  We give estimates for the approximation numbers of composition operators on
$H^2$, in terms of some modulus of continuity. For symbols whose image is
contained in a polygon, we get that these approximation numbers are dominated
by $\e^{- c \sqrt n}$. When the symbol is continuous on the closed unit disk
and has a domain touching the boundary non-tangentially at a finite number of
points, with a good behavior at the boundary around those points, we can
improve this upper estimate. A lower estimate is given when this symbol has a
good radial behavior at some point. As an application we get that, for the cusp
map, the approximation numbers are equivalent, up to constants, to $\e^{- c \,
n / \log n}$, very near to the minimal value $\e^{- c \, n}$. We also see the
limitations of our methods. To finish, we improve a result of O. El-Fallah, K.
Kellay, M. Shabankhah and H. Youssfi, in showing that for every compact set $K$
of the unit circle $\T$ with Lebesgue measure 0, there exists a compact
composition operator $C_\phi \colon H^2 \to H^2$, which is in all Schatten
classes, and such that $\phi = 1$ on $K$ and $|\phi | < 1$ outside $K$.
