Preperiodic points for families of rational maps

D. Ghioca, L. C. Hsia, T. J. Tucker

Research output: Contribution to journalArticle

12 Citations (Scopus)

Abstract

Let X be a smooth curve defined over ℚ, let a,b ε ℙ1(ℚ) and let fλ(x) ε ℚ(x) be an algebraic family of rational maps indexed by all λ ε X(ℂ). We study whether there exist infinitely many λ ε X(ℂ) such that both a and b are preperiodic for fλ. In particular, we show that if P,Q ε ℚ[x] such that deg (P)≥ 2+ deg (Q), and if a,b ε 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 ε 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.

Original languageEnglish
Pages (from-to)395-427
Number of pages33
JournalProceedings of the London Mathematical Society
Volume110
Issue number2
DOIs
Publication statusPublished - 2012 Dec 18

ASJC Scopus subject areas

  • Mathematics(all)

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