Counting Rotational Subsets of the Circle $\mathbb{R}/ \mathbb{Z}$ under the Angle-Multiplying Map $t\mapsto dt$

Authors

Ian Tan, Auburn UniversityFollow

Abstract

A rotational set is a finite subset $A$ of the unit circle $\mathbb{R}/ \mathbb{Z}$ 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 $p/q$. Lisa Goldberg introduced these sets 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.

Author Bio

Ian Tan (1994) was born in Kuala Lumpur, Malaysia. In 2018 he graduated from Lee University in Cleveland, Tennessee with a Bachelor’s of Science in Mathematics, minoring in Music. As of 2024 Ian is a graduate student at the Department of Mathematics and Statistics, Auburn University.

Faculty Sponsor

Debra Gladden

Recommended Citation

Tan, Ian (2024) "Counting Rotational Subsets of the Circle $\mathbb{R}/ \mathbb{Z}$ under the Angle-Multiplying Map $t\mapsto dt$," Rose-Hulman Undergraduate Mathematics Journal: Vol. 25: Iss. 2, Article 4.
Available at: https://scholar.rose-hulman.edu/rhumj/vol25/iss2/4