Let $X$ be a  Banach space and let $F:X\to  X$ be a $C^1$ mapping
with fixed point $0$: $F(0)=0$.  We give conditions which imply that
if the $\omega$-limit set  of a trajectory contains $0$ then either  the
$\omega$-limit  set equals $\{0\}$   or else it contains a point  of  the
unstable  manifold of  $0$  different  from  $0$.  The hypotheses on
$F$  involve the   spectrum  of $F'(0)$ (implying   the existence  of
stable, unstable, and center   manifolds of $0$) and the dynamics of
$F$  on   the center manifold   of $0$.   In a  number  of particular
cases this    result  allows one to prove    convergence of trajectories
to a single equilibrium.