Multiple partial rigidity rates in low complexity subshifts

In this paper, a topological dynamical system is a pair $(X,T)$, where $X$ is a compact metric space and $T\colon X\to X$ is a homeomorphism. We say that $(X,T)$ is minimal if for every $x\in X$ the orbit $\{T^{n}x:n\in{\mathbb{Z}}\}$ is dense in $X$. A continuous and onto map $\pi\colon X_{1}\to X_{2}$ between two topological dynamical systems $(X_{1},T_{1})$ and $(X_{2},T_{2})$ is a factor map if for every $x\in X_{1}$, $T_{2}\circ\pi(x)=\pi\circ T_{1}(x)$.

We focus on a special family of topological dynamical system, symbolic systems. To define them, let $A$ be a finite set that we call alphabet. The elements of $A$ are called letters. For $\ell\in{\mathbb{N}}$, the set of concatenations of $\ell$ letters is denoted by $A^{\ell}$ and $w=w_{1}\ldots w_{\ell}\in A^{\ell}$ is a word of length $\ell$. The length of a word $w$ is denoted by $|w|$. We set $A^{*}=\bigcup_{n\in{\mathbb{N}}}A^{\ell}$ and by convention, $A^{0}=\{\varepsilon\}$ where $\varepsilon$ is the empty word.

For a word $w=w_{1}\ldots w_{\ell}$ and two integers $1\leq i<j\leq\ell$, we write $w_{[i,j+1)}=w_{[i,j]}=w_{i}\ldots w_{j}$. 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 $|w|_{u}$ denotes the number of (possibly overleaped) occurrences of $u$ in $w$. We also write ${\rm freq}(u,w)=\frac{|w|_{u}}{|w|}$, the frequency of $u$ in $w$.

Let $A^{{\mathbb{Z}}}$ be the set of two-sided sequences $(x_{n})_{n\in{\mathbb{Z}}}$, where $x_{n}\in A$ for all $n\in{\mathbb{Z}}$. Like for finite words, for $x\in A^{{\mathbb{Z}}}$ and $-\infty<i<j<\infty$ we write $x_{[i,j]}=x_{[i,j+1)}$ for the finite word given by $x_{i}x_{i+1}\ldots x_{j}$. The set $A^{{\mathbb{Z}}}$ endowed with the product topology is a compact and metrizable space. The shift map $S\colon A^{{\mathbb{Z}}}\to A^{{\mathbb{Z}}}$ is the homeomorphism defined by $S((x_{n})_{n\in{\mathbb{Z}}})=(x_{n+1})_{n\in{\mathbb{Z}}}$. Notice that, the collection of cylinder sets $\{S^{j}[w]\colon w\in A^{*},j\in{\mathbb{Z}}\}$ where $[w]=\{x\in A^{{\mathbb{Z}}}\colon x_{[0,|w|)}=w\}$, is a basis of clopen subsets for the topology of $A^{{\mathbb{Z}}}$.

A subshift is a topological dynamical system $(X,S)$, where $X$ is a closed and $S$-invariant subset of $A^{{\mathbb{Z}}}$. In this case the topology is also given by cylinder sets, denoted $[w]_{X}=[w]\cap X$, but when there is no ambiguity we just write $[w]$. Given an element $x\in X$, the language ${\mathcal{L}}(x)$ is the set of all words appearing in $x$ and ${\mathcal{L}}(X)=\bigcup_{x\in X}{\mathcal{L}}(x)$. Notice that $[w]_{X}\neq\emptyset$ if and only if $w\in{\mathcal{L}}(X)$. Also, $(X,S)$ is minimal if and only if ${\mathcal{L}}(X)={\mathcal{L}}(x)$ for all $x\in X$.

Let $A$ and $B$ be finite alphabets and $\sigma\colon A^{*}\to B^{*}$ be a morphism for the concatenation, that is $\sigma(uw)=\sigma(u)\sigma(w)$ for all $u,w\in A^{*}$. A morphism $\sigma\colon A^{*}\to B^{*}$ is completely determined by the values of $\sigma(a)$ for every letter $a\in A$. We only consider non-erasing morphisms, that is $\sigma(a)\neq\varepsilon$ for every $a\in A$, where $\varepsilon$ is the empty word in $B^{*}$. A morphism $\sigma\colon A^{*}\to A^{*}$ is called a substitution if for every $a\in A$, $\displaystyle\lim_{n\to\infty}|\sigma^{n}(a)|=\infty$.

A directive sequence $\boldsymbol{\sigma}=(\sigma_{n}\colon A^{*}_{n+1}\to A^{*}_{n})_{n\in{\mathbb{N}}}$ is a sequence of (non-erasing) morphisms. Given a directive sequence $\boldsymbol{\sigma}$ and $n\in{\mathbb{N}}$, define ${\mathcal{L}}^{(n)}(\boldsymbol{\sigma})$, the language of level $n$ associated to $\boldsymbol{\sigma}$ by

where $\sigma_{[n,N)}=\sigma_{n}\circ\sigma_{n+1}\circ\ldots\circ\sigma_{N-1}$. For $n\in{\mathbb{N}}$, we define $X_{\boldsymbol{\sigma}}^{(n)}$, the $n$-th level subshift generated by $\boldsymbol{\sigma}$, as the set of elements $x\in A_{n}^{{\mathbb{Z}}}$ such that ${\mathcal{L}}(x)\subseteq{\mathcal{L}}^{(n)}(\boldsymbol{\sigma})$. For the special case $n=0$, we write $X_{\boldsymbol{\sigma}}$ instead of $X_{\boldsymbol{\sigma}}^{(0)}$ and we call it the ${\mathcal{S}}$-adic subshift generated by $\boldsymbol{\sigma}$.

A morphism $\sigma\colon A^{*}\to B^{*}$ has a composition matrix $M(\sigma)\in{\mathbb{N}}^{B\times A}$ given by $M(\sigma)_{b,a}=|\sigma(a)|_{b}$ for all $b\in B$ and $a\in A$. If $\tau\colon B^{*}\to C^{*}$ is another morphism, then $M(\tau\circ\sigma)=M(\tau)M(\sigma)$. Therefore, for a substitution, $\sigma\colon A^{*}\to A^{*}$, $M(\sigma^{2})=M(\sigma)^{2}$. We say that $\boldsymbol{\sigma}$ is primitive if for every $n\in{\mathbb{N}}$ there exists $k\geq 1$ such that the matrix $M(\sigma_{[n,n+k]})=M(\sigma_{n})M(\sigma_{n+1})\cdots M(\sigma_{n+k})$ has only positive entries. When $\boldsymbol{\sigma}$ is primitive, then for every $n\in{\mathbb{N}}$ $(X_{\boldsymbol{\sigma}}^{(n)},S)$ is minimal and ${\mathcal{L}}(X^{(n)}_{\boldsymbol{\sigma}})={\mathcal{L}}^{(n)}(\boldsymbol{\sigma})$.

When $\boldsymbol{\sigma}$ is the constant directive sequence $\sigma_{n}=\sigma$ for all $n\in{\mathbb{N}}$, where $\sigma\colon A^{*}\to A^{*}$ is a substitution, then $X_{\boldsymbol{\sigma}}$ is denoted $X_{\sigma}$ and it is called substitution subshift. Similarly ${\mathcal{L}}(\boldsymbol{\sigma})$ is denoted ${\mathcal{L}}(\sigma)$. Also if in that context $\boldsymbol{\sigma}$ is primitive, we say that the substitution $\sigma$ itself is primitive, which is equivalent to saying that the composition matrix $M(\sigma)$ is primitive. We also say that the substitution $\sigma$ is positive if $M(\sigma)$ only have positive entries. By definition, every positive substitution is also primitive.

A morphism $\sigma\colon A^{*}\to B^{*}$ has constant length if there exists a number $\ell\geq 1$ such that $|\sigma(a)|=\ell$ for all $a\in A$. In this case, we write $|\sigma|=\ell$. More generally, a directive sequence $\boldsymbol{\sigma}=(\sigma_{n}\colon A^{*}_{n+1}\to A^{*}_{n})_{n\in{\mathbb{N}}}$ is of constant-length if each morphism $\sigma_{n}$ is of constant length. Notice that we do not require that $|\sigma_{n}|=|\sigma_{m}|$ for distinct $n,m\in{\mathbb{N}}$.

We define the alphabet rank $AR$ of $\boldsymbol{\sigma}=(\sigma_{n}\colon A^{*}_{n+1}\to A^{*}_{n})_{n\in{\mathbb{N}}}$ as $\displaystyle AR(\boldsymbol{\sigma})=\liminf_{n\to\infty}|A_{n}|$. Having finite alphabet rank has many consequences, for instance if $AR(\boldsymbol{\sigma})<\infty$ then $X_{\boldsymbol{\sigma}}$ has zero topological entropy.

For a general subshift $(X,S)$, let $p_{X}\colon{\mathbb{N}}\to{\mathbb{N}}$ denote the word complexity function of $X$ given by $p_{X}(n)=|{\mathcal{L}}_{n}(X)|$ for all $n\in{\mathbb{N}}$. Here ${\mathcal{L}}_{n}(X)=\{w\in{\mathcal{L}}(X)\colon|w|=n\}$. If $\displaystyle\liminf_{n\to\infty}\frac{p_{X}(n)}{n}=\infty$ we say that $X$ has superlinear complexity. Otherwise we say $X$ has non-superlinear complexity.

We say that a primitive substitution $\tau\colon A^{*}\to A^{*}$ is right prolongable (resp. left prolongable) on $u\in A^{*}$ if $\tau(u)$ starts (resp. ends) with $u$. If, for every letter $a\in A$, $\tau\colon A^{*}\to A^{*}$ is left and right prolongable on $a$, then $\tau\colon A^{*}\to A^{*}$ is said to be prolongable. A word $w=w_{1}\ldots w_{\ell}\in{\mathcal{A}}^{*}$ is complete if $\ell\geq 2$ and $w_{1}=w_{\ell}$. Notice that if a substitution $\tau\colon A^{*}\to A^{*}$ is primitive and prolongable, then $\tau(a)$ is a complete word for every $a\in A$. If $W$ is a set of words, then we denote

the set of complete words in $W$. In particular, for $k\geq 2$, ${\mathcal{C}}A^{k}$ is the set of complete words of length $k$ with letters in $A$, for example, ${\mathcal{C}}\{a,b\}^{3}=\{aaa,aba,bab,bbb\}$.

Finally, when the alphabet has two letters ${\mathcal{A}}=\{a,b\}$, the complement of a word $w=w_{1}\ldots w_{\ell}\in{\mathcal{A}}^{*}$ denoted $\overline{w}$ is given by $\overline{w}_{1}\ldots\overline{w}_{\ell}$ where $\overline{a}=b$ and $\overline{b}=a$. A morphism $\tau\colon{\mathcal{A}}^{*}\to{\mathcal{A}}^{*}$ is said to be a mirror morphism if $\tau(\overline{w})=\overline{\tau(w)}$ (the name is taken from [25, Chapter 8.2] with a slight modification).

A measure preserving system 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 $A\in{\mathcal{X}}$, and we say that $\mu$ is $T$-invariant. An invariant measure $\mu$ is said to be ergodic if whenever $A\subseteq X$ is measurable and $\mu(A\Delta T^{-1}A)=0$, then $\mu(A)=0$ or $1$.

Given a topological dynamical system $(X,T)$, we denote ${\mathcal{M}}(X,T)$ (resp. ${\mathcal{E}}(X,T)$) the set of Borel $T$-invariant probability measures (resp. 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.

If $(X,S)$ is a subshift over an alphabet $A$, then any invariant measure $\mu\in{\mathcal{M}}(X,S)$ is uniquely determined by the values of $\mu([w]_{X})$ for $w\in{\mathcal{L}}(X)$. Since $X\subset A^{{\mathbb{Z}}}$, $\mu\in{\mathcal{M}}(X,S)$ can be extended to $A^{{\mathbb{Z}}}$ by $\tilde{\mu}(B)=\mu(B\cap X)$ for all $B\subset A^{{\mathbb{Z}}}$ measurable. In particular, $\tilde{\mu}([w])=\mu([w]_{X})$ for all $w\in A^{*}$. We use this extension many times, making a slight abuse of notation and not distinguishing between $\mu$ and $\tilde{\mu}$. Moreover, for $w\in A^{*}$, since there is no ambiguity with the value of the cylinder set we write $\mu(w)$ instead of $\mu([w])$. This can also be done when we deal with two alphabets $A\subset B$, every invariant measure $\mu$ in $A^{{\mathbb{Z}}}$ can be extended to an invariant measure in $B^{{\mathbb{Z}}}$, where in particular, $\mu(b)=0$ for all $b\in B\backslash A$.

A sequence of non-empty subsets of the integers, $\boldsymbol{\Phi}=(\Phi_{n})_{n\in{\mathbb{N}}}$ is a Følner sequence if for all $t\in{\mathbb{Z}}$, $\displaystyle\lim_{n\to\infty}\frac{|\Phi_{n}\Delta(\Phi_{n}+t)|}{|\Phi_{n}|}=0$. Let $(X,T)$ be a topological system and let $\mu$ be an invariant measur, an element $x\in X$ is said to be generic along $\boldsymbol{\Phi}$ if for every continuous function $f\in C(X)$

Every point in a minimal system is generic for some Følner sequence $\boldsymbol{\Phi}$, more precisely

In particular, for an ${\mathcal{S}}$-adic subshift with primitive directive sequence $\boldsymbol{\sigma}=(\sigma_{n}\colon A_{n+1}^{*}\to A_{n}^{*})_{n\in{\mathbb{N}}}$, when the infinite word $\boldsymbol{w}=\displaystyle\lim_{n\to\infty}\sigma_{0}\circ\sigma_{1}\circ\cdots\circ\sigma_{n-1}(a_{n})$ is well-defined then every invariant measure $\mu\in{\mathcal{M}}(X_{\boldsymbol{\sigma}},S)$ is given by

for some $(m_{n})_{n\in{\mathbb{N}}},(m^{\prime}_{n})_{n\in{\mathbb{N}}}\subset{\mathbb{N}}$ as before. Notice that such infinite word $\boldsymbol{w}$ is well-defined for example when $A_{n}=A$, $a_{n}=a$ and $\sigma_{n}\colon A^{*}\to A^{*}$ is prolongable, for all $n\in{\mathbb{N}}$, where $A$ and $a\in A$ are a fixed alphabet and letter respectively. Those are the condition for the construction of the system announced in Theorem 1.1.

We remark that for a primitive substitution, $\sigma\colon A^{*}\to A^{*}$ the substitution subshift $(X_{\sigma},S)$ is uniquely ergodic and the invariant measure is given by any limit of the form (3).

Every ${\mathcal{S}}$-adic subshift can be endowed with a natural sequence of Kakutani-Rokhlin partitions see for instance [4, Lemma 6.3], [16, Chapter 6] or [13, section 5]. To do this appropriately, one requires recognizability of the directive sequence $\boldsymbol{\sigma}=(\sigma_{n}\colon A_{n+1}^{*}\to A_{n}^{*})_{n\in{\mathbb{N}}}$, where we are using the term recognizable as defined in [4]. We do not define it here, but if every morphism $\sigma_{n}\colon A_{n+1}^{*}\to A_{n}^{*}$ is left-permutative, that is the first letter of $\sigma_{n}(a)$ is distinct from the first letter of $\sigma_{n}(a^{\prime})$ for all $a\neq a^{\prime}$ in $A_{n}$, then the directive sequence is recognizable. In this case we say that the directive sequence $\boldsymbol{\sigma}$ itself is left-permutative. If $\tau\colon A^{*}\to A^{*}$ is prolongable, then it is left-permutative.

Once we use the Kakutani-Rokhlin partition structure, $X^{(n)}_{\boldsymbol{\sigma}}$ can be identified as the induced system in the $n$-th basis and for every invariant measure $\mu^{\prime}$ in $X^{(n)}_{\boldsymbol{\sigma}}$, there is an invariant measure $\mu$ in $X_{\boldsymbol{\sigma}}$ such that $\mu^{\prime}$ is the induced measure of $\mu$ in $X^{(n)}_{\boldsymbol{\sigma}}$. We write $\mu^{\prime}=\mu^{(n)}$ and this correspondence is one-to-one. This is a crucial fact for computing the partial rigidity rate for an ${\mathcal{S}}$-adic subshift, for instance, if $\boldsymbol{\sigma}$ is a directive sequence of constant-length, $\delta_{\mu}=\delta_{\mu^{(n)}}$ for all $\mu\in{\mathcal{E}}(X_{\boldsymbol{\sigma}},S)$ and $n\geq 1$ (see Theorem 2.3). Since the aim of this paper is building a specific example, we give a way to characterize $\mu^{(n)}$ for a more restricted family of ${\mathcal{S}}$-adic subshift that allows us to carry out computations.

In what follows, we restrict the analysis to less general directive sequences $\boldsymbol{\sigma}$. To do so, from now on, ${\mathcal{A}}$ always denotes the two letters alphabet $\{a,b\}$. Likewise, for $d\geq 2$, ${\mathcal{A}}_{i}=\{a_{i},b_{i}\}$ for $i\in\{0,\ldots,d-1\}$ and $\Lambda_{d}=\bigcup_{i=0}^{d-1}{\mathcal{A}}_{i}$.

We cite a simplified version of [5, Theorem 4.9], the original proposition is stated for Bratelli-Vershik transformations, but under recognizability, it can be stated for ${\mathcal{S}}$-adic subshifts, see [4, Theorem 6.5].

Whenever $\boldsymbol{\sigma}=(\sigma_{n}\colon A_{n+1}^{*}\to A_{n}^{*})_{n\in{\mathbb{N}}}$ is a constant-length directive sequence, we write $h^{(n)}=|\sigma_{[0,n)}|$ where we recall that $\sigma_{[0,n)}=\sigma_{0}\circ\sigma_{1}\circ\cdots\circ\sigma_{n-1}$.

Another useful characterization of the invariant measures is given by explicit formulas between the invariant measures of $X_{\boldsymbol{\sigma}}^{(n)}$ and $X_{\boldsymbol{\sigma}}^{(n+1)}$. To do so we combine [2, Proposition 1.1, Theorem 1.4] and [1, Proposition 1.4]. In the original statements one needs to normalize the measures to get a probability measure (see [1, Proposition 1.3]), but for constant length morphisms the normalization constant is precisely the length of the morphism. Before stating the lemma, for $\sigma\colon A^{*}\to B^{*}$, $w\in A^{*}$ and $u\in B^{*}$, we define $\lfloor\sigma(w)\rfloor_{u}$, the essential occurrence of $u$ on $\sigma(w)$, that is the number of times such that $u$ occurs on $w$ for which the first letter of $u$ occurs in the image of the first letter of $w$ under $\sigma$, and the last letter of $u$ occurs in the image of last letter of $w$ under $\sigma$.