Directional complexity of the hypercubic billiard

We consider a minimal rotation on the torus TdTd of direction ωω. A natural cellular decomposition of the torus is associated to this map. We consider an infinite orbit for this map. We compute the complexity of the associated word. Under some hypothesis on the direction, we obtain an exact formula which shows that the order of magnitude is ndnd. This result is related to the billiard map inside a hypercube of Rd+1Rd+1.