Let $k \in \mathbb{N}$, let $A \in \{0,1\}^{k \times k}$, and suppose that the directed graph $G_A$ with vertex set $\{1,\dots,k\}$ and directed edge $i \to j$ exactly when $A_{ij}=1$ contains at least one directed cycle. Define the two-sided subshift of finite type
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\begin{align*}
\Sigma_A := \{x \in \{1,\dots,k\}^{\mathbb{Z}} : A_{x_m x_{m+1}} = 1 \text{ for every } m \in \mathbb{Z}\}.
\end{align*}
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Let $\sigma: \Sigma_A \to \Sigma_A$ be the shift map given by $(\sigma x)_m = x_{m+1}$. Then $\Sigma_A \neq \varnothing$ and