Fractional iteration

Summary

Fractional iteration extends the iterates $f^n$ of a map $f$ from whole numbers $n$ to real numbers $t \ge 0$. One 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. Then $f^{1/n}$ is an iterative root of order $n$ of $f$. The main tools are Schröder's equation and the Abel equation, which turn iteration into multiplication or into addition.