  We study the combinatorial types of periodic orbits of the standard covering
endomorphisms ${\mathbf m}_k(x)=k x \ (\text{mod} \ {\mathbb Z})$ of the circle
for integers $k \geq 2$ and the frequency with which they occur. For any
$q$-cycle $\sigma$ in the permutation group ${\mathcal S}_q$, we give a full
description of the set of period $q$ orbits of ${\mathbf m}_k$ that realize
$\sigma$ and in particular count how many such orbits there are. The
description is based on an invariant called the "fixed point distribution"
vector and is achieved by reducing the realization problem to finding the
stationary state of an associated Markov chain. Our results generalize earlier
work on the special case where $\sigma$ is a rotation cycle, and can be viewed
as a missing combinatorial ingredient for a proper understanding of the
dynamics of complex polynomial maps of degree $\geq 3$ and the structure of
their parameter spaces.
