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  "body": "\n## Statement\n\nLet $A$ be a finite set and let $d \\ge 1$.\nGive $A^{\\Z^d}$ the [[product topology|0Z57EC6YX3]] of the discrete topologies.\nFor $v \\in \\Z^d$ let $\\sigma_v \\colon A^{\\Z^d} \\to A^{\\Z^d}$ be the shift $\\sigma_v(x)_u = x_{u + v}$.\nLet $F \\colon A^{\\Z^d} \\to A^{\\Z^d}$ be a map.\nThen the following are equivalent:\n\n1. $F$ is [[continuous|0P27W9RW0G]] and $F \\circ \\sigma_v = \\sigma_v \\circ F$ for all $v \\in \\Z^d$.\n2. There are a finite set $N \\subseteq \\Z^d$ and a map $f \\colon A^N \\to A$ such that $F(x)_u = f\\big((x_{u + n})_{n \\in N}\\big)$ for all $x$ and all $u \\in \\Z^d$.\n\n## Remarks\n\nCondition 2 says that $F$ is a [[cellular automaton|5QXNC6NTVN]] with neighbourhood $N$ and local rule $f$.\nFor 2 implies 1, the value $F(x)_u$ depends only on finitely many coordinates of $x$, which gives continuity.\nThe same rule is used at every site, which gives commutation with shifts.\nFor 1 implies 2, the space $A^{\\Z^d}$ is compact, so $F$ is uniformly continuous.\nHence $F(x)_0$ is determined by the coordinates $x_n$ with $n$ in some finite set $N$, and this defines $f$.\nCommutation with the shifts then gives the formula at every site $u$.\nCompactness is essential: over an infinite alphabet there are continuous shift-commuting maps that are not given by a finite rule.\nThe same statement holds for maps between [[shift spaces|A1XE59ZHNM]], where condition 2 says that $F$ is a sliding block code.\n\n## Explanations\n\n### L1 human\n\nA cellular automaton updates every cell of a grid by looking at a few neighbours and applying a fixed rule.\nSuch a map treats all positions alike and changes only a little when the input changes far away.\nThe theorem says that these two properties characterise cellular automata.\nSo one can define cellular automata either by rules or by topology.\n\n### L2 human\n\nThe theorem lets one treat a cellular automaton as an endomorphism of the dynamical system $(A^{\\Z^d}, \\sigma)$.\nIt shows, for instance, that the inverse of a bijective cellular automaton is again a cellular automaton.\nA continuous bijection of a compact space onto a Hausdorff space has a continuous inverse, and that inverse commutes with the shifts.\nIn one dimension the theorem identifies the conjugacies between shift spaces with invertible sliding block codes.\nIt is a basic tool in the classification of [[subshifts of finite type|624XDXX8FN]].\n",
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