Complexity of trajectories in rectangular billiards.

*(English)*Zbl 0839.11006The 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 \(d\) \[ \rho_d (n) = \sum^{\min (d - 1,n)}_{k = 0} k! {d - 1 \choose k} {n \choose k}. \]

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 \(d\) \[ \rho_d (n) = \sum^{\min (d - 1,n)}_{k = 0} k! {d - 1 \choose k} {n \choose k}. \]

Reviewer: J.-P.Allouche (Orsay)

##### MSC:

11B83 | Special sequences and polynomials |

68R15 | Combinatorics on words |

37E99 | Low-dimensional dynamical systems |

##### Keywords:

billiard in higher dimensions; generalized Sturmian sequences; trajectories cubic billiard; cubic billiard; hypercube
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\textit{Yu. Baryshnikov}, Commun. Math. Phys. 174, No. 1, 43--56 (1995; Zbl 0839.11006)

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##### References:

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