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Preperiodic portraits for unicritical polynomials over a rational function field


Author: John R. Doyle
Journal: Trans. Amer. Math. Soc. 370 (2018), 3265-3288
MSC (2010): Primary 37P05; Secondary 37F10, 14H05
DOI: https://doi.org/10.1090/tran/7033
Published electronically: November 16, 2017
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Abstract: Let $ K$ be an algebraically closed field of characteristic zero, and let $ \mathcal {K} := K(t)$ be the rational function field over $ K$. For each $ d \ge 2$, we consider the unicritical polynomial $ f_d(z) := z^d + t \in \mathcal {K}[z]$, and we ask the following question: If we fix $ \alpha \in \mathcal {K}$ and integers $ M \ge 0$, $ N \ge 1$, and $ d \ge 2$, does there exist a place $ \mathfrak{p} \in \mathrm {Spec} K[t]$ such that, modulo $ \mathfrak{p}$, the point $ \alpha $ enters into an $ N$-cycle after precisely $ M$ steps under iteration by $ f_d$? We answer this question completely, concluding that the answer is generally affirmative and explicitly giving all counterexamples. This extends previous work by the author in the case that $ \alpha $ is a constant point.


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Additional Information

John R. Doyle
Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627
Address at time of publication: Department of Mathematics and Statistics, GTMH 330, Louisiana Tech University, Ruston, Louisiana 71272
Email: jdoyle@latech.edu

DOI: https://doi.org/10.1090/tran/7033
Keywords: Preperiodic points, abc-theorem, unicritical polynomials
Received by editor(s): April 7, 2016
Received by editor(s) in revised form: July 25, 2016
Published electronically: November 16, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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