Integer points in backward orbits

A theorem of J. Silverman states that a forward orbit of a rational map φ(z) on P1(K) contains finitely many S-integers in the number field K when (φ∘φ)(z) is not a polynomial. We state an analogous conjecture for the backward orbits using a general S-integrality notion based on the Galois conjugates of points. This conjecture is proven for the map φ(z)=zd, and consequently Chebyshev polynomials, by uniformly bounding the number of Galois orbits for zn−β when β≠0 is a non-root of unity. In general, our conjecture is true provided that the number of Galois orbits for φn(z)−β is bounded independently of n.