O. Paris-Romaskevich - Triangle tiling billiards скачать в хорошем качестве

O. Paris-Romaskevich - Triangle tiling billiards

Tiling billiards is a dynamical system in which a billiard ball moves through the tiles of some fixed tiling in a way that its trajectory is a broken line, with breaks admitted only at the boundaries of the tiles. One can think about this system as a movement of the refracted light. In this talk, I will speak about a specific kind of such billiards: negative triangle tilling billiards introduced in the work by P. Baird-Smith, D. Davis, E. Fromm and S.Iyer. The tiling is a periodic tiling of a plane by congruent triangles (obtained from the standard equilateral triangle tiling by linear transformation). The law of reflection in any side is Snell’s law with the coefficient of refraction equal to -1. This system is, in a simple and magical way, related to interval exchange transformations with flips. I will explain this relationship and some results that follow from it. First, almost all the trajectories of such tilings are either closed or escape linearly to infinity. Second, there exists an interesting measure zero set of parameters for which the trajectories escape to infinity in a non-linear fashion, and approach fractals. This set is parametrised by the Rauzy gasket. My talk will be based on a joint work with Pascal Hubert, some computer simulations by Paul Mercat, and the article of D. Davis and her coauthors.