Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37F10, 37P05, 11R99

Retrieve articles in all journals with MSC (2010): 37F10, 37P05, 11R99


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