Classification of rotations on the torus ${\mathbb{T}}^2$

We consider rotations on the torus ${\mathbb{T}}^2$, and we classify them with respect to the complexity functions. In dimension one, a minimal rotation can be coded by a sturmian word. A sturmian word has complexity $n+1$ by the Morse-Hedlund theorem. Here we make a generalization in dimension two.