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  "body": "\n## Definition\n\nLet $\\mathcal{A}$ be a finite set, the alphabet, with the discrete topology, and give $\\mathcal{A}^{\\Z}$ the [[product topology|0GBG8B49AA]].\nThe *shift map* $\\sigma \\colon \\mathcal{A}^{\\Z} \\to \\mathcal{A}^{\\Z}$ is $\\sigma(x)_i = x_{i+1}$ for $x = (x_i)_{i \\in \\Z}$.\nA *shift space* (or *subshift*) over $\\mathcal{A}$ is a subset $X \\subseteq \\mathcal{A}^{\\Z}$ such that:\n\n1. $X$ is closed in $\\mathcal{A}^{\\Z}$.\n2. $\\sigma(X) = X$.\n\n## Remarks\n\nThe space $\\mathcal{A}^{\\Z}$ is [[compact|0TVJZYB6SH]], with the metric $d(x, y) = 2^{-\\min\\set{\\abs{i} : x_i \\neq y_i}}$ for $x \\neq y$, and $\\sigma$ is a [[homeomorphism|0QT00PB5G2]]; so $\\sigma$ restricts to a homeomorphism of each shift space.\nThe whole space $\\mathcal{A}^{\\Z}$ is the *full shift*.\nEquivalent form: $X$ is a shift space if and only if it is defined by a set $\\mathcal{F}$ of forbidden words.\nHere $\\mathcal{F}$ is a set of finite words, elements of the [[free monoid|CV013KHVJG]] on $\\mathcal{A}$.\nThen $X$ is the set of sequences in which no word of $\\mathcal{F}$ occurs as a block of consecutive symbols.\nOne-sided shift spaces are defined in the same way in $\\mathcal{A}^{\\N}$, with $\\sigma(X) \\subseteq X$.\nExample: the golden mean shift $X \\subseteq \\set{0, 1}^{\\Z}$, the sequences with no block $11$.\nA limit of such sequences has no block $11$, since a block $11$ at positions $i, i+1$ would appear in the sequences close to it.\nSo $X$ is closed, and shifting does not create or remove blocks, so $\\sigma(X) = X$.\nNon-example: the set of sequences in $\\set{0, 1}^{\\Z}$ with exactly one symbol $1$ satisfies clause 2, but not clause 1.\nIts elements with the $1$ at position $n$ tend to the all-$0$ sequence as $n \\to \\infty$, which is not in the set.\n\n## Explanations\n\n### L1 human\n\nA shift space is a closed, shift-invariant set of two-sided infinite sequences over a finite alphabet.\n",
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