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arXiv · 2207.03594

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

Yee Ern Tan
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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.

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Yee Ern Tan. 2022-07-07. Counting rotational subsets of the circle $\mathbb{R}/\mathbb{Z}$ under the angle multiplying map $t\mapsto dt$. https://arxiv.org/abs/2207.03594

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