2 Preliminaries

In this section, we recall some preliminaries concerning symbolic dynamics and topological dynamics in general. Good references to these topics are [Reference Kůrka10, Reference Lind and Marcus12].

A (possibly infinite) non-empty set A is called an alphabet. For $n\in {\mathbb {N}_{+}}$ , we have a special alphabet $\Sigma _{n}=\{0,1,\ldots ,n-1\}$ . The set $A^{\mathbb {Z}}$ of bi-infinite sequences (configurations) over A is called a full shift. Formally, any $x\in A^{\mathbb {Z}}$ is a function $\mathbb {Z}\to A$ and the value of x at $i\in \mathbb {Z}$ is denoted by $x[i]$ . It contains finite, right-infinite, and left-infinite subsequences denoted by $x[i,j]=x[i]x[i+1]\cdots x[j]$ , $x[i,\infty ]=x[i]x[i+1]\cdots $ , and $x[-\infty ,i]=\cdots x[i-1]x[i]$ .

A configuration $x\in A^{\mathbb {Z}}$ (respectively, $x\in A^{\mathbb {N}}$ ) is periodic if there is a $p\in {\mathbb {N}_{+}}$ such that $x[i+p]=x[i]$ for all $i\in \mathbb {Z}$ (respectively, $i\in \mathbb {N}$ ). Then we may also say that x is p-periodic or that x has period p. If x is $1$ -periodic, we call it a fixed point. We say that $x\in A^{\mathbb {Z}}$ (respectively, $x\in A^{\mathbb {N}}$ ) is eventually periodic if there are $p\in {\mathbb {N}_{+}}$ and $i_{0}\in \mathbb {Z}$ (respectively, $i_{0}\in \mathbb {N}$ ) such that $x[i+p]=x[i]$ holds for all $i\geq i_{0}$ .

A subword of $x\in A^{\mathbb {Z}}$ is any finite sequence $x[i,j]$ where $i,j\in \mathbb {Z}$ , and we interpret the sequence to be empty if $j<i$ . Any finite sequence $w=w[1] w[2]\cdots w[n]$ (also the empty sequence, which is denoted by $\epsilon $ ), where $w[i]\in A$ , is a word over A. Unless we consider a word w as a subword of some configuration, we start indexing the symbols of w from $1$ as we have done here. Similarly, right-infinite sequences are indexed $x=x[0]x[1]x[2]\cdots\kern-1pt $ and left-infinite sequences are indexed $x=\cdots x[-3]x[-2]x[-1]$ . If $w\neq \epsilon $ , we say that w occurs in x at position i if $x[i]\cdots x[i+n-1] = w[1]\cdots w[n]$ . The concatenation of a word or a left-infinite sequence u with a word or a right-infinite sequence v is denoted by $uv$ . A word u is a prefix of a word or a right-infinite sequence x if there is a word or a right-infinite sequence v such that $x=uv$ . Similarly, u is a suffix of a word or a left-infinite sequence x if there is a word or a left-infinite sequence v such that $x=vu$ . The set of all words over A is denoted by $A^{*}$ , and the set of non-empty words is $A^{+}=A^{*}\setminus \{\epsilon \}$ . The set of words of length n is denoted by $A^{n}$ . For a word $w\in A^{*}$ , ${\vert w \vert }$ denotes its length, that is, ${\vert w \vert }=n{\iff}w\in A^{n}$ . For any word $w\in A^{+}$ , we denote by $w^{\infty }$ the right-infinite sequence obtained by infinite repetitions of the word w. We denote by $w^{\mathbb {Z}}$ the configuration defined by $w^{\mathbb {Z}}[in,(i+1)n-1]=w$ (where $n={\vert w \vert }$ ) for every $i\in \mathbb {Z}$ .

Any collection of words $L\subseteq A^{*}$ is called a language. For any set S of configurations, right-infinite sequences, left-infinite sequences, and finite words, the collection of words appearing as subwords of elements of S is the language of S, denoted by $L(S)$ . For a set $S=\{x\}$ with one element, we may omit the braces and write $L(S)=L(x)$ . For $n\in \mathbb {N}$ , we denote $L_{n}(S)=L(S)\cap A^{n}$ . For any $L,K\subseteq A^{*}$ , let $LK=\{uv\mid u\in L, v\in K\}$ and

$$ \begin{align*}L^{*}=\{w_{1}\cdots w_{n}\mid n\geq 0,w_{i}\in L\},\quad L^{+}=\{w_{1}\cdots w_{n}\mid n\geq 1,w_{i}\in L\}.\end{align*} $$

To consider topological dynamics on subsets of the full shifts, the set $A^{\mathbb {Z}}$ is endowed with the product topology (with respect to the discrete topology on A). This is a metrizable space, which is also compact when A is finite. The shift map $\sigma :A^{\mathbb {Z}}\to A^{\mathbb {Z}}$ is defined by $\sigma (x)[i]=x[i+1]$ for $x\in A^{\mathbb {Z}}$ , $i\in \mathbb {Z}$ , and it is a homeomorphism. If $X\subseteq A^{\mathbb {Z}}$ is such that $\sigma (X)=X$ , we say that X is shift-invariant. If A is finite, any topologically closed shift-invariant non-empty subset $X\subseteq A^{\mathbb {Z}}$ is called a subshift or just a shift. Alternatively, any subshift $X\subseteq A^{\mathbb {Z}}$ can be characterized by a list of forbidden words $\mathcal {F}\subseteq A^{+}$ such that

$$ \begin{align*}X=\{x\in A^{\mathbb{Z}}\mid\mbox{No element of }w\in\mathcal{F}\mbox{ occurs in }x\}.\end{align*} $$

Any $w\in L(X)\setminus {\epsilon }$ and $i\in \mathbb {Z}$ determine a cylinder of X

$$ \begin{align*}\operatorname{\mathrm{Cyl}}_{X}(w,i)=\{x\in X\mid w \mbox{ occurs in }x\mbox{ at position }i\}.\end{align*} $$

If A and B are alphabets, the elements of $A^{n}\times B^{n}$ can be naturally identified with the elements of $(A\times B)^{n}$ for all $n>0$ , and elements of $A^{\mathbb {Z}}\times B^{\mathbb {Z}}$ can be identified with the elements of $(A\times B)^{\mathbb {Z}}$ . Using these identifications, we may say that $X=Y\times Z$ is a subshift whenever Y and Z are subshifts.

Note that both memory and anticipation can be either positive or negative. Note also that if F has memory m and anticipation a with the associated local rule $f:A^{a-m+1}\to A$ , then F is also a radius-r CA for $r=\max \{{\vert m \vert },{\vert a \vert }\}$ , with possibly a different local rule $f^{\prime }:A^{2r+1}\to A$ .

If X and Y are subshifts and if there is a bijective sliding block code $F:X\to Y$ , we say that F is a conjugacy and that X is conjugate with Y (via F). It is known that then the inverse map of F is also a sliding block code. In particular, the inverse map of a bijective CA is also a CA, which is why they are also known as a reversible CA (RCA).

The notions of almost equicontinuity and sensitivity are defined for dynamical systems more general than cellular automata. For completeness, we mention the general definitions.

Definition 2.2. Let $T:X\to X$ be a continuous map on a compact metric space X defined by a metric $\operatorname {\mathrm {dist}}:X\times X\to \mathbb {R}$ . For $x\in X$ and $\delta> 0$ , let $B_{\delta }(x)$ denote the ball of radius $\delta $ with center x. We say that $x\in X$ is an equicontinuity point of T if

$$ \begin{align*} & \text{for all } \epsilon {{\kern-1pt}>{\kern-1pt}}0, \text{ there exists }\delta {{\kern-1pt}>{\kern-1pt}} 0, \text{ such that for all } y\in B_{\delta}(x) \nonumber \\ &\quad \text{and for all } n\in\mathbb{N},\ \operatorname{\mathrm{dist}}(T^{n}(y), T^{n}(x)){{\kern-1pt}<{\kern-1pt}}\epsilon. \end{align*} $$

If the set of equicontinuity points is residual, we say that T is almost equicontinuous. On the other hand, T is sensitive if

$$ \begin{align*} &\text{there exists } \epsilon>0,\ \text{such that for all } x\in X\ \text{and for all }\delta>0,\ \text{there exist } y\in B_{\delta}(x)\\ &\quad\text{and } n\in\mathbb{N}, \text{ such that}\operatorname{\mathrm{dist}}(T^{n}(y), T^{n}(x))>\epsilon. \end{align*} $$

In particular, then T has no equicontinuity points. These notions do not depend on the choice of the metric as long as the underlying topology remains unchanged.

For cellular automata on transitive subshifts, there is a combinatorial characterization for these notions using blocking words. We present this alternative characterization in Proposition 2.4.

Definition 2.3. Let $F:X\to X$ be a radius-r CA and $w\in L(X)$ . We say that w is a blocking word if there is an integer $s\in [0,{\vert w \vert }-r]$ such that

$$ \begin{align*}\text{for all } x,y\in\operatorname{\mathrm{Cyl}}_{X}(w,0), \text{ for all } n\in\mathbb{N},\quad F^{n}(x)[s,s+r-1]=F^{n}(y)[s,s+r-1].\end{align*} $$

The notion of sensitivity is refined by Sablik’s framework of directional dynamics [Reference Sablik16].

The notions of sensitive and almost equicontinuous directions are well defined in the sense that for $k,p,q\in \mathbb {Z}$ , $k,q>0$ , the map $\sigma ^{p}\circ F^{q}$ is sensitive if and only if $\sigma ^{kp}\circ F^{kq}$ is sensitive. The paper [Reference Sablik16] also considers directional dynamics in the case when directions can be any real numbers, but for notational simplicity, we only consider explicitly the case of rational directions.

For a subshift X and a configuration $x\in X$ , denote $x^{+}=x[0,\infty ]$ and $x^{-}=x[-\infty ,-1]$ , so $x=x^{-}x^{+}$ . Denote $X^{+}=\{x^{+}\mid x\in X\}$ and $X^{-}=\{x^{-}\mid x\in X\}$ . For any $x\in X$ , we define the follower set of $x^{-}$ by $\omega _{X}(x^{-})=\{y^{+}\in X^{+}\mid x^{-} y^{+}\in X\}$ and for any $w\in L(X)$ , we define the follower set of w by $\omega _{X}(w)=\{x^{+}\in X^{+}\mid wx^{+}\in X^{+}\}$ . The subscript X may be dropped when it is clear from the context. By making use of follower sets, we can define some natural classes of transitive subshifts.

The definition of a half-synchronized subshift (credited to Krieger) is [Reference Fiebig and Fiebig5, Definition 0.9]. The notion of a synchronized subshift is originally from [Reference Blanchard and Hansel2] and the characterization we present for it is essentially [Reference Fiebig and Fiebig5, Definition 0.8].

In particular, synchronized subshifts are half-synchronized. More commonly (e.g. in [Reference Fiebig and Fiebig5, Definition 0.8]), a word $w\in L(X)$ is called synchronizing if $uw\in L(X)$ and $wv\in L(X)$ implies that $uwv\in L(X)$ . The definitions are easily seen to be equivalent, but our definition makes the connection to half-synchronizing words more transparent.

More commonly (e.g. in [Reference Lind and Marcus12, §3]), a subshift X is called sofic if X is the set of labels of bi-infinite paths on some finite labeled graph. This is indeed equivalent to our definition: by [Reference Krieger9, Lemma 2.1], the existence of such a graph implies that there are finitely many follower sets, and alternatively from finitely many follower sets, one can construct a suitable finite labeled graph (the Krieger graph, see the next section). By yet another characterization [Reference Lind and Marcus12, Theorem 3.2.1], X is sofic if it is the image of an SFT under a sliding block code.

As mentioned in the introduction, we say that a subshift Y is a direct factor of a subshift X if there is a subshift Z such that X is conjugate to $Y\times Z$ . We also say that a subshift X is direct prime if X being conjugate to $Y\times Z$ implies that either Y or Z is a trivial subshift (that is, contains only one configuration). We make the simple observation that the class of half-synchronized subshifts is closed under taking direct factors.

3 Canonical covers

To any subshift X, it is possible to associate covers, that is, labeled directed graphs such that the labels of all bi-infinite paths on the graph form a dense set on X. Some of these covers turn out to be canonical in the sense that any RCA on X can be ‘lifted’ in a unique way to a sliding block code on the set of labels of bi-infinite paths of the cover. Two such covers are Krieger graphs and Fischer graphs from [Reference Fiebig and Fiebig5]. We will recall the definitions and basic results.

A (directed) graph is a pair ${\mathcal {G}}=(V,E)$ , where V is the set of vertices and E is the set of edges or arrows. Both of these sets may be infinite. Each edge $e\in E$ starts at an initial vertex denoted by $\iota (e)$ and ends at a terminal vertex denoted by $\tau (e)$ . A word $p\in E^{+}$ is a path on ${\mathcal {G}}$ if for every $1\leq i<{\vert p \vert }$ , it holds that $\tau (p[i])=\iota (p[i+1])$ . Similarly, one defines right-infinite, left-infinite, and bi-infinite paths, and the collection of all bi-infinite paths on ${\mathcal {G}}$ is denoted by $P({\mathcal {G}})$ . The initial vertex $\iota (p)$ of a finite or right-infinite path p is equal to the initial vertex of the first edge of p, and the terminal vertex $\tau (p)$ of a finite or a left-infinite path p is equal to the terminal vertex of the last edge of p. The graph ${\mathcal {G}}$ is strongly connected if for every pair of vertices $v,w\in V$ , there is a finite path p with $\iota (p)=v$ and $\tau (p)=w$ .

The tensor product of directed graphs ${\mathcal {G}}_{1}=(V_{1},E_{2})$ and ${\mathcal {G}}_{2}=(V_{2},E_{2})$ is ${\mathcal {G}}_{1}\times {\mathcal {G}}_{2}=(V_{1}\times V_{2},E_{1}\times E_{2})$ , where $\iota (e_{1},e_{2})=(\iota (e_{1}),\iota (e_{2}))$ and $\tau (e_{1},e_{2})=(\tau (e_{1}),\tau (e_{2}))$ for $e_{i}\in E_{i}$ . If each $e_{i}$ has a label $a_{i}$ , then the edge $(e_{1},e_{2})$ has the label $(a_{1},a_{2})$ .

The Krieger graph ${\mathcal {K}}_{X}=(V_{X},E_{X})$ of a subshift $X\subseteq A^{\mathbb {Z}}$ is the graph with the vertex set $V_{X}=\{\omega (x^{-})\mid x\in X\}$ and for all $x^{-}\in X^{-}$ and $a\in A$ such that $x^{-} a\in X^{-}$ , there is an edge from $\omega (x^{-})$ to $\omega (x^{-} a)$ labeled by a. We denote the label of any edge e by $\unicode{x3bb} _{X}(e)$ . The map $\unicode{x3bb} _{X}$ naturally extends to a map $\unicode{x3bb} _{X}:P({\mathcal {K}}_{X})\to X$ , which replaces each edge of a bi-infinite path by its label. It is easy to see that $\unicode{x3bb} _{X}(P({\mathcal {K}}_{X}))=X$ . It is also easy to see that for subshifts X, Y, and Z satisfying $X=Y\times Z$ , it holds that ${\mathcal {K}}_{X}={\mathcal {K}}_{Y}\times {\mathcal {K}}_{Z}$ .

If X is half-synchronized with a half-synchronizing word $w\in L(X)$ , then the Fischer graph ${\mathcal {F}}_{X}$ of X is the maximal strongly connected subgraph of ${\mathcal {K}}_{X}$ containing the vertex $\omega (w)$ (which, by Definition 2.7, is indeed a vertex of ${\mathcal {K}}_{X}$ ). The labeling map for this graph is the restriction of the map $\unicode{x3bb} _{X}$ of the previous paragraph. It is shown in [Reference Fiebig and Fiebig5, pp. 146–147] that for any pair $w,w^{\prime }\in L(X)$ of half-synchronizing words, the Krieger graph of X contains a path from $\omega (w)$ to $\omega (w^{\prime })$ . From this, it follows that ${\mathcal {F}}_{X}$ does not depend on the choice of the half-synchronizing word w, so we may speak of the Fischer graph of X.

If $w\in L(X)$ is half-synchronizing, then a vertex v of ${\mathcal {K}}_{X}$ belongs to ${\mathcal {F}}_{X}$ precisely if there is a path from $\omega (w)$ to v. Namely, if p is a finite path from $\omega (w)$ on ${\mathcal {K}}_{X}$ , then $w\unicode{x3bb} _{X}(p)$ is also a half-synchronizing word and $\tau (p)=\omega (w\unicode{x3bb} _{X}(p))$ , so by the previous paragraph, there is also a path from $\tau (p)$ to $\omega (w)$ . By transitivity, every word of $L(X)$ is a label of some path starting at $\omega (w)$ , so the set $\unicode{x3bb} _{X}({\mathcal {F}}_{X})$ is dense in X.

If ${\mathcal {F}}_{X}$ is finite, then $\unicode{x3bb} _{X}({\mathcal {F}}_{X})$ is closed and therefore equal to X. Sofic subshifts are characterized as those subshifts that are equal to the set of labels of some finite directed graph, so a non-sofic half-synchronized subshift necessarily has an infinite Fischer graph.

It turns out that Fischer graphs respect products.

Proof. Let $w_{1}$ and $w_{2}$ be equal length half-synchronizing words of Y and Z, respectively, such that $(w_{1},w_{2})$ is a half-synchronizing word of X. Let also $y^{-}\in Y^{-}$ and $z^{-}\in Z^{-}$ be such that they have suffixes $w_{1}$ and $w_{2}$ , respectively, and that $L(x^{-})=L(X)$ for $x^{-}=(y^{-},z^{-})\in X^{-}$ .

To show that ${\mathcal {F}}_{X}$ is a subgraph of ${\mathcal {F}}_{Y}\times {\mathcal {F}}_{Z}$ , let $(v_{1},v_{2})$ be a vertex of ${\mathcal {F}}_{X}$ , meaning that there is a path $p=(p_{1},p_{2})$ from $\omega _{X}(w_{1},w_{2})=(\omega _{Y}(w_{1}),\omega _{Z}(w_{2}))$ to $(v_{1},v_{2})$ in ${\mathcal {K}}_{X}$ . Therefore, $p_{1}$ and $p_{2}$ are paths from $\omega _{Y}(w_{1})$ to $v_{1}$ and $\omega _{Z}(w_{2})$ to $v_{2}$ in ${\mathcal {K}}_{Y}$ and ${\mathcal {K}}_{Y}$ , respectively, so $(v_{1},v_{2})$ is a vertex of ${\mathcal {F}}_{Y}\times {\mathcal {F}}_{Z}$ .

To show that ${\mathcal {F}}_{Y}\times {\mathcal {F}}_{Z}$ is a subgraph of ${\mathcal {F}}_{X}$ , let $(v_{1},v_{2})$ be a vertex of ${\mathcal {F}}_{Y}\times {\mathcal {F}}_{Z}$ , meaning that there are paths $p_{1}$ and $p_{2}$ from $\omega _{Y}(w_{1})$ to $v_{1}$ and $\omega _{Z}(w_{2})$ to $v_{2}$ in ${\mathcal {K}}_{Y}$ and ${\mathcal {K}}_{Z}$ , respectively. Without loss of generality, assume that ${\vert p_{1} \vert }\leq {\vert p_{2} \vert }$ . Let $w_{1}^{\prime }$ and $w_{2}^{\prime }$ be suffixes of length ${\vert w_{2} \unicode{x3bb} _{Z}(p_{2}) \vert }$ of $y^{-}\unicode{x3bb} _{Y}(p_{1})$ and $z^{-}\unicode{x3bb} _{Z}(p_{2})$ , respectively: $w_{1}^{\prime }$ and $w_{2}^{\prime }$ in turn have suffixes $w_{1}\unicode{x3bb} _{Y}(p_{1})$ and $w_{2}\unicode{x3bb} _{Z}(p_{2})$ , respectively. Then

To show that $(v_{1},v_{2})$ is a vertex of ${\mathcal {F}}_{X}$ , it is therefore sufficient to show that $(w_{1}^{\prime },w_{2}^{\prime })$ is a half-synchronizing word of X. This in turn follows after we show that $L(x^{{\prime }-})=L(X)$ for $x^{{\prime }-}=(y^{-}\unicode{x3bb} _{Y}(p_{1}),z^{-}\unicode{x3bb} _{Z}(p_{2}))\in X^{-}$ . Let therefore $u=(u_{1},u_{2})\in L(X)$ be arbitrary, with $u_{1}\in L(Y)$ and $u_{2}\in L(Z)$ of equal length n, and choose any $u_{1}^{\prime }\in L(Y)$ and $u_{2}^{\prime }\in L(Z)$ of length $k={\vert p_{2} \vert }-{\vert p_{1} \vert }$ such that $u_{1} u_{1}^{\prime }\in L(Y)$ and $u_{2}^{\prime }u_{2}\in L(Z)$ . Because $X=Y\times Z$ , it follows that $u^{\prime }=(u_{1} u_{1}^{\prime },u_{2}^{\prime }u_{2})\in L(X)$ and therefore $u^{\prime }=x^{-}[i,i+n+k-1]$ for some $i\in \mathbb {Z}$ such that $i+n+k-1<0$ . Thus,

so u occurs in $x^{{\prime }-}$ at position $i-{\vert p_{1} \vert }$ .

Let $C,D$ be disjoint alphabets (not necessarily finite), interpret $CD$ and $DC$ to be new alphabets and let $X\subseteq (CD)^{\mathbb {Z}}$ , $Y\subseteq (DC)^{\mathbb {Z}}$ be shift-invariant (not necessarily closed or compact). A bijective map $F:X\to Y$ is a forward bipartite code (for partition $(C,D)$ ) if the image of any $x\in X$ with $x[i]=c_{i} d_{i}$ ( $c_{i}\in C$ , $d_{i}\in D$ ) satisfies $F(x)[i]=d_{i}c_{i+1}$ . Similarly, F is a backward bipartite code (for partition $(C,D)$ ) if always $F(x)[i]=d_{i-1}c_{i}$ . If X and Y are subshifts, then a bipartite code $F:X\to Y$ is a conjugacy. We recall the following.

We say that a given map $F:X\to X$ lifts to a map $F^{\prime }:P({\mathcal {K}}_{X})\to P({\mathcal {K}}_{X})$ (respectively, $F^{\prime }:P({\mathcal {F}}_{X})\to P({\mathcal {F}}_{X})$ ) if $\unicode{x3bb} _{X}(F^{\prime }(x))=F(\unicode{x3bb} _{X}(x))$ for $x\in P({\mathcal {K}}_{X})$ (respectively, for $x\in P({\mathcal {F}}_{X})$ ). It is known that bipartite codes between subshifts lift to bipartite codes between the path sets of Krieger graphs and Fischer graphs. We will recall the details of this.

The following lemma is essentially from [Reference Nasu15].

Proof. Assume without loss of generality that F is a forward bipartite code for partition $(A,B)$ . Define the subshift

$$ \begin{align*} Z&=\{z\in (A\cup B)^{\mathbb{Z}}\mid\mbox{For some }x\in X\mbox{ and for all }i\in\mathbb{Z}, z[2i,2i+1]=x[i] \} \\ &\qquad\cup\{z\in (A\cup B)^{\mathbb{Z}}\mid\mbox{For some }y\in Y\mbox{ and for all }i\in\mathbb{Z}, z[2i,2i+1]=y[i] \}. \end{align*} $$

In other words, we get Z from configurations of X and Y by interpreting the symbols of $AB$ and $BA$ as pairs of symbols. It is then easily seen that ${\mathcal {K}}_{X}$ and ${\mathcal {K}}_{Y}$ are an induced pair of graphs of the Krieger graph ${\mathcal {K}}_{Z}$ . Let C be the set of edges of ${\mathcal {K}}_{Z}$ from ${\mathcal {K}}_{X}$ to ${\mathcal {K}}_{Y}$ and let D be the set of edges from ${\mathcal {K}}_{Y}$ to ${\mathcal {K}}_{X}$ : then the edges of ${\mathcal {K}}_{X}$ and ${\mathcal {K}}_{Y}$ can be renamed by elements of $CD$ and $DC$ , respectively, and the forward bipartite code $F^{\prime }:P({\mathcal {K}}_{X})\to P({\mathcal {K}}_{Y})$ between these graphs with renamed edges is clearly a lift of F.

The map $F^{\prime }$ of the previous lemma is in fact the unique continuous surjective lift of F, which is a consequence of the following theorem.

4 A sufficient criterion for direct primeness

In this section, we present a sufficient criterion of direct primeness for half-synchronized subshifts based on their Fischer graphs. First we define a few special types of paths on graphs.

It turns out that the properties of being a strictly proximal pair or an eventually geodesic path are preserved under bipartite codes.

From this lemma, it then follows that the properties of the Fischer graph having an eventually geodesic path or a strictly proximal pair are conjugacy invariant.

First we prove a sufficient criterion for when a non-sofic half-synchronized subshift is ‘almost’ direct prime in the sense that it cannot be represented as a product of two infinite subshifts.

After applying the previous theorem, to prove direct primeness, it is sufficient to rule out the possibility of non-trivial finite direct factors. The existence of fixed points gives one way to do this, as seen in the next corollary.

In the previous corollary it would be possible to replace the assumption of X having a fixed point by the assumption of X being mixing (by using the fact that a finite mixing subshift can contain only one configuration). We will not make use of this alternative criterion.

5 Direct primeness of well-known classes of subshifts

In this section, we prove that several natural classes of subshifts are direct prime. First we consider the so-called n-Dyck shifts. It was shown in [Reference Meyerovitch14] that the n-Dyck shift is topologically direct prime at least when n is a prime number. We generalize this for all $n>1$ . We first recall the basic definition from [Reference Meyerovitch14].

For a natural number $n>1$ , we define the symbol sets ${\Sigma _{(n,\ell )}}=\{\alpha _{1},\ldots ,\alpha _{n}\}$ , ${\Sigma _{(n,{r})}}=\{\beta _{1},\ldots ,\beta _{n}\}$ (the left and right brackets), and ${\Sigma _{(n)}}={\Sigma _{(n,\ell )}}\cup {\Sigma _{(n,{r})}}$ . Let M be the monoid generated by ${\Sigma _{(n)}}\cup \{0\}$ , with identity element $1$ and zero element $0$ , subject to the relations $\alpha _{i}\beta _{i}=1$ and $\alpha _{i}\beta _{j}=0$ for $i,j\in \{1,\ldots ,n\}$ , $i\neq j$ .

The n-Dyck shift is defined by

$$ \begin{align*}D_{n}=\{x\in{\Sigma_{(n)}}^{\mathbb{Z}}\mid x[i]\cdot x[i+1]\cdot \cdots \cdot x[i+k]\neq 0\quad\mbox{ for all }i\in\mathbb{Z}, k\in\mathbb{N}\};\end{align*} $$

intuitively, this means that configurations of $D_{n}$ do not contain mismatched brackets. As an example, we consider the simplest Dyck shift $D_{2}$ , in which we replace the symbols $\alpha _{1},\alpha _{2},\beta _{1},\beta _{2}$ by the brackets $(,[,),$ and $]$ , respectively. For example, $\cdots ((()))\cdots \in D_{2}$ , because any subword simplifies to an element of either $(^{*}$ or $)^{*}$ . As another example, no element of $D_{2}$ can contain $((^{n})^{n}]$ for any $n\in \mathbb {N}$ , because $((^{n})^{n}]=0$ as an element of M. This is seen by using the relation $()=1\ n$ times and then the relation $(]=0$ .

Let $w_{i}$ , $i\in \mathbb {N}$ be an enumeration of all the words in $L(D_{n})$ that are equal to $1$ as elements of the monoid M (that is, all the brackets are matched in $w_{i}$ ). Every word of $L(D_{n})$ is a subword of some $w_{i}$ . Any $w\in L(D_{n})$ is a half-synchronizing word of $D_{n}$ , since it is easily seen that $L(y^{-})=L(w)$ for $y^{-}=\cdots w_{2} w_{1} w_{0} w$ .

Note that the Fischer graph of $D_{n}$ consists of all the vertices and edges along paths starting from $\omega (\epsilon )$ in ${\mathcal {K}}_{D_{n}}$ . It turns out that ${\mathcal {F}}_{D_{n}}=(V,E)$ , where $V=\{\omega (w)\mid w\in {\Sigma _{(n,\ell )}}^{*}\}$ and for $i\in \{1,\ldots ,n\}$ :

• • from the vertex $\omega (\epsilon )$ : there are self-loops $\beta _{i}$ and edges labeled by $\alpha _{i}$ to the vertex $\omega (\alpha _{i})$ ;

• • from any vertex of the form $\omega (w\alpha _{i})$ : there are edges labeled by $\alpha _{j}$ to the vertex $\omega (w\alpha _{i}\alpha _{j})$ and an edge labeled by $\beta _{i}$ to the vertex $\omega (w)$ .

We show a part of the Fischer graph ${\mathcal {F}}_{D_{2}}$ in Figure 1.

Figure 1

The Fischer graph ${\mathcal {F}}_{D_{2}}$ .

Next we will consider the so-called S-gap shifts. It is a previous result that every non-sofic S-gap shift is topologically direct prime. (In fact, such S-gap shifts do not have RCA without almost equicontinuous directions [Reference Kopra8]: this is a stronger statement as seen in [Reference Kopra7].) We present a new argument of this fact in the Fischer graph framework.

For non-empty $S\subseteq \mathbb {N}$ , the S-gap shift $X_{S}\subseteq \Sigma _{2}^{\mathbb {Z}}$ is the subshift whose language $L(X_{S})$ consists of all the subwords of elements of $\{01^{n}\mid n\in S\}^{*}$ . Every $X_{S}$ is synchronized, because $0$ is a synchronizing word. By [Reference Dastjerdi and Jangjoo3, Theorem 3.4], an S-gap shift is sofic if and only if the characteristic function $\chi _{S}\in \Sigma _{2}^{\mathbb {N}}$ ( $\chi _{S}[n]=1$ if and only if $n\in S$ ) is eventually periodic. In particular, for non-sofic $X_{S}$ , the set S is infinite, $L(X_{S})$ contains $01^{n}$ for arbitrarily large $n\in \mathbb {N}$ , and therefore $1^{\mathbb {Z}}\in X_{S}$ .

Note that the Fischer graph of $X_{S}$ consists of all the vertices and edges along paths starting from $\omega (0)$ in ${\mathcal {K}}_{X_{S}}$ . It turns out that the Fischer graph ${\mathcal {F}}_{X_{S}}$ of a non-sofic $X_{S}$ has the vertex set $\{\omega (01^{n})\mid n\in \mathbb {N}\}$ , an edge labeled by $1$ from $\omega (01^{n})$ to $\omega (01^{n+1})$ , and for $n\in S$ , an edge labeled by $0$ from $\omega (01^{n})$ to $\omega (0)$ . In particular, from the fact that $\chi _{S}$ is not eventually periodic, it follows that the follower sets of all $01^{n}$ are different. We show a part of ${\mathcal {F}}_{X_{S}}$ in Figure 2 in the case $S=\{2^{i}\mid i\in \mathbb {N}\}$ .

Figure 2

The Fischer graph ${\mathcal {F}}_{X_{S}}$ .

We conclude this section by considering the so-called beta-shifts. It is a previous result that every non-sofic beta-shift is topologically direct prime [Reference Kopra7], but we can give a new proof of this fact in the Fischer graph framework. As in [Reference Blanchard1] and [Reference Lothaire13, §7.2.2], any real number $\beta>1$ can be associated in a certain way with a sequence $x_{\beta }\in \Sigma _{n}^{\mathbb {N}}$ (for some $n>1$ ) such that $x_{\beta }[0]>0$ , $x_{\beta }\neq 10^{\infty }$ , and every suffix of $x_{\beta }$ is lexicographically smaller than $x_{\beta }$ (with the lexicographical ordering $\leq $ induced from the usual ordering of $\Sigma _{n}$ ). Then the beta-shift (in base $\beta $ ) is

$$ \begin{align*}X_{\beta}=\{x\in\Sigma_{n}^{\mathbb{Z}}\mid x[i,\infty]\leq x_{\beta}\mbox{ for every }i\in\mathbb{Z}\}.\end{align*} $$

This is non-sofic precisely when $x_{\beta }$ is not eventually periodic. We now consider only such cases.

It is stated in [Reference Fiebig and Fiebig5] that $X_{\beta }$ is half-synchronized and that every word in $L(X_{\beta })$ is half-synchronized. The graph ${\mathcal {F}}_{X_{\beta }}$ then consists of all the vertices and edges along paths starting from $\omega (\epsilon )$ in ${\mathcal {K}}_{X_{\beta }}$ . As mentioned in [Reference Dastjerdi and Shahamat4], it turns out that the Fischer graph ${\mathcal {F}}_{X_{\beta }}$ of a non-sofic $X_{\beta }$ has the vertex set $\{\omega (x_{\beta }[0,n])\mid n\geq -1\}$ , an edge labeled by $x_{\beta }[n+1]$ from $\omega (x_{\beta }[0,n])$ to $\omega (x_{\beta }[0,n+1])$ , and for $0\leq i<x_{\beta }[n+1]$ , an edge labeled by i from $\omega (x_{\beta }[0,n])$ to $\omega (\epsilon )$ . In particular, from the fact that $x_{\beta }$ is not eventually periodic, it follows that the follower sets of all $x_{\beta }[0,n]$ are different. We show a part of ${\mathcal {F}}_{X_{\beta }}$ in Figure 3 in the case $x_{\beta }=22102\cdots $ .

Figure 3

The Fischer graph ${\mathcal {F}}_{X_{\beta }}$ .

6 A new class of direct prime subshifts

Up to this point, we have applied Theorem 4.5 to previously known classes of subshifts. The theorem can also be used in constructing new direct prime subshifts with some additional desirable properties. As an example, we present a construction that shows that a direct prime non-sofic synchronized subshift can have RCA with only sensitive directions: recall from the introduction that the existence of such RCA is trivial if the subshift is a product of two infinite subshifts.

Definition 6.1. For any subshift $X\subseteq A^{\mathbb {Z}}$ , there is a star-studded version ${X^{(*)}}$ containing a new symbol $*$ , consisting of all configurations created by taking an arbitrary $x\in X$ and interleaving the symbol $*$ so that no two consecutive $*$ occur. To be more precise, let $h:(A\cup \{*\})^{*}\to A^{*}$ be the substitution determined by $h(*)=\epsilon $ and $h(a)=a$ for $a\in A$ . If X is characterized by a set $\mathcal {F}\subseteq A^{*}$ of forbidden words, then ${X^{(*)}}$ is the subshift over $A\cup \{*\}$ characterized by the set of forbidden words

$$ \begin{align*}\mathcal{F}^{\prime}=\{**\}\cup\{w\in (A\cup\{*\})^{*}\mid h(w)\in\mathcal{F}\}.\end{align*} $$

Star-studded subshifts have RCA that resemble partial shift maps but do not rely on direct factorizations. These turn out to have all directions sensitive.

Definition 6.2. For any subshift X, define the RCA $F_{X}:{X^{(*)}}\to {X^{(*)}}$ by

$$ \begin{align*} F_{X}(x)[i]= \begin{cases} *\quad&\text{when } x[i]=*, \\ x[i+1]\quad&\text{when } x[i]x[i+1]\in A^{2}, \\ x[i+2]\quad&\mbox{when } x[i]\in A \mbox{ and } x[i+1]=*. \end{cases} \end{align*} $$

The CA $F_{X}$ fixes all occurrences of the symbol $*$ in configurations and shifts any occurrence of any symbol $a\in A$ to the left so that the symbol a skips over any occurrence of $*$ .

Now that we have seen that star-studded subshifts have RCA with all directions sensitive, it remains to show that this class of subshifts contains direct prime non-sofic synchronized subshifts. First we show that, to some extent, the star-studded subshift inherits properties from the original one.

Proof. Let $x\in X$ be such that $x^{+}\in \omega _{X}(x_{1}^{-})$ . Then $*x^{+}\in \omega _{X^{(*)}}(x_{1}^{-})$ but $*x^{+}\notin \omega _{X^{(*)}}(x_{2}^{-}*)$ .

If $\omega _{X}(x_{1}^{-})\neq \omega _{X}(x_{2}^{-})$ , then

$$ \begin{align*} \begin{aligned} & \omega_{X^{(*)}}(x_{1}^{-})\cap A^{\mathbb{N}}=\omega_{X}(x_{1}^{-})\neq\omega_{X}(x_{2}^{-})=\omega_{X^{(*)}}(x_{2}^{-})\cap A^{\mathbb{N}}\quad \mbox{ and}\\ & \omega_{X^{(*)}}(x_{1}^{-}*)\cap A^{\mathbb{N}}=\omega_{X}(x_{1}^{-})\neq\omega_{X}(x_{2}^{-})=\omega_{X^{(*)}}(x_{2}^{-}*)\cap A^{\mathbb{N}}.\\[-28pt] \end{aligned} \end{align*} $$

We make the following definition to make sense of what Krieger graphs and Fischer graphs of star-studded shifts look like.

This definition has the nice property that the non-existence of eventually geodesic strictly proximal pairs in a graph is inherited by the star-studded version of the graph.

To apply the previous lemma to Fischer graphs of star-studded subshifts, we need to show that star-studded versions of Fischer graphs are isomorphic to Fischer graphs of star-studded subshifts.

By Lemma 6.4, the vertices $\omega _{X}(x^{-})$ of $V_{X}$ are in bijective correspondence with the vertices in $V_{X,1}=\{\omega _{X^{(*)}}(x^{-})\mid x\in X\}\subseteq V_{{X^{(*)}}}$ and the vertices ${\omega _{X}(x^{-})^{(*)}}$ of ${V_{X}^{(*)}}$ (copies of $\omega _{X}(x^{-})$ ) are in bijective correspondence with the vertices in $V_{X,2}=\{\omega _{X^{(*)}}(x^{-}*)\mid x\in X\}\subseteq V_{{X^{(*)}}}$ . Also by Lemma 6.4, the sets $V_{X,1}$ and $V_{X,2}$ are disjoint. Therefore, the vertices $V_{X,1}\cup V_{X,2}$ of ${\mathcal {K}}_{X^{(*)}}$ are in a natural bijective correspondence with $V_{X}\cup {V_{X}^{(*)}}$ . The edges between the vertices of $V_{X,1}\cup V_{X,2}$ are clearly such that ${{\mathcal {K}}_{X}^{(*)}}$ is a subgraph of ${\mathcal {K}}_{X^{(*)}}$ (up to renaming the vertices). It is also clear that ${\mathcal {K}}_{X^{(*)}}$ contains no other types of vertices, so in fact ${{\mathcal {K}}_{X}^{(*)}}$ is isomorphic to ${\mathcal {K}}_{X^{(*)}}$ (up to renaming the vertices). We denote this graph isomorphism by $\phi _{X}:{{\mathcal {K}}_{X}^{(*)}}\to {\mathcal {K}}_{X^{(*)}}$ and we may denote the labeling function of both of these graphs by $\unicode{x3bb} _{X}$ .

Now we may apply Lemma 6.7 to Fischer graphs. After that, we will prove the main results of this section.

Concrete examples can be obtained e.g. from S-gap shifts.