Cutting sequence and Sturmian sequence in billiard
Dynamical Systems
2022-10-11 v1
Abstract
The winning rule of billiards is to drive the billiard ball on the table into the designated holes. We try to study the trajectory of the billiard ball, so that we can predict the direction of the ball. For rational slopes, we got cutting sequence by setting up the square torus. We simplified cutting sequence using shearing and flipping and we obtain the transformation between trajectory slope and cutting sequence. For irrational slopes, we look at some properties of Sturmian sequence, which help us distinguish between cutting sequence and Sturmian sequence. In conclusion, in the case of different slopes, we use different sequences to do research.
Cite
@article{arxiv.2210.04283,
title = {Cutting sequence and Sturmian sequence in billiard},
author = {Zhiyu Liu},
journal= {arXiv preprint arXiv:2210.04283},
year = {2022}
}
Comments
29 pages,11 figures. arXiv admin note: text overlap with arXiv:1507.02571 by other authors