  This paper studies homeomorphisms of the closed annulus that are isotopic to
the identity from the viewpoint of rotation theory, using a newly developed
forcing theory for surface homeomorphisms. Our first result is a solution to
the so called strong form of Boyland's Conjecture on the closed annulus: Assume
$f$ is a homeomorphism of $\overline{\mathbb{A}}:=(\mathbb{R}/\mathbb{Z})\times
[0,1]$ which is isotopic to the identity and preserves a Borel probability
measure $\mu$ with full support. We prove that if the rotation set of $f$ is a
non-trivial segment, then the rotation number of the measure $\mu$ cannot be an
endpoint of this segment. We also study the case of homeomorphisms such that
$\mathbb{A}=(\mathbb{R}/\mathbb{Z})\times (0,1)$ is a region of instability of
$f$. We show that, if the rotation numbers of the restriction of $f$ to the
boundary components lies in the interior of the rotation set of $f$, then $f$
has uniformly bounded deviations from its rotation set. Finally, by combining
this last result and recent work on realization of rotation vectors for annular
continua, we obtain that if $f$ is any area-preserving homeomorphism of
$\overline{\mathbb{A}}$ isotopic to the identity, then for every real number
$\rho$ in the rotation set of $f$, there exists an associated Aubry-Mather set,
that is, a compact $f$-invariant set such that every point in this set has a
rotation number equal to $\rho$. This extends a result by P. Le Calvez
previously known only for diffeomorphisms.
