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In this paper, we study small data solutions to the Vlasov-Poisson systemwith the simplest external potential, for which unstable trapping holds for theassociated Hamiltonian flow. First, we provide a new proof of global existencefor small data solutions to the Vlasov-Poisson system with the trappingpotential $\frac{-|x|^2}{2}$ in dimension two. We exploit the uniformhyperbolicity of the Hamiltonian flow, by making use of the commuting vectorfields contained in the stable and unstable invariant distributions of phasespace for the linearized system. In contrast with the proof in \cite{VV23}, wedo not use modified vector field techniques. Moreover, we obtain small datamodified scattering for this non-linear system. We show that the distributionfunction converges to a new regular distribution function along modificationsto the characteristics of the linearized problem. We define the linearlygrowing corrections to the characteristic system, in terms of a preciseeffective asymptotic force field. We make use of the scattering state to obtainthe late-time asymptotic behavior of the spatial density. Finally, we provethat the distribution function (up to normalization) converges weakly to aDirac mass on the unstable manifold of the origin.

Modified scattering of small data solutions to the Vlasov-Poisson system with a trapping potential