Let $N \in \mathbb{N}$, let $M \in \{0,1\}^{N \times N}$ be irreducible, and let $\lambda > 0$ be the Perron eigenvalue of $M$. Let $l \in (0,\infty)^N$ and $r \in (0,\infty)^N$ be left and right Perron eigenvectors normalized by
\begin{align*}
\Sigma_M := \{x = (x_0,x_1,\dots) \in \{1,\dots,N\}^{\mathbb{N}_0} : M_{x_k x_{k+1}} = 1 \text{ for every } k \in \mathbb{N}_0\}
\end{align*}
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and let $\sigma: \Sigma_M \to \Sigma_M$ be the left shift map $\sigma(x)_k = x_{k+1}$. Define the Parry transition matrix $P \in [0,1]^{N \times N}$ and stationary vector $\pi \in [0,1]^N$ by
Moreover, $\mu_P$ is the unique $\sigma$-invariant Borel probability measure on $\Sigma_M$ whose measure-theoretic entropy is $h_{\mathrm{top}}(\sigma|_{\Sigma_M})$.