We investigate the effects of round-off errors on
quasi-periodic motions in a linear symplectic planar map.
By discretizing coordinates uniformly we transform this
map into a permutation of $\Z^2$, and study motions near
infinity, which correspond to a fine discretization.
We provide numerical evidence that all orbits are periodic
and that the average order of the period grows linearly
with the amplitude.
The discretization induces fluctuations of the invariant
of the continuum system.
We investigate the associated transport process for
time scales shorter than the period, and we
provide numerical evidence that the limiting behaviour
is a random walk where the step size is modulated by a
quasi-periodic function. For this stochastic process we compute
the transport coefficients explicitly, by constructing their
generating function.
These results afford a probabilistic description of motions
on a classical invariant torus.