On Poonen's conjecture concerning rational preperiodic points of quadratic maps

Abstract

Abstract. The purpose of this note is to give some evidence in support of conjectures of Poonen and Morton and Silverman on the periods of rational numbers under the iteration of quadratic polynomials. In particular, for the family of maps fc(x) = x 2 + c for c ∈ Q, Poonen conjectured that the exact period of a Q-rational periodic points is at most 3. Using good reduction information, we verify this conjecture over Q for c values up to height 10 8 . For K/Q a quadratic number field, we provide evidence that the upper bound on the exact period of a Qrational periodic point is 6. We also show that the largest exact period of a rational periodic point increases at least linearly in the degree of the number field.

Bounds for preperiodic points for maps with good reduction

“…Let A be the set of all K-rational tail points excluding the initial point in each maximal orbit. Using equation (10) and Remark 2.6 every point P ∈ A admits a normalized form with respect to S ′ 2 . Now applying Proposition 4.1 to A and S ′ 2 , we get |A | ≤ max (5 * 10 6 (d 3 + 1)) s+4 , 4(2 64(s+3) ) .…”

Canonical heights for Hénon maps

“…The proof of Proposition 1.6 is essentially a repeated application of this algorithm. We note that the computations here are in spirit the same as those in [9], although there are some slight differences in the details. Our first lemma gives a method for computing a list of possible periods of Q-rational periodic points for ϕ, based on the dynamics modulo a prime of good reduction.…”

Scarcity of finite orbits for rational functions over a number field

“…Then | PrePer(φ c , Q)| ≤ 9. B. Hutz and P. Ingram [HI13] have shown that Poonen's conjecture holds when the numerator and denominator of c don't exceed 10 8 .…”