Preperiodic portraits for unicritical polynomials over a rational function field
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- by John R. Doyle PDF
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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.References
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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
- MR Author ID: 993361
- ORCID: 0000-0001-6476-0605
- Email: jdoyle@latech.edu
- Received by editor(s): April 7, 2016
- Received by editor(s) in revised form: July 25, 2016
- Published electronically: November 16, 2017
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 3265-3288
- MSC (2010): Primary 37P05; Secondary 37F10, 14H05
- DOI: https://doi.org/10.1090/tran/7033
- MathSciNet review: 3766849