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The study of fractional self-iterations of a map. A basic example is the analysis of functional square roots of a map $g$, i.e. solutions $f$ to the functional equation $f \circ f = g$. The continuous version of fractional iteration concerns maps which have flows. This case is also known as continuous iteration. A classic example is the problem of extending tetration to the real and complex numbers.
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Does the exponential function have a (compositional) square root?
Re: second question.
Yes, there are many of them on a half-line. Just take any smooth increasing function on $[0,1]$ with the property that $E(1)$ and $E'1(1)$ have desired values and extend it by t …