Lazutkin coordinates and the conjugation problem for billiard maps

The conjugation problem for billiard maps conjectures that if two strictly convex billiards have conjugated billiard maps, the billiard tables must be homothetic to each other. We show that if two billiard maps are conjugated, the conjugation diffeomorphism is tangent to a Lazutkin change of coordinates. We also recompute the coefficients in the billiard map expansion as the angle ttends to zero, correcting some incorrect expressions in Lazutkin’s computations.