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Proceedings of the American Mathematical Society

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On roots of unity in orbits of rational functions


Author: Alina Ostafe
Journal: Proc. Amer. Math. Soc. 145 (2017), 1927-1936
MSC (2010): Primary 11R18, 37F10
DOI: https://doi.org/10.1090/proc/13433
Published electronically: November 3, 2016
MathSciNet review: 3611309
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Abstract: We consider a large class of univariate rational functions over a number field $\mathbb {K}$, including all polynomials over $\mathbb {K}$, and give a precise description of the exceptional set of such functions $h$ for which there are infinitely many initial points in the cyclotomic closure $\mathbb {K}^c$ for which the orbit under iterations of $h$ contains a root of unity. Our results are similar to previous results of Dvornicich and Zannier describing all polynomials having infinitely many preperiodic points in $\mathbb {K}^c$. We also pose several open questions.


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Additional Information

Alina Ostafe
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia
MR Author ID: 884181
Email: alina.ostafe@unsw.edu.au

Received by editor(s): May 17, 2016
Received by editor(s) in revised form: July 3, 2016
Published electronically: November 3, 2016
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2016 American Mathematical Society