DEVIATIONS FROM S-INTEGRALITY IN ORBITS ON P-N

Silverman proved that when the second iterate of a rational function phi is not a polynomial, there are only finitely many S-integral points in each orbit of a rational point. We will survey prior results that attempt to generalize this result to higher-dimensions, and then we will discuss some extensions. More specifically, one new result incorporates geometric properties from multiple iterates simultaneously, while another generalizes to maps with some indeterminancy. All of these general theories assume some version of a very deep Diophantine conjecture by Vojta, but we will give explicit examples for which this conjecture can be avoided. We will also give some examples of maps for which these general theories do not apply directly but for which deviations from S-integrality in orbits can be analyzed unconditionally. We will end by posing many questions still to be answered.