### Abstract

Let X be a smooth curve defined over ℚ, let a,b ε ℙ^{1}(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 f_{λ}. 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 P^{2}. 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 language | English |
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Pages (from-to) | 395-427 |

Number of pages | 33 |

Journal | Proceedings of the London Mathematical Society |

Volume | 110 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2012 Dec 18 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Proceedings of the London Mathematical Society*,

*110*(2), 395-427. https://doi.org/10.1112/plms/pdu051