Resumen de Cycles for Rational Maps of Good Reduction Outside a Prescribed Set

Jung Kyu Canci

• 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 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 are equivalent if there exists an automorphism A ? PGL2(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 , 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