  In this paper, we study small data solutions to the Vlasov-Poisson system
with the simplest external potential, for which unstable trapping holds for the
associated Hamiltonian flow. First, we provide a new proof of global existence
for small data solutions to the Vlasov-Poisson system with the trapping
potential $\frac{-|x|^2}{2}$ in dimension two. We exploit the uniform
hyperbolicity of the Hamiltonian flow, by making use of the commuting vector
fields contained in the stable and unstable invariant distributions of phase
space for the linearized system. In contrast with the proof in \cite{VV23}, we
do not use modified vector field techniques. Moreover, we obtain small data
modified scattering for this non-linear system. We show that the distribution
function converges to a new regular distribution function along modifications
to the characteristics of the linearized problem. We define the linearly
growing corrections to the characteristic system, in terms of a precise
effective asymptotic force field. We make use of the scattering state to obtain
the late-time asymptotic behavior of the spatial density. Finally, we prove
that the distribution function (up to normalization) converges weakly to a
Dirac mass on the unstable manifold of the origin.
