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  "body": "\n## Definition\n\nLet $A$ be a $k \\times k$ matrix with entries in $\\set{0, 1}$ that is [[irreducible|ZEZPYJC06B]].\nLet $\\Sigma_A = \\set{x \\in \\set{1, \\ldots, k}^{\\Z} : A_{x_i x_{i+1}} = 1 \\text{ for all } i}$ be its vertex shift, a [[subshift of finite type|M7RPDE39FM]], with shift map $\\sigma$.\nBy the [[Perron–Frobenius theorem|62NS7PV1M9]], $A$ has a positive eigenvalue $\\lambda$ of largest modulus, with eigenvectors $u, v$ with positive entries such that $u^T A = \\lambda u^T$ and $A v = \\lambda v$.\nNormalise them so that $\\sum_i u_i v_i = 1$.\nPut $p_i = u_i v_i$ and $P_{ij} = A_{ij} v_j / (\\lambda v_i)$.\nThe *Parry measure* is the [[probability measure|0VGMEBVW1M]] $\\mu$ on $\\Sigma_A$ such that, for all $n \\in \\Z$, $m \\ge 0$ and states $i_0, \\ldots, i_m$,\n$$\\mu\\set{x : x_{n} = i_0, \\ldots, x_{n+m} = i_m} = p_{i_0} P_{i_0 i_1} \\cdots P_{i_{m-1} i_m}.$$\n\n## Remarks\n\nThe matrix $P$ is stochastic and $p^T P = p^T$, so these values are consistent and $\\mu$ is a $\\sigma$-[[invariant measure|B232EZ1FMB]].\nIts entropy is $\\log \\lambda$, which is the topological entropy of $\\sigma$ on $\\Sigma_A$, and it is the only invariant probability measure with this entropy.\nExample: the golden mean shift has $A = \\begin{pmatrix} 1 \u0026 1 \\\\ 1 \u0026 0 \\end{pmatrix}$ and $\\lambda = \\varphi = (1 + \\sqrt5)/2$.\nHere $u = v = (\\varphi, 1)$ up to scaling, since $\\varphi + 1 = \\varphi^2$.\nAfter normalisation, $p_1 = \\varphi^2/(\\varphi^2 + 1)$ and $p_2 = 1/(\\varphi^2 + 1)$, and $P_{11} = 1/\\varphi$, $P_{12} = 1/\\varphi^2$, $P_{21} = 1$, $P_{22} = 0$.\nEach row of $P$ sums to $1$, because $1/\\varphi + 1/\\varphi^2 = 1$.\nNon-example: on the same shift, the Markov measure with $P_{11} = P_{12} = 1/2$ is invariant for its stationary vector but is not the Parry measure, and its entropy is smaller than $\\log \\varphi$.\n",
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