A geometrical characterization of factors of multidimensional Billiard words and some applications

We consider Billiard words in alphabets with k>2 letters. Such words are associated with some k-dimensional positive vector . The language of these words is already known in the usual case, i.e. when the αj are linearly independent over Q and so for their inverses. Here we study the language of these words when there exist some linear relationships. We give a new geometrical characterization of the factors of Billiard words. As a consequence, we get some results on the associated language, and on the complexity and palindromic complexity of these words. The situation is quite different from the usual case. The languages of two distinct Billiard words with the same direction generally have a finite intersection. As examples, we get some Standard Billiard words of three letters without any palindromic factor of even length, or Billiard words of three letters whose palindromic factors have a bounded length. These results are obtained by geometrical methods.