Integral points and relative sizes of coordinates of orbits in $${{\mathbb P}^N}$$ P N

Abstract

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 \(\mathbb {P}^N\) 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.