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arXiv · 1210.7715

Preperiodic points for families of rational map

Dragos Ghioca·Liang-Chung Hsia·Thomas J. Tucker
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Abstract

Let X be a smooth curve defined over the algebraic numbers, let a,b be algebraic numbers, and let f_l(x) be an algebraic family of rational maps indexed by all l in X. We study whether there exist infinitely many l in X such that both a and b are preperiodic for f_l. In particular we show that if P,Q are polynomials over the algebraic numbers such that deg(P) >= 2+deg(Q), and there exists l such that a is periodic for P(x)/Q(x) + l, but b is not preperiodic for P(x)/Q(x) + l, then there exist at most finitely many l such that both a and b are preperiodic for P(x)/Q(x)+l. We also prove a similar result for certain two-dimensional families of endomorphisms of P^2.

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Dragos Ghioca, Liang-Chung Hsia, Thomas J. Tucker. 2012-10-29. Preperiodic points for families of rational map. https://doi.org/10.1112/plms%2Fpdu051

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