The complexity of tangent words

A smooth curve is a map γ from a compact interval I of the real line to the plane, which is C∞ and such that ||γ ′(t)|| > 0 for any t ∈ I (this last property is called regularity). Any such curve can (and will be considered to) be arc-length reparametrised (i.e. ∀t ∈ I, ||γ ′(t)||= 1). We can approximate such a curve by drawing a square grid of mesh h on the plane, and look at the sequence of squares that the curve meets. For a generic position of the grid, the curve γ does not hit any corner and crosses the grid transversally, hence the curve passes from a square to a square that is located either right, up, left or down of it. We record this sequence of moves and define the cutting sequence of the curve γ with respect to this grid as a word w on the alphabet {r,u, l,d} which tracks the lines of the grid crossed by the curve γ . The following picture shows a curve γ with cutting sequence rruuldrrrd.