On the existence of three closed magnetic geodesics for subcritical energies

TL;DR

This paper proves that for almost every energy level below a critical value, there are at least three closed magnetic geodesics on surfaces, and infinitely many if the energy level is non-degenerate.

Contribution

It establishes the existence of multiple closed magnetic geodesics for subcritical energies, extending previous results to a broader energy range on surfaces.

Findings

At least three closed magnetic geodesics exist for almost every subcritical energy.

Infinitely many closed magnetic geodesics exist at non-degenerate energy levels.

Results apply to exact magnetic flows on closed orientable surfaces.

Abstract

We consider exact magnetic flows on closed orientable surfaces. We show that for almost every energy $\kappa$ below Ma\~n\'e's critical value of the universal covering there are always at least three distinct closed magnetic geodesics with energy $\kappa$. If in addition the energy level is assumed to be non-degenerate we prove existence of infinitely many closed magnetic geodesics.