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| Format: | Preprint |
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2026
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| Accès en ligne: | https://arxiv.org/abs/2605.19493 |
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| _version_ | 1866913144811552768 |
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| author | Lamas, José Marò, Stefano |
| author_facet | Lamas, José Marò, Stefano |
| contents | We consider the breathing circle billiard, in which a point particle moves freely inside a disk. The radius varies periodically in time, with elastic reflections at the moving boundary. In this system the angular momentum is preserved, and fixing its value $c$ reduces the dynamics to a two-dimensional exact symplectic map on a cylinder. In the high-energy regime this map is a twist map generated by a diagonally periodic generating function $h_c$.
We study the small angular momentum regime as a perturbation of the limiting case $c=0$, which corresponds to the Fermi-Ulam dynamics along a diameter. Using this perturbative structure and a quantitative version of Mather's converse-KAM criterion, we exclude invariant Lipschitz graphs for suitable rotation numbers. Combined with Aubry-Mather theory and Forni's theorem, this yields positive topological entropy for sufficiently small $c\geq0$. Our result improves previous similar results obtained via the standard Mather's converse-KAM criterion by giving a sharper quantitative threshold for the destruction of invariant curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_19493 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Breaking of invariant curves: from the Fermi-Ulam map to the breathing circle billiard Lamas, José Marò, Stefano Dynamical Systems We consider the breathing circle billiard, in which a point particle moves freely inside a disk. The radius varies periodically in time, with elastic reflections at the moving boundary. In this system the angular momentum is preserved, and fixing its value $c$ reduces the dynamics to a two-dimensional exact symplectic map on a cylinder. In the high-energy regime this map is a twist map generated by a diagonally periodic generating function $h_c$. We study the small angular momentum regime as a perturbation of the limiting case $c=0$, which corresponds to the Fermi-Ulam dynamics along a diameter. Using this perturbative structure and a quantitative version of Mather's converse-KAM criterion, we exclude invariant Lipschitz graphs for suitable rotation numbers. Combined with Aubry-Mather theory and Forni's theorem, this yields positive topological entropy for sufficiently small $c\geq0$. Our result improves previous similar results obtained via the standard Mather's converse-KAM criterion by giving a sharper quantitative threshold for the destruction of invariant curves. |
| title | Breaking of invariant curves: from the Fermi-Ulam map to the breathing circle billiard |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2605.19493 |