  The semi-Lagrangian discontinuous Galerkin method, coupled with a splitting
approach in time, has recently been introduced for the Vlasov--Poisson
equation. Since these methods are conservative, local in space, and able to
limit numerical diffusion, they are considered a promising alternative to more
traditional semi-Lagrangian schemes. In this paper we study the conservation of
important physical invariants and the long time behavior of the semi-Lagrangian
discontinuous Galerkin method. To that end we conduct a theoretical analysis
and perform a number of numerical simulations. In particular, we find that the
entropy is nondecreasing for the discontinuous Galerkin scheme, while
unphysical oscillations in the entropy are observed for the traditional method
based on cubic spline interpolation.
