Rational periodic points for quadratic maps

Abstract

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.

Bounds for preperiodic points for maps with good reduction

“…In 2007, Benedetto [2] proved for the case of polynomial maps of degree d ≥ 2 that | PrePer(φ, K)| is bounded by O(|S| log |S|) , where S is the set of places of K at which φ has bad reduction, including all archimedean places of K. The big-O is essentially d 2 −2d+2 log d for large |S|. Many other results have been proven in recent years [3], [6] , [11], [15].…”

The abcd conjecture, uniform boundedness, and dynamical systems

“…We state the number field version of Canci-Paladino's result for the forward orbit O f (P ) = f (j) (P ) j ∈ N of a preperiodic point P of an endomorphism f . Specializing to periodic points of quadratic rational maps f , Canci [7] used Morton-Silverman's bound to show that quadratic rational maps with good reduction outside a set of places S typically do not have periodic points of large period.…”

Moduli Spaces of Quadratic Rational Maps with a Marked Periodic Point of Small Order

“…Now the proposition follows from the fact that Φ is a one-to-one map. where x, y, z are S-units and f is a quadratic map defined over K with good reduction outside S. See [Sil07] or [Ca10] for the definition of good reduction, but roughly speaking it means that the homogeneous resultant of the two p-coprime polynomials defining f is a p-unit for any p / ∈ S. In particular, to an S-integral point of M 2 (6) corresponds a rational map f defined over K, with good reduction outside S, which admits a K-rational periodic point of minimal period 6; and this set is finite by […”

“…Now the proposition follows from the fact that Φ is a one-to-one map. where x, y, z are S-units and f is a quadratic map defined over K with good reduction outside S. See [Sil07] or [Ca10] for the definition of good reduction, but roughly speaking it means that the homogeneous resultant of the two p-coprime polynomials defining f is a p-unit for any p / ∈ S. In particular, to an S-integral point of M 2 (6) corresponds a rational map f defined over K, with good reduction outside S, which admits a K-rational periodic point of minimal period 6; and this set is finite by [Ca10, Theorem 1.2]. Now Proposition 4.12 follows from the previous argument because for any point [W : X : Y : Z] ∈ M 2 (6) there exists a unique f that admits the cycle ([0 : 1], [1 : 0], [1 : 1], [x : 1], [y : 1], [z : 1]), where (x, y, z) = φ −1 ([W : X : Y : Z]) and the map φ −1 is the one defined in Lemma 4.1.…”