Tiling billiards and Dynnikov’s helicoid

Paris-Romaskevich, O.

doi:10.1090/mosc/317

Tiling billiards and Dynnikov’s helicoid
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by O. Paris-Romaskevich

Trans. Moscow Math. Soc. 2021, 133-147

DOI: https://doi.org/10.1090/mosc/317

Published electronically: March 15, 2022

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Abstract:

Here are two problems. First, understanding the dynamics of a tiling billiard in a cyclic quadrilateral periodic tiling. Second, describing the topology of connected components of plane sections of a centrally symmetric subsurface $S \subset \mathbb {T}^3$ of genus $3$. In this paper 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 at the end of this paper.

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