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  "body": "\n## Summary\n\nAn Abel function of a map $f$ is a solution $\\alpha$ of the [[Abel equation|68SHB7Z721]] $\\alpha(f(x)) = \\alpha(x) + 1$.\nIt is a coordinate in which one step of $f$ becomes a shift by $1$.\nWhen $\\alpha$ is injective, it defines a whole family of iterates $f^t = \\alpha^{-1} \\circ (\\alpha + t)$, one for each real $t$ in a suitable range.\nAbel functions are the standard tool for fractional iteration, for example in the construction of iterates of the exponential function.\n\n## Explanations\n\n### L1 human\n\nAn Abel function is a clock that ticks once each time the map $f$ is applied.\nFor the map $f(x) = x + 3$ on the real line, the function $\\alpha(x) = x/3$ is such a clock.\nReading the clock, one can run the map for any amount of time: running for time $t$ moves $x$ to $x + 3t$.\nFor more complicated maps the clock is harder to find, but the idea is the same.\n",
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