TY - JOUR

T1 - Preperiodic points for families of rational maps

AU - Ghioca, D.

AU - Hsia, L. C.

AU - Tucker, T. J.

N1 - Funding Information:
The first author was partially supported by NSERC. The second author was partially supported by NSC Grant 102-2115-M-003-002-MY2 and he also acknowledges the support from NCTS. The third author was partially supported by NSF Grants DMS-0854839 and DMS-1200749.
Publisher Copyright:
© 2014 London Mathematical Society.

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.

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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 -