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Preperiodic portraits for unicritical polynomials


Author: John R. Doyle
Journal: Proc. Amer. Math. Soc. 144 (2016), 2885-2899
MSC (2010): Primary 37F10; Secondary 37P05, 11R99
DOI: https://doi.org/10.1090/proc/13075
Published electronically: March 16, 2016
MathSciNet review: 3487222
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Abstract: Let $ K$ be an algebraically closed field of characteristic zero, and for $ c \in K$ and an integer $ d \ge 2$, define $ f_{d,c}(z) := z^d + c \in K[z]$. We consider the following question: If we fix $ x \in K$ and integers $ M \ge 0$, $ N \ge 1$, and $ d \ge 2$, does there exist $ c \in K$ such that, under iteration by $ f_{d,c}$, the point $ x$ enters into an $ N$-cycle after precisely $ M$ steps? We conclude that the answer is generally affirmative, and we explicitly give all counterexamples. When $ d = 2$, this answers a question posed by Ghioca, Nguyen, and Tucker.


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Additional Information

John R. Doyle
Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627
Email: john.doyle@rochester.edu

DOI: https://doi.org/10.1090/proc/13075
Keywords: Preperiodic points, generalized dynatomic polynomials, unicritical polynomials
Received by editor(s): February 12, 2015
Published electronically: March 16, 2016
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2016 American Mathematical Society

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