Fractional linear minimal models of rational functions

Many arithmetic geometric results have an arithmetic dynamic analogue. For instance, Siegel's theorem that an elliptic curve has only finitely many integer points, is analogous to the fact that any orbit under a rational function whose second iterate has a non-constant denominator has only finitely many distinct integer values. A conjecture of Lang states that the number of integer points on a minimal Weierstrass model of an elliptic curve is uniformly bounded. In order to translate this conjecture, one needs a dynamic concept of minimality. We present two such notions, affine minimality and full PGL_2(Q)-minimality, and prove they are equivalent. We also present an algorithm to test minimality. Finally, we present the results of an exhaustive search for rational functions with many integers in an orbit. These provide the best known minima for uniform bounds on the number of integers in orbits.