Bounds for preperiodic points for maps with good reduction

Scarcity of finite orbits for rational functions over a number field

“…The second author proved in [Tro17] an inverse relationship between periodic points and tail points. Using Three-Point Lemma A, and combining with results from [CV], we can improve the result in [Tro17]. Before stating the result we recall some definitions: A point [a : b] ∈ P 1 (K) is critical for a rational function φ :…”

“…The second author proved in [Tro17] an inverse relationship between periodic points and tail points. Using Three-Point Lemma A, and combining with results from [CV], we can improve the result in [Tro17].…”

“…The second author proved in [Tro17] an inverse relationship between periodic points and tail points. Using Three-Point Lemma A, and combining with results from [CV], we can improve the result in [Tro17]. Before stating the result we recall some definitions: A point [a : b] ∈ P 1 (K) is critical for a rational function φ : P 1 → P 1 if the order of zero at [a : b] of the algebraic condition φ([x : y]) = φ([a : b]) is greater than 1 (if the point P and its image φ(P ) are finite, then P is critical if and only φ ′ (P ) = 0).…”

“…A natural relaxation of the uniform boundedness conjecture is to restrict our study to families of rational functions given in terms of good reduction. A rational map φ : P 1 → P 1 of degree d ≥ 2 defined over a number field K is said to have good reduction at a non zero prime p of K if φ can be written as φ = [F (X, Y ) : G(X, Y )] where F, G ∈ R p [X, Y ] are homogeneous polynomials of degree d, such that the resultant of F and G is a p-unit, where R p is the localization of the ring of integers of K at p. The map φ is said to have bad reduction at a prime p of K if φ does not have good reduction at p. For a fixed finite set S of places of K containing all the archimedean ones, we say that φ has good reduction outside of S if it has good reduction at each place p / ∈ S. In the special case of rational functions φ : P 1 → P 1 , there are several results giving a uniform bound on the number of periodic/preperiodic points of φ depending on the cardinality of a finite set of places S, which includes all archimedean places, together with the constants [K : Q] and deg(φ), under the assumption that φ has good reduction outside of S (e.g., [Nar89,MS94,Ben07,Can07,Can10,CP16,CV,Tro17]).…”

“…Furthermore, the hypothesis of the second statement in the result is optimal in the sense that it is impossible to get any result of this form (i.e. independent of d) when |Per(φ, K)| < 4, as shown by the examples in [Tro17,§5].…”