# American Institute of Mathematical Sciences

June  2017, 37(6): 3161-3182. doi: 10.3934/dcds.2017135

## Minimality of p-adic rational maps with good reduction

 1 School of Mathematics and Statistics, Central China Normal University, 430079 Wuhan, China & LAMFA, UMR 7352, CNRS, Université de Picardie Jules Verne, 33, Rue Saint Leu, 80039 Amiens Cedex 1, France 2 School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China 3 LAMA, UMR 8050, CNRS, Université Paris-Est Créteil, 61 Avenue du Général de Gaulle, 94010, Créteil Cedex, France 4 Institute of Mathematics, AMSS, Chinese Academy of Sciences, 100190 Beijing, China

* Corresponding author: Shilei Fan

Received  March 2016 Revised  February 2017 Published  February 2017

Fund Project: A. H. FAN was supported by NSF of China (Grant No. 11471132); S. L. FAN was supported by NSF of China (Grant No. 11401236) and self-determined research funds of CCNU (Grant No. CCNU17QN0009); Y. F. WANG was supported by NSF of China (Grant No. 11231009).

A rational map with good reduction in the field $\mathbb{Q}_p$ of $p$-adic numbers defines a $1$-Lipschitz dynamical system on the projective line $\mathbb{P}^1(\mathbb{Q}_p)$ over $\mathbb{Q}_p$. The dynamical structure of such a system is completely described by a minimal decomposition. That is to say, $\mathbb{P}^1(\mathbb{Q}_p)$ is decomposed into three parts: finitely many periodic orbits; finite or countably many minimal subsystems each consisting of a finite union of balls; and the attracting basins of periodic orbits and minimal subsystems. For any prime $p$, a criterion of minimality for rational maps with good reduction is obtained. When $p=2$, a condition in terms of the coefficients of the rational map is proved to be necessary for the map being minimal and having good reduction, and sufficient for the map being minimal and $1$-Lipschitz. It is also proved that a rational map having good reduction of degrees $2$, $3$ and $4$ can never be minimal on the whole space $\mathbb{P}^1(\mathbb{Q}_2)$.

Citation: Aihua Fan, Shilei Fan, Lingmin Liao, Yuefei Wang. Minimality of p-adic rational maps with good reduction. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3161-3182. doi: 10.3934/dcds.2017135
##### References:
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##### References:
 [1] S. Albeverio, U. A. Rozikov and I. A. Sattarov, p-adic (2, 1)-rational dynamical systems, J. Math. Anal. Appl., 398 (2013), 553-566.  doi: 10.1016/j.jmaa.2012.09.009. [2] V. S. Anashin, Uniformly distributed sequences of p-adic integers, Mat. Zametki, 55 (1994), 3-46, 188.  doi: 10.1007/BF02113290. [3] V. S. Anashin, Ergodic transformations in the space of p-adic integers, In p-adic mathematical physics, volume 826 of AIP Conf. Proc. , pages 3-24. Amer. Inst. Phys. , Melville, NY, 2006. doi: 10.1063/1.2193107. [4] V. S. Anashin and A. Khrennikov, Applied Algebraic Dynamics, de Gruyter Expositions in Mathematics. 49. Walter de Gruyter & Co. , Berlin, 2009. doi: 10.1515/9783110203011. [5] V. S. Anashin, A. Khrennikov and E. Yurova, Ergodicity criteria for non-expanding transformations of 2-adic spheres, Discrete Contin. Dyn. Syst., 34 (2014), 367-377. [6] M. Baker and R. Rumely, Potential Theory and Dynamics on the Berkovich Projective Line volume 159 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/surv/159. [7] R. L. Benedetto, Hyperbolic maps in p-adic dynamics, Ergodic Theory Dynam. Systems, 21 (2001), 1-11.  doi: 10.1017/S0143385701001043. [8] J. Chabert, A. H. Fan and Y. Fares, Minimal dynamical systems on a discrete valuation domain, Discrete Contin. Dyn. Syst., 25 (2009), 777-795.  doi: 10.3934/dcds.2009.25.777. [9] Z. Coelho and W. Parry, Ergodicity of p-adic multiplications and the distribution of Fibonacci numbers, In Topology, ergodic theory, real algebraic geometry, volume 202 of Amer. Math. Soc. Transl. Ser. 2, pages 51-70. Amer. Math. Soc. , Providence, RI, 2001. doi: 10.1090/trans2/202/06. [10] D. L. Desjardins and M. E. Zieve, Polynomial mappings mod pn, arXiv: math/0103046v1. [11] B. Dragovich, A. Khrennikov and D. Mihajlović, Linear fractional p-adic and adelic dynamical systems, Rep. Math. Phys., 60 (2007), 55-68.  doi: 10.1016/S0034-4877(07)80098-X. [12] F. Durand and F. Paccaut, Minimal polynomial dynamics on the set of 3-adic integers, Bull. Lond. Math. Soc., 41 (2009), 302-314.  doi: 10.1112/blms/bdp003. [13] A. H. Fan, M. T. Li, J. Y. Yao and D. Zhou, Strict ergodicity of affine p-adic dynamical systems on $\mathbb{Z}_p$, Adv. Math., 214 (2007), 666-700.  doi: 10.1016/j.aim.2007.03.003. [14] A. H. Fan and L. M. Liao, On minimal decomposition of $p$-adic polynomial dynamical systems, Adv. Math., 228 (2011), 2116-2144.  doi: 10.1016/j.aim.2011.06.032. [15] A. H. Fan, S. L. Fan, L. M. Liao and Y. F. Wang, On minimal decomposition of p-adic homographic dynamical systems, Adv. Math., 257 (2014), 92-135.  doi: 10.1016/j.aim.2014.02.007. [16] S. L. Fan and L. M. Liao, Dynamics of convergent power series on the integral ring of a finite extension of $\mathbb{Q}_p$, J. Differential Equations, 259 (2015), 1628-1648.  doi: 10.1016/j.jde.2015.03.042. [17] S. L. Fan and L. M. Liao, Dynamics of the square mapping on the ring of p-adic integers, Proc. Amer. Math. Soc., 144 (2016), 1183-1196.  doi: 10.1090/proc12777. [18] S. L. Fan and L. M. Liao, Dynamics of Chebyshev polynomials on $\mathbb{Z}_2$, J. Number Theor., 169 (2016), 174-182.  doi: 10.1016/j.jnt.2016.05.014. [19] M. Gundlach, A. Khrennikov and K. Lindahl, On ergodic behavior of p-adic dynamical systems, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 4 (2001), 569-577.  doi: 10.1142/S0219025701000632. [20] L. C. Hsia, Closure of periodic points over a non-Archimedean field, J. London Math. Soc. (2), 62 (2000), 685-700.  doi: 10.1112/S0024610700001447. [21] S. Jeong, Toward the ergodicity of p-adic 1-Lipschitz functions represented by the van der Put series, J. Number Theory, 133 (2013), 2874-2891.  doi: 10.1016/j.jnt.2013.02.006. [22] M. Khamraev and F. M. Mukhamedov, On a class of rational p-adic dynamical systems, J. Math. Anal. Appl., 315 (2006), 76-89.  doi: 10.1016/j.jmaa.2005.08.041. [23] M. V. Larin, Transitive polynomial transformations of residue rings, Diskret. Mat., 14 (2002), 20-32. [24] F. M. Mukhamedov and U. A. Rozikov, On rational p-adic dynamical systems, Methods Funct. Anal. Topology, 10 (2004), 21-31. [25] R. Oselies and H. Zieschang, Ergodische Eigenschaften der Automorphismen p-adischer Zahlen, Arch. Math. (Basel), 26 (1975), 144-153.  doi: 10.1007/BF01229718. [26] J. Rivera-Letelier, Dynamique des fonctions rationnelles sur des corps locaux, Astérisque, 287 (2003), 147-230. [27] I. A. Sattarov, p-adic (3, 2)-rational dynamical systems, p-Adic Numbers Ultrametric Anal. Appl., 7 (2015), 39-55.  doi: 10.1134/S2070046615010045. [28] J. Silverman, The Arithmetic of Dynamical Systems volume 241 of Graduate Texts in Mathematics, Springer, New York, 2007. doi: 10.1007/978-0-387-69904-2. [29] M. Zieve, Cycles of Polynomial Mappings, Ph. D thesis, UC Berkley 1996.
Tree structure of $\mathbb{P}^1(\mathbb{Q}_2)$. The points of $\mathbb{P}^1(\mathbb{Q}_2)$ are considered as the boundary points of the infinite tree
Tree structure of $\mathbb{P}^1(\mathbb{Q}_3)$. The points of $\mathbb{P}^1(\mathbb{Q}_3)$ are considered as the boundary points of the infinite tree.
 Coefficients $a_0, a_1, a_2, a_3, b_1, b_2, b_3$ Induced periodic orbits at level 3 1, 0, 1, 3, 3, 1, 01, 1, 1, 2, 3, 2, 31, 2, 1, 1, 3, 3, 21, 2, 3, 3, 3, 3, 01, 3, 1, 0, 3, 0, 11, 3, 3, 2, 3, 0, 33, 2, 1, 3, 1, 1, 03, 3, 1, 2, 1, 2, 3 $0\to \widetilde{0}\to 1\to 6 \to \widetilde{6}\to 3 $$\to 4\to \widetilde{4}\to 5\to 2\to \widetilde{2}\to 7 1, 0, 1, 3, 1, 1, 21, 1, 1, 2, 1, 2, 11, 2, 1, 1, 1, 3, 01, 2, 3, 3, 1, 3, 21, 3, 1, 0, 1, 0, 31, 3, 3, 2, 1, 0, 13, 2, 1, 3, 3, 1, 23, 3, 1, 2, 3, 2, 1 0\to\widetilde{0}\to1\to6\to\widetilde{2}\to3$$\to4\to\widetilde{4}\to5\to2\to\widetilde{6}\to7$ 1, 0, 1, 3, 1, 1, 01, 1, 1, 2, 1, 2, 31, 2, 1, 1, 1, 3, 21, 2, 3, 3, 1, 3, 01, 3, 1, 0, 1, 0, 11, 3, 3, 2, 1, 0, 33, 2, 1, 3, 3, 1, 03, 3, 1, 2, 3, 2, 3 $0\to \widetilde{0}\to1\to2\to\widetilde{6}\to3$$\to4\to\widetilde{4}\to5\to6\to\widetilde{2}\to7 1, 0, 1, 3, 3, 1, 21, 1, 1, 2, 3, 2, 11, 2, 1, 1, 3, 3, 01, 2, 3, 3, 3, 3, 21, 3, 1, 0, 3, 0, 31, 3, 3, 2, 3, 0, 13, 2, 1, 3, 1, 1, 23, 3, 1, 2, 1, 2, 1 0\to\widetilde{0}\to1\to2\to\widetilde{2}\to3$$\to4\to\widetilde{4}\to5\to6\to\widetilde{6}\to7$ 1, 0, 3, 1, 3, 1, 01, 1, 3, 0, 3, 2, 33, 0, 1, 1, 1, 3, 03, 0, 3, 3, 1, 3, 23, 1, 1, 0, 1, 0, 33, 1, 3, 2, 1, 0, 13, 2, 3, 1, 1, 1, 03, 3, 3, 0, 1, 2, 3 $0\to \widetilde{0}\to1\to 6\to \widetilde{6}\to 7$$\to 4\to \widetilde{4}\to 5\to 2\to \widetilde{2}\to 3 1, 0, 3, 1, 1, 1, 21, 1, 3, 0, 1, 2, 13, 0, 1, 1, 3, 3, 23, 0, 3, 3, 3, 3, 03, 1, 1, 0, 3, 0, 13, 1, 3, 2, 3, 0, 33, 2, 3, 1, 3, 1, 23, 3, 3, 0, 3, 2, 1 0\to\widetilde{0}\to1\to6\to\widetilde{2}\to7$$\to4\to\widetilde{4}\to5\to2\to\widetilde{6}\to3$ 1, 0, 3, 1, 1, 1, 01, 1, 3, 0, 1, 2, 33, 0, 1, 1, 3, 3, 03, 0, 3, 3, 3, 3, 23, 1, 1, 0, 3, 0, 33, 1, 3, 2, 3, 0, 13, 2, 3, 1, 3, 1, 03, 3, 3, 0, 3, 2, 3 $0\to\widetilde{0}\to1\to2\to\widetilde{6}\to7$$\to4\to\widetilde{4}\to5\to6\to\widetilde{2}\to3 1, 0, 3, 1, 3, 1, 21, 1, 3, 0, 3, 2, 13, 0, 1, 1, 1, 3, 23, 0, 3, 3, 1, 3, 03, 1, 1, 0, 1, 0, 13, 1, 3, 2, 1, 0, 33, 2, 3, 1, 1, 1, 23, 3, 3, 0, 1, 2, 1 0\to\widetilde{0}\to1\to2\to\widetilde{2}\to7$$\to4\to\widetilde{4}\to5\to6\to\widetilde{6}\to3$
 Coefficients $a_0, a_1, a_2, a_3, b_1, b_2, b_3$ Induced periodic orbits at level 3 1, 0, 1, 3, 3, 1, 01, 1, 1, 2, 3, 2, 31, 2, 1, 1, 3, 3, 21, 2, 3, 3, 3, 3, 01, 3, 1, 0, 3, 0, 11, 3, 3, 2, 3, 0, 33, 2, 1, 3, 1, 1, 03, 3, 1, 2, 1, 2, 3 $0\to \widetilde{0}\to 1\to 6 \to \widetilde{6}\to 3 $$\to 4\to \widetilde{4}\to 5\to 2\to \widetilde{2}\to 7 1, 0, 1, 3, 1, 1, 21, 1, 1, 2, 1, 2, 11, 2, 1, 1, 1, 3, 01, 2, 3, 3, 1, 3, 21, 3, 1, 0, 1, 0, 31, 3, 3, 2, 1, 0, 13, 2, 1, 3, 3, 1, 23, 3, 1, 2, 3, 2, 1 0\to\widetilde{0}\to1\to6\to\widetilde{2}\to3$$\to4\to\widetilde{4}\to5\to2\to\widetilde{6}\to7$ 1, 0, 1, 3, 1, 1, 01, 1, 1, 2, 1, 2, 31, 2, 1, 1, 1, 3, 21, 2, 3, 3, 1, 3, 01, 3, 1, 0, 1, 0, 11, 3, 3, 2, 1, 0, 33, 2, 1, 3, 3, 1, 03, 3, 1, 2, 3, 2, 3 $0\to \widetilde{0}\to1\to2\to\widetilde{6}\to3$$\to4\to\widetilde{4}\to5\to6\to\widetilde{2}\to7 1, 0, 1, 3, 3, 1, 21, 1, 1, 2, 3, 2, 11, 2, 1, 1, 3, 3, 01, 2, 3, 3, 3, 3, 21, 3, 1, 0, 3, 0, 31, 3, 3, 2, 3, 0, 13, 2, 1, 3, 1, 1, 23, 3, 1, 2, 1, 2, 1 0\to\widetilde{0}\to1\to2\to\widetilde{2}\to3$$\to4\to\widetilde{4}\to5\to6\to\widetilde{6}\to7$ 1, 0, 3, 1, 3, 1, 01, 1, 3, 0, 3, 2, 33, 0, 1, 1, 1, 3, 03, 0, 3, 3, 1, 3, 23, 1, 1, 0, 1, 0, 33, 1, 3, 2, 1, 0, 13, 2, 3, 1, 1, 1, 03, 3, 3, 0, 1, 2, 3 $0\to \widetilde{0}\to1\to 6\to \widetilde{6}\to 7$$\to 4\to \widetilde{4}\to 5\to 2\to \widetilde{2}\to 3 1, 0, 3, 1, 1, 1, 21, 1, 3, 0, 1, 2, 13, 0, 1, 1, 3, 3, 23, 0, 3, 3, 3, 3, 03, 1, 1, 0, 3, 0, 13, 1, 3, 2, 3, 0, 33, 2, 3, 1, 3, 1, 23, 3, 3, 0, 3, 2, 1 0\to\widetilde{0}\to1\to6\to\widetilde{2}\to7$$\to4\to\widetilde{4}\to5\to2\to\widetilde{6}\to3$ 1, 0, 3, 1, 1, 1, 01, 1, 3, 0, 1, 2, 33, 0, 1, 1, 3, 3, 03, 0, 3, 3, 3, 3, 23, 1, 1, 0, 3, 0, 33, 1, 3, 2, 3, 0, 13, 2, 3, 1, 3, 1, 03, 3, 3, 0, 3, 2, 3 $0\to\widetilde{0}\to1\to2\to\widetilde{6}\to7$$\to4\to\widetilde{4}\to5\to6\to\widetilde{2}\to3 1, 0, 3, 1, 3, 1, 21, 1, 3, 0, 3, 2, 13, 0, 1, 1, 1, 3, 23, 0, 3, 3, 1, 3, 03, 1, 1, 0, 1, 0, 13, 1, 3, 2, 1, 0, 33, 2, 3, 1, 1, 1, 23, 3, 3, 0, 1, 2, 1 0\to\widetilde{0}\to1\to2\to\widetilde{2}\to7$$\to4\to\widetilde{4}\to5\to6\to\widetilde{6}\to3$
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