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

TL;DR

This paper classifies and counts the types of periodic orbits for the map $x o kx mod 1$ on the circle, using combinatorial and Markov chain methods, extending previous results and aiding understanding of complex polynomial dynamics.

Contribution

It provides a complete combinatorial description of periodic orbit types for the map, including a counting method based on fixed point distributions and Markov chains.

Findings

Classifies all period $q$ orbits realizing a given permutation $\sigma$.

Introduces the fixed point distribution vector as an invariant.

Generalizes previous results on rotation cycles.

Abstract

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.