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 ${\tau }_{+}^{\ast }(g,\sigma )$ less than or equal to the “Ma~n'e critical value of the universal cover” there are infinitely many closed magnetic geodesics with energy $k$.