Counting rotational subsets of the circle $\mathbb{R}/\mathbb{Z}$ under the angle multiplying map $t\mapsto dt$

TL;DR

This paper characterizes rotational subsets of the circle under angle multiplication maps, providing conditions for their existence, recovering classical results, and enumerating such sets based on their rotation number and orbit count.

Contribution

It offers a necessary and sufficient condition for a set to be rotational under the angle multiplication map with a given rotation number, advancing the understanding of these sets.

Findings

Derived a condition for rotational sets with a specific rotation number

Recovered classical results as special cases

Enumerated rotational sets with fixed orbit counts

Abstract

A rotational set is a finite subset $A$ of the unit circle $\mathbb{T}=\mathbb{R}/ \mathbb{Z}$ such that the angle-multiplying map $\sigma_{d}:t\mapsto dt$ maps $A$ onto itself by a cyclic permutation of its elements. Each rotational set has a geometric rotation number $p/q$. These sets were introduced by Lisa Goldberg to study the dynamics of complex polynomial maps. In this paper we provide a necessary and sufficient condition for a set to be $\sigma_{d}$-rotational with rotation number $p/q$. As applications of our condition, we recover two classical results and enumerate $\sigma_d$-rotational sets with rotation number $p/q$ that consist of a given number of orbits.