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Complexity of trajectories in rectangular billiards. (English) Zbl 0839.11006
The Sturmian sequences are the binary sequences that are a coding of a billiard trajectory in a \((2D)\) square, where the vertical sides are coded by 1 and the horizontal sides by 0. In particular, the (block) complexity of a Sturmian sequence is given by \(\rho (n) = n + 1\), where \(\rho (n)\) is the number of factors (subblocks) of the sequence with length \(n\).
What happens if one plays billiard in a cube or hypercube? A conjecture of Rauzy stated that the complexity of the trajectories for the cubic billiards is given by \(\rho (n) = n^2 + n + 1\). This conjecture has been proved by P. Arnoux, C. Mauduit, I. Shiokawa and J.-I. Tamura who published two papers [Bull. Soc. Math. Fr. 122, No. 1, 1-12 (1994; Zbl 0791.58034) and Tokyo J. Math. 17, No. 1, 211-218 (1994; Zbl 0814.11014)]. These four authors also conjectured a general formula for the hypercube, the formula presenting a mysterious symmetry in \(n\) (the length of blocks) and \(d-1\) (where \(d\) is the dimension).
The author of the paper under review solves the question completely stating in particular that, for reasonable starting angles, one has in dimension \(d\) \[ \rho_d (n) = \sum^{\min (d - 1,n)}_{k = 0} k! {d - 1 \choose k} {n \choose k}. \]

MSC:
11B83 Special sequences and polynomials
68R15 Combinatorics on words
37E99 Low-dimensional dynamical systems
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