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  "body": "\n## Summary\n\nFractional iteration extends the [[iterates|AYVTR2ES5B]] $f^n$ of a map $f$ from whole numbers $n$ to real numbers $t \\ge 0$.\nOne asks for maps $f^t$ with $f^0 = \\id$, $f^1 = f$ and $f^s \\circ f^t = f^{s+t}$, so that $f$ becomes the time-one map of a flow.\nThen $f^{1/n}$ is an [[iterative root|27C6J3YNZY]] of order $n$ of $f$.\nThe main tools are [[Schröder's equation|VTJV7JVJEA]] and the [[Abel equation|68SHB7Z721]], which turn iteration into multiplication or into addition.\n\n## Explanations\n\n### L1 human\n\nApplying $f$ twice is easy to picture, but what does applying $f$ one and a half times mean?\nFractional iteration gives an answer: a family of maps $f^t$, one for each time $t$, that fit together like the positions of a moving object.\nFor $f(x) = 2x$, the natural choice is $f^t(x) = 2^t x$.\nFor $f(x) = x + c$, it is $f^t(x) = x + tc$.\nFor most maps there is no single natural choice, and one adds conditions such as continuity or analyticity in $t$.\n",
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