  In this paper, we study small data solutions for the Vlasov-Poisson system
with the simplest external potential, for which unstable trapping holds for the
associated Hamiltonian flow. We prove sharp decay estimates in space and time
for small data solutions to the Vlasov-Poisson system with the unstable
trapping potential $\frac{-|x|^2}{2}$ in dimension two or higher. The proofs
are obtained through a commuting vector field approach. 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 dimension two, we make use of modified
vector field techniques due to the slow decay estimates in time. Moreover, we
show an explicit teleological construction of the trapped set in terms of the
non-linear evolution of the force field.

Analysis of PDEs;Dynamical Systems