Preperiodic points for families of rational maps

• Published in: Proceedings of the London Mathematical Society 2015-02, Vol.110 (2), p.395-427
• Main Authors: Ghioca, D., Hsia, L.‐C., Tucker, T. J.
• Format: Article
• Language: English
• Subjects: Algebra; Byproducts; Formulas (mathematics); Mathematical analysis; Polynomials
• ISSN: 0024-6115; 1460-244X
• DOI: 10.1112/plms/pdu051

Let X be a smooth curve defined over Q¯, let a,b∈P1(Q¯) and let fλ(x)∈Q¯(x) be an algebraic family of rational maps indexed by all λ∈X(C). We study whether there exist infinitely many λ∈X(C) such that both a and b are preperiodic for fλ. In particular, we show that if P,Q∈Q¯[x] such that deg(P)⩾2+deg(Q), and if a,b∈Q¯ such that a is periodic for P(x)/Q(x), but b is not preperiodic for P(x)/Q(x), then there exist at most finitely many λ∈C such that both a and b are preperiodic for P(x)/Q(x)+λ. We also prove a similar result for certain two‐dimensional families of endomorphisms of P2. As a by‐product of our method, we extend a recent result of Ingram [‘Variation of the canonical height for a family of polynomials’, J. reine. angew. Math. 685 (2013), 73–97] for the variation of the canonical height in a family of polynomials to a similar result for families of rational maps.