Duke Mathematical Journal

Preperiodic points and unlikely intersections

Matthew Baker and Laura Demarco

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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.

Article information

Duke Math. J., Volume 159, Number 1 (2011), 1-29.

First available in Project Euclid: 11 July 2011

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Zentralblatt MATH identifier

Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04] 11G50: Heights [See also 14G40, 37P30]
Secondary: 31A99: None of the above, but in this section 11S80: Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.)


Baker, Matthew; Demarco, Laura. Preperiodic points and unlikely intersections. Duke Math. J. 159 (2011), no. 1, 1--29. doi:10.1215/00127094-1384773. https://projecteuclid.org/euclid.dmj/1310416361

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