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  "body": "\n## Definition\n\nLet $X$ be a set and let $f \\colon X \\to X$ be a map.\nAn *Abel function* of $f$ is a [[function|0EPAT51DXQ]] $\\alpha \\colon X \\to \\C$ that satisfies the [[Abel equation|68SHB7Z721]]\n$$\\alpha(f(x)) = \\alpha(x) + 1 \\quad \\text{for all } x \\in X.$$\nMany authors also require $\\alpha$ to be injective, or, for real $X$, continuous and strictly monotone.\n\n## Remarks\n\nExample: for $f(x) = 3x$ on $X = (0, \\infty)$, $\\alpha(x) = \\log_3 x$ is an Abel function, since $\\log_3(3x) = \\log_3 x + 1$.\nNon-example: for the same $f$, $\\alpha(x) = \\ln x$ is not an Abel function, since $\\ln(3x) = \\ln x + \\ln 3$ and $\\ln 3 \\ne 1$.\nIf $\\alpha$ is an injective Abel function and $t$ is real with $\\alpha(X) + t \\subseteq \\alpha(X)$, then $f^t(x) = \\alpha^{-1}(\\alpha(x) + t)$ satisfies $f^1 = f$ and $f^s \\circ f^t = f^{s + t}$, so $(x, t) \\mapsto f^t(x)$ solves the translation equation where it is defined.\nIn the example, $f^t(x) = 3^t x$.\n",
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