  We study the long time behaviour of a dynamical system strongly linked to the
anti-diffusive scheme of Despr\'es and Lagoutiere for the $1$-dimensional
transport equation. This scheme is nondiffusive in the sens that
discontinuities are not smoothened out through time. Numerical simulations
indicates that the scheme error's is uniformly bounded with time. We prove that
this scheme is overcompressive when the Courant--Friedrichs--Levy number is
1/2: when the initial data is nondecreasing, the approximate solution becomes a
Heaviside function. In a special case, we also understand how plateaus are
formed in the solution and their stability, a distinctive feature of the
Despr\'es and Lagoutiere scheme.
