@article {ASSELLE2024109826,
	title = {Zoll magnetic systems on the two-torus: A Nash{\textendash}Moser construction},
	journal = {Advances in Mathematics},
	volume = {452},
	year = {2024},
	pages = {109826},
	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{\textendash}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 dSodS⁎ which, unlike in Guillemin{\textquoteright}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 dSodS⁎ is a sum of Fourier integral operators. We then use a resolvent identity decomposition which reduces the problem to the invertibility of dSodS⁎ 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.},
	keywords = {Bessel functions, Fourier integral operators, Hamiltonian systems, Magnetic geodesics, Nash{\textendash}Moser implicit function theorem, Zoll flows},
	issn = {0001-8708},
	doi = {https://doi.org/10.1016/j.aim.2024.109826},
	url = {https://www.sciencedirect.com/science/article/pii/S0001870824003414},
	author = {Luca Asselle and Gabriele Benedetti and Massimiliano Berti}
}
@article {FONDA20171064,
	title = {An avoiding cones condition for the Poincar{\'e}{\textendash}Birkhoff Theorem},
	journal = {Journal of Differential Equations},
	volume = {262},
	number = {2},
	year = {2017},
	pages = {1064 - 1084},
	abstract = {<p>We provide a geometric assumption which unifies and generalizes the conditions proposed in [11], [12], so to obtain a higher dimensional version of the Poincar{\'e}{\textendash}Birkhoff fixed point Theorem for Poincar{\'e} maps of Hamiltonian systems.</p>},
	keywords = {Avoiding cones condition, Hamiltonian systems, Periodic solutions, Poincar{\'e}{\textendash}Birkhoff theorem},
	issn = {0022-0396},
	doi = {https://doi.org/10.1016/j.jde.2016.10.002},
	url = {http://www.sciencedirect.com/science/article/pii/S0022039616303278},
	author = {Alessandro Fonda and Paolo Gidoni}
}
@mastersthesis {1994,
	title = {Analysis of Singularity Structures for Quasi-Integrable Hamiltonian Systems},
	year = {1994},
	school = {SISSA},
	keywords = {Hamiltonian systems},
	url = {http://hdl.handle.net/1963/5685},
	author = {Simonetta Abenda}
}
