The closure of the periodic points of rational maps over a non‐archimedean field is studied. An analogue of Montel's theorem over non‐archimedean fields is first proved. Then, it is shown that the (nonempty) Julia set of a rational map over a non‐archimedean field is contained in the closure of the periodic points.
Let a(λ), b(λ) ∈ C[λ] and let f λ (x) ∈ C[x] be a one-parameter family of polynomials indexed by all λ ∈ C. We study whether there exist infinitely many λ ∈ C such that both a(λ) and b(λ) are preperiodic for f λ .
Let fðzÞ be a polynomial of degree at least 2 with coe‰cients in a number field K. Iterating f gives rise to a dynamical system and a corresponding canonical height functionĥ h f , as defined by Call and Silverman. We prove a simple product formula relating the transfinite diameters of the filled Julia sets of f over various completions of K, and we apply this formula to give a generalization of Bilu's equidistribution theorem for sequences of points whose canonical heights tend to zero.
Let ϕ(z) ∈ K(z) be a rational function of degree d ≥ 2 defined over a number field whose second iterate ϕ 2 (z) is not a polynomial, and let α ∈ K. The second author previously proved that the forward orbit O ϕ (α) contains only finitely many quasi-Sintegral points. In this note we give an explicit upper bound for the number of such points.
Given a smooth projective curve C defined over Q and given two elliptic surfaces E1 −→ C and E2 −→ C along with sections Pi, Qi of Ei (for i = 1, 2), we prove that if there exist infinitely many t ∈ C(Q) such that for some integers m1,t, m2,t, we have that [mi,t](Pi)t = (Qi)t on Ei (for i = 1, 2), then at least one of the following conclusions must hold: either (i) there exists a nontrivial isogeny ψ : E1 −→ E2 and also there exist nontrivial endomorphisms ϕi of Ei (for i = 1, 2) such that ϕ2(P2) = ψ(ϕ1(P1)); or (ii) Qi is a multiple of Pi for some i = 1, 2. A special case of our result answers a conjecture made by Silverman.
PreliminariesFrom now on, we fix an elliptic surface π : E −→ C, where C is a projective, smooth curve defined over Q. We denote by E the generic fiber of E; this is an elliptic curve defined over Q(C). For all but finitely many t ∈ C(Q), we have that E t := π −1 ({t}) is an elliptic curve defined over Q.
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