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  "body": "\n## Definition\n\nLet $X$ be a set.\nThe *translation equation* on $X$ is the [[functional equation|SRAN4EKWS0]]\n$$F(F(x, s), t) = F(x, s + t) \\quad \\text{for all } x \\in X \\text{ and } s, t \\in \\R,$$\nfor an unknown [[function|0EPAT51DXQ]] $F \\colon X \\times \\R \\to X$.\nA solution that also satisfies $F(x, 0) = x$ for all $x \\in X$ is called a *flow* or *one-parameter group of transformations* of $X$.\n\n## Remarks\n\nExample: $F(x, t) = x e^t$ on $X = \\R$ is a solution, since $F(F(x, s), t) = x e^s e^t = x e^{s + t}$.\nNon-example: $F(x, t) = x + t^2$ is not a solution, since $F(F(0, 1), 1) = 2$ while $F(0, 2) = 4$.\nFor a flow, each map $f^t = F(\\cdot, t)$ is a bijection of $X$ with inverse $f^{-t}$, and $f^s \\circ f^t = f^{s + t}$.\nIf $\\alpha \\colon X \\to \\R$ is a bijection, then $F(x, t) = \\alpha^{-1}(\\alpha(x) + t)$ is a flow, and this is how [[Abel functions|2QBWT7T02Y]] produce fractional iterates.\nThe same equation is studied with $\\R$ replaced by $[0, \\infty)$, $\\Z$, $\\C$ or any group or semigroup.\n",
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