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  "body": "\n## Definition\n\nLet $X$ be a set and let $f \\colon X \\to X$ be a [[map|0EPAT51DXQ]].\nThe *discrete-time dynamical system* generated by $f$ is the pair $(X, f)$, viewed as the [[dynamical system|BBZP1R872N]] $\\Phi \\colon \\N \\times X \\to X$ with $\\Phi(n, x) = f^n(x)$.\nHere $f^n$ is the $n$-th [[iterate|AYVTR2ES5B]] of $f$, with $f^0 = \\id_X$.\nIf $f$ is a bijection, the system is *invertible*, and $\\Phi(n, x) = f^n(x)$ is defined for all $n \\in \\Z$.\nHere $f^{-n}$ is the $n$-th iterate of the [[inverse map|0J2WAR83WP]] $f^{-1}$.\n\n## Remarks\n\nThe forward orbit of $x$ is the sequence $x, f(x), f^2(x), \\dots$.\nEvery dynamical system with time $\\N$ arises in this way, with $f = \\Phi(1, \\cdot)$, since clause 2 of the definition of a dynamical system forces $\\Phi(n, \\cdot) = f^n$.\n\nExample: $X = \\R$ and $f(x) = 3x$.\nThen $\\Phi(n, x) = 3^n x$, and since $f$ is a bijection the system is invertible with $\\Phi(-n, x) = 3^{-n} x$.\n\nNon-example of an invertible system: $f(x) = x^2$ on $\\R$ generates a discrete-time system, but it is not invertible, because $f(-1) = f(1)$.\n\nUsually $X$ is a topological space and $f$ is continuous, or $X$ is a measure space and $f$ is measurable.\n",
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