  We consider a one-dimensional system with particles having either positive or
negative velocity, which annihilate on contact. To the ballistic motion of the
particle, a diffusion is superimposed. The annihilation may represent a
reaction in which the two particles yield an inert species. This model has been
the object of previous work, in which it was shown that the particle
concentration decays faster than either the purely ballistic or the purely
diffusive case. We report on previously unnoticed behaviour for large times,
when only one of the two species remains and also unravel the underlying
fractal structure present in the system. We also consider in detail the case in
which the initial concentration of right-going particles is $1/2+\varepsilon$,
with $\varepsilon\neq0$. It is shown that a remarkably rich behaviour arises,
in which two crossover times are observed as $\varepsilon \to 0$.
