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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2304.02765 |
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| _version_ | 1866910547245531136 |
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| author | Asselle, Luca Benedetti, Gabriele Berti, Massimiliano |
| author_facet | Asselle, Luca Benedetti, Gabriele Berti, Massimiliano |
| contents | 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 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 $\mathrm{d} S$ 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 $\mathrm{d} S\circ \mathrm{d} S^*$ 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 $\mathrm{d} S \circ \mathrm{d}S^*$ is a sum of Fourier integral operators. We then use a resolvent identity decomposition which reduces the problem to the invertibility of $\mathrm{d} S \circ \mathrm{d} S^*$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_02765 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Zoll magnetic systems on the two-torus: a Nash-Moser construction Asselle, Luca Benedetti, Gabriele Berti, Massimiliano Differential Geometry Analysis of PDEs Symplectic Geometry 70H12 (Primary) 58E10, 44A12, 58J40 (Secondary) 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 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 $\mathrm{d} S$ 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 $\mathrm{d} S\circ \mathrm{d} S^*$ 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 $\mathrm{d} S \circ \mathrm{d}S^*$ is a sum of Fourier integral operators. We then use a resolvent identity decomposition which reduces the problem to the invertibility of $\mathrm{d} S \circ \mathrm{d} S^*$ 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. |
| title | Zoll magnetic systems on the two-torus: a Nash-Moser construction |
| topic | Differential Geometry Analysis of PDEs Symplectic Geometry 70H12 (Primary) 58E10, 44A12, 58J40 (Secondary) |
| url | https://arxiv.org/abs/2304.02765 |