Integral Points and Relative Sizes of Coordinates of Orbits in P^N

摘要

We give a generalization to higher dimensions of Silverman's result onfiniteness of integer points in orbits. Assuming Vojta's conjecture, we prove asufficient condition for morphisms on P^N so that (S,D)-integral points in eachorbit are Zariski-non-dense. This condition is geometric, and for dimension 1it corresponds precisely to Silverman's hypothesis that the second iterate ofthe map is not a polynomial. In fact, we will prove a more precise formulationcomparing local heights outside S to the global height. For hyperplanes, thisamounts to comparing logarithmic sizes of the coordinates, generalizingSilverman's precise version in dimension 1. We also discuss a variant where wecan conclude that integral points in orbits are finite, rather than justZariski-non-dense. Further, we show unconditional results and examples, usingSchmidt's subspace theorem and known cases of Lang--Vojta conjecture. We endwith some extensions to the case of rational maps and to the case when thearithmetic of the orbit under one map is controlled by the geometric propertiesof another. We include many explicit examples to illustrate different behaviorsof integral points in orbits in higher dimensions.