# Periodic One Way Function

Is there any notion of a periodic OWF?
I.e. an OWF $f$ that on every input $x$ in its domain it holds that $f(f(f\ldots f(x))\ldots )=x$ when $f$ applied many times, each time on the previous output.

Also, if we take some One Way Permutation, it must be periodic of course. So is it guaranteed that its period is longer than any polynomial (in the security parameter)? (that is, if its period is of some polynomial length - is its security broken?)