Zoll magnetic systems on the two-torus: A Nash–Moser construction

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 magne...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) Vol. 452; p. 109826
Main Authors: Asselle, Luca, Benedetti, Gabriele, Berti, Massimiliano
Format: Journal Article
Language:English
Published: Elsevier Inc 01.08.2024
Subjects:
Bessel functions
Fourier integral operators
Hamiltonian systems
Magnetic geodesics
Nash–Moser implicit function theorem
Zoll flows
Hamiltonian systems
Magnetic geodesics
Nash–Moser implicit function theorem
Zoll flows
Fourier integral operators
Bessel functions
ISSN:0001-8708, 1090-2082
Online Access:Get full text
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Summary: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.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2024.109826