Integral points in two-parameter orbits

Let K be a number field, let f: P1 → P1 be a nonconstant rational map of degree greater than 1, let S be a finite set of places of K, and suppose that u,w ∈ P1 (K) are not preperiodic under f. We prove that the set of (m,n) ∈N2such that f° m(u) is S-integral relative to f° n(w) is finite and effectively computable. This may be thought of as a two-parameter analog of a result of Silverman on integral points in orbits of rational maps. This issue can be translated in terms of integral points on an open subset of P12 then one can apply a modern version of the method of Runge, after increasing the number of components at infinity by iterating the rational map. Alternatively, an ineffective result comes from a well-known theorem of Vojta. © 2015 by De Gruyter.