Preperiodic points and unlikely intersections

Matthew Baker*, Laura Demarco

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

36 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)1-29
Number of pages29
JournalDuke Mathematical Journal
Volume159
Issue number1
DOIs
StatePublished - Jul 15 2011

ASJC Scopus subject areas

  • Mathematics(all)

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