Infinitely many periodic orbits in non-exact oscillating magnetic fields on surfaces with genus at least two for almost every low energy level

In this paper we consider oscillating non-exact magnetic fields on surfaces with genus at least two and show that for almost every energy level k below a certain value τ*+(g,σ) less than or equal to the Mañé critical value of the universal cover there are infinitely many closed magnetic geodesics with energy k.