  In this paper we consider torus homeomorphisms $f$ homotopic to Dehn twists.
We prove that if the vertical rotation set of $f$ is reduced to zero, then
there exists a compact connected essential "horizontal" set K, invariant under
$f$. In other words, if we consider the lift $\hat{f}$ of $f$ to the cylinder,
which has zero vertical rotation number, then all points have uniformly bounded
motion under iterates of $\hat{f}$. Also, we give a simple explicit condition
which, when satisfied, implies that the vertical rotation set contains an
interval and thus also implies positive topological entropy. As a corollary of
the above results, we prove a version of Boyland's conjecture to this setting:
If $f$ is area preserving and has a lift $\hat{f}$ to the cylinder with zero
Lebesgue measure vertical rotation number, then either the orbits of all points
are uniformly bounded under $\hat{f}$, or there are points in the cylinder with
positive vertical velocity and others with negative vertical velocity.
