2 Definitions

2.2 Shift spaces and languages

Let $\mathscr {A}$ be a finite set, henceforth referred to as the alphabet. The full shift $\mathscr {A}^{\infty }$ is the set of all $\mathscr {A}$ -valued infinite sequences indexed by non-negative integers. We endow $\mathscr {A}^\infty $ with the product topology coming from the discrete topology on $\mathscr {A}$ . A metric on $\mathscr {A}^\infty $ compatible with the product topology is

$$ \begin{align*} \rho(x,y)=\begin{cases} 0 & \mbox{if } x=y, \\ 2^{-\min\{j:x_j\neq y_j\}} & \mbox{otherwise}. \end{cases} \end{align*} $$

We consider $\mathscr {A}^\infty $ a dynamical system under the action of the shift transformation $\sigma \colon \mathscr {A}^\infty \to \mathscr {A}^\infty $ , where $\sigma (x)_j=x_{j+1}$ for every $x=(x_i)_{i=0}^\infty \in \mathscr {A}^\infty $ and $j\ge 0$ . A shift space over $\mathscr {A}$ is a non-empty, closed and shift invariant subset of $\mathscr {A}^\infty $ . A block or a word over $\mathscr {A}$ is a finite sequence of symbols from $\mathscr {A}$ . The length of a word $w$ over $\mathscr {A}$ is denoted by $|w|$ . The concatenation of words $u$ and $v$ is denoted simply as $uv$ . We agree that the empty word has length $0$ . Given $x \in X$ and $i,j\in \mathbb {N}_0$ with $i<j$ , we set $x_{[i,j)}$ to be the word $w=w_1,\ldots , w_n\in \mathscr {A}^n$ such that $n=j-i$ and $x_{i+k-1} = w_k$ for all $1 \leq k \leq n$ . We say that a word $\omega \in \mathscr {A}^n$ appears in a shift space $X \subseteq \mathscr {A}^\infty $ if there exist some $x\in X$ and $\ell \in \mathbb {N}_0$ such that $\omega =x_{[\ell ,\ell +n)}$ . For $n\in \mathbb {N}$ , we write $\mathcal {B}_n(X) \subseteq \mathscr {A}^n$ for the set of blocks of length n appearing in X. The language of a shift space $X\subseteq \mathscr {A}^\infty $ is the set $\mathcal {B}(X)$ of finite words over $\mathscr {A}$ that appear in X. A shift space X is transitive if for every word $u,w\in \mathcal {B}(X)$ there is a word $v$ such that $uvw\in \mathcal {B}(X)$ . We say that a shift space X is topologically mixing if for every word $u,w\in \mathcal {B}(X)$ there is $N\in \mathbb {N}_0$ such that for every $n\ge N$ there is a word $v$ satisfying $uvw\in \mathcal {B}(X)$ and $|v|=n$ .

2.4 Ergodic properties of shift spaces

The set of all Borel probability measures on a shift space X is denoted by $\mathcal {M}(X)$ . We equip $\mathcal {M}(X)$ with the weak $^*$ topology, which is known to be metrizable and compact. The set of shift invariant probability measures supported on a shift space $X\subseteq \mathscr {A}^\infty $ is denoted by $\mathcal {M}_{\sigma } (X)$ , and $\mathcal {M}_{\sigma }^{\rm e}(X)$ stands for the set of ergodic $\sigma $ -invariant measures on X.

A point $x\in \mathscr {A}^\infty $ is generic for an ergodic measure $\mu \in \mathcal {M}_{\sigma }^{\rm e}(\mathscr {A}^\infty )$ , if for every continuous function $f\colon \mathscr {A}^\infty \to \mathbb {R}$ the sequence

$$ \begin{align*} \frac{1}{N}\sum_{j=0}^{N-1}f(\sigma^n(x)) \end{align*} $$

converges as $N\to \infty $ to $\int _{\mathscr {A}^\infty }f\,\text {d}\mu $ . Every ergodic measure has a generic point. Given $\mu \in \mathcal {M}_{\sigma } (\mathscr {A}^\infty )$ , we let $h(\mu )$ denote the Kolmogorov–Sinai entropy of $\mu $ .

The ergodic measures are entropy-dense if for every measure $\mu \in \mathcal {M}_{\sigma } (X)$ , every neighbourhood U of $\mu $ in $\mathcal {M}_{\sigma } (X)$ and every $\varepsilon>0$ there is an ergodic measure $\nu \in U$ with $|h(\nu )-h(\mu )|<\varepsilon $ .

Let us point out that property of having entropy-dense ergodic measures is preserved by conjugacy. In other words, two conjugated dynamical systems either both have entropy-dense ergodic measures or both do not have this property.

3 Alternative (pseudo)metrics for $\mathscr {A}^\infty $ and $\mathcal {M}_{\sigma } (\mathscr {A}^\infty )$ and their hyperspaces

3.1 The functions ${\underline d}$ , ${\bar d}$ and $\bar {d}_{\mathcal {M}}$

We now discuss the pre- and pseudometrics ${\underline d}$ and ${\bar d}$ for the space $\mathscr {A}^\infty $ and Ornstein’s metric $\bar {d}_{\mathcal {M}}$ on $\mathcal {M}_{\sigma } (\mathscr {A}^\infty )$ . These functions are not compatible with the natural topologies on $\mathscr {A}^\infty $ and $\mathcal {M}_{\sigma } (\mathscr {A}^\infty )$ , which are the product topology and the weak $^*$ topology, respectively. Note that usually $\bar {d}_{\mathcal {M}}$ is also denoted by ${\bar d}$ , but as we will juggle between ${\bar d}$ and $\bar {d}_{\mathcal {M}}$ a lot, we decided to change the notation to avoid confusion. We call $\bar {d}_{\mathcal {M}}$ the ‘d-bar distance for measures’, and its pseudometric version for $\mathscr {A}^\infty $ we call ‘pointwise d-bar’.

Given $x=(x_n)_{n=0}^\infty ,y=(y_n)_{n=0}^\infty \in \mathscr {A}^\infty $ , we define

$$ \begin{gather*} {\bar d}(x,y)=\limsup_{n\to\infty}\frac{1}{n}|\{0\le j < n:x_j\neq y_j\}|, \\ {\underline d}(x,y)=\liminf_{n\to\infty}\frac{1}{n}|\{0\le j < n:x_j\neq y_j\}|. \end{gather*} $$

The function ${\underline d}$ is only a premetric, that is, ${\underline d}$ is a real-valued, non-negative, symmetric function on $\mathscr {A}^\infty \times \mathscr {A}^\infty $ vanishing on the diagonal $\{(x,y)\in \mathscr {A}^\infty \times \mathscr {A}^\infty : x=y\}$ . It is easy to see that the triangle inequality fails for ${\underline d}$ on $\mathscr {A}^\infty $ . The function ${\bar d}$ is a pseudometric, that is, ${\bar d}$ is a premetric satisfying the triangle inequality and the implication ${\bar d}(x,y)=0\implies x=y$ fails, so ${\bar d}$ is not a metric on $\mathscr {A}^\infty $ if $\mathscr {A}$ has at least two elements. Furthermore, ${\bar d},{\underline d}\colon \mathscr {A}^\infty \times \mathscr {A}^\infty \to [0,1]$ are both shift invariant, that is, ${\underline d}(x,y)={\underline d}(\sigma (x),\sigma (y))$ and ${\bar d}(x,y)={\bar d}(\sigma (x),\sigma (y))$ for all $x,y\in \mathscr {A}^\infty $ .

The distance function $\bar {d}_{\mathcal {M}}$ is defined using joinings. We say that a measure

on $\mathscr {A}^\infty \times \mathscr {A}^\infty $ is a joining of shift invariant measures $\mu $ and $\nu $ , if

is $\sigma \times \sigma $ -invariant and $\mu $ and $\nu $ are the marginal measures for

under the projection to the first, respectively the second, coordinate. We denote by $J(\mu ,\nu )$ the set of all joinings of $\mu $ and $\nu $ . It is a non-empty set because the product measure $\mu \times \nu $ is always a joining of $\mu $ and $\nu $ . We can now define $\bar {d}_{\mathcal {M}}$ distance on $\mathcal {M}_{\sigma } (\mathscr {A}^\infty )$ as follows:

where $d_0(x,y)=1$ if $x_0\neq y_0$ and $d_0(x,y)=0$ otherwise. It is well known that $\bar {d}_{\mathcal {M}}$ is a metric on $\mathcal {M}_{\sigma } (\mathscr {A}^\infty )$ and the convergence in this metric implies weak $^*$ convergence. This metric was introduced by Ornstein (for more details, see [Reference Ornstein44]). Furthermore, the space $\mathcal {M}_{\sigma } (\mathscr {A}^\infty )$ under the $\bar {d}_{\mathcal {M}}$ metric is complete but not separable; in particular, the space $\mathcal {M}_{\sigma } (\mathscr {A}^\infty )$ endowed with $\bar {d}_{\mathcal {M}}$ is not compact. The space $\mathcal {M}_{\sigma }^{\rm e}(\mathscr {A}^\infty )\subseteq \mathcal {M}_{\sigma } (\mathscr {A}^\infty )$ of ergodic measures is $\bar {d}_{\mathcal {M}}$ -closed, as are the spaces of strong-mixing and Bernoulli measures on $\mathscr {A}^\infty $ . The entropy function $\mu \mapsto h(\mu )$ is continuous on $\mathcal {M}_{\sigma } (\mathscr {A}^\infty )$ under $\bar {d}_{\mathcal {M}}$ .

3.2 Hausdorffification of ${ \underline {d}}$ , ${ \bar d}$ , and ${ \bar {d}}_{\mathcal {M}}$

In order to introduce the notion of approximation of one shift with another, we extend the functions $\bar {d}_{\mathcal {M}}$ , ${\bar d}$ and ${\underline d}$ so that they make sense for pairs of subsets of $\mathscr {A}^\infty $ and $\mathcal {M}_{\sigma } (\mathscr {A}^\infty )$ following the well-known construction of the Hausdorff metric.

In general, a Hausdorff premetric may be defined on the powerset of any bounded premetric (or pseudometric, or metric) space. If $(Z,\rho )$ is a set equipped with a bounded premetric, then we define the Hausdorff premetric $\rho ^H$ induced by $\rho $ on the space of all non-empty subsets of Z. For a point $z\in Z$ and non-empty $A,B\subseteq Z$ , we put

$$ \begin{align*}\rho(z,B)=\inf_{b\in B}\rho(z,b)\quad\text{and}\quad \rho^H(A,B)=\max\Big\{\sup_{a\in A}\rho(a,B),\ \sup_{b\in B}\rho(b,A)\Big\}.\end{align*} $$

The function $\rho ^H$ is a premetric, which becomes a pseudometric whenever Z is a bounded pseudometric space. Even if $\rho $ is a bounded metric on Z, its Hausdorff counterpart induced on the powerset of Z might still be only a pseudometric. This is because for non-empty $A,B\subset Z$ we have $\rho (A,B)=\rho (\overline {A},\overline {B})$ , where $\overline {(\cdot )} $ is the closure operator naturally defined on the powerset of $(Z,\rho )$ . Nevertheless, if $\rho $ is a bounded metric, then the Hausdorff pseudometric $\rho ^H$ induced by $\rho $ becomes a metric when restricted to the set $\operatorname {{\rm CL}}(Z,\rho )$ of closed non-empty subsets of $(Z,\rho )$ . We apply these ideas to the metric space $(\mathcal {M}_{\sigma } (\mathscr {A}^\infty ),\bar {d}_{\mathcal {M}})$ , the pseudometric space $(\mathscr {A}^\infty ,{\bar d})$ , and the premetric space $(\mathscr {A}^\infty ,{\underline d})$ , obtaining a metric $\bar {d}^H_{\mathcal {M}}$ on the space $\operatorname {{\rm CL}}(\mathcal {M}_{\sigma } (\mathscr {A}^\infty ),\bar {d}_{\mathcal {M}})$ of non-empty closed subsets of $(\mathcal {M}_{\sigma } (\mathscr {A}^\infty ),\bar {d}_{\mathcal {M}})$ together with a pseudometric ${\bar d}^H$ and a premetric ${\underline d}^H$ on the set of all non-empty subsets of $\mathscr {A}^\infty $ .

Below, we discuss some properties of the convergence of sets with respect to the Hausdorff metric $\rho ^H$ in the case when $(Z,\rho )$ is not necessarily compact bounded pseudometric space. In this setting some properties, well known in the compact case, fail. For example, equivalent metrics, that is, metrics $\rho ,\tilde {\rho }$ inducing the same topology on Z, may induce non-homeomorphic spaces $(\operatorname {{\rm CL}}(Z),\rho ^H)$ and $(\operatorname {{\rm CL}}(Z),\tilde {\rho }^H)$ .

The following propositions gather some properties of the Hausdorff metric. Proofs can be found in the literature (see [Reference Illanes and Nadler25, §2.15]).

Proposition 1. Let $(Z,\rho )$ be a bounded metric space. If $(Z,\rho )$ is a complete metric space, then so is $(\operatorname {{\rm CL}}(Z),\rho ^H)$ .

It follows from Proposition 1 that the Hausdorff metric $\bar {d}^H_{\mathcal {M}}$ induced for $\operatorname {{\rm CL}}(\mathcal {M}_{\sigma } (\mathscr {A}^\infty ))$ by $\bar {d}_{\mathcal {M}}$ is complete. Therefore the usual Cauchy condition provides a criterion for convergence, but even if we know that $(\mathcal {M}_{\sigma } (X_k))_{k=1}^\infty $ converges in $\bar {d}_{\mathcal {M}}$ to some $\mathcal {M}\in \operatorname {{\rm CL}}(\mathcal {M}_{\sigma } (\mathscr {A}^\infty ),\bar {d}_{\mathcal {M}})$ , it is not clear if there exists a shift space $X\subseteq \mathscr {A}^\infty $ such that $\mathcal {M}=\mathcal {M}_{\sigma } (X)$ . Nevertheless, if $X_1\supseteq X_2 \supseteq \cdots $ , then we can set X to be the intersection of all $X_n$ .

Proposition 2. If $(X_n)_{n=1}^\infty $ is a decreasing sequence of shift spaces over $\mathscr {A}$ , then $X=\bigcap _{n=1}^\infty X_n$ is a non-empty shift space such that $\mathcal {M}_{\sigma } (X)=\bigcap _{n=1}^\infty \mathcal {M}_{\sigma } (X_n)$ .

Proof. Since $X_1\supseteq X_2\supseteq \cdots \supseteq X$ , we have

$$ \begin{align*} \mathcal{M}_{\sigma} (X)=\mathcal{M}_{\sigma} \bigg(\bigcap_{n=1}^\infty X_n\bigg)\subseteq\bigcap_{n=1}^\infty\mathcal{M}_{\sigma} (X_n) .\end{align*} $$

On the other hand, if $\mu \in \bigcap _{n=1}^\infty \mathcal {M}_{\sigma } (X_n)$ , then $\mu (X_n)=1$ for every $n\ge 1$ . Therefore $\mu (X)=1$ . It follows that $\mu \in \mathcal {M}_{\sigma } (X)$ .

Lemma 3. If $(X_n)_{n=1}^\infty $ is a decreasing sequence of shift spaces over $\mathscr {A}$ such that the sequence $(\mathcal {M}_{\sigma } (X_n))_{n=1}^\infty $ is $\bar {d}^H_{\mathcal {M}}$ -Cauchy, then $X=\bigcap _{n=1}^\infty X_n$ is a shift space such that $\bar {d}^H_{\mathcal {M}}(\mathcal {M}_{\sigma } (X_n),\mathcal {M}_{\sigma } (X)) \to 0$ as $n \to \infty $ .

Proof. It follows from Proposition 1 that $(\mathcal {M}_{\sigma } (X_k))_{k=1}^\infty $ converges to some $\mathcal {M}\in \operatorname {{\rm CL}}(\mathscr {A}^\infty ,\bar {d}_{\mathcal {M}})$ with respect to $\bar {d}^H_{\mathcal {M}}$ . Using Proposition 2, we can easily identify this limit with $\mathcal {M}_{\sigma } (X)$ .

4 ${\bar d}$ -approachability

In this section we introduce ${\bar d}$ -approachable shift spaces which are limits of their own canonical finite type approximations not only in the ‘usual’ Hausdorff metric topology, but also in the topology given by the ${\bar d}^H$ pseudometric.

Clearly, every shift of finite type is ${\bar d}$ -approachable. More examples of ${\bar d}$ -approachable shift spaces will follow from the characterization of chain-mixing ${\bar d}$ -approachable shift spaces by a notion we call ${\bar d}$ -shadowing. The ${\bar d}$ -shadowing property is closely related to the average shadowing property introduced by Blank [Reference Blank6] and discussed in more detail in [Reference Kulczycki, Kwietniak and Oprocha35]. Actually, following the ideas presented in [Reference Kulczycki, Kwietniak and Oprocha35], one can prove that a shift space X with the average shadowing property also has ${\bar d}$ -shadowing. Furthermore, for a surjective shift space X (i.e. when $\sigma (X)=X$ ), ${\bar d}$ -shadowing implies the average shadowing property. We leave the details to the interested reader. We decided to use ${\bar d}$ -shadowing because it is much easier to use for symbolic systems and allows us to keep our paper self-contained by avoiding a very technical detour into topological dynamics. We will apply a similar strategy to several other notions: instead of presenting a general (i.e. stated for continuous maps acting on compact metric spaces) definition of some properties (e.g. chain transitivity, chain mixing, specification and its variants), we will present equivalent definitions adapted to the symbolic dynamics setting.

It is easy to see that ${\bar d}$ -shadowing implies ${\bar d}$ -approachability, but to prove the converse we need an additional assumption. We say that a shift space is chain transitive (respectively, chain mixing) if $X^M_n$ is transitive (respectively, topologically mixing) for all except finitely many n. For shift spaces this definition is equivalent to the usual one phrased in terms of $\delta $ -chains (see [Reference Kůrka36]).

Our main result in this section characterizes chain-mixing ${\bar d}$ -approachable shift spaces. As mentioned above, this characterization is a topological counterpart of the result saying that a totally ergodic shift invariant probability measure is Bernoulli if and only if its canonical k-step Markov approximations converge to $\mu $ with respect to $\bar {d}_{\mathcal {M}}$ .

As a consequence of Theorem 6 we obtain that a transitive but not topologically mixing shift of finite type is an example of a ${\bar d}$ -approachable shift space which is not chain mixing, hence it does not have the ${\bar d}$ -shadowing property. In Example 11 we show that there are non- ${\bar d}$ -approachable transitive sofic shifts. It follows that condition (1) in Theorem 6 cannot be relaxed.

The proof of the following lemma is straightforward. We will later generalize it in §6; see Lemma 22 there.

Proof. Recall that the condition saying that X is a mixing sofic shift space implies that there exists $k\ge 0$ such that for every two words $u,w\in \mathcal {B}(X)$ one can find (this property is called the specification property and we discuss it in more detail in §6) a word $v$ of length k such that $uvw\in \mathcal {B}(X)$ . Fix $\varepsilon>0$ . Let N be such that $k/N<\varepsilon $ . Let $(w^{(j)})_{j=1}^\infty $ be a sequence of words in $\mathcal {B}(X)$ satisfying $|w^{(j)}|\ge N$ for $j=1,2,\ldots .$ For $j=1,2,\ldots ,$ we set $\ell (j)=|w^{(j)}|$ and let $u^{(j)}=w^{(j)}_1\ldots w^{(j)}_{\ell (j)-k}$ be the prefix of $w^{(j)}$ of length $\ell (j)-k$ . Using the above-mentioned property of mixing sofic shifts, we can find a sequence $(v^{(j)})_{j=1}^\infty $ of words of length k such that

$$ \begin{align*} x=u^{(1)}v^{(1)}u^{(2)}v^{(2)}\ldots u^{(n)}v^{(n)}\ldots \end{align*} $$

belongs to X. We also have

$$ \begin{align*} {\bar d}(x,w^{(1)}w^{(2)}\ldots w^{(n)}\ldots)\le k/N<\varepsilon. \\[-42pt] \end{align*} $$

The proof of the next lemma is adapted from the proof of [Reference Kwietniak, Łącka and Oprocha39, Lemma 25].

Proof. Fix $\varepsilon>0$ . Let $N\in \mathbb {N}$ be such that for every $n\ge N$ we have

(1) $$ \begin{align} {\bar d}^H(X,X_n)<\varepsilon/2. \end{align} $$

Use ${\bar d}$ -shadowing of $X_N$ to find $M\ge 0$ such that if $y=w^{(1)}w^{(2)}w^{(3)}\ldots \in \mathscr {A}^\infty $ satisfies $w^{(j)}\in \mathcal {B}(X_N)$ and $|w^{(j)}|\ge M$ for every $j\in \mathbb {N}$ , then one can find $\bar {x}\in X_N$ satisfying

(2) $$ \begin{align} {\bar d}(\bar{x},y)<\varepsilon/2. \end{align} $$

Since $\mathcal {B}(X)\subseteq \mathcal {B}(X_N)$ , we can apply (1) and (2) to $y=w^{(1)}w^{(2)}w^{(3)}\ldots \in \mathscr {A}^\infty $ satisfying $w^{(j)}\in \mathcal {B}(X)$ and $|w^{(j)}|\ge M$ for every $j\in \mathbb {N}$ . It follows that there is $x\in X$ such that ${\bar d}(x,y)<\varepsilon $ .

The proof of the following lemma is adapted from the proof of [Reference Kulczycki, Kwietniak and Oprocha35, Lemma 3.1].

Proof. Assume that $\sigma (X)=X$ . Recall that X is chain mixing if and only if $X\times X$ is chain transitive. Note that it is enough to prove that X is chain transitive, because if X has ${\bar d}$ -shadowing and $\sigma (X)=X$ , then the same holds for $X\times X$ . Fix n. We will show that the Markov approximation $X_n$ is transitive, that is, for every $u,w\in \mathcal {B}_{n-1}(X)$ there is a word $v$ such that every word of length n appearing in $uvw$ belongs to $\mathcal {B}_n(X)$ . Fix $0<\varepsilon <1/(4n)$ . Use ${\bar d}$ -shadowing to find N for that $\varepsilon $ . Set $n_0:=\max \{N,n\}$ and $n_j=2^jn_0$ for $j\in \mathbb {N}$ . For $j\ge 0$ let $w^{(2j)}$ be any word in $\mathcal {B}(X)$ such that $|w^{(2j)}|=n_{2j}$ and $u$ is the prefix of $w^{(2j)}$ , and let $w^{(2j+1)}$ be any word in $\mathcal {B}(X)$ such that $|w^{(2j+1)}|=n_{2j+1}$ and $w$ is the suffix of $w^{(2j+1)}$ (the latter word exists because $\sigma (X)=X$ ). Thus for every $j\in \mathbb {N}_0$ we have words $u^{(2j)}$ , $v^{(2j+1)}$ such that

$$ \begin{align*} w^{(2j)}=uu^{(2j)}\quad\text{and}\quad w^{(2j+1)}=v^{(2j+1)}w. \end{align*} $$

Let $y=w^{(0)} w^{(1)}w^{(2)}\ldots .$ Write $\xi (0)=0$ and

$$ \begin{align*}\xi(k+1)=|w^{(0)}|+\cdots+|w^{(k)}|.\end{align*} $$

Therefore

$$ \begin{align*} y_{[\xi(2j),\xi(2j+1))}y_{[\xi(2j+1),\xi(2j+2))}=w^{(2j)}w^{(2j+1)}=uu^{(2j)}v^{(2j+1)}w. \end{align*} $$

By the ${\bar d}$ -shadowing property there is $x\in X$ such that ${\bar d}(x,y)<\varepsilon $ . We claim that there exists $j\ge 0$ such that for some $\xi (2j)+n\le p <\xi (2j+1)-n$ and $\xi (2j+1) \le q <\xi (2j+2)-2n$ we have

$$ \begin{align*} x_{[p,p+n)}= y_{[p,p+n)}\quad\text{and}\quad x_{[q,q+n)}= y_{[q,q+n)}. \end{align*} $$

It follows that $y_{[\xi (2j)p)}x_{[p,q+n)}y_{[q+n,\xi (2j+2))}\in \mathcal {B}(X_n)$ . If the claim is false, then easy computations show that ${\bar d}(x,y)\ge \varepsilon $ , which would be a contradiction. This completes our proof.

Proof. Let X be a shift space with the ${\bar d}$ -shadowing property. We need to show that ${\bar d}^H(X^M_n,X)\to 0$ as $n\to \infty $ . Fix $\varepsilon>0$ . For this $\varepsilon $ we choose N using the ${\bar d}$ -shadowing property. Let $n\ge N$ . Since $X\subseteq X^M_n$ it is enough to check that if $x\in X^M_n$ then one can find $y\in X$ with ${\bar d}(x,y)<\varepsilon $ . Hence we fix $x\in X^M_n$ and define the sequence $w^{(j+1)}=x_{[nj,n(j+1))}=x_{nj}x_{nj+1}\ldots x_{n(j+1)-1}$ for $j=0,1,2,\ldots .$ Since for each $k\ge 1$ we have $w^{(k)}\in \mathcal {B}_n(X)$ , because $\mathcal {B}_n(X_n)=\mathcal {B}_n(X)$ , we can apply ${\bar d}$ -shadowing to find $y\in X$ such that

$$ \begin{align*} {\bar d}(x,y)={\bar d}(w^{(1)}w^{(2)}w^{(3)}\ldots,y)<\varepsilon. \end{align*} $$

It follows ${\bar d}^H(X^M_n,X)<\varepsilon $ for $n\ge N$ as needed.

We finish this section with the proof of the characterization theorem.

In the next section we will see that if a shift space $X \subseteq \mathscr {A}^\infty $ is ${\bar d}$ -approachable and chain transitive, then its ergodic measures are entropy-dense.

Example 11. We show that even very simple transitive sofic shifts may have Markov approximations that are away in ${\bar d}$ . This example also shows that we cannot replace mixing sofic shifts in Theorem 6 by transitive sofic shifts.

The shift space X is defined as the set of all sequences $x\in \{0,1\}^\infty $ such that $x\in X$ if and only if $x_{2i}=0$ for every $i\ge 0$ or $x_{2i+1}=0$ for every $i\ge 0$ . Clearly, $X=\tilde {X}$ is a hereditary transitive sofic shift, which is not topologically mixing. Given $k\in \mathbb {N}$ , we define a periodic point

$$ \begin{align*}y^{(k)}=y_0^{(k)}y_1^{(k)}\ldots=((10)^{k+1} 0^{2k+1})^\infty.\end{align*} $$

We note that $y^{(k)}\in X^M_{2k}$ for every $k\in \mathbb {N}$ . By definition, for every $k\in \mathbb {N}$ and $j>0$ , the symbol $1$ occurs $2j(k+1)$ times in the block $y^{(k)}_{[0,2j(4k+3))}$ . These occurrences are evenly distributed among even and odd positions. Therefore for every $x\in \tilde X=X$ and $j\in \mathbb {N}$ we have

$$ \begin{align*} d_{\textrm{Ham}}(y^{(k)}_{[0,2j(4k+3)} x_{[0,2j(4k+3)})\ge \frac{j(k+1)}{j(8k+6)}\geq \frac18,\end{align*} $$

It follows immediately that for every $k\in \mathbb {N}$ we have

$$ \begin{align*}{\underline d}(y^{(k)},x)\geq \tfrac18. \end{align*} $$

Therefore for each $k\in \mathbb {N}$ ,

$$ \begin{align*}{\underline d}^H(X,X^M_{2k})\ge\tfrac18,\end{align*} $$

that is, X is an example of a non- ${\bar d}$ -approachable transitive sofic shift. In addition, since every point $y^{(k)}$ is generic for an ergodic measure $\nu ^{(k)}$ on $X^M_{2k}$ we can conclude that for each $k\in \mathbb {N}$ ,

$$ \begin{align*}\bar{d}^H_{\mathcal{M}}(\mathcal{M}_{\sigma} (X),\mathcal{M}_{\sigma} (X^M_{2k}))\ge \tfrac18.\end{align*} $$

This means that we can replace Markov approximation by the approximation by sofic shifts only if these sofic shift are mixing as in condition (1) of Theorem 6.

5 Approximation and entropy density

In this section we discuss approximation schemes of shift spaces related to the pseudometric ${\bar d}$ and the metric $\bar {d}_{\mathcal {M}}$ . Then we show that these approximation schemes allow for a transfer of entropy density.

Recall that a sequence of shift spaces $(X_n)_{n=1}^\infty \subseteq \mathscr {A}^\infty $ converges to a shift space $X\subseteq \mathscr {A}^\infty $ in the usual hyperspace topology if $\rho ^H(X_n,X)\to 0$ as $n\to \infty $ , where $\rho ^H$ is the Hausdorff metric corresponding to any metric $\rho $ compatible with the product topology on $\mathscr {A}^\infty $ . We similarly define the convergence in the usual hyperspace topology of $\mathcal {M}_{\sigma } (\mathscr {A}^\infty )$ , and we have that $\mathcal {M}_{\sigma } (X_n)$ converges to $\mathcal {M}_{\sigma } (X)$ as $n\to \infty $ if $D^H(X_n,X)\to 0$ as $n\to \infty $ , where $D^H$ is the Hausdorff metric corresponding to an arbitrary metric D compatible with the weak $^*$ topology on $\mathcal {M}_{\sigma } (\mathscr {A}^\infty )$ (we can replace D by any other compatible metric).

We will consider four new ways in which a sequence of shift spaces $(X_n)_{n=1}^\infty \subseteq \mathscr {A}^\infty $ can approximate a shift space $X\subseteq \mathscr {A}^\infty $ . These ways are given by the conditions

(3) $$ \begin{align} \lim_{n\to\infty}\bar{d}^H_{\mathcal{M}}(\mathcal{M}_{\sigma} (X_n),\mathcal{M}_{\sigma} (X))=0, \end{align} $$

(4) $$ \begin{align} \lim_{n\to\infty}\bar{d}^H_{\mathcal{M}}(\mathcal{M}_{\sigma}^{\rm e}(X_n),\mathcal{M}_{\sigma}^{\rm e}(X))=0, \end{align} $$

(5) $$ \begin{align}\lim_{n\to\infty}{\underline d}^H(X_n,X)=0, \end{align} $$

(6) $$ \begin{align} \lim_{n\to\infty}{\bar d}^H(X_n,X)=0. \end{align} $$

Our aim is to show that

$$ \begin{align*} (6){\implies}(5){\implies}((4)\Longleftrightarrow\,(3)), \end{align*} $$

and any of these modes of convergence allows us to infer entropy density for X provided that entropy density holds for $X_n$ for all n. We stress that our scheme does not require the approximation to be monotone, that is, we do not require $X_1\supseteq X_2 \supseteq \cdots $ and $X=\bigcap X_n$ , although in practice these often hold. Bearing in mind future applications, we also include a result (Corollary 17) dealing with monotone limits of shift spaces.

Our first main result states that $\bar {d}^H_{\mathcal {M}}$ -convergence for simplices of invariant measures given by (3) preserves entropy density.

Theorem 12. Let $(X_k)_{k=1}^\infty $ and X be shift spaces over $\mathscr {A}$ such that

$$ \begin{align*}\bar{d}^H_{\mathcal{M}}(\mathcal{M}_{\sigma} (X_k),\mathcal{M}_{\sigma} (X)) \to 0 \quad\text{as } k \to \infty.\end{align*} $$

If ergodic measures are entropy-dense in $\mathcal {M}_{\sigma } (X_k)$ for each $k \in \mathbb {N}$ , then ergodic measures are entropy-dense in $\mathcal {M}_{\sigma } (X)$ .

A key component of the proof of Theorem 12 is the fact that for every shift space X, we have the equality

$$ \begin{align*}\bar{d}^H_{\mathcal{M}}(\mathcal{M}_{\sigma} (X_k),\mathcal{M}_{\sigma} (X))=\bar{d}^H_{\mathcal{M}}(\mathcal{M}_{\sigma}^{\rm e}(X_k),\mathcal{M}_{\sigma}^{\rm e}(X)),\end{align*} $$

which implies the equivalence (4) $\,\Longleftrightarrow \,$ (3). This is established in the following two lemmas.

Proof. It is enough to show that for every $\mu \in \mathcal {M}_{\sigma }^{\rm e}(\mathscr {A}^\infty )$ and $\nu \in \mathcal {M}_{\sigma } (Y)$ there exists $\nu ' \in \mathcal {M}_{\sigma }^{\rm e}(Y)$ such that $\bar {d}_{\mathcal {M}}(\mu ,\nu ')\leq \bar {d}_{\mathcal {M}}(\mu ,\nu )$ . Let

be a joining which realizes the $\bar {d}_{\mathcal {M}}$ distance between $\mu \in \mathcal {M}_{\sigma }^{\rm e}(\mathscr {A}^\infty )$ and $\nu \in \mathcal {M}_{\sigma } (Y)$ , that is,

Let $\bar \xi $ be the ergodic decomposition of

. We have

Let E be the set of all

such that

It is clear that E satisfies $\bar \xi (E)>0$ . Additionally, since $\mu $ is ergodic, for $\bar \xi $ -almost every

the pushforward of

through the projection onto the first coordinate is $\mu $ . Hence, there exists

such that

and

Let

. Then $\nu '$ is an ergodic measure supported on Y and

is a joining of $\mu $ and $\nu '$ . Consequently, $\bar {d}_{\mathcal {M}}(\mu ,\nu ')\leq \bar {d}_{\mathcal {M}}(\mu ,\nu )$ .

Lemma 14. If X and Y are shift spaces over $\mathscr {A}$ , then

$$ \begin{align*}\bar{d}^H_{\mathcal{M}}(\mathcal{M}_{\sigma} (X),\mathcal{M}_{\sigma} (Y)) = \bar{d}^H_{\mathcal{M}}(\mathcal{M}_{\sigma}^{\rm e}(X),\mathcal{M}_{\sigma}^{\rm e}(Y)).\end{align*} $$

Proof. It follows from Lemma 13 that

(7) $$ \begin{align} &{\bar d}^H(\mathcal{M}_{\sigma}^{\rm e}(X),\mathcal{M}_{\sigma}^{\rm e}(Y))\nonumber\\ &\quad=\max\Big( \sup_{\mu\in\mathcal{M}_{\sigma}^{\rm e}(X)}\bar{d}_{\mathcal{M}}(\mu,\mathcal{M}_{\sigma} (Y)),\ \sup_{\nu\in\mathcal{M}_{\sigma}^{\rm e}(Y)}\bar{d}_{\mathcal{M}}(\nu,\mathcal{M}_{\sigma} (X)) \Big). \end{align} $$

Since $\mathcal {M}_{\sigma }^{\rm e}(X)\subset \mathcal {M}_{\sigma } (X)$ we have

$$ \begin{align*} \sup_{\mu\in\mathcal{M}_{\sigma}^{\rm e}(X)}\bar{d}_{\mathcal{M}}(\mu,\mathcal{M}_{\sigma} (Y))\le \sup_{\mu\in\mathcal{M}_{\sigma} (X)}\bar{d}_{\mathcal{M}}(\mu,\mathcal{M}_{\sigma} (Y)), \end{align*} $$

and the same inequality with the roles of X and Y reversed. These inequalities, together with (7), give us

$$ \begin{align*} &{\bar d}^H(\mathcal{M}_{\sigma}^{\rm e}(X),\mathcal{M}_{\sigma}^{\rm e}(Y))\nonumber\\ &\quad=\max\Big( \sup_{\mu\in\mathcal{M}_{\sigma}^{\rm e}(X)}\bar{d}_{\mathcal{M}}(\mu,\mathcal{M}_{\sigma} (Y)),\ \sup_{\nu\in\mathcal{M}_{\sigma}^{\rm e}(Y)}\bar{d}_{\mathcal{M}}(\nu,\mathcal{M}_{\sigma} (X)) \Big) \\ &\quad\leq \bar{d}^H_{\mathcal{M}}(\mathcal{M}_{\sigma} (X),\mathcal{M}_{\sigma} (Y)). \end{align*} $$

It remains to prove that for every $\mu \in \mathcal {M}_{\sigma } (X)$ , there exists $\nu \in \mathcal {M}_{\sigma } (Y)$ such that $\bar {d}_{\mathcal {M}}(\mu ,\nu )\leq \bar {d}^H_{\mathcal {M}}(\mathcal {M}_{\sigma }^{\rm e}(X),\mathcal {M}_{\sigma }^{\rm e}(Y))$ . This is sufficient, since the roles of X and Y are interchangeable, meaning that

$$ \begin{align*} \bar{d}^H_{\mathcal{M}}(\mathcal{M}_{\sigma} (X),\mathcal{M}_{\sigma} (Y))\le {\bar d}^H(\mathcal{M}_{\sigma}^{\rm e}(X),\mathcal{M}_{\sigma}^{\rm e}(Y)). \end{align*} $$

Recall that finite convex combinations of ergodic measures are weak $^*$ dense in $\mathcal {M}_{\sigma } (X)$ . Therefore we have

$$ \begin{align*} \mu=\lim_{n\to\infty}\sum_{j\in I(n)}\alpha^{(n)}_j\mu^{(n)}_j, \end{align*} $$

where $I(n)$ is a finite set for every $n\in \mathbb {N}$ ; furthermore, for a fixed n and for each $j\in I(n)$ we have $\mu ^{(n)}_j\in \mathcal {M}_{\sigma }^{\rm e}(X)$ , $\alpha ^{(n)}_j>0$ and $\sum _{j\in I(n)}\alpha ^{(n)}_j=1$ . By our assumption, to every $\mu ^{(n)}_j$ there correspond an ergodic measure $\nu ^{(n)}_j\in \mathcal {M}_{\sigma }^{\rm e}(Y)$ and a joining

with

(8)

For each $n\in \mathbb {N}$ define

Using compactness of $\mathcal {M}_{\sigma \times \sigma } (X\times Y)$ . we may assume that the sequence

weak $^*$ converges to some

. By the definition of weak $^*$ topology, it follows immediately that

(9)

It is easy to see that

is a joining of $\mu $ with

. Definition of

yields that for every $n\in \mathbb {N}$ we have

(10)

Combining (8), (9), and (10), we obtain

Hence, for each $\mu \in \mathcal {M}_{\sigma } (X)$ we have $\nu \in \mathcal {M}_{\sigma } (Y)$ such that $\bar {d}_{\mathcal {M}}(\mu ,\nu )\leq \bar {d}^H_{\mathcal {M}}(\mathcal {M}_{\sigma }^{\rm e}(X), \mathcal {M}_{\sigma }^{\rm e}(Y))$ .

Having Lemma 14 at our disposal, we now proceed with the proof of Theorem 12.

Proof of Theorem 12

Fix $\varepsilon>0$ and $\nu \in \mathcal {M}_{\sigma } (X)$ . Let D be any metric compatible with the weak $^*$ topology on $\mathcal {M}_{\sigma } (\mathscr {A}^\infty )$ . Since the entropy function $\mu \mapsto h(\mu )$ is uniformly continuous on $\mathcal {M}_{\sigma } (\mathscr {A}^\infty )$ endowed with the metric $\bar {d}_{\mathcal {M}}$ [Reference Rudolph51], we can find $\delta _1>0$ such that $|h(\mu )-h(\mu ')|<\varepsilon /3$ for any $\mu ,\mu '\in \mathcal {M}_{\sigma } (\mathscr {A}^\infty )$ with ${\bar d}(\mu ,\mu ')\leq \delta _1$ . By [Reference Rudolph51, Theorem 7.7], there exists $\delta _2> 0$ such that $D(\mu ,\mu ')<\varepsilon /3$ for any $\mu ,\mu '\in \mathcal {M}_{\sigma } (\mathscr {A}^\infty )$ with $\bar {d}_{\mathcal {M}}(\mu ,\mu ')\leq \delta _2$ . Put $\delta = \min (\delta _1,\delta _2)$ and fix $k\in \mathbb {N}$ such that (cf. Lemma 14)

(11) $$ \begin{align} \bar{d}^H_{\mathcal{M}}(\mathcal{M}_{\sigma} (X_k),\mathcal{M}_{\sigma} (X)) = \bar{d}^H_{\mathcal{M}}(\mathcal{M}_{\sigma}^{\rm e}(X_k),\mathcal{M}_{\sigma}^{\rm e}(X)) < \delta. \end{align} $$

By (11), there exists $\mu \in \mathcal {M}_{\sigma } (X_k)$ such that $\bar {d}_{\mathcal {M}}(\nu ,\mu )< \delta $ , whence

(12) $$ \begin{align} D(\nu,\mu)<\varepsilon/3\quad\text{and}\quad|h(\nu)-h(\mu)|<\varepsilon/3. \end{align} $$

Since $\mathcal {M}_{\sigma }^{\rm e}(X_k)$ is entropy-dense in $\mathcal {M}_{\sigma } (X_k)$ , there exists $\mu '\in \mathcal {M}_{\sigma }^{\rm e}(X_k)$ that satisfies

(13) $$ \begin{align} D(\mu,\mu')<\varepsilon/3\quad\text{and}\quad |h(\mu)-h(\mu')|<\varepsilon/3. \end{align} $$

By another application of (11), there exists $\nu '\in \mathcal {M}_{\sigma }^{\rm e}(X)$ such that $\bar {d}_{\mathcal {M}}(\nu ',\mu ')< \delta $ , so

(14) $$ \begin{align} D(\nu',\mu')<\varepsilon/3\quad\text{and}\quad |h(\nu')-h(\mu')|<\varepsilon/3. \end{align} $$

Combining (12), (13) and (14), we conclude that

$$ \begin{align*}D(\nu,\nu')<\varepsilon\quad\text{and}\quad |h(\nu)-h(\nu')|<\varepsilon.\\[-39pt]\end{align*} $$

Note that Theorem 12 only transfers entropy density from sequences of simplices to their $\bar {d}^H_{\mathcal {M}}$ -limits. It turns out that very useful bounds for $\bar {d}^H_{\mathcal {M}}$ are provided by ${\underline d}^H$ and ${\bar d}^H$ . These bounds yield immediately the implications (6) $\implies $ (5) $\implies $ (3). This allows us to work directly with shift spaces bypassing the need to determine their simplices of invariant measures.

Proposition 15. If X and Y are shift spaces over $\mathscr {A}$ , then

$$ \begin{align*}\bar{d}^H_{\mathcal{M}}(\mathcal{M}_{\sigma}^{\rm e}(X),\mathcal{M}_{\sigma}^{\rm e}(Y))\leq {\underline d}^H(X,Y)\le {\bar d}^H(X,Y).\end{align*} $$

Proof. The second inequality is obvious. We turn to the proof of the first. Let $\mu \in \mathcal {M}_{\sigma }^{\rm e}(X)$ and fix $\Delta>{\underline d}^H(X,Y)$ . Pick a $\mu $ -generic point $\bar x$ . By our assumptions, there is a point $\bar y\in Y$ such that ${\underline d}(\bar x,\bar y)<\Delta $ . Let $(n_k)_{k\ge 1}$ be a strictly increasing sequence of integers such that

$$ \begin{align*} {\underline d}(\bar x,\bar y)=\lim_{k\to\infty}\frac{1}{n_k}|\{0\le j<n_k:\bar x_j\neq\bar y_j\}|<\Delta. \end{align*} $$

Passing to a subsequence if necessary, we assume that the point $(\bar x,\bar y)\in X\times Y$ generates a measure

, that is, the sequence of measures

weak $^*$ converges to

as $k \to \infty $ . Let

. Then

is a joining of $\mu $ and $\nu $ . It follows from Lemma 14 that

Since $\Delta>{\underline d}(X,Y)$ and $\mu \in \mathcal {M}_{\sigma }^{\rm e}(X)$ were arbitrary, we have

$$ \begin{align*} \sup_{\mu\in\mathcal{M}_{\sigma}^{\rm e}(X)}\bar{d}_{\mathcal{M}}(\mu,\mathcal{M}_{\sigma}^{\rm e}(Y))\le {\underline d}^H(X,Y). \end{align*} $$

Noting that the roles of X and Y are interchangeable, we see that

$$ \begin{align*} &\hspace{-15pt}\bar{d}^H_{\mathcal{M}}(\mathcal{M}_{\sigma}^{\rm e}(X),\mathcal{M}_{\sigma}^{\rm e}(Y))\\ &\hspace{-15pt}\quad=\max\Big( \sup_{\mu\in\mathcal{M}_{\sigma}^{\rm e}(X)}\bar{d}_{\mathcal{M}}(\mu,\mathcal{M}_{\sigma}^{\rm e}(Y)),\ \sup_{\nu\in\mathcal{M}_{\sigma}^{\rm e}(Y)}\bar{d}_{\mathcal{M}}(\nu,\mathcal{M}_{\sigma}^{\rm e}(X)) \Big) \le {\underline d}^H(X,Y). \\[-3.9pc] \end{align*} $$

In practice, we will not apply Theorem 12 directly, instead we will invoke one of the corollaries obtained from Theorem 12 and Proposition 15. The first of these is immediate.

Corollary 16. Let $(X_n)_{n=1}^\infty $ be a sequence of shift spaces over $\mathscr {A}$ with entropy density such that for a shift space $X\subseteq \mathscr {A}^\infty $ we have

$$ \begin{align*} \lim_{n\to\infty}{\underline d}^H(X_n,X)=0. \end{align*} $$

Then $\bar {d}^H_{\mathcal {M}}(\mathcal {M}_{\sigma }^{\rm e}(X_n),\mathcal {M}_{\sigma }^{\rm e}(X)) \to 0$ as $n \to \infty $ and the ergodic measures are entropy-dense for X.

Before stating the second practical corollary to Theorem 12, note that if $(X_n)_{n=1}^\infty $ is a sequence of shift spaces satisfying the Cauchy condition with respect to ${\bar d}^H$ or ${\underline d}^H$ , then using Proposition 15 we get that $\mathcal {M}_{\sigma } (X_n)_{n=1}^\infty $ satisfies the Cauchy condition for $\bar {d}^H_{\mathcal {M}}$ , so the latter sequence converges in $\bar {d}^H_{\mathcal {M}}$ to some non-empty compact subset of $\mathcal {M}_{\sigma } (\mathscr {A}^\infty )$ . Yet, it is not easy to identify the limit with $\mathcal {M}_{\sigma } (X)$ for some shift space X, unless $X_1\supseteq X_2\supseteq \cdots $ . Under this additional assumption the following result immediately follows from Lemma 3 and Corollary 16.

Corollary 17. Let $(X_n)_{n=1}^\infty $ be a decreasing sequence of shift spaces over $\mathscr {A}$ such that

$$ \begin{align*} \sum_{n=1}^\infty{\underline d}^H(X_n,X_{n+1})<\infty. \end{align*} $$

Then $X=\bigcap _{n=1}^\infty X_n$ is a shift space such that $\bar {d}^H_{\mathcal {M}}(\mathcal {M}_{\sigma }^{\rm e}(X_n),\mathcal {M}_{\sigma }^{\rm e}(X)) \to 0$ as $n \to ~\infty $ . Furthermore, if ergodic measures are entropy-dense for $X_n$ and each $n\ge 1$ , then entropy density holds also for X.

We also find the following observation useful. We stress that neither ${\underline d}$ nor ${\underline d}^H$ obeys the triangle inequality, so the assumptions of Corollary 18 do not guarantee that ${\underline d}^H(X,Y) \,{=}\, 0$ .

Corollary 18. Suppose that $(X_k)_{k=1}^\infty $ and $X,Y$ are shift spaces over $\mathscr {A}$ such that ${\underline d}^H(X_k,X)$ and ${\underline d}^H(X_k,Y)$ tend to zero as $k \to \infty $ . Then

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

In order to apply Theorem 12 or one of its corollaries we still need to single out a class of shift spaces for which entropy density is easily verifiable and then describe a family of shift spaces X such that for some sequence $(X_k)_{k=1}^\infty $ of shift spaces in the aforementioned class of shift spaces we have

$$ \begin{align*}\bar{d}^H_{\mathcal{M}}(\mathcal{M}_{\sigma}^{\rm e}(X_k),\mathcal{M}_{\sigma}^{\rm e}(X)) \to 0 \quad\text{as } k \to \infty.\end{align*} $$

A class of shifts where entropy density is easy to demonstrate is the family of transitive sofic shifts. The result is known even in a greater generality; see [Reference Eizenberg, Kifer and Weiss20, Reference Pfister and Sullivan48]. The proofs given in [Reference Eizenberg, Kifer and Weiss20, Reference Pfister and Sullivan48] simplify in the special case needed here (cf. the proof of Theorem 7.12 in [Reference Rudolph51]).

In applications, we will use either Corollary 16 or Corollary 17 together with Proposition 19. For further reference we formulate our observation as a corollary.

As the first application of Corollary 20 we may now prove entropy density of all ${\bar d}$ -approachable and chain transitive shift spaces.

6 ${\bar d}$ -approachability and specification

We prove that shift spaces satisfying popular variants of the specification property (specification and almost specification) are ${\bar d}$ -approachable. Note that the definitions below are adapted to the symbolic setting. These formulations are equivalent to but slightly different than the ones usually used for general compact dynamical systems. For more background on the specification property and its relatives we refer the reader to the survey paper [Reference Kwietniak, Łącka, Oprocha, Kolyada, Möller, Moree and Ward38].

A shift space $X\in \mathscr {A}^\infty $ has the specification property if there is $k\in \mathbb {N}$ such that for any words $u,w\in \mathcal {B}(X)$ one can find a word $v$ of length k such that $uvw\in \mathcal {B}(X)$ . A shift space X has the almost specification property if there is a mistake function $g\colon \mathbb {N}\to \mathbb {N}$ , which is a non-decreasing function with $g(n)/n\to 0$ as $n\to \infty $ and such that for any words $u,w\in \mathcal {B}(X)$ there are words $u',w'$ satisfying $|u|=|u'|$ , $|v|=|v'|$ , $d_{\textrm {Ham}}(u,u')\le g(|u|)$ and $d_{\textrm {Ham}}(w,w')\le g(|w|)$ , such that $u'w'\in \mathcal {B}(X)$ . Here and elsewhere, $d_{\textrm {Ham}}$ stands for the normalized Hamming distance, that is,

$$ \begin{align*}d_{\textrm{Ham}}(u,w)=\frac1n|\{1\le j\le n: u_j\neq w_j\}|, \end{align*} $$

where $n=|u|=|w|$ .

A prototype for the almost specification property was the g-almost product property introduced in the context of general (not necessarily symbolic) dynamical systems by Pfister and Sullivan in [Reference Pfister and Sullivan49]. Later, Thompson [Reference Thompson55] proposed to slightly modify this notion and renamed it the almost specification property. We follow Thompson, hence our almost specification property is logically weaker (less restrictive) than the notion introduced by Pfister and Sullivan. The standard examples of shift spaces with the almost specification property are $\beta $ -shifts (see [Reference Pfister and Sullivan48]). For shift spaces it is easy to see that the specification property implies almost specification. For a generic $\beta>1$ the $\beta $ -shift $X_\beta $ is an example of a shift space with almost specification but without specification (see [Reference Schmeling53]).

Both specification properties defined above are known to imply entropy density. For shifts with the specification property this was first proved in [Reference Eizenberg, Kifer and Weiss20]. The almost specification property implies entropy density because it implies the approximate product property, and the latter implies entropy density by Theorem 2.1 of [Reference Pfister and Sullivan48].

It is also easy to see that each of the specification properties considered here implies the ${\bar d}$ -shadowing property (actually, the proof of Lemma 7 applies almost verbatim).

By Theorem 6, every shift space X with the almost specification property such that $\sigma (X)=X$ is ${\bar d}$ -approachable. Unfortunately, this the latter condition (i.e. surjectivity) is not necessarily satisfied by a shift space with almost specification. For example, the shift space $X=\{0^\infty ,10^\infty \}\subseteq \{0,1\}^\infty $ has the almost specification property, but $\sigma (X)\neq X$ . Therefore we need to assume $\sigma (X)=X$ in our next result.

We can now show that the entropy density results from [Reference Eizenberg, Kifer and Weiss20, Reference Pfister and Sullivan48] are simple consequences of entropy density of transitive sofic shifts combined with ${\bar d}$ -approachability.

Proof. Let $X^+$ be the measure center of X, that is,

$$ \begin{align*}X^+=X\setminus\bigcup\{U\subseteq\mathscr{A}^\infty:U\text{ open and }\mu(U)=0\text{ for every }\mu\in\mathcal{M}_{\sigma} (X)\}.\end{align*} $$

Clearly, $\mathcal {M}_{\sigma } (X)=\mathcal {M}_{\sigma } (X^+)$ . Furthermore, a shift space X has the almost specification property if and only if $X^+$ does (see [Reference Wu, Oprocha and Chen58, Theorem 6.7] and [Reference Kulczycki, Kwietniak and Oprocha35, Theorem 5.1]). To finish the proof, note that $\sigma (X^+)=X^+$ because recurrent points are dense in $X^+$ by the Poincaré recurrence theorem.

7 Entropy density of $\mathscr {B}$ -free shifts

In this section we demonstrate the utility of our approach by showing that for every $\mathscr {B}$ -free shift $X_{\mathscr {B}}$ , its hereditary closure $\tilde X_{\mathscr {B}}$ has entropy-dense set of ergodic measures. Since it is quite common that $X_{\mathscr {B}}=\tilde X_{\mathscr {B}}$ , we obtain many examples of entropy-dense $\mathscr {B}$ -free shifts. In fact, we show more, namely that the invariant measures on $\tilde X_{\mathscr {B}}$ are also the space of shift invariant measures on a potentially larger shift space $\tilde {X}^*_{\mathscr {B}}$ .

Throughout this section we work with the alphabet $\mathscr {A} = \{0,1\}$ . Recall that the hereditary closure of a shift space $X\subseteq \{0,1\}^\infty $ is defined as

$$ \begin{align*}\tilde X=\{ y \in \{0,1\}^\infty\mid \text{ there exists } x \in X, \ y \leq x\},\end{align*} $$

where the order $\leq $ on $\{0,1\}^\infty $ is defined coordinatewise, meaning that $y \leq x$ if $y_i \leq x_i$ for all $i \in \mathbb {N}_0$ . A shift space is hereditary if it coincides with its own hereditary closure. Given a set $\mathscr {B}\subseteq \mathbb {N}$ , we say that a positive integer number is $\mathscr {B}$ -free if it is not a multiple of any member of $\mathscr {B}$ . The set of all $\mathscr {B}$ -free numbers is denoted by $\mathscr {F}_{\mathscr {B}}\subseteq \mathbb {N}_0$ , that is,

$$ \begin{align*} \mathscr{F}_{\mathscr{B}}=\mathbb{N}_0\setminus\bigcup_{b\in\mathscr{B}}b\mathbb{N}_0. \end{align*} $$

We say that $\mathscr {B}$ is primitive if for each $b,b'\in \mathscr {B}$ with $b\neq b'$ we have that b does not divide $b'$ . A set $\mathscr {B}\subseteq \mathbb {N}$ is taut if for every $b_0\in \mathscr {B}$ we have

$$ \begin{align*} {\underline d}\bigg(\bigcup_{b\in\mathscr{B}\setminus\{b_0\}}b\mathbb{N}_0\bigg) < {\underline d}\bigg(\bigcup_{b\in\mathscr{B}}b\mathbb{N}_0\bigg). \end{align*} $$

The characteristic sequence of $\mathscr {F}_{\mathscr {B}}$ is denoted by $\eta _{\mathscr {B}} :=1_{\mathscr {F}_{\mathscr {B}}}$ , and we think of it as an element of the full-shift $\{0,1\}^\infty $ . The orbit closure $X_{\mathscr {B}}\subseteq \{0,1\}^\infty $ of $\eta _{\mathscr {B}} $ is called the $\mathscr {B}$ -free shift. Given an enumeration of $\mathscr {B} = \{b_1,b_2,\ldots ,\}$ with $b_i<b_j$ for $i<j$ and $k\in \mathbb {N}$ , we let $\mathscr {B}|k = \{b_1,\ldots ,b_k\}$ denote the set of the k smallest elements of $\mathscr {B}$ and let $\mathscr {F}_{\mathscr {B}|k} \subseteq \mathbb {N}_0$ stand for the set $\mathscr {B}|k$ -free integers. Let us write $\eta _{\mathscr {B}|k}\in \{0,1\}^\infty $ for the characteristic function of $\mathscr {F}_{\mathscr {B}|k}$ . Note that $\eta _{\mathscr {B}|k}$ is a periodic point in $\{0,1\}^\infty $ , hence the $\mathscr {B}|k$ -free shift $X_{\mathscr {B}|k}$ is just the orbit of $\eta _{\mathscr {B}|k}$ . We clearly have $\mathscr {F}_{\mathscr {B}}\subseteq \mathscr {F}_{\mathscr {B}|k}$ , hence $\eta _{\mathscr {B}}\le \eta _{\mathscr {B}|k}$ and $\tilde {X}_{\mathscr {B}}\subseteq \tilde {X}_{\mathscr {B}|k}$ . It also turns out that for every $k\in \mathbb {N}$ the hereditary shifts $\tilde {X}_{\mathscr {B}|k}$ are transitive and sofic. This was first noticed in [Reference Dymek, Kasjan, Kułaga-Przymus and Lemańczyk19] (transitivity in Proposition 3.17 and soficity in §3.4.1). We provide an independent proof of a slightly more general fact.

A key fact about $\mathscr {B}$ -free integers which allows our argument to work is that the periodic sets $\mathscr {F}_{\mathscr {B}|k}$ approximate $\mathscr {F}_{\mathscr {B}}$ with respect to the premetric ${\underline d}$ . This is a consequence of a classical result of Davenport and Erdős.

Theorem 27. (Davenport and Erdős; see [Reference Davenport and Erdős14, Reference Davenport and Erdős15])

Let $\mathscr {B} = \{b_1,b_2,\ldots \} \subseteq \mathbb {N}$ . Then

$$ \begin{align*}{\underline d}(\mathscr{F}_{\mathscr{B}|k}\setminus \mathscr{F}_{\mathscr{B}}) \to 0 \quad\text{as } k \to \infty.\end{align*} $$

Observing that ${\underline d}(\mathscr {F}_{\mathscr {B}|k}\setminus \mathscr {F}_{\mathscr {B}}) ={\underline d}(\eta _{\mathscr {B}|k},\eta _{\mathscr {B}})$ , we see that the characteristic function $\eta _{\mathscr {B}}$ is the ${\underline d}$ -limit of a sequence of periodic points $\eta _{\mathscr {B}|k}$ . Furthermore, we can reformulate the Davenport–Erdős theorem in terms of the premetric ${\underline d}^H$ .

Corollary 28. Let $\mathscr {B} = \{b_1,b_2,\ldots \} \subseteq \mathbb {N}$ . Then

$$ \begin{align*}{\underline d}^H(\tilde{X}_{\mathscr{B}|k},\tilde{X}_{\mathscr{B}}) \to 0 \quad\text{as } k \to \infty.\end{align*} $$

Proof. Fix $k\in \mathbb {N}$ . We claim that ${\underline d}^H(\tilde {X}_{\mathscr {B}|k},\tilde {X}_{\mathscr {B}}) \le {\underline d}(\eta _{\mathscr {B}|k},\eta _{\mathscr {B}})$ . Since $\tilde {X}_{\mathscr {B}}\subseteq \tilde {X}_{\mathscr {B}|k}$ it is enough to show that for every $x\in \tilde {X}_{\mathscr {B}|k}$ there is $y\in \tilde {X}_{\mathscr {B}}$ satisfying ${\underline d}(x,y)\le {\underline d}(\eta _{\mathscr {B}|k},\eta _{\mathscr {B}})$ . To this end, take $x\in \tilde {X}_{\mathscr {B}|k}$ . By Proposition 26 there exists $m \geq 0$ such that $\sigma ^m(\eta _{\mathscr {B}|k} )\geq x$ coordinatewise. Consider $y\in \{0,1\}^\infty $ defined by the formula

$$ \begin{align*} y_i = \begin{cases} x_i & \text{ if } (\eta_{\mathscr{B}|k})_{i+m}=(\eta_{\mathscr{B}})_{i+m}, \\ 0 & \text{ otherwise, i.e.{} if } (\eta_{\mathscr{B}|k})_{i+m}=1 \neq 0 = (\eta_{\mathscr{B}})_{i+m}. \end{cases} \end{align*} $$

We immediately see that $y\le \sigma ^m(\eta _{\mathscr {B}} )$ coordinatewise, hence $y\in \tilde {X}_{\mathscr {B}}$ . It is also clear that ${\underline d}(x,y)\le {\underline d}(\eta _{\mathscr {B}|k},\eta _{\mathscr {B}})={\underline d}(\mathscr {F}_{\mathscr {B}|k}\setminus \mathscr {F}_{\mathscr {B}})$ . We use the Davenport–Erdős theorem to conclude that ${\underline d}^H(\tilde {X}_{\mathscr {B}|k},\tilde {X}_{\mathscr {B}}) \to 0$ as $k \to \infty $ .

Before we state our main result regarding $\mathscr {B}$ -free shifts, let us first discuss some technical issues caused by the fact that the ‘ ${\underline d}^H$ -approximation’ appearing in (5) and Corollary 28 does not uniquely determine its ‘limit’.

Remark 29. Observe that we can consider a whole spectrum of intermediate shift spaces associated to $\mathscr {B}$ . Namely, for $k \in \mathbb {N}$ we put

$$ \begin{align*} \tilde X^{(k)}_{\mathscr{B}}=\bigcap_{\substack{\mathscr{B}'\subseteq\mathscr{B}\\ \#\mathscr{B}'=k}} \tilde X_{\mathscr{B}'}. \end{align*} $$

These sets are again shift invariant and hereditary. Moreover, setting

$$ \begin{align*}\tilde X^*_{\mathscr{B}}=\bigcap^\infty_{k=1}\tilde X^{(k)}_{\mathscr{B}},\end{align*} $$

we have the sequence of inclusions

(15) $$ \begin{align} X_{\mathscr{B}}\subseteq \tilde X_{\mathscr{B}}\subseteq \tilde X^*_{\mathscr{B}} \subseteq\cdots\subseteq \tilde X^{(2)}_{\mathscr{B}}\subseteq \tilde X^{(1)}_{\mathscr{B}}. \end{align} $$

We will see later (see Remark 31) that the second inclusion in (15) may be strict. As far as we know, only the sets $X_{\mathscr {B}}$ , $\tilde X_{\mathscr {B}}$ and $\tilde X^{(1)}_{\mathscr {B}}$ have appeared in the literature before. The shift space $\tilde X^{(1)}_{\mathscr {B}}$ is the largest shift space related to $\mathscr {B}$ -free constructions. It is usually called the $\mathscr {B}$ -admissible shift and its elements are $\mathscr {B}$ -admissible sequences (see [Reference Dymek, Kasjan, Kułaga-Przymus and Lemańczyk19]). Let us also point out that the hierarchy introduced above is still rather rough: given any countable family $\mathcal C$ of subsets of $\mathscr {B}$ , the intersection $\bigcap _{\mathscr {B}'\in \mathcal C} \tilde X_{\mathscr {B}'}$ is a hereditary shift space that contains $X_{\mathscr {B}}$ .

Lemma 30. For $\mathscr {B}\subseteq \mathbb {N}$ we have

$$ \begin{align*}\tilde{X}^*_{\mathscr{B}}=\bigcap_{\substack{\mathscr{B}'\subseteq\mathscr{B}\\ \mathscr{B}'\text{ finite}}}\tilde X_{\mathscr{B}'}=\bigcap^\infty_{k=1}\tilde X_{\mathscr{B}|k}. \end{align*} $$

Proof. Fix $n \in \mathbb {N}$ . We have

$$ \begin{align*} \bigcap_{\substack{\mathscr{B}'\subseteq\mathscr{B}\\ \mathscr{B}'\text{ finite}}}\tilde X_{\mathscr{B}'}\subseteq\tilde{X}^*_{\mathscr{B}}=\bigcap^\infty_{k=1}\tilde X^{(k)}_{\mathscr{B}}\subseteq \tilde X^{(n)}_{\mathscr{B}}=\bigcap_{\substack{\mathscr{B}'\subseteq\mathscr{B}\\ \#\mathscr{B}'=n}} \tilde X_{\mathscr{B}'} \subseteq \tilde{X}_{\mathscr{B}|n}.\end{align*} $$

It follows that

$$ \begin{align*} \bigcap_{\substack{\mathscr{B}'\subseteq\mathscr{B}\\ \mathscr{B}'\text{ finite}}}\tilde X_{\mathscr{B}'}\subseteq \tilde{X}^*_{\mathscr{B}}\subseteq\bigcap^\infty_{k=1}\tilde X_{\mathscr{B}|k}. \end{align*} $$

Assume that $x\in \tilde X_{\mathscr {B}|k}$ for every $k\in \mathbb {N}$ . Take any finite set $\mathscr {B}'\subseteq \mathscr {B}$ . Then there exists $n\in \mathbb {N}$ such that $\mathscr {B}'\subseteq (\mathscr {B}\mid n)$ , hence $\tilde X_{\mathscr {B}|n}\subseteq \tilde X_{\mathscr {B}'}$ . We conclude that $x\in \tilde X_{\mathscr {B}'}$ for every finite set $\mathscr {B}'\subseteq \mathscr {B}$ . Therefore

$$ \begin{align*} \bigcap^\infty_{k=1}\tilde X_{\mathscr{B}|k}\subseteq\bigcap_{\substack{\mathscr{B}'\subseteq\mathscr{B}\\ \mathscr{B}'\text{ finite}}}\tilde X_{\mathscr{B}'}, \end{align*} $$

which finishes the proof.

Remark 31. The entire hierarchy (15) collapses to a single shift space ( $X_{\mathscr {B}}=\tilde X^{(1)}_{\mathscr {B}}$ ) whenever $\mathscr {B}$ is a taut set containing an infinite set of pairwise coprime integers (see [Reference Dymek, Kasjan, Kułaga-Przymus and Lemańczyk19, Theorem B] and [Reference Keller29, Corollary 2]; cf. the proof of Corollary 35 below). However, in general, each inclusion in (15) can be strict. For the example showing that the first inclusion may be strict see [Reference Dymek, Kasjan, Kułaga-Przymus and Lemańczyk19]. We show that $\tilde X^{*}_{\mathscr {B}}$ can differ from the $\mathscr {B}$ -admissible shift $\tilde X^{(1)}_{\mathscr {B}}$ . Indeed, if $\mathscr {B}=\{4,6\}$ , then $\tilde X^{*}_{\mathscr {B}}$ and $\tilde X^{(1)}_{\mathscr {B}}$ differ. More generally, if $\mathscr {B} = \{ q p_1, q p_2, \ldots , q p_k \}$ where $q \geq k$ and $p_1,\ldots ,p_k$ are distinct primes then $\tilde X^{*}_{\mathscr {B}} = \tilde X^{(k)}_{\mathscr {B}}$ and $\tilde X^{(k-1)}_{\mathscr {B}}$ differ.

The following example shows that $\tilde X_{\mathscr {B}}$ and $\tilde X^{*}_{\mathscr {B}}$ can be different as well.

Example 32. Let p be a large prime number (e.g. $p = 107$ ), and put $l = (p-2)(p-1)$ , and $\mathscr {B} = \{p-2, p-1,p \} \cup \{n \geq l \ : \ p \nmid n\}$ . Consider the characteristic sequence $x = 1_A \in \{0,1\}^\infty $ of the set $A \subseteq \mathbb {N}_0$ given by

$$ \begin{align*} A = \{j\in\mathbb{N}_0:0\le j <l\} \setminus ( (p-2) \mathbb{N} \cup (p-1) \mathbb{N} \cup (p \mathbb{N} - 1) ). \end{align*} $$

Then $x \in \tilde X^{*}_{\mathscr {B}}$ , because for any finite $\mathscr {B}' \subseteq \mathscr {B}$ we can find $n \in \mathbb {N}$ with $n \equiv 1 \bmod {p}$ and $n \equiv 0 \bmod {b}$ for all $b \in \mathscr {B}' \setminus \{p\}$ .

On the other hand, we claim that $x \notin \tilde {X}_{\mathscr {B}}$ . For the sake of contradiction, suppose that $x \in \tilde {X}_{\mathscr {B}}$ . Since $\tilde {X}_{\mathscr {B}}$ is finite, there exists $m \in \mathbb {N}_0$ such that $x \leq \sigma ^m(\eta _{\mathscr {B}})$ coordinatewise. Since $n \not \in \mathscr {F}_{\mathscr {B}}$ for $n \geq l$ and $l - 1 \in A$ , it follows that $m = 0$ . However, $p \in A$ while $p \not \in \mathscr {F}_{\mathscr {B}}$ so $m \neq 0$ , which is the sought contradiction.

Theorem 33. Let $\mathscr {B}\subseteq \mathbb {N}$ . Then ergodic measures are entropy-dense for $\tilde {X}_{\mathscr {B}}$ and

$$ \begin{align*}\mathcal{M}_{\sigma} (\tilde{X}_{\mathscr{B}})=\mathcal{M}_{\sigma} (\tilde X^{*}_{\mathscr{B}})=\bigcap_{k=1}^\infty\mathcal{M}_{\sigma} (\tilde{X}_{\mathscr{B}|k}).\end{align*} $$

Proof. The second equality is a consequence of Proposition 2 and Lemma 30. Fix $k \in \mathbb {N}$ . We have

$$ \begin{align*}\tilde{X}_{\mathscr{B}}\subseteq\tilde{X}^*_{\mathscr{B}} \subseteq \tilde{X}_{\mathscr{B}|k}.\end{align*} $$

It follows that we have

$$ \begin{align*} {\underline d}(\tilde{X}_{\mathscr{B}|k},\tilde{X}^*_{\mathscr{B}}) \leq {\underline d}(\tilde{X}_{\mathscr{B}|k},\tilde{X}_{\mathscr{B}}). \end{align*} $$

In particular, by Corollary 28 all ‘distances’ above tend to $0$ as $k \to \infty $ and $\mathcal {M}_{\sigma } (\tilde {X}_{\mathscr {B}})=\mathcal {M}_{\sigma } (\tilde {X}^*_{\mathscr {B}})$ by Corollary 18. We finish the proof applying Corollary 20.

Example 34. Note that for $\tilde X_{\mathscr {B}}$ we have only a ${\underline d}^H$ -approximation by transitive sofic shifts, which are not mixing. This raises the question whether $X_{\mathscr {B}}$ or $\tilde X_{\mathscr {B}}$ can be ${\bar d}$ -approachable. Note that the shift space in Example 11 can be equivalently defined as $\tilde X_{\mathscr {B}}$ , where $\mathscr {B}=\{2\}$ , showing that hereditary closures of some $\mathscr {B}$ -free shifts are non- ${\bar d}$ -approachable. We will extend this example and prove that there exists an infinite set $\mathscr {B}$ such that $\tilde X_{\mathscr {B}}$ is not ${\bar d}$ -approachable. Let $b_1=2$ and pick $b_2,b_3,\ldots $ such that

(16) $$ \begin{align} \sum^\infty_{k=2}\frac1{b_k}< \frac{1}{32}. \end{align} $$

Note that (16) ensures that ${\bar d}^H(\tilde X_{\{2\}},\tilde X_{\mathscr {B}})< 1/32$ . For n large enough, there is a word $u\in \mathcal {B}(\tilde X_{\mathscr {B}})$ with $|u|=2n$ such that $d_{\textrm {Ham}}((01)^n,u)<1/4$ . In particular, at least half of the even entries in $u$ are occupied by $1$ . In other words,

$$ \begin{align*} |\{1\le j\le n: u_{2j}=1\}|\ge \tfrac12 n. \end{align*} $$

By heredity of $\tilde X_{\mathscr {B}}$ , the periodic point $y=(u0^{2n-1})^\infty $ belongs to $(\tilde X_{\mathscr {B}})^M_{2n}$ . Reasoning as in Example 11, we obtain ${\bar d}(y,x)\geq 1/16$ for every $x\in \tilde X_{\{2\}}$ . This implies that ${\bar d}^H((\tilde X_{\mathscr {B}})^M_{2n},\tilde X_{\{2\}})\ge 1/16$ , yielding

$$ \begin{align*}{\bar d}^H((\tilde X_{\mathscr{B}})^M_{2n},\tilde X_{\mathscr{B}})> 1/32,\end{align*} $$

for n large enough. So the Markov approximations are ${\bar d}$ -far from $\tilde X_{\mathscr {B}}$ . Note that we can choose $\mathscr {B}$ that consists of pairwise prime numbers and still satisfies (16). For such a set $\mathscr {B}$ we have $\tilde X_{\mathscr {B}}=X_{\mathscr {B}}$ and the $\mathscr {B}$ -free shift $X_{\mathscr {B}}$ is not ${\bar d}$ -approachable.

So far we have discussed only the hereditary closure $\tilde X_{\mathscr {B}}$ of $X_{\mathscr {B}}$ . In general, $X_{\mathscr {B}}$ does not need to be hereditary, as shown by the example $\mathscr {B} = \mathbb {N}_{\geq 3}$ where $X_{\mathscr {B}}$ is the (finite) orbit of the point $011000\ldots $ and hence does not contain the point $01000\ldots .$ Nevertheless, for many examples of the sets $\mathscr {F}_{\mathscr {B}}$ studied in the literature the associated $\mathscr {B}$ -free shift turns out to be hereditary (e.g. the square-free shift or shifts associated with abundant numbers; see [Reference Dymek, Kasjan, Kułaga-Przymus and Lemańczyk19]). For the record, we note a version of Theorem 33 for these shifts. But first, note that every hereditary shift contains the fixed point $0^\infty $ , and since every $\mathscr {B}$ -free shift has a unique minimal subsystem (by Theorem A in [Reference Dymek, Kasjan, Kułaga-Przymus and Lemańczyk19]), the set $\{0^\infty \}$ must be the unique minimal subset for a hereditary $\mathscr {B}$ -free shift. In other words, a hereditary $\mathscr {B}$ -free shift must be proximal. For $\mathscr {B}$ -free shifts proximality is equivalent to the fact that the set $\mathscr {B}$ contains an infinite pairwise coprime subset. Therefore, hoping for $X_{\mathscr {B}}=\tilde X_{\mathscr {B}}$ , we have to assume the latter condition.

Remark 37. Note that Theorem 33 remains true if we replace $\tilde {X}_{\mathscr {B}}$ by any shift space which is the hereditary closure of the orbit closure of a characteristic function of a set $A\subseteq \mathbb {N}_0$ such that there exists a sequence of periodic sets $(A_k)_{k=1}^\infty $ satisfying

$$ \begin{gather*} A\subseteq A_k\quad\text{for every }k\in\mathbb{N},\\ {\underline d}(A_k,A) \to 0\quad (k \to \infty). \end{gather*} $$

Such shift spaces have been already considered in the literature [Reference Kulczycki, Kwietniak and Li34, Reference Kwietniak, Łącka and Oprocha39]. A closely related notion of a rational set (a subset of $\mathbb {N}$ whose characteristic function can be ${\bar d}$ -approximated by periodic characteristic functions) has also attracted attention; see [Reference Bergelson, Kułaga-Przymus, Lemańczyk and Richter4, Reference Bergelson, Ruzsa, Halasz, Lovasz, Simonovits and Sós5, Reference Deka16].

8 Inner ${\bar d}$ -approachability

So far we have considered shift spaces, which are approximated from the outside, that is, approximating subshifts are not contained in the approximated shift space. Inspired by an idea of approximation given by Dan Thompson in an unpublished manuscript [Reference Thompson56], we will show that every topologically mixing S-gap shift is approximated in the ${\bar d}^H$ sense by a sequence of shift spaces of finite type contained in $X_S$ . We call this property inner ${\bar d}$ -approachability.

We say that a shift space $X\subseteq \mathscr {A}^\infty $ is inner ${\bar d}$ -approachable if it contains a sequence $(X_n)_{n=1}^\infty $ of mixing shifts of finite type such that

$$ \begin{align*} \overline{\bigcup_{n=1}^\infty X_n}=X\quad\text{and}\quad \lim_{n\to\infty}{\bar d}^H(X_n,X)=0. \end{align*} $$

Given $S\subseteq \mathbb {N}$ , we write $S=\{n_1,n_2,\ldots \}$ with $n_i<n_{i+1}$ for $i < |S|$ , where $|S|$ stands for the cardinality of S ( $|S|=\infty $ if S is infinite). The S-gap shift $X_S$ is a shift space over $\{0,1\}$ consisting of all sequences such that the number of 0s between any two successive occurrences of the symbol 1 belongs to S. Equivalently, $\{10^n1:n\notin S\}$ is the collection of forbidden sequences for $X_S$ . By [Reference Jung26, Example 3.4], $X_S$ is topologically mixing if and only if $\gcd \{n+1:n\in S\}=1$ , and topologically mixing $X_S$ has the specification property if and only if $\sup _i |n_i - n_{i+1}|< \infty $ . Note that all sequences in $X_S$ are labels of infinite paths in the labelled directed graph $G_S = (V,E_S)$ , where $V=\{v_j:0\le j\le |S|\}$ is the set of vertices and there is an edge $v_i\to v_j$ in $E_S$ if and only if $j =i+1$ or $i \in S$ and $j = 0$ . We label each edge $v_i\to v_{i+1}$ with 0 and all other edges with 1.

Proof. Assume that $S=\{s_1,s_2,\ldots \}$ is infinite, $s_i<s_{i+1}$ for every $i\in \mathbb {N}$ , and $\gcd \{n+1:n\in S\}=1$ (if S is finite, then $X_S$ is a shift of finite type). Let N be such that $\gcd \{s_1+1,\ldots ,s_N+1\}=1$ . Let $L>0$ be the smallest integer such that for every $\ell \ge L$ there are non-negative integers $\alpha _1,\ldots ,\alpha _N$ such that

$$ \begin{align*} \alpha_1(s_1+1)+\cdots+\alpha_N(s_N+1)=\ell. \end{align*} $$

It follows that for every $\ell \ge L$ there are $r\in \mathbb {N}$ and $q_1,\ldots ,q_r\in \{s_1,\ldots ,s_N\}$ such that the word

(17) $$ \begin{align} w=0^{q_1}10^{q_2}1\ldots 0^{q_r}1 \end{align} $$

satisfies $w\in \mathcal {B}(X_S)$ and $|w|=\ell $ . Set $S[n]=\{s_1,\ldots ,s_n\}$ and let $X_n$ be the S-gap shift generated by $S[n]$ . Clearly $X_1\subseteq X_2\subseteq X_3\subseteq \cdots $ and

$$ \begin{align*} \overline{\bigcup_{n=1}^\infty X_n}=X_S. \end{align*} $$

Furthermore, for every $n\ge N$ the shift space $X_n$ is a mixing shift of finite type. Fix $\varepsilon>0$ . Let $M\ge N$ be such that $1/s_M<\varepsilon /2$ . Let $K\ge M$ satisfy $(L+s_M)/s_K<\varepsilon /2$ . We claim that for every $k\ge K$ we have

$$ \begin{align*}{\bar d}^H(X_k,X)\le \varepsilon.\end{align*} $$

Note that our proof is finished once we show that the claim holds. For a proof of our claim fix $k\ge K$ . Let $x\in X$ . Since ${\bar d}$ is $\sigma $ -invariant we assume that

$$ \begin{align*} x=10^{t_1}10^{t_2}1\ldots 10^{t_j}1\ldots, \end{align*} $$

that is, we assume that x begins with 1 and contains infinitely many 1s. The proof is similar, if the symbol 1 occurs in x only finitely many times. Note that $x\notin X_k$ means that for some $j\in \mathbb {N}$ we have $t_j\in S\setminus \{s_1,\ldots ,s_k\}$ .

We will construct, for each j such that $t_j \in S \setminus \{s_1,\ldots ,s_k\}$ , a word $v_j$ with $|v_j| = t_j$ such that $d_{\textrm {Ham}}(v_j, 0^{t_j}) < \varepsilon $ and $1v_j1 \in \mathcal {B}(X_S[k])$ . Put also $v_j = 0^{t_j}$ if $t_j \in \{s_1,\ldots ,s_k\}$ . Once this is accomplished, we set $x' = 1 v_1 1 v_2 1 \ldots .$ It is now enough to observe that $x' \in X_{S[k]}$ and ${\bar d}(x,x') \leq \sup _j d_{\textrm {Ham}}(v_j, 0^{t_j})$ .

Fix j such that $t_j \in S \setminus \{s_1,\ldots ,s_k\}$ . Let $\ell $ be the unique integer such that $L\le \ell <L+s_M$ and $t_j+1-\ell $ is divisible by $(s_M+1)$ . We replace the suffix $0^{\ell -1}1$ of $10^{t_j}1$ by a word $w$ with $|w|=\ell $ having the form as in (17). The remaining prefix of $ 10^{t_j}1$ now has the form

$$ \begin{align*} 10^{t_j+1-\ell} \end{align*} $$

where $t_j+1-\ell =p(s_M+1)$ for some $p\ge 1$ . Therefore we may replace $0^{t_j+1-\ell }$ by

$$ \begin{align*} (0^{s_M}1)^p. \end{align*} $$

We have found a word $v$ with $|v|=t_j$ such that

$$ \begin{align*} u=1v1=1(0^{s_M}1)^p0^{q_1}10^{q_2}1\ldots 0^{q_r}1\in\mathcal{B}(X_S[k]). \end{align*} $$

Note that

$$ \begin{align*} d_{\textrm{Ham}}(0^{t_j},v)\le \frac{1}{t_j}(t_j/(s_M+1)+(L+s_M))<\varepsilon, \end{align*} $$

which completes the proof.

Entropy density for mixing S-gap shifts follows from Theorems B and 3.5 in [Reference Climenhaga, Thompson and Yamamoto11]. Using Proposition 38 and Corollary 16, we obtain a new proof of this fact.