A Geometrical Approach to Palindromic Factors of Standard Billiard Words

Consider the grid G on the first quadrant of the plane : it is the set of vertical half-lines with integer x coordinate and of horizontal half-lines with integer y coordinate. The half-line D starting at the origin O with slope ρ divides G into two parts. It will be more convenient for a multidimensional generalization, to consider a positive vector α = (α1, α2) parallel to D, thus we have ρ = α2 α1 . We construct the Christoffel word uρ = uα1,α2 and the Standard billiard word cρ = cα1,α2 as shown in Fig.1.a: Coding by a1 the horizontal black segment on the grid and by a2 the vertical one as in Fig.1.a ,uα1,α2 encodes the discrete path immediately under the half line D; hence we have uα1,α2 = a1a2a1a2a1a2a2a1a2 . . . Looking at the squares crossed by D, i.e., the grey squares in Fig.1.a, the Standard billiard word cα1,α2 encodes the sequence of horizontal and vertical segments joining the two centers (i.e., white points in Fig. 1.a) of the consecutive grey unit squares. Consider the †research partially supported by Région Limousin.