  In this paper we prove the existence of a large class of periodic solutions
of the Vlasov-Poisson in one space dimension that decay exponentially as t goes
to infinity. The exponential decay is well known for the linearized version of
the Landau damping problem and it has been proved in [4] for a class of
solutions of the Vlasov-Poisson system that behaves asymptotically as free
streaming solutions and are sufficiently flat in the space of velocities. The
results in this paper enlarge the class of possible asymptotic limits,
replacing the fatness condition in [4] by a stability condition for the
linearized problem.
