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Complexity of sequences defined by billiard in the cube. (English) Zbl 0791.58034
Summary: 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 \(n^ 2 + n + 1\).

MSC:
37E99 Low-dimensional dynamical systems
05A15 Exact enumeration problems, generating functions
05B45 Combinatorial aspects of tessellation and tiling problems
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