2.1 Measure-preserving and topological systems

A measure-preserving system (or a system for simplicity) is a tuple $(X,\mathcal {X},\mu ,T)$ , where $(X,\mathcal {X},\mu )$ is a probability space and $T\colon X\to X$ is a measurable and measure-preserving transformation. That is, $T^{-1}A\in \mathcal {X}$ and $\mu (T^{-1}A)=\mu (A)$ for all $A\in {\mathcal X}$ . In this paper, we assume without loss of generality that T is invertible, with a measurable and measure-preserving inverse $T^{-1}$ . We will also assume without loss of generality that $(X, {\mathcal X}, \mu )$ is a standard probability space.

We say that $(X,{\mathcal X},\mu ,T)$ is ergodic (or simply that T or $\mu $ is ergodic) if whenever $A \in {\mathcal X}$ verifies that $\mu (A\Delta T^{-1}A)=0$ , then $\mu (A)=0$ or 1. A strictly stronger notion is that of mixing. We say that $(X,{\mathcal X},\mu ,T)$ is mixing if $\mu (A\cap T^{-n}B)\to \mu (A)\mu (B)$ as n goes to infinity for all measurable sets $A,B \in {\mathcal X}$ .

A measure-preserving system $(X,\mathcal {X},\mu ,T)$ is partially rigid if there exists an increasing sequence of positive integers $(n_k)_{k \in {\mathbb N}}$ and a constant $\gamma>0$ such that $\liminf _{k\to \infty } \mu (A \cap T^{-n_k}A) \geq \gamma \mu (A)$ for any measurable set $A \in {\mathcal X}$ . In such a case, we also say that the system, or just $\mu $ , is $\gamma $ -rigid and the sequence $(n_k)_{k \in {\mathbb N}}$ is said to be a partial rigidity sequence. It is not complicated to prove that to state partial rigidity, one only needs to consider sets A in a semi-algebra that generates ${\mathcal X}$ and that any subsequence of a partial rigidity sequence remains a partial rigidity sequence for the same constant $\gamma>0$ .

The case $\gamma =1$ has been extensively studied in the past. In this case, the measure-preserving system $(X,\mathcal {X},\mu ,T)$ is said to be rigid and the limit inferior is actually a limit. By standard density arguments, the system is rigid if and only if for any $f\in L^{2}(\mu )$ , we have that $\|f-f\circ T^{n_k}\|_{2}$ goes to 0 as k goes to infinity for an increasing sequence of positive integers $(n_k)_{k\in {\mathbb N}}$ . In this context, the sequence $(n_k)_{k \in {\mathbb N}}$ is said to be a rigidity sequence for $(X, {\mathcal X}, \mu ,T)$ . Note that for every positive integer c, the sequence $(c n_k)_{k \in {\mathbb N}}$ is also a rigidity sequence for $(X, {\mathcal X}, \mu ,T)$ . For recent developments on the rigidity notion and the sequences one can obtain, we refer to [Reference Bergelson, del Junco, Lemańczyk and Rosenblatt3].

A measure-preserving system is mildly mixing if for every measurable set $A \in {\mathcal X}$ with $\mu (A)> 0$ , $\inf _{n \in {\mathbb N}} \mu (A \Delta T^{-n}A)>0$ . A measure-preserving system $(X, {\mathcal X}, \mu , T)$ is weakly mixing if the product system $(X\times X,{\mathcal X}\otimes {\mathcal X},\mu \times \mu ,T\times T)$ is ergodic. It is well known that mixing implies mildly mixing, mildly mixing implies weakly mixing, and weakly mixing implies ergodicity, and all implications are strict. For example, there are weakly mixing and rigid systems (see [Reference Bergelson, del Junco, Lemańczyk and Rosenblatt3]), and also partially rigid and mildly mixing systems (see [Reference Chacon9]).

Let $(X, {\mathcal X}, \mu ,T)$ and $(Y, {\mathcal Y}, \nu , S)$ be two measure-preserving systems. If there exists a measurable map $\pi \colon X \to Y$ such that $\pi \circ T = S \circ \pi $ and $\mu (\pi ^{-1}A) = \nu (A)$ for all $A \in {\mathcal Y}$ , we say that $\pi $ is a measurable factor map, $(Y, {\mathcal Y}, \nu , S)$ is a measurable factor of $(X, {\mathcal X}, \mu ,T)$ , and $(X, {\mathcal X}, \mu ,T)$ is a measurable extension of $(Y, {\mathcal Y}, \nu , S)$ . When $\pi $ is invertible and $\pi ^{-1}$ is a measurable map, $(X, {\mathcal X}, \mu ,T)$ and $(Y, {\mathcal Y}, \nu , S)$ are said to be measurably isomorphic. We have that ergodicity, partial rigidity, rigidity, and mixing properties are inherited via measurable factor maps.

A topological dynamical system is a pair $(X,T)$ , where X is a compact metric space and T is a self-homeomorphism. The orbit of a point $x\in X$ is the set $\mathrm {orb}(x)=\{T^n x: n\in {\mathbb Z}\}$ . We say that $(X,T)$ is minimal if the orbit of any point is dense in X. A point $x \in X$ is periodic if its orbit is finite and aperiodic otherwise.

Let $(X,T)$ and $(Y,S)$ be two topological dynamical systems. If there is an onto continuous map $\pi \colon X \to Y$ such that $\pi \circ T = S \circ \pi $ , then we say that $\pi $ is a factor map, $(Y,S)$ is a factor of $(X,T)$ , and $(X,T)$ is an extension of $(Y,S)$ . When $\pi $ is a homeomorphism, $(X,T)$ and $(Y,S)$ are said to be topologically conjugate. Remark that minimality is preserved under factor maps.

Given a topological dynamical system $(X,T)$ , we let ${\mathcal M}(X,T)$ (respectively ${\mathcal E}(X,T)$ ) denote the set of Borel T-invariant probability (respectively the set of ergodic probability measures). For any topological dynamical system, ${\mathcal E}(X,T)$ is nonempty and when ${\mathcal E}(X,T) = \{ \mu \}$ , the system is said to be uniquely ergodic. For every $\nu \in {\mathcal M}(X,T)$ , there is a probability measure $\rho $ in ${\mathcal M}(X,T)$ such that $\nu (A) = \int _{{\mathcal E}(X,T)} \mu (A)\, d \rho (\mu )$ for every Borel measurable set A. This is called the ergodic decomposition of $\nu $ .

Remark 2.2. If every ergodic measure $\mu \in {\mathcal E}(X,T)$ is $\gamma $ -rigid for the same partial rigidity sequence $(n_k)_{k \in {\mathbb N}}$ , then every invariant measure $\nu \in {\mathcal M}(X,T)$ is $\gamma $ -rigid. Indeed, for any Borel measurable set A,

$$ \begin{align*} \liminf_{k \to \infty} \nu(A \cap T^{-n_k}A) &= \liminf_{k \to \infty} \int_{{\mathcal E}(X,T)} \mu(A \cap T^{-n_k}A)\, d \rho(\mu) \\&\geq \int_{{\mathcal E}(X,T)} \liminf_{k \to \infty} \mu(A \cap T^{-n_k}A) \geq \int_{{\mathcal E}(X,T)} \gamma \mu(A)\, d \rho(\mu) = \gamma \nu(A). \end{align*} $$

2.2 Cantor and symbolic systems

A Cantor system $(X,T)$ is a topological dynamical system such that X is a Cantor space, that is, the topology of X has a countable basis of clopen sets (closed and open sets) and it has no isolated points. Cantor spaces are compact and metric. An important class of Cantor systems is symbolic systems.

Let ${\mathcal A}$ be a finite set that we call alphabet. The elements in ${\mathcal A}$ are called letters or symbols. For $\ell \in {\mathbb N}$ , the set of concatenations of $\ell $ letters is denoted by ${\mathcal A}^{\ell }$ and $w = w_1 \cdots w_{\ell } \in {\mathcal A}^{\ell }$ is said to be a word of length $\ell $ . The length of a word w is denoted by $|w|$ . We set ${\mathcal A}^* = \bigcup _{n \in {\mathbb N}} {\mathcal A}^{\ell }$ , where, by convention, ${\mathcal A}^0 = \{ \varepsilon \}$ and $\varepsilon $ is the empty word.

Given two words u and v, we say that they are respectively a prefix and a suffix of the word $uv$ . For a word $w = w_1 \cdots w_{\ell }$ and two integers $1 \leq i < j \leq \ell $ , we write $w_{[i, j+1)} = w_{[i, j]} = w_i \cdots w_j$ . We say that u appears or occurs in $w $ if there is an index $ 1 \leq i \leq |w|$ such that $u=w_{[i,i+|u|)}$ and we denote this by $u \sqsubseteq w$ . The index i is an occurrence of u in w and $|w|_u$ is the number of occurrences of u in w.

The set of one-sided sequences $(x_n)_{n \in {\mathbb N}}$ in ${\mathcal A}$ is denoted by ${\mathcal A}^{{\mathbb N}}$ and the set of two-sided sequences $(x_n)_{n \in {\mathbb Z}}$ in ${\mathcal A}$ is denoted by ${\mathcal A}^{{\mathbb Z}}$ . The notation and concepts introduced for words can be naturally extended for sequences in ${\mathcal A}^{{\mathbb N}}$ and ${\mathcal A}^{{\mathbb Z}}$ .

The shift map $S\colon {\mathcal A}^{{\mathbb Z}} \to {\mathcal A}^{{\mathbb Z}}$ is defined by $S((x_n)_{n \in {\mathbb Z}})= (x_{n+1})_{n \in {\mathbb Z}}$ . A subshift is a topological dynamical system $(X,S)$ , where X is a closed and S-invariant subset of ${\mathcal A}^{{\mathbb Z}}$ endowed with the product topology. Usually, X itself is said to be a subshift. Let X be a subshift on the alphabet ${\mathcal A}$ . Given $x \in X$ , the language ${\mathcal L}(x)$ is the set of all words appearing in x and ${\mathcal L}(X) = \bigcup _{x \in X} {\mathcal L}(x)$ . For two words $u,v \in {\mathcal L}(X)$ , the cylinder set $[u \cdot v]_X$ is given by $\{x \in X : x_{[-|u|,|v|)} = uv \}$ . When u is the empty word, we only write $[v]_X$ . Cylinder sets are clopen sets that form a base for the topology of the subshift.

A word $w\in {\mathcal A}^*$ is said to be complete if $|w|\geq 2$ and $w_1 = w_{|w|}$ . The set of complete words in ${\mathcal L}(X)$ is denoted by ${\mathcal C} {\mathcal L} (X)$ . For $u \in {\mathcal L}(X)$ , a right return word to u is a non-empty word w such that $uw \in {\mathcal L}(X)$ , $uw$ has u as proper suffix, and $uw$ does not have an occurrence of u which is not a prefix or suffix. Symmetrically, one defines left return words to u. We denote by ${\mathcal R}_X(u)$ (respectively ${\mathcal R}_X'(u)$ ) the set of right (respectively left) return words to u. Every complete word v starting with a letter $a \in {\mathcal A}$ satisfies that $v = a w_1 \cdots w_r$ , where $w_1,\ldots , w_r \in {\mathcal R}_X(a)$ . If $(X,S)$ is minimal, then $|{\mathcal R}_X(u)| < \infty $ for every $u \in {\mathcal L}(X)$ .

The non-decreasing map $p_X \colon {\mathbb N} \to {\mathbb N}$ defined by $p_X (n) = |{\mathcal L}_n (X)|$ is called the complexity function of X, where ${\mathcal L}_n(X) = {\mathcal L}(X) \cap {\mathcal A}^n$ . If $\liminf _{n \to \infty } {p_X(n)}/{n} = \infty $ , we say that X has superlinear complexity. On the contrary, X has non-superlinear complexity.

Let ${\mathcal A}$ and ${\mathcal B}$ be finite alphabets and $\sigma \colon {\mathcal A}^* \to {\mathcal B}^*$ be a morphism for the concatenation. We say that $\sigma $ is erasing whenever there exists $a \in {\mathcal A}$ such that $\sigma (a)$ is the empty word. Otherwise, we say that it is non-erasing. The morphism $\sigma $ is proper if each word $\sigma (a)$ starts and ends with the same letter independently of a. When it is non-erasing, it extends naturally to maps from ${\mathcal A}^{{\mathbb N}}$ to ${\mathcal B}^{{\mathbb N}}$ and from ${\mathcal A}^{{\mathbb Z}}$ to ${\mathcal B}^{{\mathbb Z}}$ in the obvious way by concatenation. To the morphism $\sigma $ , we associate an incidence matrix $M_{\sigma }$ indexed by ${\mathcal B} \times {\mathcal A}$ such that its entry at position $(b,a)$ is the number of occurrences of b in $\sigma (a)$ for every $a \in {\mathcal A}$ and $b \in {\mathcal B}$ (that is, $(M_{\sigma })_{b,a} = | \sigma (a)|_b$ ). If $\tau \colon {\mathcal B}^* \to {\mathcal C}^*$ is another morphism, then $\tau \circ \sigma \colon {\mathcal A}^* \to {\mathcal C}^* $ is a morphism and $M_{\tau \circ \sigma } = M_{\tau } M_{\sigma }$ .

A directive sequence $\boldsymbol \sigma = (\sigma _n\colon {\mathcal A}^*_{n+1} \to {\mathcal A}^*_n )_{n \in {\mathbb N}}$ is a sequence of morphisms, where we only consider non-erasing ones. When all the morphisms $\sigma _n$ , $n \geq 1$ , are proper, we say that $\boldsymbol \sigma $ is proper, and, when all incidence matrices $M_{\sigma _n}$ , $n \geq 1$ , are positive, we say that $\boldsymbol \sigma $ is positive. Here, we stress the fact that in the definition, there is no assumption (more that non-erasingness) on the first morphism $\sigma _0$ , since in many cases of interest, this morphism is neither proper nor positive.

For $0 \leq n < N$ , we denote $\sigma _{[n,N)} = \sigma _{[n,N-1]} = \sigma _n \circ \sigma _{n+1} \circ \cdots \circ \sigma _{N-1}$ . We say that $\boldsymbol {\sigma '} = (\sigma ^{\prime }_k \colon {\mathcal B}_{k+1}^* \to {\mathcal B}_k^*)_{k \in {\mathbb N}} $ is a contraction of $\boldsymbol \sigma = (\sigma _n \colon {\mathcal A}_{n+1}^* \to {\mathcal A}_n^*)_{n \in {\mathbb N}} $ if there is a sequence $(n_k)_{k \in {\mathbb N}}$ such that $n_0 = 0$ , ${\mathcal A}_{n_k} = {\mathcal B}_k $ and $\sigma _k' = \sigma _{[n_k, n_{k+1})} $ for all $k \in {\mathbb N}$ . A directive sequence $\boldsymbol \sigma $ is primitive if it has a positive contraction $\boldsymbol {\sigma '}$ .

For $n \in {\mathbb N}$ , the language ${\mathcal L}^{(n)}(\boldsymbol \sigma ) $ of level n associated with $\boldsymbol \sigma $ is defined by

$$ \begin{align*} {\mathcal L}^{(n)}(\boldsymbol \sigma) = \{ w \in {\mathcal A}_n^* : w \sqsubseteq \sigma_{[n,N)}(a) \text{ for some } a \in {\mathcal A}_N \text{ and } N>n \}, \end{align*} $$

and $X_{\boldsymbol \sigma }^{(n)}$ is the set of points $x \in {\mathcal A}_n^{{\mathbb Z}}$ such that ${\mathcal L}(x) \subseteq {\mathcal L}^{(n)}(\boldsymbol \sigma )$ . This set is the subshift generated by ${\mathcal L}^{(n)}(\boldsymbol \sigma )$ . It may happen that ${\mathcal L}(X^{(n)}_{\boldsymbol \sigma }) $ is strictly contained in ${\mathcal L}^{(n)}(\boldsymbol \sigma )$ , but in the primitive case, both sets coincide and $X^{(n)}_{\boldsymbol \sigma }$ is a minimal subshift. Finally, $X_{\boldsymbol \sigma } = X_{\boldsymbol \sigma }^{(0)}$ is the ${\mathcal S}$ -adic subshift generated by the directive sequence $\boldsymbol \sigma $ . Note that if $\boldsymbol {\sigma '}$ is a contraction of $\boldsymbol \sigma $ , then $X_{\boldsymbol \sigma } = X_{\boldsymbol {\sigma '}} $ .

We define the alphabet rank of $\boldsymbol \sigma $ as

$$ \begin{align*} AR(\boldsymbol \sigma) = \liminf_{n \to \infty} |{\mathcal A}_n|. \end{align*} $$

When $AR(\boldsymbol \sigma )$ is finite, via contraction and relabeling, $(X_{\boldsymbol \sigma }, S)$ can be defined by a directive sequence $\boldsymbol \sigma '$ , where for every $n \geq 1$ , the morphism $\sigma ^{\prime }_n$ is an endomorphism on a free monoid with $AR(\boldsymbol \sigma )$ generators.

Let $\sigma \colon {\mathcal A}^* \to {\mathcal B}^*$ be a morphism and $x \in {\mathcal B}^{{\mathbb Z}}$ . If $x = S^k \sigma (y)$ for some $y \in {\mathcal A}^{{\mathbb Z}} $ and $k \in {\mathbb Z}$ , then $(k,y)$ is a $\sigma $ -representation of x. If y belongs to some subshift $Y \subseteq {\mathcal A}^{{\mathbb Z}}$ , we say $(k,y)$ is a $\sigma $ -representation of x in Y. Also, if $0 \leq k < |\sigma (y_0)|$ , then $(k,y)$ is a centered $\sigma $ -representation of x in Y. Following [Reference Berthé, Steiner, Thuswaldner and Yassawi5], we say that $\sigma $ is recognizable in Y if each $x \in {\mathcal B}^{{\mathbb Z}}$ has at most one centered $\sigma $ -representation in Y. If any aperiodic point $x \in {\mathcal B}^{{\mathbb Z}}$ has at most one centered $\sigma $ -representation in Y, we say that $\sigma $ is recognizable in Y for aperiodic points. A directive sequence $\boldsymbol \sigma $ is recognizable at level n if $\sigma _n$ is recognizable in $X^{(n+1)}_{\boldsymbol \sigma }$ . The sequence $\boldsymbol \sigma $ is recognizable if it is recognizable at level n for each $n \in {\mathbb N}$ .

An endomorphism $\sigma \colon {\mathcal A}^* \to {\mathcal A}^*$ is called a substitution. The ${\mathcal S}$ -adic subshift $X_{\sigma }$ generated by the constant directive sequence $\sigma $ is determined by $ {\mathcal L}( X_{\sigma })={\mathcal L}(\sigma )$ , where ${\mathcal L}(\sigma ) = \{ w \in {\mathcal A}^* : w \sqsubseteq \sigma ^n(a) \text { for some } a\in {\mathcal A} \text { and } n \geq 1\}$ . The subshift $(X_{\sigma },S)$ is called the substitution subshift associated with $\sigma $ . A substitution is primitive if $M_{\sigma }$ is primitive.

A substitution $\sigma \colon {\mathcal A}^* \to {\mathcal A}^*$ has constant length if there exists a number $\ell \geq 1$ such that $|\sigma (a)| = \ell $ for all $a \in {\mathcal A}$ . More generally, a directive sequence $\boldsymbol \sigma = (\sigma _n\colon {\mathcal A}^*_{n+1} \to {\mathcal A}^*_n)_{n \in {\mathbb N}}$ is said to be of constant length if there exists a sequence of positive integers $(\ell _n)_{n \in {\mathbb N}}$ such that $|\sigma _n(a)| = \ell _n$ for every $n \in {\mathbb N}$ and $a \in {\mathcal A}_{n+1}$ . In that case, we also say that the ${\mathcal S}$ -adic subshift given by $\boldsymbol \sigma $ has constant length.

Another important family of subshifts is the linearly recurrent subshifts. Using the result of Durand [Reference Durand20, Proposition 1.1], we say that an ${\mathcal S}$ -adic subshift $(X_{\boldsymbol \sigma }, S)$ is linearly recurrent if there exists a finite set of morphisms ${\mathcal S}$ such that the directive sequence $\boldsymbol \sigma = (\sigma _n \colon {\mathcal A}_{n+1}^* \to {\mathcal A}_n^*)_{n \in {\mathbb N}}$ is proper, positive, and $\sigma _n \in {\mathcal S}$ for all $n \geq 1$ . Minimal substitution subshifts and linearly recurrent subshifts are uniquely ergodic.

For a minimal Cantor system $(X, T)$ and $U \subseteq X$ a clopen subset, we say that $(U,T_U)$ is the induced system on U, where $T_Ux = T^{n(x)}x$ and $n(x) = \inf \{ k> 0 : T^kx \in U \}$ , which is finite by minimality. The system $(U, T_U)$ is also a minimal Cantor system (see [Reference Durand and Perrin21, Ch. 1.1.3]). If $\mu \in {\mathcal M}(X,T)$ is an invariant (ergodic) measure of the original system and $\mu (U) \neq 0$ , then the induced measure $\mu _U = {\mu }/{\mu (U)}$ is an invariant (ergodic) measure of $(U,T_U)$ (see [Reference Petersen38, Ch. 2]).

2.3 Kakutani–Rokhlin partitions

Let $(X, {\mathcal X}, \mu , T)$ be a measure-preserving system. We say that ${\mathscr P}$ is a Kakutani–Rokhlin partition of the system if it is a partition of X of the form

$$ \begin{align*} {\mathscr P} = \{ T^j B_a : 1 \leq a \leq d, 0 \leq j < h_a \}, \end{align*} $$

where d is a positive integer, $B_1, \ldots , B_d$ are measurable subsets of X, and $h_1,\ldots ,h_d$ are positive integers. For $a \in \{1, \ldots ,d\}$ , ${\mathscr T}_a = \bigcup _{j=0}^{h_a - 1} T^j B_a$ is the ath tower of ${\mathscr P}$ , $B_a$ is the base of this tower, and $B= \bigcup _{a=1}^{d} B_a$ is the base of ${\mathscr P}$ . It follows from the definition of ${\mathscr P}$ that $\mu ( \bigcup _{a=1} ^d {\mathscr T}_a) =1$ .

Now, consider a sequence of Kakutani–Rokhlin partitions

$$ \begin{align*} ({\mathscr P}^{(n)} = \{ T^j B_a^{(n)} : 1 \leq a \leq d^{(n)}, 0 \leq j < h_a^{(n)} \} )_{n \in {\mathbb N}} \end{align*} $$

with ${\mathscr P}^{(0)} = \{ X \}$ . For every $n \in {\mathbb N}$ and $a \in \{1,\ldots ,d^{(n)}\}$ , we denote by ${\mathscr T}^{(n)}_a$ and $B^{(n)}$ the ath tower and the base of ${\mathscr P}^{(n)}$ , respectively.

We say that $({\mathscr P}^{(n)})_{n \in {\mathbb N}}$ is nested if for every $n \in {\mathbb N}$ , it satisfies the following two properties:

1. (KR1) $B^{(n+1)} \subseteq B^{(n)} $ ;

2. (KR2) ${\mathscr P}^{(n+1)} \succeq {\mathscr P}^{(n)} $ ; that is, for all $A \in {\mathscr P}^{(n+1)}$ , there exists $A' \in {\mathscr P}^{(n)}$ such that $A \subseteq A'$ .

We consider mostly nested sequences of Kakutani–Rokhlin partitions which also have the following extra properties:

1. (KR3) $\lim _{n \to \infty }\mu (B^{(n)}) = 0$ ;

2. (KR4) $\bigcup _{n \in {\mathbb N}} {\mathscr P}^{(n)}$ generates the $\sigma $ -algebra ${\mathcal X}$ .

5.1 Kakutani–Rokhlin partitions in ${\mathcal S}$ -adic subshifts

Let $\boldsymbol \sigma = (\sigma _n \colon {\mathcal A}_{n+1}^* \to {\mathcal A}_n^*)_{n \in {\mathbb N}}$ be a primitive recognizable directive sequence and let $(X_{\boldsymbol \sigma }, S)$ be the ${\mathcal S}$ -adic subshift generated by $\boldsymbol \sigma $ . We define its natural sequence of Kakutani–Rohklin partitions $({\mathscr P}^{(n)})_{n \in {\mathbb N}}$ as follows:

$$ \begin{align*} {\mathscr P}^{(n)} = \{ S^k \sigma_{[0,n)}([a]) : a \in {\mathcal A}_n, 0 \leq k < \lvert \sigma_{[0,n)} (a) \rvert \}. \end{align*} $$

It is straightforward from [Reference Berthé, Steiner, Thuswaldner and Yassawi5, Lemma 6.3] that if $\mu $ is an S-invariant probability measure and we consider the measure-preserving system $(X_{\boldsymbol \sigma }, {\mathcal B}, \mu ,S)$ , then $({\mathscr P}^{(n)})_{n \in {\mathbb N}}$ satisfies properties (KR1)–(KR4) (here, ${\mathcal B}$ is the Borel sigma-algebra). Moreover, sets in ${\mathscr P}^{(n)}$ are not only measurable, but also clopen and, when the directive sequence $\boldsymbol \sigma $ is proper, then $\bigcap _{n \geq 1} B^{(n)}$ is equal to a single point and $\bigcup _{n \in {\mathbb N}} {\mathscr P}^{(n)}$ generates the topology (see [Reference Durand and Perrin21, Ch. 5.3] for more details). Another important aspect is that the induced system on the base $B^{(n)}$ coincides with $X^{(n)}_{\boldsymbol \sigma }$ . So, each invariant measure on $X^{(n)}_{\boldsymbol \sigma }$ can be treated as the induced measure $\mu (\cdot )/\mu (B^{(n)})$ of an invariant measure $\mu $ on $X_{\boldsymbol \sigma }$ .

If $\boldsymbol \sigma $ is primitive and recognizable, for any $w \in {\mathcal A}_{n}^*$ , ${\mathscr T}^{(n)}_w$ has a positive measure if and only if $w \in {\mathcal L}^{(n)}(\boldsymbol \sigma )$ . This follows from the fact that $B_w^{(n)}$ coincides with the cylinder set $[w]_{X_{\boldsymbol \sigma }^{(n)}}$ on the induced system on $B^{(n)}$ and $[w]_{X_{\boldsymbol \sigma }^{(n)}}$ has positive measure whenever $w \in {\mathcal L}^{(n)}(\boldsymbol \sigma )$ .

The following two lemmas will be very useful in the next sections.

Lemma 5.1. Let $\boldsymbol \sigma = ( \sigma _n \colon {\mathcal A}_{n+1}^* \to {\mathcal A}_n^*)_{n \in {\mathbb N}}$ be a primitive and recognizable directive sequence. For $a \in {\mathcal A}_{n+1}$ , if $\sigma _n(a) = p w^r s$ with $r \geq 1$ , $p,w,s \in {\mathcal A}_n^*$ , and $w_1 = b$ , then

$$ \begin{align*} \mu({\mathscr T}^{(n)}_w \cap {\mathscr T}^{(n+1)}_a) \geq \frac{r}{|\sigma_n(a)|_b} \mu({\mathscr T}^{(n)}_b \cap {\mathscr T}^{(n+1)}_a) \end{align*} $$

and

$$ \begin{align*} \mu({\mathscr T}^{(n)}_{wb} \cap {\mathscr T}^{(n+1)}_a) \geq \frac{r-1}{|\sigma_n(a)|_b} \mu({\mathscr T}^{(n)}_b \cap {\mathscr T}^{(n+1)}_a). \end{align*} $$

Proof. It is clear that $\mu ({\mathscr T}^{(n)}_b \cap {\mathscr T}^{(n+1)}_a) = |\sigma _n(a)|_b h_b^{(n)} \mu (B_a^{(n+1)})$ and that for every $j \in \{0,\ldots ,r-1\}$ , $S^{\sum _{i =1}^{|p|} h_{p_i}^{(n)} + j \sum _{i=1}^{|w|} h_{w_i}^{(n)} } B_a^{(n+1)} \subseteq B_w^{(n)}$ . Therefore, $\mu ({\mathscr T}^{(n)}_w \cap {\mathscr T}^{(n+1)}_a) \geq r h_b^{(n)} \mu (B_a^{(n+1)})$ and

$$ \begin{align*} \mu({\mathscr T}^{(n)}_w \cap {\mathscr T}^{(n+1)}_a) \geq r \frac{|\sigma_n(a)|_b}{|\sigma_n(a)|_b} h_b^{(n)} \mu(B_a^{(n+1)}) = \frac{r}{|\sigma_n(a)|_b} \mu({\mathscr T}^{(n)}_b \cap {\mathscr T}^{(n+1)}_a). \end{align*} $$

For the second inequality, the argument is analogous.

More generally, we have the following lemma.

Lemma 5.2. Let $\boldsymbol \sigma = ( \sigma _n \colon {\mathcal A}_{n+1}^* \to {\mathcal A}_n^*)_{n \in {\mathbb N}}$ be a primitive and recognizable directive sequence. For $a \in {\mathcal A}_{n+1}$ and $w \in {\mathcal A}_{n}^*$ with $w_1 = b$ , we have that

(12) $$ \begin{align} \mu({\mathscr T}^{(n)}_w \cap {\mathscr T}^{(n+1)}_a) \geq \frac{|\sigma_n(a)|_w}{|\sigma_n(a)|_b} \mu({\mathscr T}^{(n)}_b \cap {\mathscr T}^{(n+1)}_a). \end{align} $$

Proof. Fix $n \in {\mathbb N}$ , $a \in {\mathcal A}_{n+1}$ , $b \in {\mathcal A}_n$ , and $w \in {\mathcal A}_n^*$ as above. Let I be the set of occurrences of the word w in $u = \sigma _n(a)$ , that is, $I = \{ i \in \{ 1, \ldots , |u|\} : u_{[i,i+|w|)} = w \}$ . Thus, for every $i \in I$ , $S^{\sum _{j =1}^{i-1} h_{u_j}^{(n)} } B_a^{(n+1)} \subseteq B_w^{(n)}$ . Therefore, $\mu ({\mathscr T}^{(n)}_w \cap {\mathscr T}^{(n+1)}_a) \geq |I| h_b^{(n)} \mu (B_a^{(n+1)})$ and $|I| = |\sigma _n(a)|_w$ . We get that

$$ \begin{align*} \mu({\mathscr T}^{(n)}_w \cap {\mathscr T}^{(n+1)}_a) \geq |\sigma_n(a)|_w \frac{|\sigma_n(a)|_b}{|\sigma_n(a)|_b} h_b^{(n)} \mu(B_a^{(n+1)}) = \frac{|\sigma_n(a)|_w}{|\sigma_n(a)|_b} \mu({\mathscr T}^{(n)}_b \cap {\mathscr T}^{(n+1)}_a). \end{align*} $$

Finally, we define a clean directive sequence. This notion has different versions in the literature (see, for example, [Reference Bezuglyi, Kwiatkowski, Medynets and Solomyak6, Reference Bressaud, Durand and Maass7] or [Reference Arbulú and Durand1] in the ${\mathcal S}$ -adic context). Let $\boldsymbol \sigma $ be a primitive and recognizable directive sequence, and let $\mu $ be an ergodic measure of $(X_{\boldsymbol \sigma },S)$ . We say that $\boldsymbol \sigma $ is clean with respect to $\mu $ if:

1. (1) there exists $n_0 \in {\mathbb N}$ such that ${\mathcal A}_n = {\mathcal A}_{n_0}$ for all $n \geq n_0$ . Set ${\mathcal A} = {\mathcal A}_{n_0}$ ;

2. (2) there exists a constant $\eta>0 $ and ${\mathcal A}_{\mu } \subseteq {\mathcal A}$ such that

   $$ \begin{align*} \mu({\mathscr T}^{(n)}_a) \geq \eta & \quad \text{for all } a \in {\mathcal A}_{\mu},\ n \geq n_0 \quad \text{and } \\ \lim_{n \to \infty } \mu({\mathscr T}^{(n)}_a) = 0 & \quad \text{for all } a \in {\mathcal A} \backslash {\mathcal A}_{\mu}. \end{align*} $$

Combining the results from [Reference Berthé, Steiner, Thuswaldner and Yassawi5, Reference Bezuglyi, Kwiatkowski, Medynets and Solomyak6], it is known that any recognizable and primitive directive sequence $\boldsymbol \sigma $ of finite alphabet rank can be contracted to become clean with respect to $\mu $ . More generally, we say that $\boldsymbol \sigma $ is clean if it is clean with respect to every ergodic measure $\mu $ . When $\boldsymbol \sigma $ is clean, if $\mu $ and $\nu $ are distinct ergodic measures of $(X_{\boldsymbol \sigma },S)$ , then ${\mathcal A}_{\mu } \cap {\mathcal A}_{\nu } = \emptyset $ .

5.2 Sufficient conditions for partial rigidity

In the following, we provide a set of sufficient conditions that guarantee partial rigidity of an $\mathcal {S}$ -adic subshift. These conditions are of different nature: algebraic, combinatorial, and related to the order in which the towers of level n intersect the towers of level $n+1$ . This allows retrieving results for primitive substitutive systems, linearly recurrent systems [Reference Cortez, Durand, Host and Maass11], among others.

5.2.1 Conditions regarding tower heights

We start by extending the result concerning exact finite rank subshifts, that is, ${\mathcal S}$ -adic subshifts such that ${\mathcal A} = {\mathcal A}_{\mu }$ . The concept of exact finite rank subshift was introduced in [Reference Bezuglyi, Kwiatkowski, Medynets and Solomyak6], where it was proved that these systems are not mixing. Subsequently, partial rigidity was established in [Reference Danilenko13] and part of the proof of the next lemma shares the approach used in that paper. However, the conditions of Theorem 5.5 are also satisfied by some superlinear complexity and non-finite exact rank ${\mathcal S}$ -adic subshifts.

Lemma 5.4. Let $\boldsymbol \sigma = (\sigma _n \colon {\mathcal A}_{n+1}^* \to {\mathcal A}_n^*)_{n \in {\mathbb N}}$ be a clean, primitive, and recognizable directive sequence, with ${\mathcal A}_n={\mathcal A}$ for all $n\in {\mathbb N}$ , and $\mu $ an ergodic measure on $(X_{\boldsymbol \sigma }, S)$ . If ${\mathcal A}_{\mu } \neq {\mathcal A}$ and for every letter $b \in {\mathcal A} \backslash {\mathcal A}_{\mu }$ ,

$$ \begin{align*} \frac{\mu(B_b^{(n)})}{\mu(B^{(n)})} \xrightarrow[n \rightarrow \infty]{} 0, \end{align*} $$

then $(X_{\boldsymbol \sigma },{\mathcal B},\mu ,S)$ is partially rigid.

Proof. Let $n \in {\mathbb N}$ and $a_0 \in {\mathcal A}$ be such that $\mu (B_{a_0}^{(n)}) = \max _{a \in {\mathcal A} } \mu (B_{a}^{(n)})$ . We can take $a_1 \in {\mathcal A}$ such that

$$ \begin{align*} \mu(B_{a_1}^{(n)} \cap S^{h_{a_0}^{(n)}} B_{a_0}^{(n)}) \geq \frac{\mu(B_{a_0}^{(n)})}{d}, \end{align*} $$

where $d = |{\mathcal A}|$ . Similarly, we can take $a_2 \in {\mathcal A}$ such that

$$ \begin{align*} \mu(B_{a_2}^{(n)} \cap S^{h_{a_1}^{(n)}} (B_{a_1}^{(n)} \cap S^{h_{a_0}^{(n)}} B_{a_0}^{(n)})) \geq \frac{\mu(B_{a_0}^{(n)})}{d^2}. \end{align*} $$

We can iterate this process to form a word $w = a_0 \cdots a_d \in {\mathcal A}^*$ such that $\mu (B_w^{(n)}) \geq {\mu (B_{a_0}^{(n)})}/{d^d}$ . Since the length of w is $d+1$ , there are two integers $0 \leq i < j \leq d$ such that $a_i = a_j$ . We will refer to this letter as a and we set $u = a_i a_{i+1} \cdots a_j$ . Therefore, since $S^{h_{a_0}^{(n)}+\cdots +h_{a_i-1}^{(n)}}B_w^{(n)}\subseteq B_u^{(n)} $ , we have

$$ \begin{align*} \mu(B_u^{(n)}) \geq \mu(S^{h_{a_0}^{(n)}+\cdots+h_{a_i-1}^{(n)}}B_w^{(n)}) = \mu(B_w^{(n)}) \geq \frac{\mu(B_{a_0}^{(n)})}{d^d} \geq \frac{\mu(B_{a}^{(n)})}{d^d}. \end{align*} $$

Multiplying by $h_a^{(n)}$ , we get

(13) $$ \begin{align} \mu({\mathscr T}_{u}^{(n)}) \geq \frac{\mu({\mathscr T}_{a}^{(n)})}{d^d}. \end{align} $$

Now, assume that for all n, in an infinite subset ${\mathcal N} \subseteq {\mathbb N}$ , letter $a_0$ such that $\mu (B_{a_0}^{(n)}) = \max _{a \in {\mathcal A} } \mu (B_{a}^{(n)})$ , the word u and the letter a that fulfill equation (13) are the same.

Claim. $a_0, a \in {\mathcal A}_{\mu }$ .

Indeed, if $a_0 \in {\mathcal A} \backslash {\mathcal A}_{\mu }$ , then

(14) $$ \begin{align} 1 = \frac{\mu(B^{(n)})}{\mu(B^{(n)})} = \frac{\sum_{b \in {\mathcal A}} \mu(B_b^{(n)})}{\mu(B^{(n)})} \leq d \cdot \frac{\mu(B_{a_0}(n))}{\mu(B^{(n)})} \xrightarrow[n \rightarrow \infty, n\in {\mathcal N}]{} 0, \end{align} $$

which is a contradiction. Now, if $a \in {\mathcal A} \backslash {\mathcal A}_{\mu }$ , then

$$ \begin{align*} \frac{1}{d^d}\frac{\mu(B_{a_0}^{(n)})}{\mu(B^{(n)})} \leq \frac{\mu(B_u^{(n)}) }{\mu(B^{(n)}) } \leq \frac{\mu(B_a^{(n)})}{\mu(B^{(n)})} \xrightarrow[n \rightarrow \infty, n\in {\mathcal N}]{} 0, \end{align*} $$

which produce the same contradiction given by inequalities in equation (14).

To conclude, we have that for n large enough,

$$ \begin{align*} \frac{\eta}{d^d} \leq \frac{\mu({\mathscr T}_a^{(n)})}{d^d} \leq \mu({\mathscr T}_{u}^{(n)}), \end{align*} $$

where $\eta>0$ is the constant stating whether the directive sequence is clean. The conclusion follows from Corollary 3.7.

Theorem 5.5. Let $\boldsymbol \sigma = (\sigma _n : {\mathcal A}_{n+1}^* \to {\mathcal A}_n^*)_{n \in {\mathbb N}}$ be a clean, primitive, and recognizable directive sequence, with ${\mathcal A}_n={\mathcal A}$ for all $n\in {\mathbb N}$ , and $\mu $ an ergodic measure on $(X_{\boldsymbol \sigma }, S)$ . If one of the following hypotheses holds:

1. (1) ${\mathcal A}_{\mu } = {\mathcal A}$ ;

2. (2) there is $a \in {\mathcal A}_{\mu }$ such that $\limsup _{n\to \infty }{h^{(n)}_{a}}/{h^{(n)}_{b}} < \infty $ for every $ b \in {\mathcal A} \backslash {\mathcal A}_{\mu }$ ,

then $(X_{\boldsymbol \sigma },{\mathcal B},\mu ,S)$ is partially rigid.

Proof. For the first assumption, it is sufficient to repeat the proof of Lemma 5.4 and observe that we only need to prove that $a \in {\mathcal A}_{\mu }$ . However, since ${\mathcal A}_{\mu } = {\mathcal A}$ , this is straightforward.

For the second one, we will prove that the hypotheses of Lemma 5.4 are fulfilled, that is, for all $b \in {\mathcal A} \backslash {\mathcal A}_{\mu }$ , ${\mu (B_b^{(n)})}/{\mu (B^{(n)})} \xrightarrow [n \rightarrow \infty ]{} 0$ . Indeed, as $\limsup _{n\to \infty }{h^{(n)}_{a}}/{h^{(n)}_b} < \infty $ for every $ b \in {\mathcal A} \backslash {\mathcal A}_{\mu }$ , we can fix $c \in (0,\infty )$ such that for infinitely many $n \in {\mathbb N}$ and for every $ b \in {\mathcal A} \backslash {\mathcal A}_{\mu }$ , $h^{(n)}_a \leq c h^{(n)}_b$ . Then, for an arbitrary $ b \in {\mathcal A} \backslash {\mathcal A}_{\mu }$ ,

$$ \begin{align*} \limsup_{n \to \infty} \frac{\mu(B_b^{(n)}) }{\mu(B^{(n)}) } &\leq \limsup_{n \to \infty} \frac{\mu(B_b^{(n)}) }{\mu(B_a^{(n)}) } \\ &\leq c \limsup_{n \to \infty} \bigg( \frac{h^{(n)}_b}{h^{(n)}_a } \frac{\mu(B_b^{(n)}) }{\mu(B_a^{(n)}) } \bigg) \\ &= c \limsup_{n \to \infty} \frac{\mu({\mathscr T}_b^{(n)}) }{\mu({\mathscr T}_a^{(n)}) } \\ &\leq \frac{c}{\eta} \limsup_{n \to \infty} \mu({\mathscr T}_b^{(n)})= 0, \end{align*} $$

where $\eta> 0$ is the constant defining whether the directive sequence is clean. Finally, we conclude by Lemma 5.4.

Example 5.6. Theorem 5.5 allows to ensure the partial rigidity for systems that are not of exact finite rank. Fix ${\mathcal A}_n = \{0,1\}$ for all $n \geq 0$ and recall that for $\sigma _n : {\mathcal A}_{n+1}^* \to {\mathcal A}_n^*$ , $(M_{\sigma _n})_{b,a} = | \sigma _n(a)|_b$ , where $b \in {\mathcal A}_n^*$ and $a\in {\mathcal A}_{n+1}^*$ . Let $\boldsymbol \sigma $ be a directive sequence such that for all $n \geq 1$ , the incidence matrix of $\sigma _n$ is of the form

$$ \begin{align*} M_{\sigma_n} = \left( \begin{array}{@{}cc@{}} a_n & b_n \\ c & d \end{array} \right)\!, \end{align*} $$

where $c,d$ are two positive integers and $(a_n)_{n \geq 0}$ , $(b_n)_{n \geq 0}$ are integer sequences such that $\lim _{n \to \infty } a_n = \infty $ and there is a constant $K>0$ , $K b_n \geq a_n$ for all $n \geq 1$ . Then, $(X_{\boldsymbol {\sigma }}, S)$ is uniquely ergodic with invariant measure $\mu $ satisfying $\lim _{n \to \infty } \mu ( {\mathscr T}^{(n)}_0 ) = 1$ and $\lim _{n \to \infty } \mu ( {\mathscr T}^{(n)}_1 ) = 0$ . That means ${\mathcal A}_{\mu } = \{0\}$ and, in particular, it is not of exact finite rank. Also notice that

$$ \begin{align*} \limsup_{n \to \infty} \frac{h_0^{(n+1)}}{h_1^{(n+1)}} = \limsup_{n \to \infty} \frac{a_n h_0^{(n)} + c h_1^{(n)}}{b_n h_0^{(n)} + d h_1^{(n)}} \leq \limsup_{n \to \infty} K \frac{b_n h_0^{(n)} + (c/K) h_1^{(n)}}{b_n h_0^{(n)} + d h_1^{(n)}} < \infty \end{align*} $$

and, by Theorem 5.5, $(X_{\boldsymbol \sigma },{\mathcal B},\mu ,S)$ is partially rigid.

Corollary 5.7. Let $\boldsymbol \sigma = (\sigma _n \colon {\mathcal A}_{n+1}^* \to {\mathcal A}_n^*)_{n \in {\mathbb N}}$ be a clean, primitive, and recognizable directive sequence, with ${\mathcal A}_n={\mathcal A}$ for all $n\in {\mathbb N}$ , and $\mu $ an ergodic measure on $(X_{\boldsymbol \sigma }, S)$ . If there is a constant $c>0$ such that $h^{(n)}_a \leq c h^{(n)}_b$ for every $a,b \in {\mathcal A}$ and infinitely many $n \in {\mathbb N}$ , then $(X_{\boldsymbol \sigma },{\mathcal B},\mu ,S)$ is partially rigid.

Proof. By assumption, $\limsup _{n\to \infty } {h^{(n)}_a}/{h^{(n)}_b} \leq c$ for every $a, b \in {\mathcal A}$ , so one of the hypotheses of Theorem 5.5 is satisfied.

5.2.2 Conditions regarding the order

In what follows, we study partial rigidity of ${\mathcal S}$ -adic subshifts that we call m-consecutive. Danilenko proved that consecutive Bratteli–Vershik systems with a technical, but crucial, extra assumption are partially rigid (we refer to [Reference Danilenko13, Theorem 7.6]). The m-consecutive property is slightly more general than this family of systems and, as we shall see, is sufficiently rich to build interesting examples. In addition, we show that the lower bound for the partial rigidity rate given for general m-consecutive sequences can be improved if we know that the directive sequence is of constant length.

Definition 5.8. A morphism $\sigma \colon {\mathcal A}^* \to {\mathcal B}^*$ is m-consecutive, where $m\geq 2$ is an integer, if for every $a \in {\mathcal A}$ , $\sigma (a)=b_1^{k_1}b_2^{k_2} \cdots b_r^{k_r}$ , where $k_1,\ldots , k_r \geq m$ and $b_1,\ldots ,b_r \in {\mathcal B}$ . In that case, we can assume that consecutive $b_i$ terms are different.

A directive sequence $\boldsymbol \sigma $ is m-consecutive if there is $n_0 \in {\mathbb N}$ such that $\sigma _n$ is m-consecutive for every $n \geq n_0$ .

Remark 5.9. If $\tau \colon {\mathcal A}^* \to {\mathcal B}^*$ is a morphism and $\sigma \colon {\mathcal B}^* \to {\mathcal C}^*$ a m-consecutive morphism, then $\sigma \circ \tau $ is an m-consecutive morphism. Therefore, every contraction of an m-consecutive directive sequence is also m-consecutive.

Theorem 5.10. Let $m\geq 2$ and $\boldsymbol \sigma = (\sigma _n \colon {\mathcal A}_{n+1}^* \to {\mathcal A}_n^*)_{n \in {\mathbb N}}$ be a primitive, recognizable, and m-consecutive directive sequence with finite alphabet rank d. Then, every ergodic measure for $(X_{\boldsymbol \sigma }, S) $ is $(({m-1})/{m} \cdot {1}/{d})$ -rigid.

In the following proof, for a letter b, we say that $b^k$ is a b-block on w if one of the following holds: $w_{[1,k]} = b^k$ and $w_{k+1} \neq b$ ; or $w_{(|w|-k,|w|]}= b^k$ and $w_{|w|-k} \neq $ b; or there exists $i \in \{2,\ldots ,|w|-k\}$ such that $w_{[i,i+k)} = b^k$ , $w_{i-1}\neq b$ , and $w_{i+k} \neq b$ .

Proof. Fix $\mu $ an ergodic measure and $n \geq n_0$ , where $n_0$ is taken as in the definition of an m-consecutive directive sequence. Furthermore, since the alphabet rank of $\boldsymbol \sigma $ is d, except for contraction, as m-consecutivity is preserved, we can assume that $|{\mathcal A}_n| = d$ for every $n \geq n_0$ .

Consider $a \in {\mathcal A}_{n+1}$ and $b \in {\mathcal A}_n$ such that b appears in $\sigma _n(a)$ . Observe that $|\sigma _n(a)|_{bb} = \sum _{i=1}^{r} (k_i-1)$ , where $k_i$ is the length of the ith b-block in $\sigma _n(a)$ in the order of occurrence and r is the number of b-blocks in $\sigma _n(a)$ . By m-consecutivity, $k_i \geq m$ . So,

$$ \begin{align*} \frac{|\sigma_n(a)|_{b} - |\sigma_n(a)|_{bb}}{|\sigma_n(a)|_{b}} = \frac{r}{\sum_{i=1}^{r}k_i} \leq \frac{r}{rm} = \frac{1}{m}. \end{align*} $$

Therefore, by Lemma 5.2,

$$ \begin{align*} \mu({\mathscr T}_{bb}^{(n)} \cap {\mathscr T}^{(n+1)}_a) \geq \frac{|\sigma_n(a)|_{bb}}{|\sigma_n(a)|_{b}} \mu({\mathscr T}_{b}^{(n)} \cap {\mathscr T}^{(n+1)}_a) \geq \frac{m-1}{m} \mu({\mathscr T}_{b}^{(n)} \cap {\mathscr T}^{(n+1)}_a). \end{align*} $$

Thus, summing over $a \in {\mathcal A}_{n+1}$ , we conclude that $\mu ({\mathscr T}_{bb}^{(n)}) \geq ({m-1})/{m}\cdot \mu ({\mathscr T}_{b}^{(n)})$ (note that this conclusion only requires the m-consecutive condition, and the finite alphabet range condition is not necessary). Since $\sum _{b \in {\mathcal A}_n} \mu ({\mathscr T}^{(n)}_b) =1 $ , there is at least one letter $b \in {\mathcal A}_n$ such that $\mu ({\mathscr T}^{(n)}_b) \geq {1}/{d}$ .

In summary, for every $n \geq n_0$ , there is a letter $b \in {\mathcal A}_n$ such that $\mu ({\mathscr T}_{bb}^{(n)}) \geq ({m-1})/{m} \cdot {1}/{d}$ and we conclude by Corollary 3.7.

Example 5.11. As mentioned before, m-consecutive ${\mathcal S}$ -adic subshifts are a useful family for constructing examples of partially rigid systems with a variety of desired properties. In particular, by employing a construction similar to that in [Reference Donoso, Durand, Maass and Petite16, Example 6.4] (see also [Reference Donoso, Durand, Maass and Petite15, §4]), we can create a partially rigid finite rank ${\mathcal S}$ -adic subshift with superlinear complexity. Note that to achieve superlinear complexity, we necessarily need to build a system which is not a substitution or linearly recurrent subshift.

Let $(a_n)_{n \in {\mathbb N}}$ be an increasing sequence of integers larger than $4$ and let $\boldsymbol \sigma = (\sigma _n\colon \{0,1\}^* \to \{0,1\}^*)_{n \in {\mathbb N}}$ be a directive sequence defined by

$$ \begin{align*} \sigma_n(0) = 00111 \quad \text{and} \quad \sigma_n(1) = 0^3 11 0^4 11 \cdots 11 0^{a_n+1} 11. \end{align*} $$

Let $(X_{\boldsymbol {\sigma }},S)$ be the ${\mathcal S}$ -adic subshift generated by $\boldsymbol \sigma $ . It is not complicated to see that this directive sequence is recognizable, the proof is left to the reader.

To prove that $\liminf _{n\to \infty } {p_{X_{\boldsymbol \sigma }} (n)}/{n} = \infty $ , we show the stronger condition $ \lim _{n\to \infty } p_{X_{\boldsymbol \sigma }} (n+1) - p_{X_{\boldsymbol \sigma }} (n)= \infty $ . Notice that for every $n \geq 1$ , $ p_{X_{\boldsymbol \sigma }} (n+1) - p_{X_{\boldsymbol \sigma }} (n) = r(n)$ , where $r(n)$ is the number of right special words of length n, that is, $r(n) = |\{w \in {\mathcal L}_n(X_{\boldsymbol \sigma }): w0,w1 \in {\mathcal L}_{n+1}(X_{\boldsymbol \sigma })\}|$ .

Claim 1. Fix $n \geq 1$ and $k \in \{4,\ldots , a_{n+1}\}$ . Then, the word

$$ \begin{align*} W_n(k) = \sigma_{[0,n]}(0^{k-1}110^k) \sigma_{[0,n-1]}(00) \cdots \sigma_0(00)00 \end{align*} $$

is right special in $X_{\boldsymbol \sigma }$ .

Proof. Indeed, $0^{k-1}110^k0$ and $0^{k-1}110^k1$ belong to ${\mathcal L}(X_{\boldsymbol \sigma }^{(n+1)})$ , so $\sigma _{[0,n]}(0^{k-1}110^k0)$ and $\sigma _{[0,n]}(0^{k-1}110^k1)$ belong to ${\mathcal L}(X_{\boldsymbol \sigma })$ . However, a simple computation yields to

$$ \begin{align*} \sigma_{[0,n]}(0^{k-1}110^k0)=\sigma_{[0,n]}(0^{k-1}110^k) \sigma_{[0,n-1]}(00) \cdots \sigma_0(00)00 a u \end{align*} $$

and

$$ \begin{align*} \sigma_{[0,n]}(0^{k-1}110^k1)=\sigma_{[0,n]}(0^{k-1}110^k) \sigma_{[0,n-1]}(00) \cdots \sigma_0(00)00 b v , \end{align*} $$

where $a, b \in \{0,1\}$ are different and $u,v \in \{0,1\}^*$ . Therefore, $W_n(k)0, W_n(k)1 \in {\mathcal L}(X_{\boldsymbol \sigma })$ , which allows us to conclude.

For $n\in {\mathbb N}$ , set $A_n = \lvert \sigma _{[0,n]}(1) \rvert $ , $B_n = \lvert \sigma _{[0,n]}(0) \rvert $ , and $C_{n}=|\sigma _{[0,n-1]}(00) \cdots \sigma _0(00)00| = 2(B_{n-1} + \cdots + B_0 + 1 )$ , where $C_0=2$ by convention. A simple but tedious computation yields $\lfloor A_n/B_n \rfloor \geq c \ a_n$ for some positive constant c. Thus, by adjusting the values of the $a_n$ terms, we can assume that $\lfloor A_n/B_n \rfloor \geq 4$ .

Claim 2. Given $n \geq 1$ , $k \in \{4,\ldots , a_{n+1}\}$ and $N \in \{kB_n+ C_n +1,\ldots , (k+1) B_n +C_n \}$ , there are at least $\min (k, \lfloor A_n/B_n \rfloor )$ different right special words in ${\mathcal L}_N(X_{\boldsymbol \sigma })$ .

Proof. First, a simple computation shows that

$$ \begin{align*} |W_n(i)|=C_n+2A_n+(2i-1)B_n \geq C_n + (k+1)B_n \geq N \end{align*} $$

for each $i \in \{ 4+ k - \min (k,\lfloor A_n/B_n\rfloor ),\ldots , k \}$ . Let $w_i$ be the suffix of length N of $W_n(i)$ . Then,

$$ \begin{align*} w_i = p_i \sigma_{[0,n]}(0^i) \sigma_{[0,n-1]}(00) \cdots \sigma_1(00)00, \end{align*} $$

where $p_i$ is a suffix of $\sigma _{[0,n]}(0^{i-1}11)$ . From Claim 1, these words are right special. We are left to prove that they are distinct. Consider $i, j\in \{ 4+ k - \min (k,\lfloor A_n/B_n\rfloor ),\ldots , k \}$ with $i<j$ , and notice that

$$ \begin{align*} p_j \sigma_{[0,n]}(0^{j-i}) \ \sigma_{[0,n]}(0^i) \sigma_{[0,n-1]}(00) \cdots \sigma_1(00)00 &= w_j \quad \text{and }\\ p_i \ \sigma_{[0,n]}(0^i) \sigma_{[0,n-1]}(00) \cdots \sigma_1(00)00 &= w_i, \end{align*} $$

then $w_i \neq w_j$ if and only if $p_i \neq p_j \sigma _{[0,n]}(0^{j-i})$ .

As $w_i$ and $w_j$ have the same length, $p_i$ and $p_j \sigma _{[0,n]}(0^{j-i})$ also have the same length, which is at least $|\sigma _{[0,n]}(0)|$ . However, as $\sigma _{[0,n]}(0)$ is not a suffix of $\sigma _{[0,n]}(1)$ , it cannot be a suffix of $p_i$ . Nonetheless, $\sigma _{[0,n]}(0)$ is clearly a suffix of $p_j\sigma _{[0,n]}(0^{j-i})$ . Thus, $p_i \neq p_j\sigma _{[0,n]}(0^{j-i})$ and we conclude.

Claim 3. If the integer sequence $(a_n)_{n \in {\mathbb N}}$ satisfies $(a_{n+1}+1)> (5n + 7)A_n/B_n$ , then $ \lim _{n\to \infty } p_{X_{\boldsymbol \sigma }} (n+1) - p_{X_{\boldsymbol \sigma }} (n)= \lim _{n\to \infty } r(n) = \infty $ .

Proof. From Claim 2, we know that for every $n \geq 1$ , $k \in \{4,\ldots ,a_{n+1}\}$ , and $N \in \{ kB_n + C_n +1, \ldots , (k+1)B_n + C_n \}$ , $r(N) \geq \min (k, \lfloor A_n/B_n \rfloor )$ . Also, since $A_n/B_n \geq c \ a_n$ , $ \lim _{n \to \infty } \min (n,\lfloor A_n/B_n \rfloor ) = \infty $ .

Now, we have that

$$ \begin{align*} \{m,m+1,\ldots \} &= \bigcup_{n\geq 4}\bigcup_{k=n}^{a_{n+1}} \{kB_n + C_n +1, \ldots, (k+1)B_n + C_n\} \\ &=\bigcup_{n\geq 4} \{nB_n + C_n +1, \ldots, (a_{n+1}+1)B_n + C_n\} \end{align*} $$

for some integer m (which can be computed explicitly). Indeed, as $B_n \leq A_n$ ,

$$ \begin{align*} (n+1)B_{n+1} + C_{n+1} +1 &= (n+1)(2B_n + 3A_n) + 2B_n + C_n +1 \\ &\leq (5n+7)A_n + C_n + 1 \leq (a_{n+1}+1) B_n + C_n. \end{align*} $$

Therefore, for any $N \geq m$ , there exist $n\geq 4$ , $k\in \{n,\ldots ,a_{n+1}\}$ such that $N\in \{nB_n + C_n +1, \ldots , (a_{n+1}+1)B_n + C_n\}$ , and thus

$$ \begin{align*}r(N)\geq \min(k,\lfloor A_n/B_n \rfloor ) \geq \min(n,\lfloor A_n/B_n \rfloor ),\end{align*} $$

and we conclude.

In conclusion, $(X_{\boldsymbol \sigma },S)$ has superlinear complexity and $\boldsymbol \sigma $ is $2$ -consecutive, so for $\mu \in {\mathcal E}(X_{\boldsymbol \sigma },S)$ , by Theorem 5.10, $(X_{\boldsymbol \sigma },{\mathcal B}, \mu , S)$ is $\tfrac 14$ -rigid.

When specializing Theorem 5.10 to constant length m-consecutive directive sequences, we get the following theorem that does not require the hypothesis of finite topological rank.

Theorem 5.12. Let $m\geq 2$ and $\boldsymbol \sigma = (\sigma _n \colon {\mathcal A}_{n+1}^* \to {\mathcal A}_n^*)_{n \in {\mathbb N}}$ be a primitive, recognizable, constant length, and m-consecutive directive sequence. Then, every invariant measure for $(X_{\boldsymbol \sigma }, S) $ is $({m-1})/{m}$ -rigid.

Proof. We can use the first part of the proof of Theorem 5.10 which does not require the system to be of finite alphabet rank. Thus, let $\mu $ be an ergodic measure and take $n \geq n_0$ , then for every $b \in {\mathcal A}_n$ , we have

$$ \begin{align*} \mu({\mathscr T}_{bb}^{(n)}) \geq \frac{m-1}{m} \mu({\mathscr T}_{b}^{(n)}). \end{align*} $$

However, as the directive sequence has constant length, for every $a,b \in {\mathcal A}_n$ , $aa \sim _n bb$ . Therefore,

$$ \begin{align*} \mu({\mathscr T}_{[aa]_n}^{(n)}) = \sum_{b \in {\mathcal A}_n} \mu({\mathscr T}_{bb}^{(n)}) \geq \frac{m-1}{m} \sum_{b \in {\mathcal A}_n} \mu({\mathscr T}_{b}^{(n)}) = \frac{m-1}{m} \end{align*} $$

and, as $\mu $ is an arbitrary ergodic measure, by Theorem A, we conclude that every ergodic measure is $({m-1})/{m}$ -rigid (recall that in this context, ergodic measures are non-atomic). Moreover, from Remark 3.6, it is known that the partial rigidity sequence is $(h^{(n)})_{n \in {\mathbb N}}$ , which, in this case, is independent of the measure, so by Remark 2.2, every invariant measure is $({m-1})/{m}$ -rigid.

Remark 5.13. It is interesting to note that for every Toeplitz subshift $(X, S)$ (see [Reference Downarowicz18] for a precise definition), there exists a minimal Cantor system that is rigid for every ergodic measure $\mu $ and is strong orbit equivalent to $(X,S)$ . A simple proof can be constructed with the results of [Reference Gjerde and Johansen29] and the elements developed in this section.

Remark 5.14. In [Reference Cecchi-Bernales and Donoso8, Theorem 1.4], there is a construction of a recognizable, infinite alphabet rank, primitive, and constant length directive sequence $\boldsymbol \sigma $ such that $(X_{\boldsymbol {\sigma }},S)$ has superlinear complexity and with infinitely many ergodic measures. A slight modification of that example allows us to ensure that the directive sequence is also $2$ -consecutive and therefore, by Theorem 5.12, the system is partially rigid for every ergodic measure. Thus, the above criteria can be applied beyond non-superlinear complexity systems and exact finite rank systems.

5.2.3 Conditions regarding return words

This condition may be one of the most restrictive. Still, it can be used to prove the partial rigidity for the natural coding of interval exchange transformations and, more generally, for dendric subshifts (see for instance [Reference Berthé, Dolce, Durand, Leroy and Perrin4, Reference Durand and Perrin21]). Although the above are classical examples where the proposition can be used, the range of applications is broader. This is because the condition only requires finiteness of the return words sets for infinitely many levels and not for every level.

Proposition 5.15. Let $\boldsymbol \sigma = (\sigma _n \colon {\mathcal A}_{n+1}^* \to {\mathcal A}_n^*)_{n \in {\mathbb N}}$ be a primitive and recognizable directive sequence. If there is a constant $c> 0$ such that $ \sum _{a \in {\mathcal A}_n} | {\mathcal R}_{X_{\boldsymbol \sigma }^{(n)}}(a)| \leq c $ for infinitely many $n \in {\mathbb N}$ , then $(X_{\boldsymbol \sigma },{\mathcal B},\mu ,S)$ is ${1}/{c}$ -rigid for every ergodic measure $\mu $ .

Proof. Observe that

(15) $$ \begin{align} {\mathscr T}^{(n)}_a = \bigcup_{w \in {\mathcal R}_{X_{\boldsymbol \sigma}^{(n)}}(a)} {\mathscr T}_{aw}^{(n)} \end{align} $$

and thus

(16) $$ \begin{align} X_{\boldsymbol \sigma} = \bigcup_{a \in {\mathcal A}_n} \bigcup_{w \in {\mathcal R}_{X_{\boldsymbol \sigma}^{(n)}}(a)} {\mathscr T}_{aw}^{(n)}. \end{align} $$

Let ${\mathcal N}\subseteq {\mathbb N}$ be infinite such that $ \sum _{a \in {\mathcal A}_n} | {\mathcal R}_{X_{\boldsymbol \sigma }^{(n)}}(a)| \leq c $ for all $n\in {\mathcal N}$ . Then, we have that there exists a sequence of complete words $(w(n))_{n \in {\mathcal N}}$ with $w(n) = a_nw'(n)$ , where $w'(n) \in {\mathcal R}_{X_{\boldsymbol \sigma }^{(n)}}(a_n)$ , such that $\mu ({\mathscr T}^{(n)}_{w(n)}) \geq {1}/{c} $ . From this, we conclude using Corollary 3.7.

We deduce the following.

Corollary 5.16. Let $(X,S)$ be a minimal subshift on an alphabet ${\mathcal A}$ . If there is a number $d>0$ such that $|{\mathcal R}_X (w) | \leq d$ for every word $w \in {\mathcal L}(X)$ , then $(X,{\mathcal B},\mu ,S)$ is ${1}/{d^2}$ -rigid for every ergodic measure $\mu $ .

Before proving the corollary, we state two lemmas whose proofs can be found in [Reference Durand and Perrin21]. To state them, note that given a word $u \in {\mathcal L}(X)$ appearing in a minimal subshift X, the set $U = {\mathcal R}_X(u)$ is a circular code, that is, every word $w \in {\mathcal A}^*$ admits at most one decomposition as a concatenation of elements of U and if $uv, vu \in U^*$ , then $u,v \in U^*$ . A circular morphism is a morphism $\varphi ^* \colon {\mathcal B}^* \to {\mathcal A}^*$ whose restriction to ${\mathcal B}$ is a bijection into a circular code $U \subseteq {\mathcal A}^*$ .

Lemma 5.17. [Reference Durand and Perrin21, Proposition 1.4.32]

A (non-erasing) morphism $\sigma \colon {\mathcal B}^* \to {\mathcal A}^*$ is recognizable on ${\mathcal B}^{{\mathbb Z}}$ if and only if $\sigma $ is a circular morphism.

Lemma 5.18. [Reference Durand and Perrin21, Lemma 6.4.11]

Let $(X,S)$ be a minimal subshift on an alphabet ${\mathcal A}$ and $(u_n)_{n \in {\mathbb N}}$ be a sequence of words in ${\mathcal L}(X)$ , with $u_0 = \varepsilon $ , such that $u_n$ is a proper suffix of $u_{n+1}$ for every $n\in {\mathbb N}$ . Let ${\mathcal A}_n = \{ 1,\ldots , |{\mathcal R}_X(u_n)| \}$ and $\alpha _n \colon {\mathcal A}_n^* \to {\mathcal A}^*$ a coding morphism for ${\mathcal R}_X(u_n)$ . Let $\boldsymbol \tau $ be the sequence of morphisms such that $\alpha _n \circ \tau _n = \alpha _{n+1}$ for every $n \in {\mathbb N}$ . Then, $\boldsymbol \tau $ is a primitive directive sequence such that $X_{\boldsymbol \tau } = X$ .

Proof of Corollary 5.16

Let $(u_n)_{n \in {\mathbb N}}$ be a sequence of words in ${\mathcal L}(X)$ such that $u_0 = \varepsilon $ , $u_{n+1} \in {\mathcal R}_X(u_n)$ , and $|u_{n+1}|>|u_n|$ . In particular, $u_n$ is a proper suffix of $u_{n+1}$ . Let $\boldsymbol \tau = (\tau _n \colon {\mathcal A}_{n+1 }^* \to {\mathcal A}_n^* )_{n \in {\mathbb N}}$ be the directive sequence associated with $(u_n)_{n \in {\mathbb N}}$ as in Lemma 5.18.

By construction of $\boldsymbol \tau $ , $\tau _n({\mathcal A}_{n+1}) = {\mathcal R}_{X_{\boldsymbol {\tau }}^{(n)}}(a_n)$ , where $a_n \in {\mathcal A}_n$ is the letter such that $\tau _{[0,n)}(a) = u_{n+1}$ . Thus, by Lemma 5.17, $\tau _n$ is recognizable for every $n \in {\mathbb N}$ and, therefore, $\boldsymbol \tau $ is recognizable.

By assumption, for every $n \in {\mathbb N}$ , $|{\mathcal R}_X(u_n)| \leq d$ , and then $|{\mathcal A}_n| \leq d $ . Moreover, for every $n\in {\mathbb N}$ and $a \in {\mathcal A}_n$ , $\tau _{[0,n)}({\mathcal R}_{X^{(n)}_{\boldsymbol \tau }} (a)) = {\mathcal R}_X (\tau _{[0,n)}(a))$ and therefore $|{\mathcal R}_{X^{(n)}_{\boldsymbol \tau }} (a)| \leq d$ . Therefore, $\sum _{a \in {\mathcal A}_n} | {\mathcal R}_{X^{(n)}_{\boldsymbol \tau }} (a) | \leq d^2$ . Applying Proposition 5.15 to the directive sequence $\boldsymbol {\tau }$ , we obtain that $X_{\boldsymbol {\tau }}$ is ${1}/{d^2}$ -rigid for every ergodic measure.

Remark 5.19. Every statement involving the set of right return words ${\mathcal R}_X(w)$ (Proposition 5.15, Corollary 5.16, and Lemma 5.18) also applies to the set of left return words ${\mathcal R}_X'(w)$ .

6.1 Necessary conditions for rigidity

We recall that for a subshift X, a set of complete words on ${\mathcal L}(X)$ is denoted by ${\mathcal C} {\mathcal L} (X)$ . We start with a general lemma.

Lemma 6.1. Let $\boldsymbol \sigma = (\sigma _n \colon {\mathcal A}_{n+1}^* \to {\mathcal A}_n^*)_{n \in {\mathbb N}}$ be a primitive and recognizable directive sequence, and $\mu $ be an ergodic measure on $X_{\boldsymbol \sigma }$ . If $(X_{\boldsymbol \sigma }, {\mathcal B}, \mu ,S)$ is rigid, then

(17) $$ \begin{align} \limsup_{n \rightarrow \infty} \bigg[ \mu \bigg( \bigcup_{w \in {\mathcal C}{\mathcal L}_n(X_{\boldsymbol{\sigma}})} [w]_{X_{\boldsymbol \sigma}} \bigg) \bigg] = 1, \end{align} $$

where ${\mathcal C}{\mathcal L}_n(X_{\boldsymbol \sigma }) = {\mathcal C}{\mathcal L}(X_{\boldsymbol \sigma }) \cap {\mathcal A}_0^{n} $ .

Proof. As $(X_{\boldsymbol \sigma }, {\mathcal B}, \mu ,S)$ is rigid, by Corollary 3.5 and Theorem 4.4, there is a sequence of complete words $(w(n))_{n\in {\mathbb N}}$ , $w(n) \in {\mathcal C} {\mathcal L}( X^{(n)}_{\boldsymbol \sigma })$ such that

$$ \begin{align*} \lim_{n \rightarrow \infty} \mu({\mathscr T}_{[w(n)]_n}^{(n)}) = 1. \end{align*} $$

Fix $n\in {\mathbb N}$ , $u = u_1 \cdots u_{r} \in [w(n)]_n$ , and $\ell = |\sigma _{[0,n)}(u_1) |$ . For $0 \leq k \leq \ell $ , let $p_{k}^{u_1,n}$ be the kth prefix of $\sigma _{[0,n)}(u_1)$ (that is, $|p_{k}^{u_1,n}| = k$ and it is a prefix of $\sigma _{[0,n)}(u_1)$ ) and $s_{k}^{u_1,n}$ be the kth suffix of $\sigma _{[0,n)}(u_1)$ (that is, $|s_{k}^{u_1,n}| = \ell - k$ and it is a suffix of $\sigma _{[0,n)}(u_1)$ ). Notice that $p_{0}^{u_1,n}$ and $s_{\ell }^{u_1,n}$ are empty words. Then,

$$ \begin{align*} {\mathscr T}_{u}^{(n)} \subseteq \bigcup_{v \in {\mathcal C}_{u}^n} [v]_{X_{\boldsymbol \sigma}}, \end{align*} $$

where ${\mathcal C}_{u}^n = \{ s_{k}^{u_1,n} \sigma _{[0,n)}(u_2 \cdots u_{r-1}) p_{k+1}^{u_1,n} : 0 \leq k < \ell \} \subseteq {\mathcal A}^*_0$ . Moreover, every word $v \in {\mathcal C}_u^n$ is a complete word because the last letter of $p_{k+1}^{u_1,n}$ is equal to the first letter of $s_{k}^{u_1,n}$ . Thus, by definition,

(18) $$ \begin{align} {\mathscr T}_{[w(n)]_n }^{(n)} \subseteq \bigcup_{u \in [w(n)]_n} \bigcup_{v \in {\mathcal C}_{u}^n} [v]_{X_{\boldsymbol \sigma}}. \end{align} $$

To conclude, notice that by the definition of $\sim _n$ , for $m_n = |\sigma _{[0,n)} (u_1 \cdots u_{r-1})| + 1$ , one has that if $ v \in \bigcup _{u \in [w(n)]_n} {\mathcal C}_{u}^n$ , then $|v| = m_n$ . Therefore, for every $n \in {\mathbb N}$ , $ \bigcup _{u \in [w(n)]_n} {\mathcal C}_{u}^n \subseteq {\mathcal C} {\mathcal L}_{m_n} ({X}) $ and then

$$ \begin{align*} 1 = \limsup_{n \rightarrow \infty} \mu ({\mathscr T}_{[w(n)]_n}^{(n)}) &\leq \limsup_{n \rightarrow \infty} \mu \bigg( \bigcup_{u \in [w(n)]_n} \bigcup_{v \in {\mathcal C}_{u}^n} [v]_{X_{\boldsymbol \sigma}} \bigg) \\ & \leq \limsup_{n \rightarrow \infty} \mu \bigg( \bigcup_{v \in {\mathcal C}{\mathcal L}_{m_n}(X)} [v]_{X_{\boldsymbol \sigma}} \bigg) \\ &\leq \limsup_{n \rightarrow \infty} \mu \bigg( \bigcup_{v \in {\mathcal C}{\mathcal L}_{n}(X)} [v]_{X_{\boldsymbol \sigma}} \bigg). \end{align*} $$

This allows us to conclude.

We recall that a linearly recurrent subshift is an ${\mathcal S}$ -adic subshift $(X_{\boldsymbol \sigma }, S)$ such that there exists a finite set of morphisms ${\mathcal S}$ such that the directive sequence $\boldsymbol \sigma = (\sigma _n \colon {\mathcal A}_{n+1}^* \to {\mathcal A}_n^*)_{n \in {\mathbb N}}$ is proper, positive, and $\sigma _n \in {\mathcal S}$ for all $n \geq 1$ . Linearly recurrent subshifts are uniquely ergodic. We will use that if $(X_{\boldsymbol \sigma }, S)$ is linearly recurrent, then there is a constant $L>0$ such that $\mu ([w]_{X_{\boldsymbol {\sigma }}}) \leq L \mu ([u]_{X_{\boldsymbol \sigma }})$ for all words $u,w \in {\mathcal L}(X_{\boldsymbol \sigma })$ of the same length (see [Reference Durand19, Proposition 13]).

Theorem 6.3. Let $(X_{\boldsymbol \sigma }, S)$ be a linearly recurrent subshift and let $\mu $ be its unique invariant measure. If $(X_{\boldsymbol \sigma }, {\mathcal B}, \mu ,S)$ is rigid, then

(19) $$ \begin{align} \limsup_{n \rightarrow \infty} \frac{q_{X_{\boldsymbol \sigma}}(n)}{p_{X_{\boldsymbol \sigma}}(n)} = 1, \end{align} $$

where $p_{X_{\boldsymbol \sigma }}(n) = | {\mathcal L}_n(X_{\boldsymbol \sigma })|$ and $q_{X_{\boldsymbol \sigma }}(n)= | {\mathcal C} {\mathcal L}_n(X_{\boldsymbol \sigma })|$ .

To prove this theorem, we use the following lemma.

Lemma 6.4. Let $(X_{\boldsymbol \sigma }, S)$ be a linearly recurrent subshift and let $\mu $ be its unique invariant measure. Let $(W_n)_{n\in {\mathbb N}}$ such that $W_n \subseteq {\mathcal L}_n(X_{\boldsymbol \sigma })$ for all $n\in {\mathbb N}$ . Then, for any increasing sequence of integers $(m_n)_{n\in {\mathbb N}}$ , we have that

$$ \begin{align*} \frac{|W_{m_n}|}{p_{X_{\boldsymbol \sigma}}(m_n)} \xrightarrow[n \rightarrow \infty]{} 0\quad \text{if and only if } \mu \bigg( \bigcup_{w \in W_{m_n}} [w]_{X_{\boldsymbol \sigma}} \bigg) \xrightarrow[n \rightarrow \infty]{} 0. \end{align*} $$

Proof. For $m\in {\mathbb N}$ , let $ w_m, u_m \in {\mathcal L}_m(X_{\boldsymbol \sigma })$ such that $\mu ( [w_m]_{X_{\boldsymbol \sigma }} ) = \max _{w \in {\mathcal L}_m (X_{\boldsymbol \sigma })} \mu ( [w]_{X_{\boldsymbol \sigma }} )$ and $\mu ( [u_m]_{X_{\boldsymbol \sigma }} ) = \min _{w \in {\mathcal L}_m (X_{\boldsymbol \sigma })}$ $ \mu ( [w]_{X_{\boldsymbol \sigma }} )$ . Let $L>0$ such that $\mu ( [w_m]_{X_{\boldsymbol \sigma }} )\leq L\mu ( [u_m]_{X_{\boldsymbol \sigma }} )$ for all $m\in {\mathbb N}$ . Combining this with the equality

$$ \begin{align*}\mu\bigg( \bigcup_{w \in W_m} [w]_{X_{\boldsymbol \sigma}} \bigg) = \frac{ \mu( \bigcup_{w \in W_m} [w]_{X_{\boldsymbol \sigma}} ) }{ \mu( \bigcup_{w \in {\mathcal L}_m({X_{\boldsymbol \sigma}})} [w]_{X_{\boldsymbol \sigma}} ) }= \frac{ \sum_{w \in W_m} \mu( [w]_{X_{\boldsymbol \sigma}} ) }{ \sum_{w \in {\mathcal L}_m({X_{\boldsymbol \sigma}})} \mu( [w]_{X_{\boldsymbol \sigma}} ) } , \end{align*} $$

we obtain that

$$ \begin{align*} \frac{1}{L}\frac{|W_m|}{p_{X_{\boldsymbol \sigma}}(m)}\leq \mu\bigg( \bigcup_{w \in W_m} [w]_{X_{\boldsymbol \sigma}} \bigg) \leq L \frac{|W_m|}{p_{X_{\boldsymbol \sigma}}(m)} \end{align*} $$

from where the conclusion follows.

Proof of Theorem 6.3

By Lemma 6.1, since $\mu $ is rigid, we have that

$$ \begin{align*} \lim_{n \rightarrow \infty} \mu \bigg( \bigcup_{w \in {\mathcal C}{\mathcal L}_{m_n}({X_{\boldsymbol \sigma}})} [w]_{X_{\boldsymbol \sigma}} \bigg) = 1 \end{align*} $$

for an increasing sequence of integers $(m_n)_{n \in {\mathbb N}}$ . Setting $W_n = {\mathcal L}_{m_n}(X_{\boldsymbol \sigma }) \backslash {\mathcal C} {\mathcal L}_{m_n} (X_{\boldsymbol \sigma }) $ , by Lemma 6.4,

$$ \begin{align*} \lim_{n \to \infty} \frac{|W_n|}{p_{X_{\boldsymbol \sigma}}(m_n)} = 0. \end{align*} $$

We conclude by noting that $|W_n| + q_{X_{\boldsymbol \sigma }}(m_n) = p_{X_{\boldsymbol \sigma }}(m_n)$ for every $n \in {\mathbb N}$ .

6.2 A rigid ${\mathcal S}$ -adic subshift with multiple ergodic measures

To illustrate some applications of the results in §5, we construct an ${\mathcal S}$ -adic subshift with $r>0$ ergodic measures, each one rigid for the same rigidity sequence. Examples with the same property can be found for some Toeplitz systems [Reference Williams44]. To introduce the ${\mathcal S}$ -adic subshift, we quote the following lemma. We need the following notion: for a vector $v \in {\mathbb R}^{{\mathcal A}}$ , set $|v| = \sum _{a \in {\mathcal A}} |v(a)|$ .

Proof. Let ${\mathcal A} = \{ a_1, b_1, a_2, b_2, \ldots , a_r, b_r\} $ . We define $\boldsymbol \sigma = ( \sigma _n \colon {\mathcal A}^* \to {\mathcal A}^* )_{n \in {\mathbb N}} $ such that

$$ \begin{align*} \sigma_n(a_i) &= a_1 a_2 \cdots a_r (a_i b_i)^{2^n} b_1 b_2 \cdots b_r, \\ \sigma_n(b_i) &= a_1 a_2 \cdots a_r ( b_i a_i)^{2^n} b_1 b_2 \cdots b_r \end{align*} $$

for $i\in \{1,\ldots ,r\}$ , which is recognizable, positive, and proper. Moreover, $\boldsymbol \sigma $ has constant length and we denote by $h^{(n)}$ the height at level n.

Claim. There are r ergodic measures $\mu _1, \ldots , \mu _r$ and for each $i \in \{1,\ldots ,r\}$ , ${\mathcal A}_{\mu _i} = \{a_i,b_i\}$ .

Notice that if $\mu $ is an ergodic measure, then it is clear that $a_i \in {\mathcal A}_{\mu }$ if and only if $b_i \in {\mathcal A}_{\mu }$ , because

$$ \begin{align*} \mu({\mathscr T}_{a_i}^{(n)}) &= \frac{h^{(n)}}{h^{(n+1)}} \bigg( 2^{n+1} (\mu({\mathscr T}_{a_i}^{(n+1)}) + \mu({\mathscr T}_{b_i}^{(n+1)})) + \sum_{j \neq i} (\mu({\mathscr T}_{a_j}^{(n+1)}) + \mu({\mathscr T}_{b_j}^{(n+1)})) \bigg) \\ &= \mu({\mathscr T}_{b_i}^{(n)}). \end{align*} $$

To conclude the claim, note that there are two possibilities: either there is a unique invariant measure $\mu $ such that ${\mathcal A}_{\mu } = {\mathcal A}$ or there are r ergodic measures $\mu _i$ such that ${\mathcal A}_{\mu _i} = \{a_i, b_i\}$ for all $i \in \{1,\ldots ,r\}$ . This is trivial when $r \in \{1,2\} $ . Suppose without loss of generality that $r\geq 3$ , $a_1, a_2,b_1,b_2 \in {\mathcal A}_{\mu }$ for some ergodic measure $\mu $ and define $\phi _i \colon X_{\boldsymbol \sigma } \to X_{\boldsymbol \sigma }$ as the homeomorphism that, for every $x \in X_{\boldsymbol \sigma }$ , interchanges the letters $a_1, b_1$ with $a_i, b_i$ , respectively, for $ i \in \{3,\ldots ,r\}$ . Then, $\phi _i$ commutes with S, and the pushforward measure $\mu ' = \phi _i \mu $ is an S-ergodic probability measure such that $a_i, b_i , a_2, b_2 \in {\mathcal A}_{\mu '}$ . In particular, ${\mathcal A}_{\mu } \cap {\mathcal A}_{\mu '} \neq \emptyset $ and, therefore, $\mu = \mu '$ and $a_i, b_i \in {\mathcal A}_{\mu }$ . Since $i \in \{3,\ldots , r\}$ is arbitrary, we conclude that ${\mathcal A}_{\mu } = {\mathcal A}$ . Finally, assuming that ${\mathcal A}_{\mu _1} = \{a_1, b_1\}$ , if we define $\mu _i = \phi _i \mu _1$ for every $i \in \{1,\ldots ,r\}$ , then ${\mathcal A}_{\mu _i} = \{a_i,b_i\}$ .

Now, using the same notation as in Lemma 6.7, for every contraction $\boldsymbol {\sigma '}$ of $\boldsymbol {\sigma }$ , ${|v_k|}/{|\sigma ^{\prime }_k|}\hspace{-1pt} =\hspace{-1pt} O({1}/{2^k})$ for every k and, therefore, $ \sum _{k \in {\mathbb N}} {|v_k|}/{|\sigma ^{\prime }_k|}\hspace{-1pt} <\hspace{-1pt} \infty $ . Thus, by Lemma 6.7, the system cannot be uniquely ergodic and we conclude the claim.

Claim. For $i \in \{1,\ldots ,r\}$ , $\mu _i$ is rigid and $(2 h^{(n)})_{n \in {\mathbb N}} $ is a rigidity sequence.

Notice that as ${\mathcal A}_{\mu _i} = \{a_i,b_i\} $ , $\mu _i(({\mathscr T}^{(n)}_{a_i} \cup {\mathscr T}^{(n)}_{b_i}) \cap ({\mathscr T}^{(n+1)}_{a_i} \cup {\mathscr T}^{(n+1)}_{b_i} )) \xrightarrow [n \to \infty ]{} 1 $ . Also, by Lemma 5.1,

$$ \begin{align*} \mu_i({\mathscr T}^{(n)}_{a_i b_i a_i}) &= \mu_i({\mathscr T}^{(n)}_{a_i b_i a_i} \cap {\mathscr T}^{(n+1)}_{a_i}) + \mu_i({\mathscr T}^{(n)}_{a_i b_i a_i} \cap {\mathscr T}^{(n+1)}_{b_i}) \\ &\geq \frac{n^2-1}{ n^2 + 1} \mu_i({\mathscr T}^{(n)}_{a_i } \cap {\mathscr T}^{(n+1)}_{a_i}) + \frac{n^2-1}{ n^2 + 1} \mu_i({\mathscr T}^{(n)}_{a_i } \cap {\mathscr T}^{(n+1)}_{b_i}), \end{align*} $$

and the same applies to $ \mu ({\mathscr T}^{(n)}_{b_i a_i b_i}) $ . As $f_{{\mathcal A}}(a_i b_i) = f_{{\mathcal A}} (b_ia_i)$ , by Lemma 3.4, $a_i b_i a_i \sim _n b_i a_i b_i$ , so

$$ \begin{align*} \mu_i({\mathscr T}^{(n)}_{[a_i b_i a_i]_n}) \geq \frac{n^2-1}{n^2+1} \bigg( \sum_{j,l \in \{a_i,b_i\} } \mu_i({\mathscr T}^{(n)}_{j } \cap {\mathscr T}^{(n+1)}_{l}) \bigg) \xrightarrow[n \to \infty]{} 1. \end{align*} $$

Then, $(X_{\boldsymbol \sigma }, {\mathcal B}, \mu _i, S)$ is rigid for every $i \in \{1,\ldots ,r\}$ and a rigidity sequence is given by $h^{(n)}_{a_i} + h^{(n)}_{b_i} = 2 h^{(n)}$ . Finally, by Remark 2.2, every invariant measure is rigid.

7.1 Partial rigidity rate in constant length ${\mathcal S}$ -adic subshifts

Theorem 7.1. Let $\boldsymbol \sigma = (\sigma _n \colon {\mathcal A}_{n+1}^* \to {\mathcal A}_n^*)_{n \in {\mathbb N}}$ be a recognizable, constant length, and primitive directive sequence. Let $\mu $ be an S-invariant ergodic measure on $X_{\boldsymbol \sigma }$ . Then,

(20) $$ \begin{align} \delta_{\mu} = \inf_{n \geq 1} \sup_{\ell \geq 2} \bigg\{\! \sum_{\substack{w \in {\mathcal C}{\mathcal L}^{(n)}(\boldsymbol \sigma) \\ \lvert w \rvert = \ell }} \mu^{(n)} ([w]_{X^{(n)}_{\boldsymbol \sigma}}) \bigg\}, \end{align} $$

where $ \mu ^{(n)} $ is the induced measure on the base $B^{(n)}$ of the natural sequence of Kakutani–Rokhlin partitions $({\mathscr P}^{(n)})_{n \in {\mathbb N}}$ associated with $(X_{\boldsymbol \sigma },S)$ .

Proof. By Theorem 4.4,

$$ \begin{align*} \delta_{\mu} = \inf_{n \geq 1} \bigg\{\! \sup_{\substack{w \in {\mathcal A}_n^* \\ w_1 = w_{|w|} }} \mu ( {\mathscr T}_{[w]_n}^{(n)} ) \bigg\}. \end{align*} $$

First, note that two complete words $u,w \in {\mathcal A}_n^*$ are $\sim _n$ equivalent if and only if $\lvert u \rvert = \lvert w \rvert $ . Indeed, for any word $v\in {\mathcal A}_n^*$ , $\sum _{i=1}^{|v|-1} h^{(n)}_{v_i} = (|v|-1) h^{(n)} $ , where $h^{(n)}$ is the height of every tower.

In addition, if $w \not \in {\mathcal L}^{(n)} (\boldsymbol \sigma ) $ , then $\mu ({\mathscr T}^{(n)}_{w})=0$ . Therefore, we can consider only the complete words of the language of $X_{\boldsymbol \sigma }^{(n)}$ . Finally, for $w \in {\mathcal C} {\mathcal L}^{(n)} (\boldsymbol \sigma )$ ,

$$ \begin{align*} \mu({\mathscr T}^{(n)}_w) &= \frac{\mu({\mathscr T}^{(n)}_w) }{ \sum_{a \in {\mathcal A}_n} \mu ({\mathscr T}^{(n)}_a) } \\ &= \frac{h^{(n)} \mu(B_w^{(n)}) }{ h^{(n)} \sum_{a \in {\mathcal A}_n} \mu (B_a^{(n)}) } \\ &= \frac{\mu(B_w^{(n)}) }{ \mu (B^{(n)}) } = \mu^{(n)} ([w]_{X^{(n)}_{\boldsymbol \sigma}}). \end{align*} $$

Thus, we have proved that $ \mu ( {\mathscr T}_{[w]_n}^{(n)} ) = \sum _{\substack {w \in {\mathcal C}{\mathcal L}^{(n)}(\boldsymbol \sigma ) \\ \lvert w \rvert = \ell }} \mu ^{(n)} ([w]_{X^{(n)}_{\boldsymbol \sigma }})$ and we are done.

Corollary 7.2. Let $\boldsymbol \sigma = (\sigma _n \colon {\mathcal A}_{n+1}^* \to {\mathcal A}_n^*)_{n \in {\mathbb N}}$ be a recognizable, constant length, and primitive directive sequence. Let $\mu $ be an S-invariant ergodic measure on $X_{\boldsymbol \sigma }$ and $ \mu ^{(m)} $ the induced measure on $X^{(m)}_{\boldsymbol {\sigma }}$ for $m \in {\mathbb N}$ . Then,

(21) $$ \begin{align} \delta_{\mu} = \delta_{ \mu^{(m)}}. \end{align} $$

Proof. As was proven in Theorem 4.4, $ (\sup _{\ell \geq 2} \{ \sum _{\substack {w \in {\mathcal C}{\mathcal L}^{(n)}(\boldsymbol \sigma ) \\ \lvert w \rvert = \ell }} \mu ^{(n)} ([w]_{X^{(n)}_{\boldsymbol \sigma }}) \} )_{n \in {\mathbb N}}$ is a decreasing sequence. Therefore,

$$ \begin{align*} \inf_{n \geq 1} \sup_{\ell \geq 2} \bigg\{\! \sum_{\substack{w \in {\mathcal C}{\mathcal L}^{(n)}(\boldsymbol \sigma) \\ \lvert w \rvert = \ell }} \mu^{(n)} ([w]_{X^{(n)}_{\boldsymbol \sigma}}) \bigg\} = \inf_{n \geq m} \sup_{\ell \geq 2} \bigg\{\! \sum_{\substack{w \in {\mathcal C}{\mathcal L}^{(n)}(\boldsymbol \sigma) \\ \lvert w \rvert = \ell }} \mu^{(n)} ([w]_{X^{(n)}_{\boldsymbol \sigma}}) \bigg\}. \end{align*} $$

So, equation (21) follows from Theorem 7.1.