## Abstract

We construct an infinite-dimensional family of smooth integrable magnetic systems on the two-torus which are Zoll, meaning that all the unit-speed magnetic geodesics are periodic. The metric and the magnetic field of such systems are arbitrarily close to the flat metric and to a given constant magnetic field. This extends to the magnetic setting a famous result by Guillemin [19] on the two-sphere. We characterize Zoll magnetic systems as zeros of a suitable action functional S, and then look for its zeros by means of a Nash–Moser implicit function theorem. This requires showing the right-invertibility of the linearized operator dS in a neighborhood of the flat metric and constant magnetic field, and establishing tame estimates for the right inverse. As key step we prove the invertibility of the normal operator dS∘dS^{⁎} which, unlike in Guillemin's case, is pseudo-differential only at the highest order. We overcome this difficulty noting that, by the asymptotic properties of Bessel functions, the lower order expansion of dS∘dS^{⁎} is a sum of Fourier integral operators. We then use a resolvent identity decomposition which reduces the problem to the invertibility of dS∘dS^{⁎} restricted to the subspace of functions corresponding to high Fourier modes. The inversion of such a restricted operator is finally achieved by making the crucial observation that lower order Fourier integral operators satisfy asymmetric tame estimates.

Original language | English |
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Article number | 109826 |

Pages (from-to) | 1-39 |

Number of pages | 39 |

Journal | Advances in Mathematics |

Volume | 452 |

Early online date | 8 Jul 2024 |

DOIs | |

Publication status | Published - Aug 2024 |

### Bibliographical note

Publisher Copyright:© 2024 The Author(s)

## Keywords

- Bessel functions
- Fourier integral operators
- Hamiltonian systems
- Magnetic geodesics
- Nash–Moser implicit function theorem
- Zoll flows