Deformational spectral rigidity of axially-symmetric symplectic billiards
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912115323830272 |
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| author | Fierobe, Corentin Sorrentino, Alfonso Vig, Amir |
| author_facet | Fierobe, Corentin Sorrentino, Alfonso Vig, Amir |
| contents | Symplectic billiards were introduced by Albers and Tabachnikov as billiards in strictly convex bounded domains of the plane with smooth boundary having a specific law of reflection. This paper proves a rigidity result for symplectic billiards which is similar to a previous result on classical billiards formulated by De Simoi, Kaloshin and Wei. Namely, it states that close to an ellipse, a sufficiently smooth one-parameter family of axially symmetric domains either contains domains with different area-spectra or is trivial, in a sense that the domains differ by area-preserving affine transformations of the plane. The paper also prove that in the general setting - that is even if the domains are not close to an ellipse - any sufficiently smooth one-parameter family of axially symmetric domains which preserves the area-spectrum is tangent to a finite dimensionnal space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_13777 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Deformational spectral rigidity of axially-symmetric symplectic billiards Fierobe, Corentin Sorrentino, Alfonso Vig, Amir Dynamical Systems Symplectic billiards were introduced by Albers and Tabachnikov as billiards in strictly convex bounded domains of the plane with smooth boundary having a specific law of reflection. This paper proves a rigidity result for symplectic billiards which is similar to a previous result on classical billiards formulated by De Simoi, Kaloshin and Wei. Namely, it states that close to an ellipse, a sufficiently smooth one-parameter family of axially symmetric domains either contains domains with different area-spectra or is trivial, in a sense that the domains differ by area-preserving affine transformations of the plane. The paper also prove that in the general setting - that is even if the domains are not close to an ellipse - any sufficiently smooth one-parameter family of axially symmetric domains which preserves the area-spectrum is tangent to a finite dimensionnal space. |
| title | Deformational spectral rigidity of axially-symmetric symplectic billiards |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2410.13777 |