On the existence of periodic invariant curves for analytic families of twist maps and billiards
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909266866077696 |
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| author | Fierobe, Corentin Sorrentino, Alfonso |
| author_facet | Fierobe, Corentin Sorrentino, Alfonso |
| contents | In this paper we prove that in any analytic one-parameter family of twist maps of the annulus, homotopically invariant curves filled with periodic points corresponding to a given rotation number, either exist for all values of the parameters or at most for a discrete subset. Moreover, we show that the set of analytic twist maps having such an invariant curve of a given rotation number is a strict analytic subset of the set of analytic twist maps. The first result extends, in dimension 2, a previous result by Arnaud, Massetti and Sorrentino. We then apply our result to rational caustics of billiards, considering several models such as Birkhoff billiards, outer billiards and symplectic billiards. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_17090 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the existence of periodic invariant curves for analytic families of twist maps and billiards Fierobe, Corentin Sorrentino, Alfonso Dynamical Systems In this paper we prove that in any analytic one-parameter family of twist maps of the annulus, homotopically invariant curves filled with periodic points corresponding to a given rotation number, either exist for all values of the parameters or at most for a discrete subset. Moreover, we show that the set of analytic twist maps having such an invariant curve of a given rotation number is a strict analytic subset of the set of analytic twist maps. The first result extends, in dimension 2, a previous result by Arnaud, Massetti and Sorrentino. We then apply our result to rational caustics of billiards, considering several models such as Birkhoff billiards, outer billiards and symplectic billiards. |
| title | On the existence of periodic invariant curves for analytic families of twist maps and billiards |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2407.17090 |