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Normal forms for strong magnetic systems on surfaces: trapping regions and rigidity of Zoll systems

Published online by Cambridge University Press:  22 March 2021

LUCA ASSELLE
Affiliation:
Justus Liebig Universität Giessen, Mathematisches Institut, Arndtstrasse 2, 35392Giessen, Germany (e-mail: [email protected])
GABRIELE BENEDETTI*
Affiliation:
Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 205, 69120Heidelberg, Germany
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Abstract

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We prove a normal form for strong magnetic fields on a closed, oriented surface and use it to derive two dynamical results for the associated flow. First, we show the existence of invariant tori and trapping regions provided a natural non-resonance condition holds. Second, we prove that the flow cannot be Zoll unless (i) the Riemannian metric has constant curvature and the magnetic function is constant, or (ii) the magnetic function vanishes and the metric is Zoll. We complement the second result by exhibiting an exotic magnetic field on a flat two-torus yielding a Zoll flow for arbitrarily weak rescalings.

Type
Original Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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