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Trudy Moskovskogo Matematicheskogo Obshchestva, 2021, Volume 82, Issue 1, Pages 157–174
(Mi mmo652)
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Tiling billiards and Dynnikov's helicoid
O. Paris-Romaskevich Aix-Marseille Université
Abstract:
Here are two problems. First, understand the dynamics of a tiling billiard in a cyclic quadrilateral periodic tiling. Second, describe the topology of connected components of plane sections of a centrally symmetric subsurface $S \subset \mathbb{T}^3$ of genus $3$. In this note we show that these two problems are related via a helicoidal construction proposed recently by Ivan Dynnikov. The second problem is a particular case of a classical question formulated by Sergei Novikov. The exploration of the relationship between a large class of tiling billiards (periodic locally foldable tiling billiards) and Novikov's problem in higher genus seems promising, as we show in the end of this note.
Key words and phrases:
Novikov's problem, tiling billiards, billiards, translation surfaces.
Received: 20.02.2021
Citation:
O. Paris-Romaskevich, “Tiling billiards and Dynnikov's helicoid”, Tr. Mosk. Mat. Obs., 82, no. 1, MCCME, M., 2021, 157–174; Trans. Moscow Math. Soc., 82 (2021), 133–147
Linking options:
https://www.mathnet.ru/eng/mmo652 https://www.mathnet.ru/eng/mmo/v82/i1/p157
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Statistics & downloads: |
Abstract page: | 78 | Full-text PDF : | 14 | References: | 22 | First page: | 6 |
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