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Dynamics of the square mapping on the ring of $p$-adic integers


Authors: Shilei Fan and Lingmin Liao
Journal: Proc. Amer. Math. Soc. 144 (2016), 1183-1196
MSC (2010): Primary 37P05; Secondary 11S82, 37B05
DOI: https://doi.org/10.1090/proc12777
Published electronically: July 29, 2015
MathSciNet review: 3447671
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Abstract: For each prime number $p$, the dynamical behavior of the square mapping on the ring $\mathbb {Z}_p$ of $p$-adic integers is studied. For $p=2$, there are only attracting fixed points with their attracting basins. For $p\geq 3$, there are a fixed point $0$ with its attracting basin, finitely many periodic points around which there are countably many minimal components and some balls of radius $1/p$ being attracting basins. All these minimal components are precisely exhibited for different primes $p$.


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Additional Information

Shilei Fan
Affiliation: School of Mathematics and Statistics, Central China Normal University, 430079, Wuhan, People’s Republic of China
MR Author ID: 1014218
Email: slfan@mail.ccnu.edu.cn

Lingmin Liao
Affiliation: LAMA, UMR 8050, CNRS, Université Paris-Est Créteil Val de Marne, 61 Avenue du Général de Gaulle, 94010 Créteil Cedex, France
Email: lingmin.liao@u-pec.fr

Keywords: $p$-adic dynamical system, minimal decomposition, square mapping
Received by editor(s): August 23, 2014
Received by editor(s) in revised form: March 7, 2015
Published electronically: July 29, 2015
Additional Notes: The first author was partially supported by self-determined research funds of CCNU (Grant No. CCNU14Z01002) and NSF of China (Grant No. 11231009). The second author was partially supported by 12R03191A - MUTADIS (France) and the project PHC Orchid of MAE and MESR of France.
Communicated by: Yingfei Yi
Article copyright: © Copyright 2015 American Mathematical Society