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Mathematics > Dynamical Systems

arXiv:1510.00152 (math)
[Submitted on 1 Oct 2015 (v1), last revised 15 Sep 2016 (this version, v2)]

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

Authors:Luca Asselle, Gabriele Benedetti
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Abstract:Let $(\mathbb T^2,g)$ be a Riemannian two-torus and let $\sigma$ 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 $(g,\sigma)$ 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.
Comments: 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
Subjects: Dynamical Systems (math.DS); Symplectic Geometry (math.SG)
MSC classes: 37J45, 58E05
Cite as: arXiv:1510.00152 [math.DS]
  (or arXiv:1510.00152v2 [math.DS] for this version)
  https://doi.org/10.48550/arXiv.1510.00152
Journal reference: Mathematische Zeitschrift (2016), 1-17
Related DOI: https://doi.org/10.1007/s00209-016-1787-6

Submission history

From: Gabriele Benedetti Mr [view email]
[v1] Thu, 1 Oct 2015 09:17:27 UTC (50 KB)
[v2] Thu, 15 Sep 2016 16:51:18 UTC (19 KB)
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