  The main result of this paper gives a topological property satisfied by any
homeomorphism of the annulus $\mathbb{A}=\mathbb{S}^1 \times [-1,1]$ isotopic
to the identity and with at most one fixed point. This generalizes the
classical Poincar\'e-Birkhoff theorem because this property certainly does not
hold for an area preserving homeomorphism of $\mathbb{A}$ with the usual
boundary twist condition. We also have two corollaries of this result. The
first one shows in particular that the boundary twist assumption may be
weakened by demanding that the homeomorphism $h$ has a lift $H$ to the strip
$\widetilde{\mathbb{A}} = \mathbb{R} \times [-1,1]$ possessing both a forward
orbit unbounded on the right and a forward orbit unbounded on the the left. As
a second corollary we get a new proof of a version of the Conley-Zehnder
theorem in $\mathbb{A}$: if a homeomorphism of $\mathbb{A}$ isotopic to the
identity preserves the area and has mean rotation zero, then it possesses two
fixed points.
