## Duke Mathematical Journal

### Preperiodic points and unlikely intersections

#### Abstract

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 $a, b\in\mathbb{C}$ and any integer $d \geq 2$, the set of $c\in\mathbb{C}$ for which both $a$ and $b$ are preperiodic for $z^d+c$ is infinite if and only if $a^d = b^d$. 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 $\varphi, \psi \in\mathbb{C}(z)$ 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 $\varphi$ and $\psi$ are defined over $\bar{\mathbb{Q}}$. 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

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

Dates
First available in Project Euclid: 11 July 2011

https://projecteuclid.org/euclid.dmj/1310416361

Digital Object Identifier
doi:10.1215/00127094-1384773

Mathematical Reviews number (MathSciNet)
MR2817647

Zentralblatt MATH identifier
1242.37062

#### Citation

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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