# Rotational subsets of the circle

## TL;DR

This paper extends the understanding of rotational subsets on the circle, proving a structure theorem for a broader class of transformations with finite fibers, using more analytic methods than previous works.

## Contribution

It generalizes the structure theorem for rotational subsets to continuous transformations with finite fibers, employing analytic techniques.

## Key findings

- Infinite rotational subsets are extensions of irrational rotations.
- The structure theorem applies to a wider class of transformations.
- Analytic methods provide new insights into rotational subsets.

## Abstract

A rotational subset, relative to a continuous transformation $T: \mathbb{T} \to \mathbb{T}$ on the unit circle, is a closed, invariant subset of $\mathbb{T}$ that is minimal and on which $T$ respects the standard orientation of the unit circle. In the case where $T$ is the standard angle doubling map, such subsets were studied by Bullet and Sentenac. The case where $T$ multiplies angles by an integer $d > 2$ was studied by Goldberg and Tresser, and Blokh, Malaugh, Mayer, Oversteegen, and Parris. These authors prove that infinite rotational subsets arise as extensions of irrational rotations of the unit circle. In this paper, we prove that such a structure theorem holds for the wider class of continuous transformations $T$ with finite fibers. Our methods are more squarely analytic in nature than the works mentioned, and hence of interest even in the cases treated by the works mentioned above. The paper concludes with an exposition of those cases from the point of view taken here.