Complexity of sequences defined by billiard in the cube

  title={Complexity of sequences defined by billiard in the cube},
  author={Pierre Arnoux and Christian Mauduit and IEKATA SHIOKAWA and Jun-ichi Tamura},
— We prove a conjecture of Gérard Rauzy related to the structure of billiard trajectories in the cube : let us associate to any such trajectory the sequence with values in {1,2,3} given by coding 1 (resp. 2, 3) any time the particle rebounds on a frontal (resp. lateral, horizontal) side of the cube. We show that, if the direction is totally irrational, the number of distinct finite words of length n appearing in this sequence is exactly n2 + n + 1. 1. Statement of the result We consider… CONTINUE READING