Let Φ be an endomorphism of P 1 Q , the projective line over the algebraic closure of Q, of degree ≥ 2 defined over a number field K. Let v be a non-archimedean valuation of K. We say that Φ has critically good reduction at v if any pair of distinct ramification points of Φ do not collide under reduction modulo v and the same holds for any pair of branch points. We say that Φ has simple good reduction at v if the map Φ v , the reduction of Φ modulo v, has the same degree of Φ. We prove that if Φ has critically good reduction at v and the reduction map Φ v is separable, then Φ has simple good reduction at v.

Let K be a number field and φ ∈ K(z) a rational function. Let S be the set of all archimedean places of K and all non-archimedean places associated to the prime ideals of bad reduction for φ. We prove an upper bound for the length of finite orbits for φ in P 1 (K) depending only on the cardinality of S .

Let k be an algebraic closed field of characteristic zero. Let K be the rational function field K = k(t). Let φ be a non isotrivial rational function in K(z). We prove a bound for the cardinality of the set of K-rational preperiodic points for φ in terms of the number of places of bad reduction and the degree d of φ.

The surface corresponding to the moduli space of quadratic endomorphisms of P 1 with a marked periodic point of order n is studied. It is shown that the surface is rational over Q when n ≤ 5 and is of general type for n = 6.An explicit description of the n = 6 surface lets us find several infinite families of quadratic endomorphisms f : P 1 → P 1 defined over Q with a rational periodic point of order 6. In one of these families, f also has a rational fixed point, for a total of at least 7 periodic and 7 preperiodic points. This is in contrast with the polynomial case, where it is conjectured that no polynomial endomorphism defined over Q admits rational periodic points of order n > 3.

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