Tiling billiards and Dynnikov’s helicoid

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 ⊂ T 3 S \subset \mathbb {T}^3 of genus  3 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.