Cycles for rational maps of good reduction outside a prescribed set

Let K be a number field and S a fixed finite set of places of K containing all the archimedean ones. Let RS be the ring of S-integers of K. In the present paper we study the cycles in P1(K) for rational maps of degree ≥2 with good reduction outside S. We say that two ordered n-tuples (P0, P1,… ,Pn−1) and (Q0, Q1,… ,Qn−1) of points of P1(K) are equivalent if there exists an automorphism A ∈ PGL2(RS) such that Pi = A(Qi) for every index i∈{0,1,… ,n−1}. We prove that if we fix two points P0,P1∈P1(K), then the number of inequivalent cycles for rational maps of degree ≥2 with good reduction outside S which admit P0, P1 as consecutive points is finite and depends only on S and K. We also prove that this result is in a sense best possible.