Rational periodic points for quadratic maps

Let K be a number field. Let S be a 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 consider endomorphisms of ℙ 1 of degree 2, defined over K, with good reduction outside S. We prove that there exist only finitely many such endomorphisms, up to conjugation by PGL 2 (R S ), admitting a periodic point in ℙ 1 (K) of order >3. Also, all but finitely many classes with a periodic point in ℙ 1 (K) of order 3 are parametrized by an irreducible curve.