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We study the combinatorial types of periodic orbits of the standard coveringendomorphisms ${\mathbf m}_k(x)=k x \ (\text{mod} \ {\mathbb Z})$ of the circlefor 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 fulldescription of the set of period $q$ orbits of ${\mathbf m}_k$ that realize$\sigma$ and in particular count how many such orbits there are. Thedescription is based on an invariant called the "fixed point distribution"vector and is achieved by reducing the realization problem to finding thestationary state of an associated Markov chain. Our results generalize earlierwork on the special case where $\sigma$ is a rotation cycle, and can be viewedas a missing combinatorial ingredient for a proper understanding of thedynamics of complex polynomial maps of degree $\geq 3$ and the structure oftheir parameter spaces.

On Combinatorial Types of Periodic Orbits of the Map $x \mapsto kx$ (mod $\mathbb Z$)