Non-resonant circles for strong magnetic fields on surfaces

Asselle, Luca; Benedetti, Gabriele

Annales Henri Lebesgue, Tome 5 (2022), pp. 1191-1211.

Nous étudions les cercles non résonants pour des champs magnétiques forts sur une surface fermée connexe orientée, et montrer comment celles-ci peuvent être utilisées pour prouver l’existence de régions piégeantes et de géodésiques magnétiques périodiques avec une vitesse faible prescrite. En corollaire, il existe une infinité de géodésiques magnétiques périodiques de petite vitesse arbitraire dans les cas suivants  : i) la surface n’est pas la sphère, ii) le champ magnétique s’annule quelque part.

We study non-resonant circles for strong magnetic fields on a closed, connected, oriented surface and show how these can be used to prove the existence of trapping regions and of periodic magnetic geodesics with prescribed low speed. As a corollary, there exist infinitely many periodic magnetic geodesics for every low speed in the following cases: i) the surface is not the two-sphere, ii) the magnetic field vanishes somewhere.

Classification : 37J99, 58E10
Mots clés : Magnetic systems, KAM tori, periodic orbits, trapping regions

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