Magnetic Katok Examples on the two-sphere

TL;DR

This paper constructs non-exact magnetic flows on the two-sphere with energy levels related to irrational ellipsoids, extending Katok's integrable Finsler metrics with minimal closed geodesics.

Contribution

It introduces new examples of magnetic flows on the two-sphere with specific contactomorphic properties, broadening the understanding of magnetic dynamics.

Findings

Existence of non-exact magnetic flows with specific contactomorphic energy levels.

Generalization of Katok's Finsler metrics to magnetic flows.

Connection between magnetic flows and irrational ellipsoids in complex space.

Abstract

We show that there exist non-exact magnetic flows on the two-sphere having an energy level whose double cover is strictly contactomorphic to an irrational ellipsoid in $\mathbb C^2$. Our construction generalizes the examples of integrable Finsler metrics on the two-sphere with only two closed geodesics due to A. Katok.