcal C(\sigma(X)) \end{align} $$

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

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

2.2. Measures on subshifts via vector towers on directive sequences

To state Theorem 2.10 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

(2.13) $$ \begin{align} \sigma_n: \mathcal A_{n+1}^* \to \mathcal A_n^* \end{align} $$

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

$$ \begin{align*}\overleftarrow \sigma = \sigma_0 \circ \sigma_{1} \circ \sigma_{2} \circ \cdots\end{align*} $$

to denote a directive sequence.

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

$$ \begin{align*}\sigma_{[n, m)} := \sigma_n \circ \sigma_{n+1} \circ \cdots \circ \sigma_{m-1}\end{align*} $$

as well as the level n truncated directive sequence

(2.14) $$ \begin{align} \overleftarrow \sigma \dagger_{\! n} = (\sigma_k)_{k \geq n}. \end{align} $$

Any directive sequence $\overleftarrow \sigma $ as in equation (2.13) above generates a subshift $X = X_{\tiny \overleftarrow \sigma }$ over the base alphabet $\mathcal A_0,$ defined by the convention that $\mathbf {x} \in \mathcal A_0^{\mathbb {Z}}$ belongs to X if and only if for any finite factor w of $\mathbf {x}$ , there exists some level $n \geq 1$ and some letter $a_j \in \mathcal A_n$ 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 $X_n \subseteq \mathcal A_n^{\mathbb {Z}}$ which is generated by the truncated sequence $\overleftarrow \sigma \dagger _{\! n}\!:$

(2.15) $$ \begin{align} X_n := X_{\tiny \overleftarrow \sigma\dagger_{\! n}}. \end{align} $$

The subshift $X_n$ is the image subshift of the analogously defined level $n+1$ intermediate subshift $X_{n+1}$ under the morphism $\sigma _n ,$ that is:

(2.16) $$ \begin{align} X_n = \sigma_n(X_{n+1}) \quad \text{for any level } n \geq 0. \end{align} $$

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

(2.17) $$ \begin{align} \beta_-(n) := \min{ \{}\,|\sigma_{[0, n-1)}(a_j)| \,\,{|} \,\,a_j \in \mathcal A_n { \}} \end{align} $$

tends to $\infty $ for $n \to \infty $ . We have the following fact (see for instance [3, Proposition 5.10]).

Fact 2.4. For every subshift $X \subseteq \mathcal A^{\mathbb {Z}}$ , there exists an everywhere growing directive sequence $\overleftarrow \sigma $ that generates X. More precisely, using the notation from equation (2.13), one has

(2.18) $$ \begin{align} \mathcal A_0 = \mathcal A \quad \text{and} \quad X_{\tiny \overleftarrow \sigma} = X. \end{align} $$

Remark 2.5. Going back to Assumption 2.1, we would like to note that the above evoked directive sequence $\overleftarrow \sigma $ from [3, Proposition 5.10] has indeed the property that every level morphism is non-erasing.

In this context, we observe that, a priori, if a directive sequence $\overleftarrow \sigma = (\sigma _n)_{n \geq 0}$ is everywhere growing, it could well be that some of the level maps $\sigma _n: \mathcal A_{n+1}^* \to \mathcal A_n^*$ map a generator $a_i \in \mathcal A_{n+1}$ to the empty word in $\mathcal A_n^*$ . However, it follows directly from the assumption ‘everywhere growing’ that this can only occur for a finite number of level maps $\sigma _n,$ so that one could easily bypass these levels by suitable telescoping and thus obtain an everywhere growing directive sequence which generates the same subshift and has only level maps which are all non-erasing.

Remark 2.6.

1. (1) A directive sequence $\overleftarrow \sigma $ that generates a subshift X is also called an S-adic development (or an S-adic expansion) of X, where S stands sometimes for an (often assumed to be finite) set of substitutions that contains all level morphisms. This concept and in particular the terminology ‘S-adic’ has been introduced by 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 $\mathcal A$ .

2. (2) If the set S consists of a single endomorphism $\sigma : \mathcal A^* \to \mathcal A^*$ , 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]).

Definition 2.7. A directive sequence $\overleftarrow \sigma = (\sigma _n)_{n \geq 0}$ is called weakly primitive if for every level $n \geq 0$ , there exists a level $m> n$ such that the telescoped incidence matrix $M(\sigma _{[n, m)})$ is positive (that is, it has all coefficients $> 0$ ).

Remark 2.8.

1. (1) One verifies easily that any directive sequence $\overleftarrow \sigma $ which is weakly primitive is in particular everywhere growing, unless all level alphabets have cardinality 1.

2. (2) Weakly primitive directive sequences $\overleftarrow \sigma $ have another important property, namely that the subshift $X_{\tiny \overleftarrow \sigma }$ generated by $\overleftarrow \sigma $ is minimal. For this conclusion, we cite [7], proved originally in [14].

In [3], to any directive sequence $\overleftarrow \sigma $ as in equation (2.13), there has been associated the set $\mathcal V(\overleftarrow \sigma )$ of vector towers $\overleftarrow v = (\vec v_n)_{n \geq 0}$ over $\overleftarrow \sigma $ . (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 in equation (2.20) for any $n \geq 0$ .) Such a vector tower consists of non-negative vectors

(2.19) $$ \begin{align} \vec v_n = {\sum_{a_j \in \mathcal A}} \vec v_n(a_j) \, \vec e_{a_j} \in \mathbb R_{\geq 0}^{\mathcal A_n} \end{align} $$

that are subject to the compatibility condition

(2.20) $$ \begin{align} \vec v_n = M(\sigma_n) \cdot \vec v_{n+1} \end{align} $$

for all $n \geq 0$ . It has been shown (see [3, Remark 9.5]) that for any word $w \in \mathcal A_0^*$ and any such vector tower $\overleftarrow v$ , the sequence of sums

$$ \begin{align*}\sum_{a_j \,\in \mathcal A_n} \vec v_n(a_j) \, |\sigma_{[0,n)}(a_j)|_w \, \end{align*} $$

is bounded above and increasing, as long as $\overleftarrow \sigma $ is everywhere growing (but no other condition is needed). This shows that the value

(2.21) $$ \begin{align} \mu^{\tiny \overleftarrow v}(w) := \lim_{n \to \infty} \sum_{a_j \,\in \mathcal A_n} \vec v_n(a_j) \, |\sigma_{[0,n)}(a_j)|_w \end{align} $$

is well defined for any $w \in \mathcal A_0^*$ . Furthermore, it is shown in [3, Propositions 7.4 and 9.4] that the issuing function $\mu ^{\tiny \overleftarrow v}: \mathcal A_0^* \to \mathbb R_{\geq 0}$ satisfies the Kirchhoff equalities in equation (2.5), so that we can summarize to get the following proposition.

Proposition 2.9. [3]

Any vector tower $\overleftarrow v$ on an everywhere growing directive sequence $\overleftarrow \sigma $ defines via equation (2.21) an invariant measure on the subshift X generated by $\overleftarrow \sigma $ , denoted by $\mu ^{\tiny \overleftarrow v} \in \mathcal M(X)$ . ${}^{}$ ${}^{}$

In terms of S-adic language, the main result of [3] translates directly into the following theorem (see also [4, §3]).

Theorem 2.10. [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

$$ \begin{align*} {\frak m}_{\tiny \overleftarrow \sigma} : \mathcal V(\overleftarrow \sigma) \to \mathcal M(X),\quad \overleftarrow v \mapsto \mu^{\tiny\overleftarrow v}\end{align*} $$

is $\mathbb R_{\geq 0}$ -linear and surjective.

For any of the level alphabets $\mathcal A_n$ of a directive sequence $\overleftarrow \sigma $ as above, we consider the projection map of the set of vector towers to the corresponding non-negative alphabet cone:

$$ \begin{align*}pr_n: \mathcal V(\overleftarrow \sigma) \to \mathbb R_{\geq 0}^{\mathcal A_n}, \quad \overleftarrow v = (\vec v_n)_{n \geq 0} \mapsto \vec v_n.\end{align*} $$

(The map $pr_n$ was denoted in [3, 4] by $\frak m_n$ , but we decided to reserve this notation here for the more telling maps introduced below in §5.) On the base level $n=0$ , this projection splits over the evaluation map $\zeta _{\mathcal A_0^{\mathbb {Z}}}$ from equation (2.10) via the map $\frak m_{\tiny \overleftarrow \sigma }$ from the last theorem. More precisely, this gives (see [3, Proposition 10.2(1) and (2)]).

Proposition 2.11. For any subshift $X \subseteq \mathcal A^{\mathbb {Z}}$ generated by an everywhere growing directive sequence $\overleftarrow \sigma $ as in equations (2.13) and (2.18), one has the following items.

1. (1) The map $\zeta _X: \mathcal M(X) \to \mathbb R_{\geq 0}^{\mathcal A}, \,\, \mu \mapsto (\mu ([a_k])_{a_k \in \mathcal A}$ satisfies

   $$ \begin{align*}pr_0 = \zeta_X \circ {\frak m}_{\tiny \overleftarrow \sigma}.\end{align*} $$

2. (2) In particular, for the letter frequency cone $\mathcal C(X) = \mathrm {im}(\zeta _X)$ (see equation (2.11)), this gives

   $$ \begin{align*}\mathcal C(X) = \zeta_X({\frak m_{\tiny \overleftarrow \sigma}}(\mathcal V(\overleftarrow\sigma))).\end{align*} $$

3. (3) Alternatively, the letter frequency cone is obtained as a nested intersection as

   $$ \begin{align*}\mathcal C(X) := \bigcap_ {n \geq 1} \, M(\sigma_{[0,n)})(\mathbb R_{\geq 0}^{\mathcal A_{n}}).\end{align*} $$

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

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

Lemma 2.12. For any vector tower $\overleftarrow v = (\vec v_n)_{n \geq 0}$ over an everywhere growing directive sequence $\overleftarrow \sigma $ as in equation (2.13), one has

$$ \begin{align*}\lim_{n \to \infty} \sum_{a_j \in \mathcal A_n} \vec v_n(a_j) = 0,\end{align*} $$

where the coefficient $\vec v_n(a_j) \in \mathbb R_{\geq 0}$ is defined in equation (2.19).

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 §3.1 below), then recall the definition and some related properties of recognizable morphisms (see §3.2 below). In §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 $\sigma : \mathcal A^* \to \mathcal B^*$ , we define the subdivision alphabet $\mathcal A_\sigma = \{a_i(k) \mid a_i \in \mathcal A \text { and} 1 \leq k \leq |\sigma (a_i)|\}$ . The morphism $\sigma $ now defines a subdivision morphism ${\pi _\sigma : \mathcal A^* \to \mathcal A_\sigma ^*}$ and a letter-to-letter morphism $\alpha _\sigma : \mathcal A_\sigma ^* \to \mathcal B^*$ , given for any $a_i \in \mathcal A$ and any $a_i(k) \in \mathcal A_\sigma $  by

$$ \begin{align*}\pi_\sigma(a_i) = a_i(1) \, a_i(2) \ldots a_i(|\sigma(a_i)|) \quad \text{and} \quad \alpha_\sigma(a_i(k)) = [\sigma(a_i)]_k.\end{align*} $$

Here, by $[\sigma (a_i)]_k$ , we mean the kth letter of the word $\sigma (a_i) \in \mathcal B^*$ . We obtain directly the following fact.

Fact 3.1. For any non-erasing monoid morphism $\sigma : \mathcal A^* \to \mathcal B^*$ , one has

$$ \begin{align*}\sigma \,\, = \,\, \alpha_\sigma \circ \pi_\sigma.\end{align*} $$

For any word $w \in \mathcal A_\sigma ^*$ , we denote by $\widehat w \in \mathcal A^*$ the shortest word such that $\pi _\sigma (\widehat w)$ contains w as a factor. If such $\widehat w$ exists, it is unique; otherwise (for notational convenience only), we treat $\widehat w$ as formal symbol for which we set

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

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

For any measure $\mu \in \mathcal M(\mathcal A^{\mathbb {Z}})$ , a measure $\mu ^{\pi _\sigma } \in \mathcal M(\mathcal A_\sigma ^{\mathbb {Z}})$ is defined in [5, §3.1] by setting $\mu ^{\pi _\sigma }([w]) := \mu ([\widehat w])$ , where $[\widehat w]$ is the cylinder associated to the word $\widehat w$ (see §2.1). However, for any measure $\mu ' \in \mathcal M(\mathcal A_\sigma ^{\mathbb {Z}})$ , the classical push-forward measure $(\alpha _\sigma )_*(\mu ')$ is an invariant measure on $\mathcal B^{\mathbb {Z}}$ , since $\alpha _\sigma $ is letter-to-letter. We thus obtain the following theorem (see [5, §3]).

Theorem 3.2. Let $\sigma : \mathcal A^* \to \mathcal B^*$ be a non-erasing morphism of free monoids.

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

   $$ \begin{align*}\mu^\sigma = (\alpha_\sigma)_*(\mu^{\pi_\sigma}).\end{align*} $$

2. (2) For any word $w' \in \mathcal B^*$ , the ‘transferred measure’ $\mu ^\sigma $ takes on the cylinder $[w']$ the value

   $$ \begin{align*}\mu^\sigma([w']) = \sum_{w_i \in \alpha_\sigma^{-1}(w')} \mu([\widehat w_i]).\end{align*} $$

3. (3) The issuing measure transfer map

   $$ \begin{align*}\sigma^{\mathcal{M}}: \mathcal M(\mathcal A^{\mathbb{Z}}) \to \mathcal M(\mathcal B^{\mathbb{Z}}), \,\, \mu \mapsto \mu^\sigma\end{align*} $$

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

1. (3a) The map $\sigma ^{\mathcal {M}}$ is linear (over $\mathbb R_{\geq 0}$ ) and continuous (with respect to the weak $^*$ -topology).

2. (3b) The map $\sigma ^{\mathcal {M}}$ is functorial.

3. (3c) If X is the support of $\mu $ , then $\sigma (X)$ is the support of $\mu ^\sigma .$ Hence, $\sigma ^{\mathcal {M}}$ induces in particular on any subshift $X \subseteq \mathcal A^{\mathbb {Z}}$ a restriction/co-restriction map

   $$ \begin{align*} \sigma^{\mathcal{M}}_X: \mathcal M(X) \to \mathcal M(\sigma(X)). \end{align*} $$

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

Proposition 3.3. Let $\sigma : \mathcal A^* \to \mathcal B^*$ be a non-erasing free monoid morphism and let $\sigma ^{\mathcal {M}}$ be the induced transfer map on the measure cones. Let $\mu \in \mathcal M(\mathcal A^{\mathbb {Z}})$ be an invariant measure on the full shift $\mathcal A^{\mathbb {Z}}$ , and denote as before by $\mu ^\sigma = \sigma ^{\mathcal {M}}(\mu )$ the transferred measure on $\mathcal B^{\mathbb {Z}}$ . Then one has the following properties.

1. (a) The total mass of the transferred measure $\mu ^\sigma $ is given by the formula

   $$ \begin{align*}\mu^\sigma(\mathcal B^{\mathbb{Z}}) = \sum_{a_k \in \mathcal A} \sum_{b_j \in \mathcal B} |\sigma(a_k)|_{b_j}\cdot \mu(a_k).\end{align*} $$
   In particular, if $\mu $ is a probability measure, then, in general, $\mu ^\sigma $ will not be probability.

2. (b) For any generator $b_j \in \mathcal B$ , we have

   $$ \begin{align*}\mu^\sigma([b_j]) = \sum_{a_k \in \mathcal A} |\sigma(a_k)|_{b_j} \cdot \mu(a_k).\end{align*} $$
   In particular, for the letter frequency vectors from equation (2.9), we obtain
   (3.2) $$ \begin{align} \vec v(\mu^\sigma) = M(\sigma) \cdot \vec v(\mu). \end{align} $$
   In other words (see [5, Proposition 4.5]), the measure transfer map $\sigma ^{\mathcal {M}}$ commutes via the evaluation maps $\zeta _{\mathcal A^{\mathbb {Z}}}$ and $\zeta _{\mathcal B^{\mathbb {Z}}}$ from equation (2.10) with the linear map induced by $\sigma $ on the non-negative cone $\mathbb R_{\geq 0}^{\mathcal {A}} :$
   $$ \begin{align*} \zeta_{\mathcal B^{\mathbb{Z}}} \circ \sigma^{\mathcal{M}} = M(\sigma) \circ \zeta_{\mathcal A^{\mathbb{Z}}}.\end{align*} $$

3. (c) For any $w \in \mathcal A^*$ , the cylinder measures satisfy

   $$ \begin{align*}\mu^\sigma([\sigma(w)]) \geq \mu([w]).\end{align*} $$

4. (d) For any word $w \in \mathcal A^*$ , the characteristic measure $\mu _w$ satisfies

   $$ \begin{align*} \sigma^{\mathcal{M}}(\mu_w) = \mu_{\sigma(w)}.\end{align*} $$

It remains to quote a useful evaluation technique for the transferred measure, derived in [5, §4] from what is stated above as part (2) of Theorem 3.2. For this purpose, we define for any non-erasing morphism $\sigma : \mathcal A^* \to \mathcal B^*$ and any $w \in \mathcal A^*, \, u \in \mathcal B^*$ , 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 the first letter of w, and the last letter of u occurs in the $\sigma $ -image of the 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 $\sigma : \mathcal A^* \to \mathcal B^*$ be any non-erasing monoid morphism and let $\mu \in \mathcal M(\mathcal A^{\mathbb {Z}})$ . Then for any $w' \in \mathcal B^*$ with $|w'| \geq 2$ , the transferred measure $\mu ^\sigma = \sigma ^{\mathcal {M}}(\mu )$ takes on the cylinder $[w']$ the value

$$ \begin{align*} \mu^\sigma([w']) = \sum_{{ \{}w_j \in \mathcal A^* \,{ |}\, |w_j| \leq ({|w'|-2})/{\langle\sigma\rangle}+2{ \}}}{\lfloor\sigma(w_j) \rfloor}_{w'} \cdot \mu([w_j]). \end{align*} $$

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

Consider biinfinite words $\mathbf {x}, \mathbf {x'} \in X \subseteq \mathcal A^{\mathbb {Z}}$ , and $\mathbf {y} \in \mathcal B^{\mathbb {Z}}$ which satisfy:

• (*) $\mathbf {y} = T^k(\sigma ^{\mathbb {Z}}(\mathbf {x}))$ and $\mathbf {y} = T^\ell (\sigma ^{\mathbb {Z}}(\mathbf {x'}))$ 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 $\mathbf {x_{[1, \infty )}} = x_1 x_2 \ldots $ of $\mathbf {x}$ and $\mathbf {x^{\prime }_{[1, \infty )}} = x^{\prime }_1 x^{\prime }_2 \ldots $ of $\mathbf {x'}$ , respectively.

Then one has $\mathbf {x} = \mathbf {x}'$ 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, §5]

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

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

2. (2) $\sigma $ is shift-period preserving in X: for any periodic biinfinite word ${w^{\pm \infty } = \cdots w w w \cdots \in X}$ , the word w can be written as a proper power (see equation (2.6)) if and only if $\sigma (w)$ can be written as a proper power. ${}^{}$

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

Lemma 3.7. Let $\sigma _1: \mathcal A^* \to \mathcal B^*$ and $\sigma _2: \mathcal B^* \to \mathcal C^*$ be two non-erasing morphisms, and consider a subshift $X \subseteq \mathcal A^{\mathbb {Z}}$ as well as its image subshift $Y = \sigma _1(X) \subseteq \mathcal B^{\mathbb {Z}}$ . Then we have the following properties.

1. (1) The composed morphism $\sigma _2 \circ \sigma _1: \mathcal A^* \to \mathcal C^*$ 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 $\sigma _2 \circ \sigma _1: \mathcal A^* \to \mathcal C^*$ 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.

3.3. Injectivity of the measure transfer for recognizable morphisms

Let $\sigma : \mathcal A^* \to \mathcal B^*$ be a non-erasing morphism of free monoids, and let $\pi _\sigma : \mathcal A^* \to \mathcal A_\sigma ^*$ and $\alpha _\sigma : \mathcal A_\sigma ^* \to \mathcal B^*$ 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 $X \subseteq \mathcal A^{\mathbb {Z}}$ , we consider the image subshift $\pi _\sigma (X) \subseteq \mathcal A_\sigma ^{\mathbb {Z}}$ and the induced restriction/co-restriction

$$ \begin{align*}\alpha_\sigma^X: \pi_\sigma(X) \to \sigma(X)\end{align*} $$

of the map $\alpha _\sigma ^{\mathbb {Z}}: \mathcal A_\sigma ^{\mathbb {Z}} \to \mathcal B^{\mathbb {Z}}$ to $\pi _\sigma (X)$ and $\sigma (X)$ , respectively.

Proposition 3.8. For any non-erasing morphism $\sigma : \mathcal A^* \to \mathcal B^*$ and any subshift ${X \subseteq \mathcal A^{\mathbb {Z}}}$ , 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 [9, Lemma 3.5], since every subdivision morphism $\pi _\sigma $ is recognizable in the full shift, as follows directly from the definition of $\pi _\sigma $ .

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 $x \in \mathcal A_\sigma ,$ in the hypothesis (*) of Definition 3.5, the integers k $\ell $ are necessarily equal to 0. However, in this case, the conclusion $\mathbf {x} = \mathbf {x}'$ 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 [5, Lemma 5.3]).

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 to derive the following corollary. In the meantime, we have obtained a result which is actually a bit stronger: it turns out (see [5, Theorem 5.5]) 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.

We can now derive Proposition 1.3 from §1, restated here for the convenience of the reader.

Corollary 3.9. For any non-erasing morphism $\sigma : \mathcal A^* \to \mathcal B^*$ and any subshift ${X \subseteq \mathcal A^{\mathbb {Z}}}$ , the measure transfer map $\sigma _X^{\mathcal {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 $\sigma _X^{\mathcal {M}} = (\alpha _\sigma ^X)^{\mathcal {M}} \circ \pi _\sigma ^{\mathcal {M}}$ . The injectivity of $\pi _\sigma ^{\mathcal {M}}$ is immediate from the definition of a subdivision morphism (see [5, Lemma 5.4]), and the injectivity of $(\alpha _\sigma ^X)^{\mathcal {M}}$ is a direct consequence of Proposition 3.8(2).

Remark 3.10. Consider any non-erasing morphism $\sigma : \mathcal A^* \to \mathcal B^*$ and any subshift ${X \subseteq \mathcal A^{\mathbb {Z}}}$ with image subshift $Y = \sigma (X) \subseteq \mathcal B^{\mathbb {Z}}$ .

1. (1) Assume that the subshift Y contains a periodic word $w^{\pm \infty }$ for some $w \in \mathcal B^* \smallsetminus \{\varepsilon \}$ , and that the morphism $\sigma $ is shift-orbit injective. Then, for $\sigma $ to be shift-period preserving in X, a necessary condition is that at least one of the letters $a_i \in \mathcal A$ 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. (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 $\mathbf { y}$ is not a periodic word.

3. (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 $ . However, 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 measure $\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 $\mu '$ on the subshift Y generated by this new sequence. The main goal of this section is to show that the new measure $\mu '$ is precisely the image of given measure $\mu $ under the transfer map $\tau ^{\mathcal {M}}$ 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 $\tau : \mathcal A^* \to \mathcal B^*$ be a non-erasing morphism of free monoids over finite alphabets $\mathcal A$ and $\mathcal B$ , and let $\overleftarrow \sigma = (\sigma _n)_{n \geq 0}$ be an everywhere growing directive sequence with base level alphabet $\mathcal A_0 = \mathcal A$ . Let $X := X_{\tiny \overleftarrow \sigma } \subseteq \mathcal A^{\mathbb {Z}}$ 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 2.2).

Remark 4.2.

1. (1) For any morphism $\tau $ and any directive sequence $\overleftarrow \sigma = (\sigma _n)_{n \geq 0}$ as in Assumption 4.1, with subshifts X and Y as defined there, there is a second ‘prolonged’ directive sequence $\overleftarrow \sigma ^\tau = (\sigma ^{\prime }_n)_{n \geq 0}$ , given by setting $\sigma ^{\prime }_n := \sigma _{n-1}$ for any level $n \geq 1$ and $\sigma ^{\prime }_0 := \tau $ . We observe from Lemma 2.3 that the subshift $X_{\tiny \overleftarrow \sigma ^\tau }$ generated by $\overleftarrow \sigma ^\tau $ agrees precisely with the $\tau $ -image subshift $Y = \tau (X) \in \mathcal B^{\mathbb {Z}}$ .

2. (2) Consider now any vector tower $\overleftarrow v = (\vec v_n)_{n \geq 0}$ over $\overleftarrow \sigma $ , and let $\mu = \frak m_{\tiny \overleftarrow \sigma }(\overleftarrow v)$ be the invariant measure on X associated to $\overleftarrow v$ via Theorem 2.10. There is a second ‘prolonged’ vector tower $\overleftarrow v^\tau = (\vec v\, ^{\prime }_{\! n})_{n \geq 0}$ over $\overleftarrow \sigma ^\tau $ , given by setting $\vec v\, ^{\prime }_{\! n} = \vec v_{n-1}$ for any level $n \geq 1$ and by setting $\vec v\, ^{\prime }_{\! 0} := M(\tau ) \cdot \vec v_0.$ We denote by $\mu '$ be the measure on Y associated to $\overleftarrow v^\tau = (\vec v\, ^{\prime }_{\! n})_{n \geq 0},$ that is,

   (4.1) $$ \begin{align} \mu' = \frak m_{\tiny\overleftarrow \sigma^\tau}(\overleftarrow v^\tau). \end{align} $$

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 $\mu = \frak m_{\tiny \overleftarrow \sigma }(\overleftarrow v)$ on X. Let ${\overleftarrow \sigma ^\tau = (\sigma ^{\prime }_n)_{n \geq 0},}$ $\overleftarrow v^\tau = (\vec v\, ^{\prime }_{\! n})_{n \geq 0}$ , and $\mu ' = \frak m_{\tiny \overleftarrow \sigma ^\tau }(\overleftarrow v^\tau )$ be as in Remark 4.2.

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

$$ \begin{align*}\mu' = \tau^{\mathcal{M}}(\mu) \,\,\, [= \mu^\tau].\end{align*} $$

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

For any word $w' \in \mathcal B^*$ , we consider in the subdivision monoid $\mathcal A_\tau ^*$ the subset $W(w')$ of preimages $w_i$ of $w'$ under the induced letter-to-letter morphism $\alpha _\tau : \mathcal A_\tau ^* \to \mathcal B^*$ . For each $w_i \in W(w')$ , consider (as in the paragraph subsequent to Fact 3.1) the word $\widehat w_i \in \mathcal A^*$ 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 $\mathcal A^*$ which satisfy condition (a). Recall from equation (3.1) that either $\widehat w_i$ exists and is unique, or else we formally set $\mu (\widehat w_i) = 0$ for any $\mu \in \mathcal M(\mathcal A^{\mathbb {Z}})$ . From Theorem 3.2(2), we know that $\mu ^\tau (w') = {\sum _{w_i \in W(w')}} \mu (\widehat w_i)\,$ (with $\mu ^\tau = \tau ^{\mathcal {M}}(\mu )$ as before).

For the purpose of using equation (2.21), we consider now the value of the approximating sum on the 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 $\overleftarrow v$ , that is, the term (see equation (2.19) for the notation)

(4.2) $$ \begin{align} \sum_{a \,\in \mathcal A_{n-1}} \vec v_{n-1}(a) \, |\sigma_{[0,n-1)}(a)|_{\widehat w_i}. \end{align} $$

We sum up the results of equation (4.2) over all $w_i \in W(w')$ to get

(4.3) $$ \begin{align} \sum_{w_i \in W(w')}\sum_{a \,\in \mathcal A_{n-1}} \vec v_{n-1}(a) \, |\sigma_{[0,n-1)}(a)|_{\widehat w_i}, \end{align} $$

and compare the obtained sum to the limit on the right-hand side of equation (2.21), when this formula is applied to $w'$ and to the vector tower $\overleftarrow v^\tau $ over the prolonged directive sequence $\overleftarrow \sigma ^\tau $ . The nth term of this limit gives the sum

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

When comparing the two sums in equations (4.3) and (4.4), we keep in mind that according to the set-up from Remark 4.2 for any $n \geq 1$ , we have $\vec v\,^{\prime }_{\! n}(a) = \vec v_{n-1}(a)$ for any $a \in \mathcal A_{n-1},$ 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 $a \in \mathcal A_{n-1}$ defines precisely an occurrence of $w_i$ in $\pi _\tau (\sigma _{[0, n-1)}(a))$ , and thus an occurrence of $w'$ 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 $w'$ 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) $$ \begin{align} \sum_{w_i \in W(w')} \, \sum_{a \,\in \mathcal A_{n-1}} \vec v_{n-1}(a) \, |\sigma_{[0,n-1)}(a)|_{\widehat w_i} \leq \sum_{a \,\in \mathcal A_{n-1}} \vec v\, ^{\prime}_n(a) \, |\sigma^{\prime}_{[0,n)}(a)|_{w'}. \end{align} $$

However, 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) $$ \begin{align} \sum_{a \,\in \mathcal A_{n-1}} \vec v\, ^{\prime}_n(a) \, |\sigma^{\prime}_{[0,n)}(a)|_{w'} \leq \sum_{w_i \in W(w')} \, \sum_{a \,\in \mathcal A_{n-1}} \vec v_{n-1}(a) \, |\sigma_{[0,n-1)}(a)|_{\widehat w_i} + K_n. \end{align} $$

Indeed, any occurrence of $w'$ 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 $m(w') \geq 0$ of all $|\widehat w_i|$ . We hence deduce:

$$ \begin{align*} K_n \leq 2 m(w') \sum_{a \,\in \mathcal A_{n-1}} \vec v_{n-1}(a). \end{align*} $$

It follows now from Lemma 2.12 that the right-hand side of the last inequality tends to 0 for $n \to \infty $ , so that we obtain from equations (4.5) and (4.6) through the above definitions $\mu = \frak m_{\tiny \overleftarrow \sigma }(\overleftarrow v) = \mu ^{\tiny \overleftarrow v}$ and $\mu ' = \frak m_{\tiny \overleftarrow \sigma ^\tau }(\overleftarrow v^\tau ) = \mu ^{\tiny \overleftarrow v^\tau }$ the desired result

$$ \begin{align*} \mu^\tau(w') = \sum_{w_i \in W(w')} \bigg( \lim_{n \to \infty} \sum_{a \,\in \mathcal A_{n-1}} \vec v_{n-1}(a) \, |\sigma_{[0,n-1)}(a)|_{\widehat w_i} \bigg)\end{align*} $$

$$ \begin{align*}= \lim_{n \to \infty}\sum_{a \,\in \mathcal A_{n-1}} \vec v\,^{\prime}_n(a) \, |\sigma^{\prime}_{[0,n)}(a)|_{w'} = \mu'(w') \end{align*} $$

for any $w' \in \mathcal B^*$ .

As a first application of the above shown ‘basic’ Proposition 4.3, we derive the following proposition.

Proposition 4.4. For any non-erasing morphism $\sigma :\mathcal A^* \to \mathcal B^*$ and any subshift $X \subseteq \mathcal A^{\mathbb {Z}}$ with image subshift $\sigma (X) $ , the induced measure transfer map

$$ \begin{align*}\sigma^{\mathcal{M}}: \mathcal M(\mathcal A^{\mathbb{Z}}) \to \mathcal M(\mathcal B^{\mathbb{Z}}), \,\, \mu \mapsto \mu^\sigma\end{align*} $$

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

Proof. We consider any everywhere growing directive sequence $\overleftarrow \sigma $ which generates X; from Fact 2.4, we know that such $\overleftarrow \sigma $ exists for any subshift X. By prolonging $\overleftarrow \sigma $ through the morphism $\sigma $ , as explained above in Remark 4.2, we obtain any everywhere growing directive sequence $\overleftarrow \sigma ' := \overleftarrow \sigma ^\sigma $ which generates $\sigma (X)$ . We then apply Theorem 2.10 to obtain for any measure $\mu ' \in \mathcal M(\sigma (X))$ , a vector tower $\overleftarrow v'$ on $\overleftarrow \sigma '$ with $\frak m_{\tiny \overleftarrow \sigma '}(\overleftarrow v') = \mu $ . Truncating now the last term of $\overleftarrow v'$ gives a vector tower $\overleftarrow v$ on $\overleftarrow \sigma $ , which by Remark 2.9(2) defines a measure $\mu := \frak m_{\tiny \overleftarrow \sigma }(\overleftarrow v)$ on X. We can now apply Proposition 4.3 to obtain $\mu ' = \mu ^\sigma \, [= \sigma ^{\mathcal {M}}(\mu )]$ .

Remark 4.5. We would like to note that as a result of the material presented in this section, we have now derived an alternative way of how to understand the transferred measure $\mu ^\sigma = \sigma ^{\mathcal {M}}(\mu ) \in \mathcal M(\mathcal B^{\mathbb {Z}})$ , for any non-erasing morphism $\sigma : \mathcal A^* \to \mathcal B^*$ and any invariant measure $\mu \in \mathcal M(\mathcal A^{\mathbb {Z}})$ .

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 with the definition as studied in [5, §§3 and 4], and also compared with the approximation method via weighted characteristic measures indicated in [5, Remark 3.9].

5. Measure towers and vector towers

Throughout this section, we will assume that

$$ \begin{align*}\overleftarrow \sigma = (\sigma_n: \mathcal A_{n+1}^* \to \mathcal A_n^*)_{n \geq 0}\end{align*} $$

is an everywhere growing directive sequence which generates a subshift $X = X_0 \subseteq \mathcal A_0^{\mathbb {Z}}$ (and where all level maps $\sigma _n$ are non-erasing, see Remark 2.5). As in equation (2.15), we denote, for any level $k \geq 0$ , by $X_k \subseteq \mathcal A_k^{\mathbb {Z}}$ the intermediate subshift of level k, which is generated by the truncated sequence $ \overleftarrow \sigma \dagger _{\! k} = (\sigma _n: \mathcal A_{n+1}^* \to \mathcal A_n^*)_{n \geq k}\, $ from equation (2.14).

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 $\mu _n \in \mathcal M(\mathcal A_n^{\mathbb {Z}})$ which satisfy

$$ \begin{align*}\mu_n = \sigma_n^{\mathcal M}(\mu_{n+1}).\end{align*} $$

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

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’ $\frak m_k := \frak m_{\tiny \overleftarrow \sigma \dagger _{\! k}}: \mathcal V( \overleftarrow \sigma \dagger _{\! k}) \to \mathcal M(X_k)$ . 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) $$ \begin{align} \mu_k := \frak m_k( \overleftarrow v\dagger_{\! k}) \end{align} $$

on the subshift $X_k \subseteq \mathcal A_k^{\mathbb {Z}}$ .

Lemma 5.2.

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

2. (2) 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 equations (2.9) and (2.10).

Proof. (1) It suffices to observe that Proposition 4.3 gives directly $\sigma _{k}^{\mathcal {M}}(\mu _{k+1}) = \mu _k$ for all $k \geq 0$ .

(2) We only need to verify that $\overleftarrow \zeta (\overleftarrow \mu )$ is indeed a vector tower, that is, that compatibility conditions in equation (2.20) are satisfied. This is a direct application of of [5, Proposition 4.5], stated above as equation (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 the following result, which has been quoted in a notationally adapted version in §1 as Proposition 1.1.

Proposition 5.3. For any everywhere growing directive sequence $\overleftarrow \sigma $ , there is a canonical $\mathbb R_{\geq 0}$ -linear bijection

$$ \begin{align*}\overleftarrow \zeta: \mathcal M(\overleftarrow \sigma) \to \mathcal V(\overleftarrow \sigma)\end{align*} $$

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

$$ \begin{align*} \overleftarrow \mu \mapsto \overleftarrow \zeta(\overleftarrow \mu) \quad\text{and its inverse} \quad \overleftarrow v \mapsto \overleftarrow{\frak m}(\overleftarrow v) .\end{align*} $$

Proof. The fact that the composition $\overleftarrow \zeta \circ \overleftarrow {\frak m} $ gives the identity on $\mathcal V(\overleftarrow \sigma )$ follows directly from Proposition 2.11(1), when applied to all truncated sequences $\overleftarrow \sigma \dagger _{\! k}$ with $k \geq 0$ . We obtain in particular that the map $\overleftarrow {\frak m}$ is injective.

However, we can apply Theorem 2.10 to each of the truncated sequences $ \overleftarrow \sigma \dagger _{\! k}$ to obtain the surjectivity of the map $\frak m_k: \mathcal V( \overleftarrow \sigma \dagger _{\! k}) \to \mathcal M(X_k)$ for any level $k \geq 0$ . It follows then directly from the definition set up in Lemma 5.2 (1) above that the map $\overleftarrow {\frak m}: \mathcal V(\overleftarrow \sigma ) \mapsto \mathcal M(\overleftarrow \sigma )$ must be surjective.

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

The linearity of the maps $\overleftarrow \zeta $ and $\overleftarrow {\frak m}$ is a direct consequence of the linearity (see §2.2) of the maps $\frak m_k$ and $\zeta _{X_n}$ used in the above definitions of the measure or vector towers $\overleftarrow {\frak m}(\overleftarrow v) = (\frak m_k( \overleftarrow v\dagger _{\! k}))_{k \geq 0}$ and $\overleftarrow \zeta (\overleftarrow \mu ) = (\zeta _{X_n}(\mu _n))_{n \geq 0}$ , respectively.

Although slightly similar in notation, the two cones $\mathcal M(\overleftarrow \sigma )$ and $\mathcal M(X_{\tiny \overleftarrow \sigma })$ should not be confused. Indeed, without further assumptions on the given set-up, the structure of the cone $\mathcal M(\overleftarrow \sigma )$ 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 the following remark.

Remark 5.4. For any everywhere growing directive sequence $\overleftarrow \sigma $ which generates a subshift $X = X_{\tiny \overleftarrow \sigma }$ , the composition

(5.2) $$ \begin{align} \frak m_{\tiny \overleftarrow \sigma} \circ \overleftarrow \zeta : \mathcal M(\overleftarrow \sigma) \to \mathcal M(X) \end{align} $$

is $\mathbb R_{\geq 0}$ -linear and surjective since $\overleftarrow \zeta $ is $\mathbb R_{\geq 0}$ -linear and bijective by Proposition 5.3, and $\frak m_{\tiny \overleftarrow \sigma }$ is $\mathbb R_{\geq 0}$ -linear and surjective by Theorem 2.10. However, in general, the map $\frak 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 [9, Definition 4.1] or [13, §3.3]).

Definition 5.5. 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.

The following result is quoted in a slightly shortened version in §1 as Theorem 1.4.

Theorem 5.6. For any everywhere growing totally recognizable S-adic development $\overleftarrow \sigma $ of a subshift X and its associated cone $\mathcal V(\overleftarrow \sigma )$ of vector towers, the canonical $\mathbb R_{\geq 0}$ -linear map

$$ \begin{align*}\frak m_{\tiny \overleftarrow \sigma}: \mathcal V(\overleftarrow \sigma) \to \mathcal M(X)\end{align*} $$

is a bijection.

In particular, for any level $n \geq 0$ , the map $\sigma _{[0, n)}^{\mathcal {M}}: \mathcal M(X_n) \to \mathcal M(X)$ is a linear bijection of cones. Similarly, the same conclusion follows for the map $\frak m_{\tiny \overleftarrow \sigma } \circ \overleftarrow \zeta $ from equation (5.2).

Proof. From the assumption that $\overleftarrow \sigma $ is totally recognizable, it follows (using statement (3d) of Theorem 3.2) that the induced $\mathbb R_{\geq 0}$ -linear map

$$ \begin{align*}(\sigma_n)^{\mathcal{M}}_{X_{n+1}}: \mathcal M(X_{n+1}) \to \mathcal M(X_{n})\end{align*} $$

is bijective for any level $n \geq 0$ . It follows that the composed map $m_{\tiny \overleftarrow \sigma } \circ \overleftarrow \zeta : \mathcal M(\overleftarrow \sigma ) \to \mathcal V(\overleftarrow \sigma ) \to \mathcal M(X)$ from equation (5.2) is bijective. Since we know from Proposition 5.3 that the map $\overleftarrow \zeta $ is a bijection, we deduce that $m_{\tiny \overleftarrow \sigma }$ must be bijective.

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. However, 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 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, that is, $X \in \mathcal A^{\mathbb {Z}}$ is a subshift over the finite alphabet $\mathcal A = \mathcal A_0$ , and $\overleftarrow \sigma = (\sigma _n: \mathcal A_{n+1}^* \to \mathcal A_n^*)_{n \geq 0}$ is an everywhere growing directive sequence which generates X. For the subsequent discussion, the following notion will be helpful.

Definition 6.1.

1. (1) For any integer $n \geq 0$ , the intermediate letter frequency cone ${\mathcal C_n = \mathcal C_n(\overleftarrow \sigma ) \subseteq \mathbb R_{\geq 0}^{\mathcal A_n}}$ of the directive sequence $\overleftarrow \sigma $ is given, through the level subshifts $X_n$ (see equation (2.15)) and their measure cones $\mathcal M(X_n)$ , via equation (2.11) by

   $$ \begin{align*}\mathcal C_n := \mathcal C(X_n) = \zeta_{X_n}(\mathcal M(X_n)).\end{align*} $$
   As set up in §2, here $\zeta _{X_n}: \mathcal M(X_n) \to \mathbb R_{\geq 0}^{\mathcal A_n}$ is given for $\mathcal A_n = \{a_{n, 1}, \ldots , a_{n, d(n)}\}$ by $\mu \mapsto ([\mu (a_{n, 1})], \ldots , [\mu (a_{n, d(n)})])$ for any $\mu \in \mathcal M(X_n)$ .

2. (2) We denote the dimension of the cone $\mathcal C_n$ by $c_n$ , that is,

   $$ \begin{align*}c_n := \dim \mathcal C_n \leq \mathrm{card}(\mathcal A_n).\end{align*} $$

Remark 6.2. From Proposition 2.11, applied to the truncated directive sequence $\overleftarrow \sigma \dagger _n$ , we observe directly that the cone $\mathcal C_n$ is the image of the set $\mathcal V(\overleftarrow \sigma )$ of vector towers under the level n projection map $pr_n,$ which amounts to stating that $\mathcal C_n$ is the intersection of the nested images of the non-negative alphabet cones of level $m \geq n$ under the telescoped level maps:

$$ \begin{align*}\mathcal C_n = \bigcap\, \{ \mathbb R_{\geq 0}^{\mathcal A_n}\supseteq \cdots \supseteq M(\sigma_{[n, m)})(\mathbb R_{\geq 0}^{\mathcal A_{m+1}}) \supseteq \cdots\}.\end{align*} $$

In particular, one always has

(6.1) $$ \begin{align} \mathcal C_n = M(\sigma_n) ( \mathcal C_{n+1}) \end{align} $$

and thus

$$ \begin{align*}c_n \leq c_{n+1}\end{align*} $$

for all $n \geq 0$ .

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

Remark 6.3. 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

   $$ \begin{align*}e(X) \leq \lim c_n \leq \liminf (\mathrm{card}\,\mathcal A_n),\end{align*} $$
   but in general, both inequalities may well be strict.

4. (4) However, if $\overleftarrow \sigma $ is totally recognizable, then we have

   $$ \begin{align*}e(X) = \lim c_n.\end{align*} $$
   In particular, we recover the well-known upper bound $e(X) \leq \liminf (\mathrm{card}\,\mathcal A_n)$ , as well as the lower bounds $c_n \leq e(X)$ for all $n \geq 0$ .

From Remark 6.3(3), we observe directly that for any directive sequence $\overleftarrow \sigma $ with finite alphabet rank (that is, $\liminf (\mathrm{card}\ \mathcal A_n) < \infty $ ), there is a critical level $n_0 \geq 0$ such that one has

(6.2) $$ \begin{align} c_n = c_{n_0}\quad\ \text{for all}\ n \geq n_0 \ \text{and}\ c_n < c_{n_0}\ \text{for all}\ n < n_0. \end{align} $$

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 6.4. [3]

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

$$ \begin{align*} \mathcal M(X) \to C(X) , \mu \mapsto (\mu(a_k))_{a_k \in \mathcal A}. \end{align*} $$

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 $a_k \in \mathcal A$ .

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

Proof of Proposition 6.4

For any two measure $\mu , \mu ' \in \mathcal M(X)$ , there exist, by Theorem 2.10, vector towers $\overleftarrow v = (\vec v_n)_{n \geq 0}$ and $\overleftarrow v' = (\vec v\, ^{\prime }_{\! n})_{n \geq 0}$ on $\overleftarrow \sigma $ with $\frak m_{\tiny \overleftarrow \sigma }(\overleftarrow v) = \mu $ and $\frak m_{\tiny \overleftarrow \sigma }(\overleftarrow v') = \mu '$ . Thus, $\mu \neq \mu '$ implies $\overleftarrow v \neq \overleftarrow v'$ and hence $\vec v_n \neq \vec v\, ^{\prime }_{\! n}$ for some $n \geq 0$ . However, then we deduce from equation (6.1) and the hypothesis that $\dim \mathcal C(X_n) = c_n = c_0 = \dim \mathcal C(X)$ 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 2.11(1), we know that $\vec v_0 = pr_0(\overleftarrow v) = \zeta _X(\mu )$ and $\vec v\, ^{\prime }_{\! 0} = pr_0(\overleftarrow v') = \zeta _X(\mu ')$ , which shows that the map $\zeta _X$ is injective. For the linearity of $\zeta _X$ and the equality $\zeta _X(\mathcal M(X)) = \mathcal C(X)$ , see equations (2.10) and (2.11).

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 Corollary 6.5 is rather frequently satisfied. This corollary has been quoted as Corollary 1.6 in §1; it is stated here again for the convenience of the reader.

Corollary 6.5. Let $X \subseteq \mathcal A^{\mathbb {Z}}$ 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, that is, by the values $\mu ([a_k])$ for all $a_k \in \mathcal A$ .

Proof. From equation (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 6.4 are satisfied, which gives directly the claimed statement.

Note that the conclusion of Corollary 6.5 has recently been proved by Berthé et al under somewhat more restrictive hypotheses (see [6, Corollary 4.2]); 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 6.6.

1. (1) If in Proposition 6.4 the hypothesis ‘thin’ is replaced by ‘thinning’, with critical level $n_0 \geq 1$ , then the conclusion that any two distinct measures $\mu \neq \mu ' \in \mathcal M(X)$ can be distinguished by the evaluation on the letter cylinders $[a_k]$ for all $a_k \in \mathcal A$ , may in some cases still hold, despite the fact that from the definition of the critical level, we have

   $$ \begin{align*}\dim \mathcal C_0 = c_0 < c_{n_0} = \dim \mathcal C_{n_0} = \dim \mathcal M(X_{n_0}).\end{align*} $$
   Here the last equality follows from Proposition 6.4, applied to the directive sequence truncated at the critical level $n_0$ . The reason why the above strict inequality does not contradict the presumed equality $c_0 = \dim \mathcal C_0 = \dim \mathcal M(X)$ is that the measure transfer map $\sigma _{[0, n_0)}^{\mathcal {M}}: \mathcal M(X_{n_0}) \to \mathcal M(X)$ 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 6.4 necessarily fails: this case is treated in Example 6.7 below.

2. (2) In view of the fact that the measure transfer map $\sigma ^{\mathcal {M}}$ induced by a non-recognizable monoid morphism $\sigma $ is in general far from being injective, it seems noteworthy that in Proposition 6.4 and Corollary 6.5, 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’, that is, all level maps above some ‘other critical level’ must be recognizable in their level subshift. However, 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 6.7. The subshift X in this example consists of two periodic words and is hence all by itself not so interesting. We chose it 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 build 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 $\mathcal M(X_n)$ and of the associated letter frequency cone $\mathcal C_n$ .

Set $\mathcal A_1 = \{a, b\}$ and let $X_1 \subseteq \mathcal A_1^{\mathbb {Z}}$ be the union of the two periodic subshifts $\mathcal O(w^{\pm \infty })$ and $\mathcal O({w'}^{\pm \infty })$ , defined by the words $w = a^2 b$ and $w' = b^2 a$ . We consider the two characteristic measures $\mu := \mu _{w}$ and $\mu ' := \mu _{w'},$ and observe that $\mathcal M(X_1)$ consists of all non-negative linear combinations of these two measures. The letter frequency map $\zeta _{X_1}: \mathcal M(X_1) \to \mathcal C(X_1) \subseteq \mathbb R_{\geq 0}^{\{a, b\}}$ is injective, in that $\zeta _{X_1}(\mu ) = 2 \vec e_a + \vec e_b$ and $\zeta _{X_1}(\mu ') = \vec e_a + 2 \vec e_b$ . This results in $c_1 = \dim (\mathcal C_1) = 2$ .

For $\mathcal A_0 = \{c, d\}$ , consider now the ‘Thue–Morse’ morphism $\sigma _0 : \mathcal A_1^{\mathbb {Z}} \to \mathcal A_0^{\mathbb {Z}}, \,\, a \mapsto cd, \,\, b \mapsto dc$ , and recall (see Proposition 3.3(d)) that $\sigma _0^{\mathcal {M}}(\mu ) = \mu _{\sigma _0(w)}$ and $\sigma _0^{\mathcal {M}}(\mu ') = \mu _{\sigma _0(w')},$ with $\sigma _0(w) = cdcddc$ and $\sigma _0(w') = dcdccd$ . Since $cdcddc$ and $dcdccd$ cannot be obtained from each other by a cyclic permutation, we have $\mathcal O((cdcddc)^{\pm \infty }) \neq \mathcal O((dcdccd)^{\pm \infty })$ , so that from equation (2.8), 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 $\mathcal M(X_0)$ , which is spanned by $\mu _{cdcddc}$ and $\mu _{dcdccd},$ is of dimension $2$ .

However, using Proposition 3.4 (or more directly, equation (2.7)), 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 $\mathcal C_0$ has dimension $c_0 = 1$ .

We now define the higher up levels of the directive sequence by setting $\mathcal A_n = \{x, y\}$ for any $n \geq 2$ , and by defining all level morphisms $\sigma _n: \mathcal A_{n+1} \to \mathcal A_n$ 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 two times the two-by-two identity matrix $I_2,$ that is, $M(\sigma _n) = 2 \cdot I_2,$ so that we have $\mathcal M(X_n) = \mathcal C_n = \mathbb R_{\geq 0}^{\{x, y\}}$ .

It remains now to define $\sigma _1: \mathcal A_2 \to \mathcal A_1$ via $x \mapsto w$ , $y \mapsto w'$ , which ensures $\sigma _1(X_2) = X_1$ , 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 $\mathcal M(X_0)$ .

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 equation (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 work involved 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 [7, Theorem 4.3] and is attributed there to Thierry Monteil; alternatively, it can be found in [8] as Lemma 6.7.1 of Ch. 6, written by Fabien Durand, who told us that the result can actually be traced back to the paper [10] by Boyle and 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 $\mathcal A_n$ . Then, for the minimal level letter image length $\beta _-(n)$ from equation (2.17), one has

$$ \begin{align*}h_X \leq \inf_{n \geq 0} \frac{\log (\mathrm{{card}} \,\mathcal A_n)}{\beta_-(n)}.\end{align*} $$

Proposition 7.2. [4, §4.1]

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 $\mathcal A_{(d)} = \{a_1, \ldots , a_d\}$ ). Assume that for any level $n \geq 0$ , the incidence matrix of the level map $\sigma _{n,d}$ is given by

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

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 below in our proof of Theorem 7.4. Proposition 7.2, however, will not be formally used in the following, 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 [4, §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 ${\mathcal A_{(d)} = \{a_1, \ldots , a_d\}}$ , the morphism $\tau _{d}: \mathcal A_{({d+1})}^* \to \mathcal A_{(d)}^*,$ 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. (1) For the morphism $\tau _d$ as given above, it is easy to see that any biinfinite word $\mathbf {y} \in \mathcal A_{(d)}^{\mathbb {Z}} \smallsetminus \{a_1^{\pm \infty }, a_2^{\pm \infty }, \ldots , a_d^{\pm \infty }\}$ can be ‘desubstituted’ in at most one way (compare [5, Remark 6.2(2)]) to give a biinfinite word $\mathbf {x} \in \mathcal A_{({d+1})}^{\mathbb {Z}}$ with $\tau _d(\mathcal O(\mathbf {x})) = \mathcal O(\mathbf {y}).$ Since for any $i = 1, \ldots , n$ , the periodic word $a_i^{\pm \infty }$ is the only element $\mathbf {x} \in \mathcal A_{(d+1)}^{\mathbb {Z}}$ with $\tau _d(\mathcal O(\mathbf {x})) = \mathcal O(a_i^{\pm \infty })$ , it follows that $\tau _d$ is recognizable in every subshift which does not contain any of the periodic words $a_i^{\pm \infty }$ .

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

   (Indeed, it suffices to check in any biinfinite word $\mathbf {y} \in \sigma _{n,d}(\mathcal A_{(d)}^{\mathbb {Z}})$ for a factor ${w \in \mathcal A_{(d)}^*}$ which is ‘distinguished’ in that some letter $a_i \in \mathcal A_{(d)}$ occurs precisely three times in w, while all other letters $a_j \in \mathcal A_{(d)}$ 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 $\mathcal A_{(d)}^*$ 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 $\mathbf {y} = \cdots y_{n-1} y_{n} y_{n+1} \cdots $ , one considers the factors $w_+ = y_s \ldots y_{t'}$ and $w_- = y_{r'} \ldots y_s$ of $\mathbf {y}$ , with $y_s = y_{s+1} = \cdots = y_{t'-1} = a_i$ and $y_{t'} \neq a_i$ , and similarly $y_s = y_{s-1} = \cdots = y_{r'+1} = a_i$ and $y_{r'} \neq a_i$ . 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 $\mathbf {y}$ can be desubstituted in precisely one way.)

3. (3) From the conditions on the incidence matrix $M(\sigma _n)$ in equation (7.1), it follows directly that every word in $\sigma _{n, d}(\mathcal A_{(d)}^*)$ must contain each of the letters of $\mathcal A_{(d)}$ . Hence, we observe that $\sigma _{n, d}(\mathcal A_{(d)}^{\mathbb {Z}})$ cannot contain any of the periodic words $a_i^{\pm \infty }$ .

4. (4) As a consequence of the above observations (1)–(3), we deduce for the following ‘alternating’ directive sequence

   (7.2) $$ \begin{align} \overleftarrow \sigma = \sigma_{2} \circ \tau_2 \circ \sigma_{3} \circ \tau_3 \circ \cdots\!\,, \end{align} $$

   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 $\mathcal A_{(d)} = \{a_1, \ldots , a_d\}$ and let $\sigma _d: \mathcal A_{(d)}^* \to \mathcal A_{(d)}^*$ be a morphism with incidence matrix $M(\sigma _d) = M_{\ell (d), d}$ from equation (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 equation (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 $\frak X$ the class of all subshifts $X \subseteq \mathcal A_{(d)}^{\mathbb {Z}}$ which satisfy all of the above conditions.)

Proof. For each integer $d \geq 2$ , we identify the finite alphabet $\mathcal A_{(d)} = \{a_1, a_2, \ldots , a_d\}$ with the corresponding subset of an infinite alphabet, via $\mathcal A_{(2)} \subseteq \mathcal A_{(3)} \subseteq \cdots \subseteq \mathcal A_{(\infty )} = \{a_1, a_2, \ldots \}$ . For the issuing infinite non-negative cone $\mathbb R_{\geq 0}^{\mathcal A_{(\infty )}}$ , 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 $\mathcal C^n := \mathbb R_{\geq 0}^{\mathcal A_{(d)}} \subseteq \mathbb R_{\geq 0}^{\mathcal A_{(\infty )}}$ and, in particular, the ‘center vector’ $\vec c_n = \sum \vec e_i$ of $\mathcal C^n$ . We observe that both families, the morphisms $\sigma _d$ as well as the morphisms $\tau _d,$ induce maps $M(\sigma _d): \mathcal C^n \to \mathcal C^n$ and $M(\tau _d): \mathcal C^{n+1} \to \mathcal C^n$ , 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 $\unicode{x3bb} _1 \vec e_i + \unicode{x3bb} _2 \vec c_d$ . Note here that (again for both $\sigma _d$ and $\tau _d$ ,)

(7.3)

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 \cdots $ with the property that $\overleftarrow v_{\! i}$ has for any level $n \geq n_0$ , a level vector $\vec v^{\, i}_n = \unicode{x3bb} _{1, n} \vec e_i + \unicode{x3bb} _{2,n} \vec c_n$ , with coefficients

(7.4)

(which must both tend to 0 for $n \to \infty $ ). From equation (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 equation (7.4) are satisfied, while the compatibility condition in equation (2.20) 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 $X_{n_0} \subseteq \mathcal A_{(d)}^{\mathbb {Z}},$ generated by the truncated sequence $\overleftarrow \sigma \dagger _{\! n_0},$ the truncated evaluation map $\frak m_{n_0} := \frak m_{\tiny \overleftarrow \sigma \dagger _{\! n_0}}\!\!\!\!: \overleftarrow {\mathcal V}(\overleftarrow \sigma \dagger _{\! n_0}) \to \mathcal M(X_{n_0})$ from equation (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:

$$ \begin{align*}\mu_i = \frak m_{n_0}(\overleftarrow v_{\! i}).\end{align*} $$

It follows from Proposition 2.11(1) that the subcone

$$ \begin{align*}\mathcal M_{n_0} := {\mathbb R_{\geq 0}}\, \langle \mu_1, \ldots, \mu_d\rangle \subseteq \mathcal M(X_{n_0})\end{align*} $$

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 $\mathcal M_{n_0}$ is mapped by $\sigma _2^{\mathcal {M}} \circ \tau _2^{\mathcal {M}} \circ \cdots \circ \sigma _{n_0 -1}^{\mathcal {M}} \circ \tau _{n_0 -1}^{\mathcal {M}}$ to a subcone of $\mathcal M(X)$ that also has dimension d.

We have thus proved that $\mathcal M(X)$ contains subcones of arbitrary large dimension, and hence must be infinite dimensional, that is, $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 2.8(2).

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 four steps, organized below as follows. In §8.1, we present our basic quotient construction in geometric language. In §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 §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 §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. (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. 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 one-to-one relation with the shift-orbits of the resulting set of words $w(\ell )$ . Indeed, for this one-to-one 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 $w_{\mathrm {{above}}}(\ell _P)$ and $w_{\mathrm {{below}}}(\ell _P)$ 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 one-to-one 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 $w_{\mathrm {{above}}}(\ell _P)$ and $w_{\mathrm {{below}}}(\ell _P)$ 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 $w_{\mathrm {{above}}}(\ell _P)$ and $w_{\mathrm {{below}}}(\ell _P)$ , for any line $\ell $ that passes through our distinguished base square Q, is a subshift—indeed, a well-known Sturmian subshift.

2. (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 $h_{\mathrm {{left}}}$ , and any transition through the right half by $h_{\mathrm {{right}}}$ , to give a new biinfinite word $w'(\ell ) \in \{h_{\mathrm {{left}}}, h_{\mathrm {{right}}}, v\}^{\mathbb {Z}}$ . The morphism $\sigma : \{h_{\mathrm {{left}}}, h_{\mathrm {{right}}}, v\}^{\mathbb {Z}} \to \{h, v\}^{\mathbb {Z}}$ defined by $h_{\mathrm {{left}}} \mapsto h, h_{\mathrm {{right}}} \mapsto h$ and $v \mapsto v$ maps any $w'(\ell )$ to $w(\ell )$ , and it will be one-to-one, 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 two words $w^{\prime }_{\mathrm {{above}}}(\ell _R)$ and $w^{\prime }_{\mathrm {{below}}}(\ell _R)$ , and both have the same image word $w(\ell _R)$ . Since $w^{\prime }_{\mathrm {{above}}}(\ell _R)$ and $w^{\prime }_{\mathrm {{below}}}(\ell _R)$ belong as above to distinct shift-orbits, the morphism $\sigma $ is not shift-orbit injective, and hence not 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. (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 $w'(\ell )$ ), 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 four 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 $\partial \tilde h$ -fixed point on $\partial \mathbb H^2$ (where $\tilde h$ is a lift of h to $\mathbb H^2$ and $\partial \tilde h$ is the canonical extension of $\tilde h$ 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 $X \subseteq \mathcal A^{\mathbb {Z}}$ , where $\mathcal A$ 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 $F(\mathcal A)$ .)

3. (H3) Each complementary component has at least two 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 $\Sigma '$ 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 $\Sigma '$ . Then h induces a pseudo-Anosov homeomorphism $h': \Sigma ' \to \Sigma '$ 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 $h'$ , respectively. The map q is one-to-one everywhere, except at points on the leaves $\ell _1$ and $\ell _2$ , which are identified by q to a single leaf $\ell ' \in \Lambda ^{\prime }s$ . The leaf $\ell '$ is fixed and expanded by $h'$ , and the sole $h'$ -fixed point on $\ell '$ 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 $\mathcal F^s$ for h.

Remark 8.2.

1. (1) There is a remarkable feature here in that both $\Lambda ^s$ and $\Lambda ^{\prime }s$ are minimal laminations (that is, 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. (2) This is translated (via the associated IETs as indicated above) into a subshift $X \subseteq \mathcal A^{\mathbb {Z}}$ that is mapped by a morphism $\sigma : \mathcal A^* \to \mathcal A^{\prime }*$ to a subshift $\sigma (X) =: X' \subseteq \mathcal A^{\prime }\mathbb Z$ (for $\mathcal A^{\prime }* \subseteq F(\mathcal A') = \pi _1 \Sigma '$ , in complete analogy to $\mathcal A$ and $\Sigma $ in the above set-up). Here both X and $X'$ are minimal, while the map induced by $\sigma $ on X is not shift-orbit injective, so that $\sigma $ is not recognizable in X.

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

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 $\sigma _0, h_0, \Lambda _0^s, \mathcal A_0$ and $X_0$ are as $\Sigma , h, \Lambda ^s, \mathcal A$ and X in §8.1 above, and that in particular the conditions (H1)–(H3) are satisfied, except that in condition (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) $$ \begin{align} h_0(\ell_0) = \ell_0. \end{align} $$

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 equation (8.1) that $h_0$ induces a homeomorphism $h_1: \Sigma _1 \to \Sigma _1$ . Again from considering the stable foliation $\mathcal F_0^s$ 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) $$ \begin{align} h_0 \circ q_0 = q_0 \circ h_1 , \end{align} $$

and $q_0$ is one-to-one 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 §8.1 when passing from $\Sigma , h$ and $\Lambda ^s$ to $\Sigma ', h'$ 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 $\mathcal A_1$ for $\Lambda _1^s$ derives from $\mathcal A_0$ by doubling one of its letters, namely the one corresponding to the subdivided interval.

For the minimal subshift $X_1 \subseteq \mathcal A_1^{\mathbb {Z}}$ associated to $\Lambda _1$ and the morphism $\sigma _0: \mathcal A_1^* \to \mathcal A_0^*$ 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 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 ${\sigma _1: \mathcal A_2^* \to \mathcal A_1^*}$ that is non-recognizable on the minimal subshift $X_2$ which satisfies $\sigma _1(X_2) = X_1.$

Hence, to be able to repeat this procedure infinitely often, with the purpose to get for any $n \geq 0$ , a morphism $\sigma _{n}: \mathcal A_{n+1}^* \to \mathcal A_{n}^*$ 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 $\overleftarrow \sigma = (\sigma _n: \mathcal A_{n+1}^* \to \mathcal A_{n}^*)_{n \geq 0}$ and subshifts $X_n \subseteq \mathcal A_n^{\mathbb {Z}},$ 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) $\mathrm{card}(\mathcal A_{n+1}) = \mathrm{card}(\mathcal A_{n}) +1$ ;

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 2.6(2)) for some primitive substitution $\tau _n: \mathcal A_n^* \to \mathcal A_n^*;$

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 property (5) is the translation of the homeomorphism $h_n$ into the monoid setting through the canonical embedding $\mathcal A_n^* \subseteq F(\mathcal A_n) = \pi _1 \Sigma _n$ . 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 property (5), and property (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 equation (8.2) together with the above replacement of $h_n$ by $h_n^{t(n)}$ .

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

$$ \begin{align*}\sigma^{\prime}_n: = \tau_n^{t'(n)} \circ \sigma_n \,\,\, ( = \sigma_n \circ \tau_{n+1}^{s(n)}) ,\end{align*} $$

where we set $t'(n) := s(n) \, t(n)$ for some suitably chosen integer $s(n) \geq 1$ which ensures that the incidence matrix $M(\tau _n^{t'(n)})$ 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) $$ \begin{align} \text{the incidence matrix}\ M(\sigma^{\prime}_n)\ \text{is positive for any index}\ n \geq 0. \end{align} $$

We now define a directive sequence $\overleftarrow \sigma ' = (\sigma ^{\prime }_n: \mathcal A_{n+1}^* \to \mathcal A_{n}^*)_{n \geq 0}$ 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) $$ \begin{align} \sigma^{\prime}_n(X_{n+1}) = X_n \end{align} $$

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 Lemma 3.7 that $\sigma ^{\prime }_n$ is not recognizable in $X_{n+1}$ and thus neither in $X^{\prime }_{n+1}$ . From equation (8.3), we obtain directly (see Remark 2.8(1)) that the sequence $\overleftarrow \sigma '$ 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 $|\mathcal A_0| = 2$ , 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 equation (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 §1 and stated there as Proposition 1.7.

Corollary 8.5.

1. (1) The directive sequence $\overleftarrow \sigma '$ 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. (2) For any integer $k \geq 0$ , there exists a directive sequence $\overleftarrow \tau _{\! k}$ , with level alphabets $\mathcal A_n$ of size $\mathrm{card}(\mathcal A_n) = k +2$ for any level $n \geq k$ , and $\mathrm{card}(\mathcal A_n) = n +2$ 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. (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.

Remark 8.6. It turns out that property (3) of Corollary 8.5 is also true for the directive sequence $\overleftarrow \sigma '$ . 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 equation (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, 13]. We note that the examples given in part (2) of Corollary 8.5 above contradict the bound claimed in [13, Theorem 3.7] as stated; to rectify that statement, additional hypotheses would need to be imposed. This error could also effect the upper bound claimed in [17, Corollary 1.5] on the number of successive factor maps, for a large class of subshifts.

In this context, we also want to point to the very recent paper [2, Example 7.5], 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 to our Corollary 8.5 above) has been communicated to us by Espinoza [18] in the final stages of the revision of this paper.