An elementary proof of a Theorem by Matsumoto

TL;DR

This paper provides a straightforward, elementary proof of Matsumoto's theorem on prime end rotation numbers and their relation to the rotation set, using only basic planar topology.

Contribution

It offers a simpler, more accessible proof of Matsumoto's theorem, avoiding complex techniques used in the original proof.

Findings

Prime end rotation numbers are contained in the rotation set.

The proof relies solely on elementary planar topology.

The result clarifies the relationship between prime end rotation numbers and the rotation set.

Abstract

Matsumoto proved in arXiv:1012.0981 that the prime end rotation numbers associated to an invariant annular continuum are contained in its rotation set. An alternative proof of this fact using only simple planar topology is presented.