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

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.