  A point-shift $F$ maps each point of a point process $\Phi$ to some point of
$\Phi$. For all translation invariant point-shifts $F$, the $F$-foliation of
$\Phi$ is a partition of the support of $\Phi$ which is the discrete analogue
of the stable manifold of $F$ on $\Phi$. It is first shown that foliations lead
to a classification of the behavior of point-shifts on point processes. Both
qualitative and quantitative properties of foliations are then established. It
is shown that for all point-shifts $F$, there exists a point-shift $F_\bot$,
the orbits of which are the $F$-foils of $\Phi$, and which are
measure-preserving. The foils are not always stationary point processes.
Nevertheless, they admit relative intensities with respect to one another.
