  Let $f:S^1\times [0,1]\to S^1\times [0,1]$ be a real-analytic annulus
diffeomorphism which is homotopic to the identity map and preserves an area
form. Assume that for some lift $\tilde {f}:\mathbb{R}\times [0,1]\rightarrow
\mathbb{R}\times [0,1]$ we have ${\rm Fix}(\tilde{f})=\mathbb{R}\times \{0\}$
and that $\tilde{f}$ positively translates points in $\mathbb{R}\times \{1\}$.
Let $\tilde{f}_\epsilon $ be the perturbation of $\tilde{f}$ by the rigid
horizontal translation $(x,y)\mapsto (x+\epsilon,y)$. We show that for all
$\epsilon >0$ sufficiently small we have ${\rm Fix}
(\tilde{f}_\epsilon)=\emptyset $. The proof follows from Ker\'ekj\'art\'o's
construction of Brouwer lines for orientation preserving homeomorphisms of the
plane with no fixed points. This result turns out to be sharp with respect to
the regularity assumption: there exists a diffeomorphism $f$ satisfying all the
properties above, except that $f$ is not real-analytic but only smooth, so that
the above conclusion is false. Such a map is constructed via generating
functions.
