  For a twist map $f$ of the annulus preserving the Lebesgue measure, we give
sufficient conditions to assure the existence of a set of positive measure of
points with non-zero asymptotic torsion. In particular, we deduce that every
bounded instability region for $f$ contains a set of positive measure of points
with non-zero asymptotic torsion. Moreover, for an exact symplectic twist map
$f$, we provide a simple, geometric proof of a result by Cheng and Sun (see
[CS96]) which characterizes $\mathcal{C}^0$-integrability of $f$ by the absence
of conjugate points.
