Algebraic billiards in the Fermat hyperbola
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917057298169856 |
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| author | Weinreich, Max |
| author_facet | Weinreich, Max |
| contents | We prove two results on the algebraic dynamics of billiards in generic algebraic curves of degree $d \geq 2$. First, the dynamical degree grows quadratically in $d$; second, the set of complex periodic points has measure 0, implying the Ivrii Conjecture for the classical billiard map in generic algebraic domains. To prove these results, we specialize to a new billiard table, the Fermat hyperbola, on which the indeterminacy points satisfy an exceptionality property. Over $\mathbb{C}$, we construct an algebraically stable model for this billiard via an iterated blowup. Over more general fields, we prove essential stability, i.e. algebraic stability for a particular big and nef divisor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16172 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Algebraic billiards in the Fermat hyperbola Weinreich, Max Dynamical Systems 37C83 (Primary) 37P05, 37F80 (Secondary) We prove two results on the algebraic dynamics of billiards in generic algebraic curves of degree $d \geq 2$. First, the dynamical degree grows quadratically in $d$; second, the set of complex periodic points has measure 0, implying the Ivrii Conjecture for the classical billiard map in generic algebraic domains. To prove these results, we specialize to a new billiard table, the Fermat hyperbola, on which the indeterminacy points satisfy an exceptionality property. Over $\mathbb{C}$, we construct an algebraically stable model for this billiard via an iterated blowup. Over more general fields, we prove essential stability, i.e. algebraic stability for a particular big and nef divisor. |
| title | Algebraic billiards in the Fermat hyperbola |
| topic | Dynamical Systems 37C83 (Primary) 37P05, 37F80 (Secondary) |
| url | https://arxiv.org/abs/2406.16172 |