15 July 2011 Preperiodic points and unlikely intersections
Matthew Baker, Laura Demarco
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Duke Math. J. 159(1): 1-29 (15 July 2011). DOI: 10.1215/00127094-1384773


In this article, we combine complex-analytic and arithmetic tools to study the preperiodic points of 1-dimensional complex dynamical systems. We show that for any fixed a,bC and any integer d2, the set of cC 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 ϕ,ψC(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 ϕ 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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Matthew Baker. Laura Demarco. "Preperiodic points and unlikely intersections." Duke Math. J. 159 (1) 1 - 29, 15 July 2011. https://doi.org/10.1215/00127094-1384773


Published: 15 July 2011
First available in Project Euclid: 11 July 2011

zbMATH: 1242.37062
MathSciNet: MR2817647
Digital Object Identifier: 10.1215/00127094-1384773

Primary: 11G50 , 37F10
Secondary: 11S80 , 31A99

Rights: Copyright © 2011 Duke University Press


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Vol.159 • No. 1 • 15 July 2011
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