Let $X$ be a set, let $F: X \to X$ be a map, and let $\Lambda \subset X$ be a subset satisfying $F(\Lambda) \subset \Lambda$. Let $F|_\Lambda: \Lambda \to \Lambda$ denote the restricted map $x \mapsto F(x)$. Let

\begin{align*} \Sigma_2 := \{0,1\}^{\mathbb Z} \end{align*}

be the set of all bi-infinite binary sequences, and let $\sigma: \Sigma_2 \to \Sigma_2$ be the left shift map defined by $(\sigma(s))_k = s_{k+1}$ for every sequence $s = (s_k)_{k \in \mathbb Z} \in \Sigma_2$ and every coordinate index $k \in \mathbb Z$. Suppose there is a bijection $\pi: \Lambda \to \Sigma_2$ satisfying the conjugacy relation

\begin{align*} \pi(F(x)) = \sigma(\pi(x)) \end{align*}

for every $x \in \Lambda$.

For a self-map $T: Y \to Y$, a point $y \in Y$ is called periodic if there exists $p \in \mathbb N$ such that $T^p(y)=y$, and it has exact period $p$ if $T^p(y)=y$ and $T^k(y) \ne y$ for every integer $k$ with $1 \le k < p$. A periodic orbit of $T$ is the finite set $\{T^k(y): k \in \mathbb N \cup \{0\}\}$ associated to a periodic point $y$.

Then every periodic bi-infinite binary sequence corresponds to a periodic point of $F|_\Lambda$: if $s \in \Sigma_2$ and $p \in \mathbb N$ satisfy $\sigma^p(s) = s$, then the unique point $x \in \Lambda$ with $\pi(x) = s$ satisfies $F^p(x) = x$. If $s$ has exact period $p$ under $\sigma$, then $x$ has exact period $p$ under $F|_\Lambda$. Consequently, $F|_\Lambda$ has periodic points of exact period $p$ for every $p \in \mathbb N$, and the collection of periodic orbits contained in $\Lambda$ is countable.