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  "body": "\n## Definition\n\nLet $A$ be a finite set, called the alphabet.\nLet $A^{\\Z}$ be the set of all sequences $x = (x_i)_{i \\in \\Z}$ with $x_i \\in A$, with the [[product topology|0Z57EC6YX3]] of the discrete topologies.\nLet $\\sigma \\colon A^{\\Z} \\to A^{\\Z}$ be the shift map, $\\sigma(x)_i = x_{i+1}$.\nA word $w = w_1 \\cdots w_n$ over $A$ occurs in $x$ if $x_{i+1} \\cdots x_{i+n} = w$ for some $i \\in \\Z$.\nFor a set $F$ of words, let $X_F$ be the set of all $x \\in A^{\\Z}$ in which no word of $F$ occurs.\nA subset $X \\subseteq A^{\\Z}$ is a *subshift of finite type* if $X = X_F$ for some finite set $F$ of words.\n\n## Remarks\n\nEach $X_F$ is closed and satisfies $\\sigma(X_F) = X_F$, so it is a [[shift space|A1XE59ZHNM]].\nExample: the golden mean shift is $X_F$ with $A = \\set{0, 1}$ and $F = \\set{11}$.\nIt is a subshift of finite type because $F$ is finite.\nNon-example: the even shift is the set of $x \\in \\set{0,1}^{\\Z}$ in which between any two $1$s there is an even number of $0$s.\nIt is a shift space, but it is not of finite type.\nEach word $1 0^{2n+1} 1$ is forbidden, while all its proper subwords occur in allowed sequences, so no finite list of forbidden words can exclude all of them.\nEvery subshift of finite type is conjugate to a vertex shift.\nGiven a finite set $B$ and a square $0$-$1$ matrix $M$ indexed by $B$, this is the set of $x \\in B^{\\Z}$ with $M_{x_i x_{i+1}} = 1$ for all $i$.\nThe same definition with $A^{\\Z^d}$ and finite forbidden patterns gives the multidimensional subshifts of finite type.\n",
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