Arithmetic Dynamics - Integer Points in Orbits

Integer Points in Orbits

The orbit of a rational map may contain infinitely many integers. For example, if F(x) is a polynomial with integer coefficients and if a is an integer, then it is clear that the entire orbit O_F(a) consists of integers. Similarly, if F(x) is a rational map and some iterate F(n)(x) is a polynomial with integer coefficients, then every nth entry in the orbit is an integer. An example of this phenomenon is the map F(x) = 1/xd, whose second iterate is a polynomial. It turns out that this is the only way that an orbit can contain infinitely many integers.

Theorem Let F(x) ∈ Q(x) be a rational function of degree at least two, and assume that no iterate of F is a polynomial. Let a ∈ Q. Then the orbit O_F(a) contains only finitely many integers.