On the periodic motions of a charged particle in an oscillating magnetic field on the two-torus

Abstract

Let $(\mathbb T^2,g)$ be a Riemannian two-torus and let $σ$ be an oscillating $2$-form on $\mathbb T^2$. We show that for almost every small positive number $k$ the magnetic flow of the pair $(g,σ)$ has infinitely many periodic orbits with energy $k$. This result complements the analogous statement for closed surfaces of genus at least $2$ [Asselle and Benedetti, Calc. Var. Partial Differential Equations, 2015] and at the same time extends the main theorem in [Abbondandolo, Macarini, Mazzucchelli, and Paternain, J. Eur. Math. Soc. (JEMS), to appear] to the non-exact oscillating case.

15 pages. Revised version incorporating the precious comments of the referee and the notion of essential family suggested to us by M. Mazzucchelli. Comments are very welcome. To appear in Mathematische Zeitschrift