3.1 Entropy and FP systems

For our result on LPSs, we will need some tools from topological dynamics, which we develop here. For a detailed introduction, we refer to [Reference WaltersWal00].

Let $(X,d)$ be a compact metric space and $f:X\to X$ our transformation of interest. The transformation f induces a new metric

$$ \begin{align*}d_n(x,y)=\max_{0\leq i < n} d(f^ix,f^iy)\end{align*} $$

for each $n\in \mathbb {N}$ . We say that a set of points E is $(n,\varepsilon )$ -separated if for $x,y\in E$ , $x\neq y$ , we have $d_n(x,y)>\varepsilon $ . Let

$$ \begin{align*}s_n(\varepsilon):=\max_{E\subset X} \{|E|: E \text{ is } (n,\varepsilon)\text{-separated}\},\end{align*} $$

which is finite for fixed $n,\varepsilon $ by compactness of X.

We can now state one way of defining the topological entropy of a dynamical system. This particular definition is due to Bowen, see [Reference WaltersWal00, Ch. 7.2].

Definition 3.1. The topological entropy of a dynamical system $(X,f)$ as above is

$$ \begin{align*}h(f)=\lim_{\varepsilon \searrow 0} \limsup_{n\to\infty} \frac{1}{n} \log s_n(\varepsilon).\end{align*} $$

The first half of the next lemma is well known and follows for example from combining Theorems 7.2 and 7.8 in [Reference WaltersWal00]. A purely topological proof of the second half can be found in [Reference RobinsonRob95, Ch. IX], Theorem 1.8, which uses [Reference BowenBow71, Theorem 17].

Lemma 3.2. If $\pi :(X,f)\to (Y,g)$ is a factor map between dynamical systems, then

$$ \begin{align*}h(g)\leq h(f).\end{align*} $$

Moreover, if $\pi $ is also bounded-to-one, then

$$ \begin{align*}h(g)=h(f).\end{align*} $$

We also use a property called entropy-minimality, which says that any proper subsystem must have strictly smaller entropy than the whole system.

Definition 3.3. A dynamical system $(X,f)$ is entropy-minimal if, whenever Y is a closed invariant proper subset of X, we have

$$ \begin{align*}h(f|_Y)<h(f).\end{align*} $$

Irreducible SFTs, for example, have this property. The following lemma is [Reference Lind and MarcusLM21, Theorem 4.4.7].

Combining this result with Lemma 3.2 immediately shows that entropy-minimality holds for any bounded-to-one factor of an irreducible SFT.

Proof. Let $Z\subsetneq \Omega $ be a proper subsystem. Since Z is closed and f-invariant, $Y:=\pi ^{-1}(Z)$ is compact and a strict subshift of X. Moreover, since $\pi $ is bounded-to-one, so is $\pi ':=\pi |_Y$ , giving the following diagram.

Thus,

$$ \begin{align*}h(f|_Z) = h(Y) <h(X)=h(f).\\[-32pt]\end{align*} $$

Finally, the following well-known lemma on entropy for expansive homeomorphisms is in [Reference BowenBow78].

Lemma 3.6. When $(\Omega ,f)$ is an expansive homeomorphism, we have

$$ \begin{align*}h(f)\geq \limsup_{n\to\infty} \frac{1}{n}\log(p_n(f)).\end{align*} $$

Proof. Let $\delta $ denote an expansive constant for f and let $\varepsilon <\delta .$ Let $x,y\in P_n(f).$ By expansiveness, for some $i\in \mathbb {Z}$ ,

$$ \begin{align*}d(f^ix,f^iy)>\delta>\varepsilon.\end{align*} $$

However, since $x,y$ are periodic with period n, this means

$$ \begin{align*}d_n(x,y)=\max_{0\leq i < n} d(f^ix,f^iy)>\varepsilon.\end{align*} $$

That is, $P_n(f)$ is $(n,\varepsilon )$ -separated for all n and we get that

$$ \begin{align*}s_n(\varepsilon)\geq p_n(f)\end{align*} $$

for all n. The conclusion follows.

Recall from the introduction that an FP system is an expansive factor of an SFT. We introduce briefly an equivalent definition of FP given by Fried in [Reference FriedFri87]. Suppose $f:\Omega \to \Omega $ is a factor of an SFT $X\subset \mathcal {A}^{\mathbb {Z}}$ on alphabet via $\pi :X\to \Omega $ . Then, we have an equivalence relation E on X given by

$$ \begin{align*}xEy \iff \pi (x)=\pi(y),\end{align*} $$

and we can write $\Omega = X/E$ . Now, if we view E as a subshift of $(\mathcal {A}\times \mathcal {A})^{\mathbb {Z}}$ , [Reference FriedFri87] introduces the term FP for such f where E is an SFT. The same reference also provides the following fundamental theorem, which in particular gives the equivalence of the two definitions.

These Markov partitions have a long history dating back to Sinai in his work on Anosov diffeomorphisms [Reference SinaiSin68], and work on toral automorphisms by Adler and Weiss [Reference Adler and WeissAW70]. They were further developed by Bowen as a tool in better understanding what are known as Axiom A diffeomorphisms [Reference BowenBow08] and generalized by Fried into FP systems. We also point out that [Reference Ashley, Kitchens and StaffordAKS92] collects many of the results on Markov partitions in generality.

Often, Markov partitions are constructed with a particular class of dynamical system in mind, so there is some variance in the definitions across the literature. For completeness, we give a definition of Markov partitions following [Reference FriedFri87], who in turn follows [Reference BowenBow08]. The definition is somewhat involved, but the main point for us is that a Markov partition yields an SFT and a factor map with properties outlined in Proposition 3.8.

Let $(\Omega ,f)$ be an expansive dynamical system, with expansive constant $c>0$ . For ${\varepsilon>0}$ , define the $\varepsilon $ -stable and $\varepsilon $ -unstable sets

$$ \begin{align*}W_\varepsilon^s(x) &= \{y\in \Omega: d(f^nx,f^ny)\leq \varepsilon:n\geq 0\},\\ W_\varepsilon^u(x) &= \{y\in \Omega: d(f^nx,f^ny)\leq \varepsilon:n\leq 0\}.\end{align*} $$

Taking $\varepsilon \leq c/2$ , $W_\varepsilon ^u(x)\cap W_\varepsilon ^s(y)$ consists of at most one point for any $x,y\in \Omega $ . Let

$$ \begin{align*}D_\varepsilon:\{(x,y)\in \Omega\times \Omega:W_\varepsilon^u(x)\cap W_\varepsilon^s(y)\neq \emptyset\}.\end{align*} $$

A set $R\subset \Omega $ is a rectangle if $R\times R\subset D_\varepsilon $ . Finally, a Markov partition for $(\Omega ,f)$ is a finite cover $\mathcal {M}=\{R_i\}_{i=1}^n$ of $\Omega $ consisting of non-empty rectangles with disjoint interiors of diameter less than $\varepsilon $ such that $\overline {\operatorname {int}R}=R$ for all $R\in \mathcal {M}$ , satisfying the Markov property: whenever $x\in \operatorname {int}R_i$ and $fx\in \operatorname {int}R_j$ ,

$$ \begin{align*}f(W_\varepsilon^s(x) \cap R_i)\subset R_j,\quad f^{-1}(W_\varepsilon^s(fx) \cap R_j)\subset R_i.\end{align*} $$

Given the Markov partition $\mathcal {M}=\{R_i\}_{i=1}^n$ , we define a transition matrix

$$ \begin{align*}A_{ij}=\begin{cases} 1& \text{if }\operatorname{int}(R_i)\cap f^{-1}(\operatorname{int}(R_j)) \neq \emptyset, \\ 0& \text{otherwise,} \end{cases}\end{align*} $$

and denote the shift space generated by this matrix $X_A$ . We then obtain a factor map $\pi :X_A\to \Omega $ defined by

$$ \begin{align*}\pi(\ldots x_{-1}x_0x_1 \ldots) = \bigcap_{i\in\mathbb{Z}} f^{-i}(R_{x_i}).\end{align*} $$

This map is well defined, continuous, and makes the following diagram commute.

In fact, $\pi $ has several other desirable properties.

Properties (A) and (B) are consequences of [Reference Ashley, Kitchens and StaffordAKS92, Lemma 1.8], sometimes known as the ‘no diamonds’ property of Markov partitions. For the latter, see [Reference ManningMan71, Corollary 11 and Proposition 12], which refers to [Reference BowenBow70].

One can also find property (C) in [Reference Ashley, Kitchens and StaffordAKS92]. From this property, we will actually only need the existence of a single point with a unique preimage in the presentation in the proof of Theorem 1.4. Surprisingly, in [Reference Marcus, Meyerovitch, Thomsen and WuMMTW25], there is significant interest in arbitrary SFT covers where at least one point has a unique preimage. We will touch on this later in this section.

The forward direction of property (D) is [Reference BowenBow08, Theorem 3.19] specifically for the Axiom A case, but the same proof generalizes to FP systems. We note also that the converse statements of property (D) are clearly true.

3.2 Proof of Theorems 1.4 and 1.5

We have collected all the necessary tools to prove the next main result. Along the way, we obtain two intermediate results that are interesting in their own right. Intuitively, we look at different ‘layers’ of the system, where the layers are separated by the number of preimages a point has with respect to the cover arising from a Markov partition. The idea of this proof was first suggested by Doering in [Reference DoeringDoe20] as a way of potentially characterizing the LPS of sofic shift spaces.

Let us begin the proof, by letting $(\Omega ,f)$ be an FP system. Recall that our goal is to show that

$$ \begin{align*}\operatorname{LPS}(f)=F\cup\bigcup_{i=1}^n d_iS_i,\end{align*} $$

where F is finite, each $S_i$ is cofinite, and either F or the big union may be empty, but not both.

Let $\mathcal {M}=\{R_i\}_{i=1}^{|\mathcal {M}|}$ be a Markov partition for $(\Omega ,f)$ , X the associated SFT with transition matrix A, and $\pi :X\to \Omega $ the corresponding factor map. Moreover, let

$$ \begin{align*}\Omega_i:=\{w\in\Omega: \pi^{-1}w \text{ has at least } i \text{ mutually separated preimages}\},\end{align*} $$

where it is clear that $\Omega _1=\Omega $ . We have that $\Omega _i$ is f-invariant and

$$ \begin{align*}\Omega_i\supset \Omega_{i+1}\end{align*} $$

for all i. Note that by Proposition 3.8(A), $\Omega _i$ is non-empty for only finitely many i. We analyze each non-empty $\Omega _i$ separately, as if we are ‘peeling back layers’. The authors of [Reference Ashley, Kitchens and StaffordAKS92] chose independently to call $\Omega _2$ the core of a Markov partition, invoking a similar image.

Denote

$$ \begin{align*}\tilde{P}_n(f,\pi):=\{w\in\Omega:f^n(w)=w \text{ and }|\pi^{-1}w|=1\},\end{align*} $$

and let

$$ \begin{align*}\tilde{Q}_n(f,\pi):=\{w\in \tilde{P}_n(f,\pi): \text{ for all } k<n,f^kw\neq w \}\end{align*} $$

be the set of all periodic points with unique preimage and least period n. It is immediate that

(5) $$ \begin{align} \tilde{Q}_n(f,\pi)=\tilde{P}_n(f,\pi)\setminus \bigcup_{d|n,d<n} \tilde{P}_d(f,\pi), \end{align} $$

which we use later.

We now construct a set of adjacency matrices $A^{(i)}$ such that $X_i:=X_{A^{(i)}}$ is a bounded-to-one factor map onto $\Omega _i$ . The idea is inspired by the set of matrices used by Manning to prove that Axiom A diffeomorphisms have rational zeta functions in [Reference ManningMan71].

We define $A^{(i)}$ to be the adjacency matrix of a directed graph $G_i$ whose vertices are the set of all

$$ \begin{align*}V\subset\{1,2,3, \ldots,|\mathcal{M}|\}\end{align*} $$

such that $|V|=i$ and

$$ \begin{align*}\bigcap_{l\in V} R_l \neq \emptyset.\end{align*} $$

We draw an edge from vertex $U=\{u_1,\ldots ,u_i\}$ to vertex $V=\{v_1,\ldots ,v_i\}$ if and only if there exists a reordering of V such that

$$ \begin{align*}A_{u_k,v_{k}}=1\end{align*} $$

for all $1\leq k\leq i$ .

Recall now that $\pi (\ldots x_{-1}x_0x_1 \ldots )$ is defined to be the unique point w such that

$$ \begin{align*}w\in \bigcap_{n\in\mathbb{Z}} f^{-i}(R_{x_n}).\end{align*} $$

There is a natural bounded-to-one factor map $\pi _i:X_i\to \Omega _i$ induced from $\pi $ . If $x\in X_i$ , then x corresponds to at least i mutually separated points $y^1,y^2, \ldots , y^i\in X_1$ such that for all $n\in \mathbb {Z}$ ,

$$ \begin{align*}\bigcap_{1\leq k\leq i} R_{y^k_n} \neq \emptyset.\end{align*} $$

Lemma 1 from [Reference ManningMan71] deduces directly from the definition of $\pi $ and expansiveness of f that

$$ \begin{align*}\pi y^1=\pi y^2=\cdots=\pi y^i=w\in \Omega,\end{align*} $$

and we may thus set

$$ \begin{align*}\pi_i x:=w\end{align*} $$

to obtain the following commutative diagram.

Since $X_i$ is compact and $\pi _i$ is continuous, $\Omega _i$ is a compact subset of $\Omega $ .

Whenever we write f in the remainder of the proof, we always mean f restricted to the appropriate invariant subset of $\Omega $ . When there is no ambiguity, we allow ourselves to refrain from explicitly stating this restriction to avoid cumbersome notation.

We briefly put the proof on hold to remark that we have effectively just shown that $(\Omega _i,f|_{\Omega _i})$ is FP for all i. Indeed, since $\Omega _i$ is a factor of the SFT $X_i$ , the only other requirement is that $f|_{\Omega _i}$ is expansive by Theorem 3.7, but this is immediate since f is expansive. This fact was already shown by a similar argument for $(\Omega _2,f|_{\Omega _2})$ in [Reference Ashley, Kitchens and StaffordAKS92, Observation 3.1.] We encode this intermediate result as a theorem.

Theorem 3.10. Let $(\Omega ,f)$ be an FP system with SFT cover X and factor map $\pi :X\to \Omega $ arising from a Markov partition. Let

$$ \begin{align*}\Omega_i:=\{w\in\Omega: \pi^{-1}w \text{ has at least } i \text{ mutually separated preimages}\}.\end{align*} $$

Then, if $\Omega _i$ is non-empty, $(\Omega _i,f|_{\Omega _i})$ is an FP system.

We return to the proof. Since $\pi $ is bounded-to-one by Proposition 3.8(A), so is $\pi _i$ for all i, although the bounds vary in i. Indeed, if

$$ \begin{align*}M(i):=\max\{|\pi_i^{-1}w|:w\in\Omega_i\},\end{align*} $$

then $M(i)$ may be as high as $\binom {M(1)}{i}$ .

If $w\in \Omega $ is periodic and has exactly i preimages under $\pi $ , then these preimages are mutually separated by Proposition 3.8(B) and w must have a unique preimage under $\pi _i$ . In other words, if we write $M:=M(1)$ , then

$$ \begin{align*}\operatorname{LPS}(f)=\bigcup_{1\leq i \leq M} \operatorname{supp}(|\tilde{Q}_n(f,\pi_i)|).\end{align*} $$

Note also that if $j<i$ , then w must have at least two mutually separated preimages with respect to $\pi _j$ , and if $j>i$ , then $w\notin \Omega _j$ .

For any i, it is possible that $X_i$ is not irreducible. However, since $X_i$ is an SFT, it consists of finitely many irreducible components and possibly some transient points, cf. [Reference Lind and MarcusLM21], §4.4. Call these irreducible components $X_i^j$ for $1\leq j \leq k_i,$ where $k_i<\infty $ . We consider the images of these irreducible components separately.

If a periodic point $w\in \Omega $ has a unique preimage x under $\pi _i$ , x must be periodic, and hence cannot be transient. That is, x must be in $X_i^j$ for some unique j. Thus, if we write

$$ \begin{align*}\pi_i^j:=\pi_i|_{X_i^j},\end{align*} $$

we get

$$ \begin{align*}\operatorname{LPS}(f)=\bigcup_{1\leq i \leq M}\bigcup_{1\leq j \leq k_i} \operatorname{supp}(|\tilde{Q}_n(f,\pi_i^j)|).\end{align*} $$

Now, it is certainly true that a periodic point $w\in \Omega $ having i preimages under $\pi $ means it has a unique preimage under $\pi _i$ and, therefore, that preimage x must lie in $X_i^j$ for a unique j. However, it may transpire that w has more than i preimages under $\pi $ , but a unique preimage with respect to $\pi _i^j$ for some (or several) j. For example, two preimages of w with respect to $\pi _i$ could lie in separate irreducible components of $X_i$ . In other words,

$$ \begin{align*}\{\tilde{Q}_n(f,\pi_i^j)\}_{1\leq i \leq M}^{1\leq j \leq k_i}\end{align*} $$

need not form a partition of $Q_n(f)$ .

Suppose a periodic point $w\in \Omega _i^j:=\pi _i^j(X_i^j)$ does not have a unique preimage with respect to $\pi _i$ (note the lack of superscript). Then, w will also be in $\Omega _{i'}^{j'}$ for some $i'> i$ and $j'\leq k_{i'}$ . We can thus ignore the period of this point when analyzing $\Omega _i$ and handle it when we get to $\Omega _{i'}$ . This means that if $\Omega _i^j$ contains no points with a unique preimage under $\pi _i$ , we can ignore the index pair $(i,j)$ , without losing any periods.

At the same time, if there is $w\in \Omega _i^j$ such that $|\pi _i^{-1}w|=1$ , then $\pi _i^{-1}w\in X_i^j$ . Thus, if we denote

$$ \begin{align*}\mathcal{G}:=\{(i,j):1\leq i\leq M,\;1\leq j\leq k_i, \text{ there exists } w\in\Omega_i^j:|\pi_i^{-1}w|=1\},\end{align*} $$

we can write

$$ \begin{align*}\operatorname{LPS}(f)=\bigcup_{(i,j)\in\mathcal{G}}\operatorname{supp}(|\tilde{Q}_n(f,\pi_i^j)|).\end{align*} $$

To ease the notation, we write

$$ \begin{align*}\tilde{p}_{n,i,j}:=|\tilde{P}_n(f,\pi_i^j)| \quad \text{and} \quad\tilde{q}_{n,i,j} := |\tilde{Q}_n(f,\pi_i^j)|\end{align*} $$

in what follows.

At this point, if we can show that for all $(i,j)\in \mathcal {G}$ ,

$$ \begin{align*}\operatorname{supp}(\tilde{q}_{n,i,j})\end{align*} $$

is either a singleton $\{d\}$ or $dS$ for a cofinite set S and a positive integer d, we are effectively done with the proof. This is exactly what we do.

Consider two cases. If $X_i^j$ is a finite orbit of length d, then $\Omega _i^j$ is also a finite orbit of length d, because $(i,j)\in \mathcal {G}$ . Hence,

$$ \begin{align*}\operatorname{supp}(\tilde{q}_{n,i,j})=\{d\}.\end{align*} $$

Otherwise, $X_i^j$ does not consist of a single orbit and some point in $\Omega _i^j$ has a unique preimage under $\pi _i$ . We will see that this means

$$ \begin{align*}\operatorname{supp}(\tilde{q}_{n,i,j})=dS,\end{align*} $$

where S is cofinite and d is the global period of $X_i^j$ .

To this end, we first show that

$$ \begin{align*}\lim_{n\to\infty} \frac{1}{dn} \log(\tilde{p}_{dn,i,j})=\log(\unicode{x3bb}),\end{align*} $$

which we abbreviate to

$$ \begin{align*}\tilde{p}_{dn,i,j}\sim \unicode{x3bb}^{dn},\end{align*} $$

where $\unicode{x3bb} $ is such that $h(X_i^j)=\log (\unicode{x3bb} )$ .

Let

$$ \begin{align*}\Omega':=\Omega_{i+1}\cap \Omega_i^j\end{align*} $$

and $X':=(\pi _i^j)^{-1}(\Omega ')$ . Since $\Omega _{i+1}$ is closed and invariant, so is $\Omega '$ . Moreover, $\Omega '$ is a proper subsystem of $\Omega _i^j$ , because $(i,j)\in \mathcal {G}$ , so $X'$ is a proper subshift of $X_i^j$ . Recall also that if $w\in \Omega _i^j$ is periodic but has strictly more than i mutually separated preimages with respect to $\pi $ , then $w\in \Omega '$ .

Let $\mu $ denote the real number such that $h(X')=\log (\mu )$ . We first observe then that $f|_{\Omega '}$ is expansive because f is and since $\pi _i$ is bounded-to-one, we get that

$$ \begin{align*} \limsup_{n\to\infty}\frac{1}{n}\log|P_n(f|_{\Omega'})| \stackrel{{3.6}}{\leq} h(f|_{\Omega'}) \stackrel{{3.2}}{=}h(X'). \end{align*} $$

Moreover, since $X'$ is a proper subsystem of $X_i^j$ , by Lemma 3.4,

(6) $$ \begin{align} \mu < \unicode{x3bb}. \end{align} $$

Now, for all $\varepsilon>0$ , we have that for all n large enough,

$$ \begin{align*}\frac{1}{n}\log|P_n(f|_{\Omega'})| < \log(\mu)+\varepsilon,\end{align*} $$

or, equivalently,

$$ \begin{align*} |P_{n}(f|_{\Omega'})| < (\mu e^\varepsilon)^n. \end{align*} $$

We pick $\varepsilon $ such that

$$ \begin{align*}\mu e^\varepsilon < \unicode{x3bb},\end{align*} $$

which is possible by (6).

As a consequence of [Reference Lind and MarcusLM21, Theorem 4.5.11], there exist $C, D>0$ such that for all large enough $n\in \mathbb {N}$ ,

$$ \begin{align*}C\cdot \unicode{x3bb}^{dn} \leq |P_{dn}(X_i^j)| \leq D \cdot \unicode{x3bb}^{dn}.\end{align*} $$

If $w\in \Omega _i^j$ has period n and a unique preimage in $X_i^j$ , this preimage must also be periodic with period n. Since d is the global period of $X_i^j$ , this means that a periodic point w can only have a unique preimage if its period is a multiple of d. This yields

$$ \begin{align*} \tilde{p}_{dn,i,j} \leq |P_{dn}(X_i^j)| \leq D\cdot \unicode{x3bb}^{dn} \end{align*} $$

for large enough n.

However, every periodic point $w\in \Omega _i^j$ that does not have a unique preimage under $\pi _i^j$ lies in $\Omega '$ . We remarked earlier that the converse need not be true, so while

$$ \begin{align*}P_{dn}(f|_{\Omega_i^j}) = P_{dn}(f|_{\Omega'}) \cup \tilde{P}_{dn}(f,\pi_i^j),\end{align*} $$

the sets on the right-hand side need not be disjoint. We get the inequality

$$ \begin{align*}\tilde{p}_{dn,i,j}+|P_{dn}(f|_{\Omega'})|\geq|P_{dn}(f|_{\Omega_i^j})|.\end{align*} $$

Next, the image of any periodic point of period $dn$ in $X_i^j$ must also have (not necessarily least) period $dn$ . Some such points in $X_i^j$ may be collapsed into a single point in $\Omega _i^j$ , but we know that $\pi _i$ is bounded-to-one with bound $M(i)$ . We get

$$ \begin{align*}\frac{1}{M(i)}|P_{dn}(X_i^j)|\leq |P_{dn}(f|_{\Omega_i^j})|.\end{align*} $$

Therefore, for all sufficiently large n,

$$ \begin{align*} \tilde{p}_{dn,i,j} &\geq |P_{dn}(f|_{\Omega_i^j})| - |P_{dn}(f|_{\Omega'})| \\ &\geq \frac{1}{M(i)}|P_{dn}(X_i^j)|-|P_{dn}(f|_{\Omega'})| \\ &\geq \frac{C}{M(i)}\cdot\unicode{x3bb}^{dn} - (\mu e^\varepsilon)^{dn} \\ &= \unicode{x3bb}^{dn}\bigg(\frac{C}{M(i)}-\bigg(\frac{\mu e^\varepsilon}{\unicode{x3bb}}\bigg)^{dn}\bigg). \end{align*} $$

By our choice of $\varepsilon $ ,

$$ \begin{align*}\bigg(\frac{\mu e^\varepsilon}{\unicode{x3bb}}\bigg)^{dn}\to0\end{align*} $$

as $n\to \infty $ , and we conclude that indeed

$$ \begin{align*}\tilde{p}_{dn,i,j}\sim\unicode{x3bb}^{dn}.\end{align*} $$

At this stage, we pause the proof once again to collect an intermediate result.

Theorem 3.11. Let $(\Omega ,f)$ be a topologically transitive FP system, and let an irreducible SFT X and factor map $\pi :X\to \Omega $ be those arising from a Markov partition. Let

$$ \begin{align*}\tilde{P}_n(f,\pi):=\{w\in\Omega:f^n(w)=w \text{ and }|\pi^{-1}w|=1\},\end{align*} $$

and let d be the global period of X. Then, the following quantities are equal:

1. (1) $h(f)$ ;

2. (2) $\lim _{n\to \infty }({1}/{dn})\log (|\tilde {P}_{dn}(f,\pi )|);$

3. (3) $\lim _{n\to \infty }({1}/{dn}) \log (p_{dn}(f))$ ;

4. (4) $\limsup _{n\to \infty } ({1}/{n})\log (p_n(f))$ .

Proof. By Proposition 3.8(C), there is at least one point in $\Omega $ with a unique preimage under $\pi $ . Hence, $\Omega _2$ is a strict subsystem of $\Omega $ . Since X is irreducible by Proposition 3.8(D), following the above argument gives that ${(1)}={(2)}$ , which we use to obtain the remaining equalities. Note that

$$ \begin{align*}P_n(f)\supset\tilde{P}_{n}(f,\pi),\end{align*} $$

and, hence,

$$ \begin{align*}p_n(f)=|P_n(f)|\geq |\tilde{P}_n(f,\pi)|.\end{align*} $$

A direct chain of inequalities completes the proof:

$$ \begin{align*} h(f)&\stackrel{{3.6}}{\geq} \limsup_{n\to\infty} \frac{1}{n} \log(p_n(f)) \\ &\geq \limsup_{n\to\infty} \frac{1}{dn}\log(p_{dn}(f)) \\ &\geq \liminf_{n\to\infty} \frac{1}{dn}\log(p_{dn}(f)) \\ &\geq \liminf_{n\to\infty} \frac{1}{dn}\log(|\tilde{P}_{dn}(f,\pi)|) \\ &=\lim_{n\to\infty}\frac{1}{dn}\log(|\tilde{P}_{dn}(f,\pi)|)=h(f).\\[-42pt] \end{align*} $$

Let us continue with the proof of Theorem 1.4. Recall that

$$ \begin{align*}\tilde{p}_{n,i,j}:=\bigg|\tilde{P}_n\bigg(f,\pi_i^j\bigg)\bigg| \quad \text{and} \quad\tilde{q}_{n,i,j} := \bigg|\tilde{Q}_n\bigg(f,\pi_i^j\bigg)\bigg|,\end{align*} $$

and our aim is to show that

$$ \begin{align*}\operatorname{supp}(\tilde{q}_{n,i,j})=dS,\end{align*} $$

where S is cofinite and d is the global period of $X_i^j$ .

We have shown that

$$ \begin{align*}\lim_{n\to\infty} \frac{1}{dn} \log(\tilde{p}_{dn,i,j})=\log(\unicode{x3bb}),\end{align*} $$

where $\unicode{x3bb} $ is such that $h(X_i^j)=\log (\unicode{x3bb} )$ and d is the global period of $X_i^j$ . We apply this fact in conjunction with a technique from [Reference Lind and MarcusLM21, proof of Corollary 4.3.8].

First, observe that from (5), we get that

(7) $$ \begin{align} \tilde{q}_{n,i,j} \geq \tilde{p}_{n,i,j}-\sum_{k|n,k<n}\tilde{p}_{k,i,j} \geq \tilde{p}_{n,i,j}-\sum_{k<n/2}\tilde{p}_{k,i,j}. \end{align} $$

Moreover, we note that $\tilde {P}_n(f,\pi _i^j)$ is empty whenever n is not a multiple of d, since a periodic point w with a unique preimage has period n if and only if its preimage has period n. Hence, $\tilde {Q}_n(f,\pi _i^j)$ is empty if $n\notin d\mathbb {N}$ .

Since also $\tilde {p}_{dn,i,j}\sim \unicode{x3bb} ^{dn}$ , we conclude from (7) that $\tilde {Q}_{dn}(f,\pi _i^j)$ is non-empty for all large enough n. That is,

$$ \begin{align*}\operatorname{supp}(\tilde{q}_{n,i,j})=dS,\end{align*} $$

where S is cofinite.

We have now shown that

$$ \begin{align*} \operatorname{LPS}(f)&=\bigcup_{(i,j)\in\mathcal{G}} \operatorname{supp}(|\tilde{Q}_n(f,\pi_i^j)|) \\ &= F\cup \bigcup_{l=1}^L d_lS_l, \end{align*} $$

where F is finite and each $S_l$ is cofinite. Since $X=X_1$ is an SFT, it has at least one periodic point and, therefore, $\Omega $ has at least one periodic point. Hence, either F or the big union may be empty, but not both. This concludes the general case.

Recall from Proposition 3.8(D) that if $(\Omega ,f)$ is FP and topologically transitive, X can be taken to be irreducible. Such an SFT X must either consist of a single finite orbit or have infinitely many periodic points. Consequently, the LPS of a topologically transitive FP system must be of the form above, but if it is finite, it must be a singleton.

Moreover, if $(\Omega ,f)$ is topologically mixing, Proposition 3.8(D) says that $X_1$ is mixing, and hence, $X_1$ has period 1 by [Reference Lind and MarcusLM21, Proposition 4.5.10]. In the notation above, this means that $d_l=1$ for some l. The union of a cofinite set with any other subset of the natural numbers is still cofinite, so the contribution of $\tilde {q}_{n,i,j}$ for $i>1$ , the ‘deeper’ layers of the system, to the LPS is effectively hidden. With that, we have proven Theorem 1.4.

Suppose now that $\Omega $ is an SFT. Then, it is straightforward to show that the cylinder sets formed by the alphabet of $\Omega $ are a Markov partition, where $\pi $ is the identity map. That is, $X=\Omega $ . We thus only need to consider the irreducible components $X_1^1, \ldots , X_1^{k_1}$ , noting again that any periodic point is not transient, so it must lie in one of these irreducible components. By the above analysis, the LPS of each component is either a singleton $\{d\}$ or $dS$ for S cofinite. Thus,

$$ \begin{align*}LPS(X)=F\cup \bigcup_{l=1}^Ld_lS_l,\end{align*} $$

where F is finite and each $S_l$ cofinite. Again, either F or the big union may be empty, but not both. This classification was first proved in [Reference Doering and PavlovDP19] by a direct combinatorial argument.

The observations we made with respect to the irreducibility hold in the same way. If X is irreducible, then we only have one component $X_1^1$ , and $LPS(X)$ is either a singleton $\{d\}$ or $dS$ for S cofinite. If X is mixing, then $d=1$ , and $LPS(X)$ is either $\{1\}$ or a cofinite set S. This concludes the proof of Theorem 1.5.

3.2.1 Receptive points

In the introduction, we mentioned that our main results are motivated by Krieger’s famed embedding theorem. It is natural to ask for generalizations of this theorem. In [Reference BoyleBoy83], Boyle proves one direction of Krieger’s theorem in the case of mixing sofic shifts, and recent work by Marcus et al in [Reference Marcus, Meyerovitch, Thomsen and WuMMTW25] explores the irreducible case for both sofic shifts and SFTs. In both cases, the main ingredient is the set of receptive periodic points.

Let Y be a sofic shift. A word v is synchronizing if for any words $w, w'$ , such that $wv$ and $vw'$ are both allowed in the shift, then $wvw'$ is also allowed. A periodic point $w^\infty $ in Y with least period $|w|$ is then said to be receptive if there exist synchronizing words $v_1,v_2$ such that the word $v_1w^kv_2$ is allowed in X for all $k\in \mathbb {N}$ .

Briefly, the intuition behind the utility of receptive periodic points is as follows. Recall that to embed a shift space X into a mixing SFT Y, Krieger’s embedding theorem requires that $h(X)<h(Y)$ and $q_n(X)\leq q_n(Y)$ for all $n\in \mathbb {N}$ . Both conditions force our embedding to behave a certain way on the set of all periodic points: the former gives rise to an embedding into $P_n(Y)$ for all but finitely many values of n, while the latter condition forces an embedding on these remaining values of n. A priori, these two embeddings need not cooperate nicely if X is a mixing sofic shift. Receptive periodic points are the key element that allow us to ‘glue together’ the two embeddings that arise from the two conditions, which is precisely what is done in [Reference BoyleBoy83, Theorem 3.6].

For our purposes, it will be convenient to use the following definition of receptive periodic points, which is equivalent to the definition above by [Reference Marcus, Meyerovitch, Thomsen and WuMMTW25, Theorem 4.7].

Definition 3.13. Let Y be a sofic shift and $w\in Y$ be periodic. We say w is receptive if there exists an irreducible SFT X and a factor map $\pi :X\to Y$ such that w has a preimage under $\pi $ with the same least period. In particular, periodic points with a unique preimage under some such factor map $\pi $ are receptive.

We also extract from Proposition 5.1 of the same reference the equivalence of two ways in which one might define the ‘period’ of a sofic shift. Recall that the global period of an irreducible SFT X is defined to be $\gcd (\operatorname {PS}(X)),$ which is equal to the period of any graph presentation of X.

Proposition 3.14. [Reference Marcus, Meyerovitch, Thomsen and WuMMTW25]

Let Y be an irreducible sofic shift. Then, the global period of any irreducible SFT X for which there exists a factor map $\pi :X\to Y$ which is one-to-one on at least one point in Y is equal to the gcd of the periods of all receptive periodic points.

This leads us to the following. Suppose Y is an irreducible sofic shift that is not a finite orbit. Then, Y is a topologically transitive FP system and, hence, admits a Markov partition. Let X be the SFT cover arising from this Markov partition for Y and denote the corresponding factor map $\pi :X\to Y$ . Denote also by $\tilde {Q}_n(Y)$ the set of points of least period n which have a unique preimage under $\pi $ .

By Proposition 3.8(C), there is at least one such point, so we know from the proof of Theorem 1.4 that

$$ \begin{align*}\operatorname{supp}(\{|\tilde{Q}_n(Y)|\})=dS',\end{align*} $$

where $S'$ is some cofinite set and d is the global period of X. By Definition 3.13, any point with a unique preimage in this particular cover of Y is receptive, and thus,

$$ \begin{align*}\operatorname{LPS}(\{y\in Y:y \text{ is periodic and receptive}\})\supset dS'.\end{align*} $$

However, by Proposition 3.14, we also know that

$$ \begin{align*}\operatorname{LPS}(\{y\in Y:y \text{ is periodic and receptive}\})\subset d\mathbb{N}.\end{align*} $$

Of course, in the case that Y is a finite orbit, all of its points are receptive and of the same period. Combining all of the above, we immediately get the following result.

Theorem 3.15. Let Y be an irreducible sofic shift. Then,

$$ \begin{align*}\operatorname{LPS}(\{y\in Y:y \text{ is periodic and receptive}\})\end{align*} $$

is either a singleton $\{d\}$ or $dS$ , where S is a cofinite set.

We point out that Definition 3.13 is not symbolic and, hence, a priori can be used to define receptive points in an FP system. In fact, although [Reference Marcus, Meyerovitch, Thomsen and WuMMTW25] focuses on sofic systems, it seems many of their results carry over directly to the FP case. We thus conjecture that Theorem 3.15 also holds for FP systems.

3.3 Gap shifts

We shall obtain a more precise characterization of the LPS of a subclass of shift spaces called gap shifts. We denote $\mathbb {N}_0:=\mathbb {N}\cup \{0\}$ .

Note that if S is finite, the symbol 1 occurs infinitely many times to the right and to the left in any point $x\in X(S)$ . Moreover, if S is infinite, infinitely many consecutive 0s are allowed and, hence, the point $0^\infty \in X(S)$ .

The intersection of S-gap shifts and sofic shifts (respectively SFTs) is easily described, due to the following result due to Dastjerdi and Jangjoo. We say a set $S\subset \mathbb {N}_0$ is eventually periodic if, up to some finite number of elements, it consists only of tails of infinite arithmetic progressions.

A subset R of the natural numbers is almost sum-closed if for any pair of distinct elements of R, their sum is also in R. Given a set S, we denote its almost sum-closure by

$$ \begin{align*} \Sigma_{S}^*:=\bigg\{\!\sum_{i=1}^kn_{i}:k\in\mathbb{N},n_{i}\in S \text{ with } k=1 \text{ or } \text{ there exists } i,j\in\{1,\ldots,k\} \text{ such that }n_{i}\neq n_{j} \bigg\}. \end{align*} $$

The following properties of almost sum-closures are immediate from the definition and Bézout’s lemma.

Proposition 3.18. If $S\subset \mathbb {N}$ is a singleton, then

$$ \begin{align*}\Sigma_S^*=S.\end{align*} $$

More generally, if $S\subset \mathbb {N}$ is almost sum-closed, then

$$ \begin{align*}\Sigma_{S}^*=S.\end{align*} $$

Finally, if S contains at least two distinct elements, then

$$ \begin{align*} \Sigma_{S}^*=\gcd(S)\cdot S', \end{align*} $$

where $S'$ is cofinite.

Recall that a word w is not purely periodic if there is no other word v and integer $k>1$ such that $w=v^k$ . In other words, if w is not purely periodic and $w^\infty $ is allowed in a shift, then the least period of $w^\infty $ is $|w|$ . Our classification of the LPS of gap shifts is then as follows.

Theorem 3.19. Let $S\subset \mathbb {N}_0$ be non-empty and let $X(S)$ be the S-gap shift. Then,

$$ \begin{align*}\operatorname{LPS}(X(S))=\Sigma_{S+1}^*\end{align*} $$

if S is finite, and

$$ \begin{align*}\operatorname{LPS}(X(S))=\Sigma_{S+1}^*\cup \{1\}\end{align*} $$

if S is infinite. In particular, $\operatorname {LPS}(X(S))$ is one of the following:

• • a singleton $\{ d \}$ ; or

• • an almost sum-closed set $dR$ with R cofinite; or

• • an almost sum-closed set $dR$ with R cofinite, together with the singleton $\{ 1 \}$ .

Proof. We consider three cases. First, if $S=\{d\}$ , then

$$ \begin{align*}X(S)=\{\sigma^i((10^d)^\infty):i\in\mathbb{Z}\},\end{align*} $$

which is a finite orbit of length $d+1$ . Hence,

$$ \begin{align*}\operatorname{LPS}(X(S)) = \{d+1\}=\Sigma_{\{d+1\}}^*.\end{align*} $$

Suppose now that S is finite with $|S|>1$ and let $x\in X(S)$ be periodic. Since S is finite, $x\neq 0^\infty $ . Then, we either have

$$ \begin{align*}x=\sigma^i((10^n)^\infty)\end{align*} $$

for some $i\in \mathbb {Z}$ , $n\in S$ , or

$$ \begin{align*}x=\sigma^i((10^{n_{1}}10^{n_{2}}1\cdots 10^{n_{k}})^\infty),\end{align*} $$

where $i\in \mathbb {Z}$ , $k>1$ , $n_j\in S$ for all $1\leq j\leq k$ , and $w:=10^{n_{1}}10^{n_{2}}1\cdots 10^{n_{k}}$ is not purely periodic. In the first case, the least period of x is

$$ \begin{align*}|10^n|=n+1\in \Sigma^*_{S+1}.\end{align*} $$

In the second case, at least two $n_j,n_{j'}$ are distinct, so the least period of x is

$$ \begin{align*}|10^{n_1}1\cdots10^{n_k}|=\sum_{j=1}^k(n_j+1)\in \Sigma^*_{S+1}.\end{align*} $$

Thus, $\operatorname {LPS}(X(S))\subset \Sigma ^*_{S+1}.$ Conversely, let $\tilde {n}\in \Sigma _{S+1}^*$ . Then, there exists a set $\{n_j\}_{j=1}^k$ such that $n_j\in S$ for all $1\leq j\leq k$ and at least two $n_j,n_{j'}$ are distinct, and

$$ \begin{align*}\sum_{j=1}^k(n_j+1)=\tilde{n}.\end{align*} $$

Then, after sorting the $n_j$ in weakly increasing order, the word

$$ \begin{align*}10^{n_1}1\cdots10^{n_k}\end{align*} $$

is not purely periodic, so

$$ \begin{align*}x=(10^{n_1}1\cdots10^{n_k})^\infty\end{align*} $$

is in $X(S)$ and has least period $\tilde {n}$ . Using Proposition 3.18, we conclude that

$$ \begin{align*}\operatorname{LPS}(X(S))=\Sigma_{S+1}^*=\gcd(S+1)\cdot R\end{align*} $$

for some cofinite set R.

Finally, suppose S is infinite. We may then repeat exactly the argument above, but must also note that $0^\infty \in X(S)$ . This yields a point of period 1. In this case, we thus get

$$ \begin{align*}\operatorname{LPS}(X(S))=\Sigma_{S+1}^*\cup\{1\}=(\gcd(S+1)\cdot R)\cup\{1\},\end{align*} $$

where again R is cofinite.

In the next section, we will show that any set of the form in Theorem 3.19 can in fact be realized by a sofic gap shift or even a gap SFT.

4.1 SFTs

The following result is a converse to Theorem 1.5, which is also due to Doering and Pavlov. For completeness, we repeat their constructive argument here. For a graph G, a simple cycle is a cycle where every vertex has exactly one incoming edge and exactly one outgoing edge. We call the edge shift of G the collection of all bi-infinite walks on G encoded as sequences of edges. We denote this shift $X_G$ .

Proof. We construct a graph G such that $X_G$ is irreducible and

$$ \begin{align*}\operatorname{LPS}(X_G)=R.\end{align*} $$

If $R=\{ d \}$ , let G be a simple cycle of length d. Then, the edge shift of G consists of d points each with least period d, so $\operatorname {LPS}(X_G)=\{d\}$ .

Suppose now that $R=dS$ with S cofinite. We can write

$$ \begin{align*}S=F\cup \{ N,N+1,N+2,\ldots \},\end{align*} $$

where $F=\{f_1,\ldots , f_k\}$ is finite, $f_j<f_{j+1}<N$ for all $1\leq j < k$ . We begin by constructing a graph G such that $X_G$ is a mixing SFT with $\operatorname {LPS}(X_G)=S$ . First, construct a simple cycle of length N. For each $f_j\in F$ , attach a simple cycle of length $f_j$ to this first cycle, so that each vertex in the N-cycle has at most one smaller cycle attached to it. Each of these smaller cycles gives a periodic point with least period $f_j$ , so

$$ \begin{align*}F\subset\operatorname{LPS}(X_G).\end{align*} $$

Recall that $f_1=\min F$ . For all $i\in \{1,2,\ldots ,f_1-1\},$ attach a simple cycle of length $N+i$ to the ith vertex in the $f_1$ -cycle, completing the graph construction. See Figure 1 for a visualization, where each cycle is labeled by its length.

Figure 1

An SFT construction.

If $n\geq N$ , we can write $n=qf_1+N+i$ for some $q\in \mathbb {N}_0, 0\leq i <f_1$ . By traversing the $N+i$ -cycle once and the $f_1$ -cycle q times, we obtain a point with least period n; hence,

$$ \begin{align*}\{N,N+1,N+2,\ldots\}\subset \operatorname{LPS}(X_G).\end{align*} $$

It remains to show that the least period of any periodic point in $X_G$ is in S. Suppose that $x\in X_G$ is periodic and consider two cases. If x only traverses a simple cycle, then this must either be an f-cycle for $f\in F$ or an $N+i$ -cycle for $0\leq i < f_1$ . In both cases, the least period of x is in S by construction.

Suppose now that x traverses more than one simple cycle from our construction. Then, necessarily, the least period of x must be at least N. However, any such natural number is in S by assumption, so we have exactly realized the desired LPS. Since G has two coprime cycles, one of length N and one of length $N+1$ , the graph period of G is 1, and hence, $X_G$ is mixing.

We have just shown the second half of the theorem statement, namely the case when $d=1$ . If $d>1$ , it suffices to replace every edge in G with a path of d vertices. Call this new graph $G'$ and note that it is necessarily irreducible. Then, for all $n\in \mathbb {N}$ , there is a bijection between the points of least period n in $X_G$ and the points of least period $dn$ in $X_G'$ . That is,

$$ \begin{align*}\operatorname{LPS}(X_{G'})=d\cdot\operatorname{LPS}(X_G),\end{align*} $$

which completes the proof.

Note that the above construction can be used to realize a set of the form

$$ \begin{align*}F\cup \bigcup_{i=1}^nd_iS_i\end{align*} $$

as the LPS of a (generally) reducible SFT. This is done by simply treating each singleton in F and each set $d_iS_i$ as above, and taking the edge shift on the graph consisting of several disconnected components.

4.2 Sofic shifts

At the end of the previous section, we noted how to construct a certain SFT to realize more complicated LPSs than those in Theorem 4.1. The cost of restricting ourselves to SFTs is then that the resulting shift is not irreducible. However, below we will realize almost all such sets as the LPS of some irreducible sofic shift on a binary alphabet.

We construct a graph G with edge labeling $\Phi $ such that the shift on the labeled graph $\mathcal {G}=(G,\Phi )$ , which we denote $X_{\mathcal {G}}$ , has the desired LPS. Since sofic shifts are FP, the following result provides a converse to Theorem 1.4.

Theorem 4.2. Suppose S is a singleton or a set of the form

$$ \begin{align*}F\cup\bigcup_{i=1}^n d_{i}S_{i},\end{align*} $$

where F is finite, each $S_i$ is cofinite, and $n\geq 1$ (that is, there is at least one cofinite $S_i$ ). Then, there is an irreducible sofic shift with alphabet $\{0,1\}$ whose LPS is S.

Proof. If $S=\{ d \}$ , let $w=10^{d-1}$ . Then, the shift consisting of the orbit of $w^\infty $ is sofic (in fact, of finite type), irreducible, and has LPS S.

Otherwise, S can be written as

$$ \begin{align*}S=F\cup\bigcup_{i=1}^nd_i(\mathbb{N} +k_{i} ),\end{align*} $$

where $k_i$ are such that

$$ \begin{align*}d_i(k_i+1)>2i.\end{align*} $$

Recalling that $\mathbb {N}:=\{1,2,3,\ldots \}$ , we begin by writing

$$ \begin{align*}\bigcap_{i=1}^n d_i(\mathbb{N}+k_i)=d^*(\mathbb{N}+k^*),\end{align*} $$

where

$$ \begin{align*}d^*=\operatorname{lcm}(\{d_i:1\leq i \leq n\}),\quad k^*=\min\{k\in\mathbb{N}:\text{ for all } i, d^*(k+1)\geq d_i(k_i+1) \}.\end{align*} $$

Our construction now goes in several steps. At each step, we will create several individual graphs dubbed sections to realize some part of the given LPS.

(1) Consider the set $d_i(\mathbb {N}+k_{i})$ for $i\geq 1$ . We will make a section called $\mathcal {T}_i$ to realize most of this set as least periods as follows.

Since we assumed $d_i(k_i+1)>2i$ , we also have $d_i(k_i+2)>2i+1$ . We set

$$ \begin{align*}u_i=1^{2i}0^{d_i(k_i+1)-2i} \quad \text{and} \quad v_i=1^{2i+1}0^{d_i(k_i+2)-2i-1},\end{align*} $$

so that $|u_i|=d_i(k_i+1),|v_i|=d_i(k_i+2).$

Let $t_i\in \mathbb {N}$ be such that $t_i>1$ and

$$ \begin{align*}t_{i}|u_i|,t_i|v_i|\in d^*(\mathbb{N}+k^*).\end{align*} $$

The exact choice of $t_i$ does not matter, but the latter condition generally requires it to be large.

The section $\mathcal {T}_i$ that we build will be a torus-section. We first make a $t_i$ by $t_i$ grid. From each node $(n,m)$ in this grid, add a path to $(n,m+1\!\! \mod t_{i})$ labeled $u_i$ and a path to $(n+1\!\! \mod t_{i},m)$ labeled $v_i$ . Within the labeled graph $\mathcal {T}_i$ , we can make a bi-infinite path labeled

$$ \begin{align*}(u_i^av_i^b)^\infty\end{align*} $$

for all $a,b\in \mathbb {N}_0=\mathbb {N}\cup \{0\}$ . Moreover, any other periodic point in $X_{\mathcal {T}_i}$ will, up to a shift, be of the form $w^\infty $ for

$$ \begin{align*}w=u_i^{n_1}v_i^{m_1}\ldots u_i^{n_k}v_i^{m_k}\end{align*} $$

not purely periodic. The least period of this point is

$$ \begin{align*}\sum_{j=1}^kn_jd_i(k_i+1)+m_jd_i(k_i+2)=ad_i(k_i+1)+bd_i(k_i+2)\end{align*} $$

for some $a,b\in \mathbb {N}_0$ .

Thus, $\operatorname {LPS}(X_{\mathcal {T}_i})$ is $d_i(\mathbb {N}+k_{i})$ , except for possibly a finite subset. We will treat these missed periods in step (2).

Figure 2 is a visual presentation of a torus-section. Two ends of a snaked arrow on the same column or row are in fact a single path, and each arrow labeled $u_i$ or $v_i$ represents a path of sufficiently many edges labeled by the symbols in these words. Call the (0,0) node the root node, indicated here by .

Figure 2

A torus-section.

(2) Next, write $F':= S\setminus \bigcup _{i=1}^n \operatorname {LPS}(X_{\mathcal {T}_i})$ and note that $F'$ is finite. We realize each $j\in F'$ by making a labeled cycle-section $\mathcal {C}_j$ . First, if $j\neq 1$ , construct a simple cycle of length $c_j\cdot j$ , where $c_j$ is such that $c_j\cdot j\in d^*(\mathbb {N}+k^*)$ and $c_j>1$ . Label this cycle with the word

$$ \begin{align*}w_j=10^{j-1}\end{align*} $$

repeated $c_j$ times.

If $j=1$ , make a simple cycle of length $c_1$ such that $c_1\in d^*(\mathbb {N}+k^*)$ and ${c_1>2n+1}$ . Label this cycle with the word $w_1=1$ repeated $c_1$ times. Thus, for all ${j\in F'}$ , we have $\operatorname {LPS}(X_{\mathcal {C}_j})=\{j\}$ .

Figure 3 is a visual representation of such a cycle-section, with the same visual indicators as in step 1. Again, call the 0 node the root node, indicated by .

Figure 3

A cycle-section.

(3) Finally, identify the root nodes of each section together, so that the graph is irreducible. Call this labeled graph $\mathcal {G}$ .

We now have an irreducible sofic shift $X_{\mathcal {G}}$ . By construction,

$$ \begin{align*}S=\bigcup_{j\in F'} \operatorname{LPS}(X_{\mathcal{C}_j}) \cup \bigcup_{i=1}^n\operatorname{LPS}(X_{\mathcal{T}_i}) \subset \operatorname{LPS}(X_{\mathcal{G}}).\end{align*} $$

It remains to show that no other least periods were created in joining the sections together.

Our construction is such that the shifts on all $\mathcal {T}_i$ and $\mathcal {C}_j$ are disjoint. Having taken $t_i,c_j>1$ , we also have the following properties for bi-infinite labeled paths $\tau $ on $\mathcal {G}$ :

1. (i) $\tau $ contains $1^{c_1}$ if and only if it passes through $\mathcal {C}_1$ ;

2. (ii) $\tau $ contains the word $010^j1$ if and only if it passes through $\mathcal {C}_j$ , $j\neq 1$ ; and

3. (iii) $\tau $ contains $01^{2i}0$ or $01^{2i+1}0$ if and only if it passes through $\mathcal {T}_i$ .

Moreover, a path that goes from one section to another must pass through the root node.

Let $\tau $ be a bi-infinite path in G that presents a periodic point $x\in X_{\mathcal {G}}$ . If $\tau $ only passes through a single section, the least period of x is in S by construction.

Otherwise, the above allows us to write, up to a shift, $x=w^\infty $ with

$$ \begin{align*}w=\overline{w}_{1}\overline{w}_{2}\cdots\overline{w}_{m},\end{align*} $$

with w not purely periodic and where each $\overline {w}_{k}$ is presented by a path in a single section, starting and ending at the root node.

Writing x in this way, the least period of x is

$$ \begin{align*}\sum_{k=1}^m |\overline{w}_{k}|,\end{align*} $$

which will be shown to be in $d^*(\mathbb {N}+k^*)$ and, hence, in S. Since $d^*(\mathbb {N}+k^*)$ is closed under addition, it suffices to show that

$$ \begin{align*}|\overline{w}_{k}|\in d^*(\mathbb{N}+k^*)\end{align*} $$

for all k.

Suppose $\overline {w}_{k}$ traverses a torus-section $\mathcal {T}_{i_k}$ . Then, $\overline {w}_{k}$ must consist of a combination of the words $u_{i_k},v_{i_k}$ such that each word appears some integer multiple of $t_{i_k}$ times. In other words,

$$ \begin{align*}|\overline{w}_{k}|=a\cdot t_{i_k}|u_{i_k}|+b\cdot t_{i_k}|v_{i_k}|\end{align*} $$

for some $a,b\in \mathbb {N}_0$ , not both 0. Recalling again that $t_{i_k}|v_{i_k}|,t_{i_k}|u_{i_k}|\in d^*(\mathbb {N}+k^*)$ and that this latter set is closed under addition, we get that

$$ \begin{align*}|\overline{w}_{k}|\in d^*(\mathbb{N}+k^*).\end{align*} $$

If $\overline {w}_k$ traverses a cycle-section $\mathcal {C}_{j_k}$ , the argument is similar. Indeed, since the path presenting $\overline {w}_{k}$ starts and ends at the root node, the length of this path must be an integer multiple of the cycle length of $\mathcal {C}_{j_k}$ . That is,

$$ \begin{align*}|\overline{w}_{k}|=a'\cdot c_{j_k} \cdot j_k \in d^*(\mathbb{N}+k^*)\end{align*} $$

for some $a'\in \mathbb {N}$ . Having shown that $|\overline {w}_{k}|\in d^*(\mathbb {N}+k^*)$ for all j, we conclude that the least period of x is in $d^*(\mathbb {N}+k^*)\subset S$ , as desired.

We have thus shown that

$$ \begin{align*}\operatorname{LPS}(X_{\mathcal{G}})=S,\end{align*} $$

which completes the proof.

4.3 Gap shifts

The inherent structure of gap shifts allows us to realize their LPS with relative ease. In fact, we obtain the desired LPS via a sofic gap shift. The majority of the proof below follows from Proposition 3.18 and our observations during the proof of Theorem 3.19.

Proof. Let Q be a set as indicated and for $S\subset \mathbb {N}_0$ , let $X(S)$ be the S-gap shift. We will first prove the second half of the theorem. If Q is a singleton $\{ n \}$ , we take $S=\{ n-1 \}$ . Then, $X(S)$ is a finite orbit of length n. Singletons are trivially almost sum-closed and $X(S)$ is an SFT, so we are done.

Next, suppose Q is an almost sum-closed set $dR$ with R cofinite. We will find a finite set of elements whose almost sum-closure is Q. Write first

$$ \begin{align*}Q=F\cup \{dN,d(N+1),d(N+2),\ldots\},\end{align*} $$

where every element of F is divisible by d and strictly smaller than $dN$ . Now, by assumption, Q is almost sum-closed, so for some $F'\subset \mathbb {N}$ ,

$$ \begin{align*}\Sigma_F^*=F\cup F'\subset Q.\end{align*} $$

Moreover, if we set

$$ \begin{align*}B=\{dN,d(N+1),d(N+2),\ldots,d\cdot2N\},\end{align*} $$

then

$$ \begin{align*}\Sigma^*_B=\{dN,d(N+1),d(N+2),\ldots\}.\end{align*} $$

Thus, in this particular case,

$$ \begin{align*}\Sigma^*_{F\cup B}=F\cup \{dN,d(N+1),d(N+2),\ldots\}=Q.\end{align*} $$

We noted in the proof of Theorem 3.19 that if S is finite,

$$ \begin{align*}\operatorname{LPS}(X(S))=\Sigma^*_{S+1}.\end{align*} $$

Thus, if we set $S=(F\cup B) - 1$ , then $X(S)$ is an SFT by Proposition 3.17 and

$$ \begin{align*}\operatorname{LPS}(X(S))=Q.\end{align*} $$

Finally, suppose

$$ \begin{align*}Q= {dR} \cup {\{ 1 \}}\end{align*} $$

with R cofinite and $dR$ almost sum-closed. Then, $dR$ is eventually periodic, so $dR-1$ is also eventually periodic. Thus, setting $S=dR-1$ , the S-gap shift $X(S)$ is sofic by Proposition 3.17. Now, we recall from the proof of Theorem 3.19 that if S is infinite,

$$ \begin{align*}\operatorname{LPS}(X(S))=\Sigma_{S+1}^*\cup\{1\}.\end{align*} $$

However, by Proposition 3.18, we have

$$ \begin{align*}\Sigma_{(dR-1)+1}^*=\Sigma_{dR}^*=dR,\end{align*} $$

and, hence, for $S=dR-1$ ,

$$ \begin{align*}\operatorname{LPS}(X(S))=dR\cup\{1\},\end{align*} $$

which completes the proof.

4.4 Arbitrary LPS

We conclude our collection of realizations by showing that any subset of the natural numbers is the LPS of some subshift of the full 2-shift.

Proof. If S is finite, take $X=\{\sigma ^n((10^{k-1})^\infty ):k\in S,n\in \mathbb {Z}\}$ . Then, X is a finite shift space.

Suppose $S=\{k_{1},k_{2},\ldots \}$ is infinite and written in strictly increasing order. We consider two cases.

If $1\in S$ , for all $i\in \mathbb {N}$ , take $u_{i}=10^{k_{i}-1}$ , which is not purely periodic. Then, the LPS of

$$ \begin{align*} X=\overline{\{ \sigma^n(u_{i}^\infty):i \in \mathbb{N}, n\in \mathbb{Z} \}} \end{align*} $$

certainly contains S. We show that it contains no other periods.

Let X be as above and take $x\in X$ periodic with least period k, where $x=w^\infty $ for w not purely periodic. If there are no 1s in w, then $x=0^\infty $ , which has least period 1. This period is in S by assumption and we are safe.

Otherwise, take $x^{(n)}$ to be a sequence of points such that for all $n\in \mathbb {N}$ ,

$$ \begin{align*}x^{(n)}=\sigma^j(u_i^\infty)\end{align*} $$

for some $i\in \mathbb {N},j\in \mathbb {Z}$ and

$$ \begin{align*}\lim_{n\to\infty} x^{(n)} = x.\end{align*} $$

Now, there exists an N such that for all $n\geq N $ , we have that

$$ \begin{align*}x^{(n)}_{[-k,k-1]} = x_{[-k,k-1]} =w^2.\end{align*} $$

Since there is a 1 somewhere in w, we know that the word $10^{l}1$ is in $w^2$ for some l. However, if this word is contained in $x^{(n)}$ , then $x^{(n)}$ must be some shift of $(10^l)^\infty $ and, therefore, $l=k_i-1$ for some $k_i\in S$ . We conclude that for some $i\in \mathbb {N},j\in \mathbb {Z}$ ,

$$ \begin{align*}x^{(n)}=\sigma^j(u_i^\infty)\end{align*} $$

for all $n\geq N$ . Hence,

$$ \begin{align*}x=\sigma^j(u_i^\infty),\end{align*} $$

and the least period of x is in S by construction.

Suppose now that $1\notin S$ . Set $u_1=10^{k_1-1}$ . For $i>1$ , take

$$ \begin{align*}u_{i}=1u_{1}^{g_i}u_{i}',\end{align*} $$

where $g_i\in \mathbb {N}_0$ and $u_{i}'$ is a strict prefix of $u_1$ such that $|u_1^{g_i}u_{i}'|=k_i-1$ . This way, $u_i$ is not purely periodic, since the subword 110 only occurs at the very start. At the same time, $u_i$ contains the subword 110 for any $i>1$ , since $k_1\neq 1.$

Now, the LPS of the shift space

$$ \begin{align*} X=\overline{\{ \sigma^n(u_{i}^\infty):i \in \mathbb{N}, n\in \mathbb{Z} \}} \end{align*} $$

again contains S. We show that no other least periods are created, using a similar, but slightly more subtle, argument as in the previous case.

Let again X be as above and take $x\in X$ periodic with least period k, where $x=w^\infty $ for w not purely periodic.

Moreover, take once again $x^{(n)}$ to be a sequence of points such that for all $n\in \mathbb {N}$ ,

$$ \begin{align*}x^{(n)}=\sigma^j(u_i^\infty)\end{align*} $$

for some $i\in \mathbb {N},j\in \mathbb {Z}$ and

$$ \begin{align*}\lim_{n\to\infty} x^{(n)} = x.\end{align*} $$

Slightly altering our previous argument, there exists an N such that for all $n\geq N $ , we have that

$$ \begin{align*}x^{(n)}_{[-k,2k-1]} = x_{[-k,2k-1]} =w^3.\end{align*} $$

Suppose that the word 110 appears in x. Then, $w^3$ contains a word of the form $110\tilde {u}110$ for some subword $\tilde {u}$ . For all n large enough, $x^{(n)}$ then also contains $110\tilde {u}110$ . This is only possible if

$$ \begin{align*}x^{(n)}=\sigma^j(u_i^\infty)\end{align*} $$

for some $i\in \mathbb {N},j\in \mathbb {Z}$ for all large n, and hence, $x=\sigma ^j(u_i^\infty )$ with least period $k_i\in S$ .

If the word 110 does not appear in x, take a subsequence $n_m$ such that

$$ \begin{align*}x^{(n_m)}_{[-mk,mk-1]} = x_{[-mk,mk-1]}=w^{2m}\end{align*} $$

for all $m\in \mathbb {N}$ . Since the word 110 does not appear in $w^{2m}$ for any m, we know that if 110 appears in $x^{(n_m)},$ it must be outside of the window $[-mk,mk-1]$ . In other words, for all m,

$$ \begin{align*}x^{(n_m)}_{[-mk,mk-1]}=\sigma^j(u_1^\infty)_{[-mk,mk-1]}\end{align*} $$

for some $j\in \mathbb {Z}$ . We then conclude that $x=\sigma ^j(u_1^\infty )$ , which has least period $k_1\in S$ .