Deviations from $S$-Integrality in Orbits on $P^N$
Vol. 9 No. 4 (2014) P.603~P.632
Silverman proved that when the second iterate of a rational function $\varnothing$ 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 an improvement of them. All of these general theories assume some version of a very deep Diophantine conjecture by Vojta. 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.
Integral points, orbits, higher-dimensional dynamics, Vojta's conjecture
MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 37P55, 11J97, 37P15
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Revised : 2014-07-07