Tiling billiards and Dynnikov's helicoid

Olga Paris-Romaskevich (1, 2)

1 CNRS - Centre National de la Recherche Scientifique
2 I2M - Institut de Mathématiques de Marseille

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.

Domaines: Mathématiques [math]

Olga Paris-Romaskevich. Tiling billiards and Dynnikov's helicoid. Transactions of the Moscow Mathematical Society, 2021, 81 (1). ⟨hal-03175411⟩