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  "body": "\n## Definition\n\nLet $\\mathcal{A}$ be a finite alphabet and let $\\sigma(x)_i = x_{i+1}$ be the shift map on $\\mathcal{A}^{\\Z}$.\nA *symbolic dynamical system* is a pair $(X, \\sigma_X)$ where $X \\subseteq \\mathcal{A}^{\\Z}$ is a [[shift space|A1XE59ZHNM]] and $\\sigma_X$ is the restriction of $\\sigma$ to $X$.\nOne-sided versions use $\\mathcal{A}^{\\N}$.\n*Symbolic dynamics* is the part of [[topological dynamics|5X6E97EMFR]] that studies symbolic dynamical systems and the maps between them that commute with the shifts.\nIt also studies codings: given a [[continuous map|0VGFHTM75Y]] $T \\colon M \\to M$ and a partition $M = P_0 \\cup \\cdots \\cup P_{k-1}$, the *itinerary* of $x \\in M$ is the sequence $(a_n)_{n \\ge 0}$ with $T^n(x) \\in P_{a_n}$.\n\n## Remarks\n\nThe itinerary map $x \\mapsto (a_n)$ turns $T$ into the shift: the itinerary of $T(x)$ is the shifted itinerary of $x$.\nA coding is rarely continuous, since a connected space cannot be split into finitely many disjoint nonempty open pieces.\nInstead one often has a continuous surjection $\\pi$ from a shift space onto $M$ with $\\pi \\circ \\sigma = T \\circ \\pi$, a [[semiconjugacy|AEXJNRBASV]] that sends a sequence to a point with that itinerary.\nExample: for the doubling map $T(x) = 2x \\bmod 1$ on $[0, 1)$ with $P_0 = [0, 1/2)$ and $P_1 = [1/2, 1)$, the itinerary of $x$ is its binary expansion $x = \\sum_{n \\ge 0} a_n 2^{-n-1}$ that does not end in all $1$s.\nIndeed $T$ deletes the first binary digit, and the first digit is $0$ exactly when $x \\in P_0$.\nHere $\\pi(a) = \\sum_{n \\ge 0} a_n 2^{-n-1}$ is a semiconjugacy from the one-sided full shift on $\\set{0, 1}$ onto the circle $\\R / \\Z$.\nNon-example: a coding with the single region $P_0 = M$ gives the constant itinerary $000\\ldots$ for every point, so it records nothing about the dynamics.\n\n## Explanations\n\n### L1 human\n\nSymbolic dynamics studies systems whose states are infinite words, moved by deleting or shifting letters, and uses them as codes for other systems.\n",
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