Non-resonant circles for strong magnetic fields on surfaces

TL;DR

This paper investigates non-resonant circles on surfaces under strong magnetic fields, demonstrating their role in establishing trapping regions and periodic magnetic geodesics, with implications for surfaces with specific properties.

Contribution

It introduces the concept of non-resonant circles for magnetic fields on surfaces and proves their utility in finding trapping regions and periodic geodesics, extending previous results.

Findings

Existence of trapping regions using non-resonant circles

Presence of infinitely many low-speed periodic magnetic geodesics under certain conditions

Results apply to surfaces not homeomorphic to the two-sphere or with magnetic field zeros

Abstract

We study non-resonant circles for strong magnetic fields on a closed, connected, oriented surface and show how these can be used to prove the existence of trapping regions and of periodic magnetic geodesics with prescribed low speed. As a corollary, there exist infinitely many periodic magnetic geodesics for every low speed in the following cases: i) the surface is not the two-sphere, ii) the magnetic field vanishes somewhere.