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A twist condition for magnetic flows on the two-sphere.

The magnetic flow on the two-dimensional sphere \(S^2\) is determined by a Riemannian metric \(g\) and a two-form \(\sigma\), each on \(S^2\). The triple \((S^2,g,\sigma)\) is called a magnetic system. The goal of this thesis is to find a condition on the magnetic system so that the magnetic flow has infinitely many periodic orbits. We will use existing results that allow us to view the problem in a contact geometric context where the dynamics of the Reeb flow corresponds to the dynamics of the magnetic flow on different kinetic energy level sets. We will then construct a global surface of section that is an annulus and find a condition for which the first return map of the surface of section satisfies the requirements of the Poincaré-Birkhoff Theorem which then implies that the first return map has infinitely many fixed points and the Reeb flow therefore has an infinite number of periodic Reeb orbits.

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Year: 2020

Author : Raphael Schlarb

Gabriele Benedetti