### 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 |
---|---|

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

**Preperiodic points for families of rational maps.** / Ghioca, D.; Hsia, L. C.; Tucker, T. J.

Research output: Contribution to journal › Article

*Proceedings of the London Mathematical Society*, vol. 110, no. 2, pp. 395-427. https://doi.org/10.1112/plms/pdu051

}

TY - JOUR

T1 - Preperiodic points for families of rational maps

AU - Ghioca, D.

AU - Hsia, L. C.

AU - Tucker, T. J.

PY - 2012/12/18

Y1 - 2012/12/18

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84928894653&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84928894653&partnerID=8YFLogxK

U2 - 10.1112/plms/pdu051

DO - 10.1112/plms/pdu051

M3 - Article

AN - SCOPUS:84928894653

VL - 110

SP - 395

EP - 427

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

SN - 0024-6115

IS - 2

ER -