TY - JOUR

T1 - Preperiodic points and unlikely intersections

AU - Baker, Matthew

AU - Demarco, Laura

N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2011/7/15

Y1 - 2011/7/15

N2 - In this article, we combine complex-analytic and arithmetic tools to study the prepe-riodic points of 1-dimensional complex dynamical systems. We show that for any fixed a,b ε C and any integer d ≥ 2, the set of c ε C for which both a and b are preperiodic for zd + c is infinite if and only if ad = bd. This provides an affirmative answer to a question of Zannier, which itself arose from questions of Masser concerning simultaneous torsion sections on families of elliptic curves. Using similar techniques, we prove that if rational functions π{variant}, ψ ε C(z) have infinitely many preperiodic points in common, then all of their preperiodic points coincide (and, in particular, the maps must have the same Julia set). This generalizes a theorem of Mimar, who established the same result assuming that π{variant} and ψ are defined over Q. The main arithmetic ingredient in the proofs is an adelic equidistribution theorem for preperiodic points over number fields and function fields, with nonarchimedean Berkovich spaces playing an essential role.

AB - In this article, we combine complex-analytic and arithmetic tools to study the prepe-riodic points of 1-dimensional complex dynamical systems. We show that for any fixed a,b ε C and any integer d ≥ 2, the set of c ε C for which both a and b are preperiodic for zd + c is infinite if and only if ad = bd. This provides an affirmative answer to a question of Zannier, which itself arose from questions of Masser concerning simultaneous torsion sections on families of elliptic curves. Using similar techniques, we prove that if rational functions π{variant}, ψ ε C(z) have infinitely many preperiodic points in common, then all of their preperiodic points coincide (and, in particular, the maps must have the same Julia set). This generalizes a theorem of Mimar, who established the same result assuming that π{variant} and ψ are defined over Q. The main arithmetic ingredient in the proofs is an adelic equidistribution theorem for preperiodic points over number fields and function fields, with nonarchimedean Berkovich spaces playing an essential role.

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U2 - 10.1215/00127094-1384773

DO - 10.1215/00127094-1384773

M3 - Article

AN - SCOPUS:79960619314

VL - 159

SP - 1

EP - 29

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 1

ER -