  We give a generalization to higher dimensions of Silverman's result on
finiteness of integer points in orbits. Assuming Vojta's conjecture, we prove a
sufficient condition for morphisms on P^N so that (S,D)-integral points in each
orbit are Zariski-non-dense. This condition is geometric, and for dimension 1
it corresponds precisely to Silverman's hypothesis that the second iterate of
the map is not a polynomial. In fact, we will prove a more precise formulation
comparing local heights outside S to the global height. For hyperplanes, this
amounts to comparing logarithmic sizes of the coordinates, generalizing
Silverman's precise version in dimension 1. We also discuss a variant where we
can conclude that integral points in orbits are finite, rather than just
Zariski-non-dense. Further, we show unconditional results and examples, using
Schmidt's subspace theorem and known cases of Lang--Vojta conjecture. We end
with some extensions to the case of rational maps and to the case when the
arithmetic of the orbit under one map is controlled by the geometric properties
of another. We include many explicit examples to illustrate different behaviors
of integral points in orbits in higher dimensions.
