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Proceedings of the American Mathematical Society

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Simultaneously preperiodic points for families of polynomials in normal form


Authors: Dragos Ghioca, Liang-Chung Hsia and Khoa Dang Nguyen
Journal: Proc. Amer. Math. Soc. 146 (2018), 733-741
MSC (2010): Primary 37P05; Secondary 37P30, 37P45
DOI: https://doi.org/10.1090/proc/13762
Published electronically: September 7, 2017
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Abstract: Let $ d>m>1$ be integers, let $ c_1,\dots , c_{m+1}$ be distinct complex numbers, and let $ \mathbf {f}(z):=z^d+t_1z^{m-1}+t_2z^{m-2}+\cdots + t_{m-1}z+t_m$ be an $ m$-parameter family of polynomials. We prove that the set of $ m$-tuples of parameters $ (t_1,\dots , t_m)\in \mathbb{C}^m$ with the property that each $ c_i$ (for $ i=1,\dots , m+1$) is preperiodic under the action of the corresponding polynomial $ \mathbf {f}(z)$ is contained in finitely many hypersurfaces of the parameter space $ \mathbb{A}^m$.


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Additional Information

Dragos Ghioca
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
Email: dghioca@math.ubc.ca

Liang-Chung Hsia
Affiliation: Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan, ROC
Email: hsia@math.ntnu.edu.tw

Khoa Dang Nguyen
Affiliation: Department of Mathematics and Statistics, University of Calgary, Calgary, AB T2N 4T4, Canada
Email: dangkhoa.nguyen@ucalgary.ca

DOI: https://doi.org/10.1090/proc/13762
Received by editor(s): November 6, 2016
Received by editor(s) in revised form: April 5, 2017
Published electronically: September 7, 2017
Additional Notes: The research of the first author was partially supported by an NSERC Discovery grant.
The second author was supported by MOST grant 105-2918-I-003-006.
The third author was partially supported by a UBC-PIMS fellowship
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2017 American Mathematical Society

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