  We construct (modified) scattering operators for the Vlasov-Poisson system in
three dimensions, mapping small asymptotic dynamics as $t\to -\infty$ to
asymptotic dynamics as $t\to +\infty$. The main novelty is the construction of
modified wave operators, but we also obtain a new simple proof of modified
scattering.
  Our analysis is guided by the Hamiltonian structure of the Vlasov-Poisson
system. Via a pseudo-conformal inversion we recast the question of asymptotic
behavior in terms of local in time dynamics of a new equation with singular
coefficients which is approximately integrated using a generating function.
