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

Abstract

Abstract. 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.

The Multiplicity Problem for Periodic Orbits of Magnetic Flows on the 2-Sphere

“…The combination of Theorem 1.1 together with the above mentioned results in [AB15a,AB15b,AM16], yields the following statement about the multiplicity of periodic orbits on general closed surfaces. Theorem 1.2.…”

“…Taimanov's result is that, given a kinetic Lagrangian L(q, v) = 1 2 g q (v, v) and an oscillating magnetic 2-form σ on a closed 2-dimensional configuration space, there exists a waist α e at the energy level e, for all e ∈ (0, e 1 (L, σ)) (see also [CMP04] for a different proof). When σ is exact, Abbondandolo, Macarini, Mazzucchelli and Paternain [AMMP14] employed Taimanov's waist α e on any energy level e belonging to a full measure subset of (0, e 1 (L, σ)) in order to construct a sequence of minmax families giving an infinite number of (geometrically distinct) periodic orbits with energy e. Short afterwards, Asselle and Benedetti extended the result to non-exact σ on surfaces of genus at least one [AB15a,AB15b]. The results in [Tai91, Tai92a, Tai92b, AMMP14, AB15a, AB15b] have been further extended by Asselle and Mazzucchelli [AM16] to the general magnetic Tonelli setting.…”

“…This paper is the last chapter of a work started in [AMP15] and further developed in [AMMP14,AB15a,AB15b,AM16] devoted to studying the multiplicity of periodic orbits on generic low energy levels in magnetic Tonelli Lagrangian systems on surfaces. Such a study was based on a generalization of Bangert's waist Theorem [Ban80, Theorem 4], classically formulated for geodesic flows on S 2 , to the magnetic Tonelli setting.…”

“…In Theorem 1.1, as well as in [AMMP14,AB15a,AB15b], a negligible subset of energies must be excluded due to a lack of compactness in the variational setting that is employed. However, energy levels with only finitely many periodic orbits can be found above [Zil83,Ben16] as well as below [AM16] …”

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

“…We already showed in [AB15] that the result proved in [AMMP14] for exact forms extends to oscillating forms when M has genus at least 2 and c u (L ϑ ) is replaced by some τ * + (g, σ) ∈ (0, c u (L ϑ )] (observe that c u (L ϑ ) is still well-defined since the lift of σ to the universal cover is exact). Implementing ideas contained in [AB16], we are now able to treat the case in which M = T 2 is the two-torus.…”