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Rational map ax + 1/x on the projective line over ℚ2

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Abstract

The dynamical structure of the rational map ax+1/x on the projective line \(\mathbb{P}^1(\mathbb{Q}_2)\) over the field ℚ2 of 2-adic numbers, is fully described.

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References

  1. Ahmad M A K, Liao L, Saburov M. Periodic p-adic Gibbs measures of q-states Potts model on Cayley tree: The chaos implies the vastness of p-adic Gibbs measures. ArXiv:1708.02152, 2017

    MATH  Google Scholar 

  2. Albeverio S, Rozikov U A, Sattarov I A. p-Adic (2; 1)-rational dynamical systems. J Math Anal Appl, 2013, 398: 553–566

    Article  MathSciNet  MATH  Google Scholar 

  3. Anashin V S, Khrennikov A. Applied Algebraic Dynamics. de Gruyter Expositions in Mathematics, vol. 49. Berlin: Walter de Gruyter, 2009

    Google Scholar 

  4. Baker M, Rumely R. Potential theory and dynamics on the Berkovich projective line. Mathematical Surveys and Monographs, vol. 159. Providence: Amer Math Soc, 2010

    Book  MATH  Google Scholar 

  5. Benedetto R L. Hyperbolic maps in p-adic dynamics. Ergodic Theory Dynam Systems, 2001, 21: 1–11

    Article  MathSciNet  MATH  Google Scholar 

  6. Dragovich B, Khrennikov A, Mihajlović D. Linear fractional p-adic and adelic dynamical systems. Rep Math Phys, 2007, 60: 55–68

    Article  MathSciNet  MATH  Google Scholar 

  7. Dremov V, Shabat G, Vytnova P. On the chaotic properties of quadratic maps over non-Archimedean fields. In: p-Adic Mathematical Physics. AIP Conference Proceedings, vol. 826. Melville: Amer Inst Phys, 2006, 43–54

    Book  MATH  Google Scholar 

  8. Fan A H, Fan S L, Liao L M, et al. On minimal decomposition of p-adic homographic dynamical systems. Adv Math, 2014, 257: 92–135

    Article  MathSciNet  MATH  Google Scholar 

  9. Fan A H, Fan S L, Liao L M, et al. Minimality of p-adic rational maps with good reduction. Discrete Contin Dyn Syst, 2017, 37: 3161–2182

    Article  MathSciNet  MATH  Google Scholar 

  10. Fan A H, Liao L M. On minimal decomposition of p-adic polynomial dynamical systems. Adv Math, 2011, 228: 2116–2144

    Article  MathSciNet  MATH  Google Scholar 

  11. Fan S L, Liao L M. Dynamics of convergent power series on the integral ring of a finite extension of ℚ2. J Differential Equations, 2015, 259: 1628–1648

    Article  MathSciNet  MATH  Google Scholar 

  12. Fan S L, Liao L M. Dynamics of the square mapping on the ring of p-adic integers. Proc Amer Math Soc, 2016, 144: 1183–1196

    Article  MathSciNet  MATH  Google Scholar 

  13. Fan S L, Liao L M. Dynamics of Chebyshev polynomials on Z2. J Number Theory, 2016, 169: 174–182

    Article  MathSciNet  MATH  Google Scholar 

  14. Fan S L, Liao L M. Rational map ax + 1/x on the projective line over ℚp. ArXiv:1612.01881, 2016

    Google Scholar 

  15. Fan A H, Liao L M, Wang Y F, et al. p-Adic repellers in ℚ2 are subshifts of finite type. C R Acad Sci Paris Ser I, 2007, 344: 219–224

    Article  MathSciNet  MATH  Google Scholar 

  16. Hsia L C. Closure of periodic points over a non-Archimedean field. J Lond Math Soc (2), 2000, 62: 685–700

    Article  MathSciNet  MATH  Google Scholar 

  17. Khamraev M, Mukhamedov F M. On a class of rational p-adic dynamical systems. J Math Anal Appl, 2006, 315: 76–89

    Article  MathSciNet  MATH  Google Scholar 

  18. Mahler K. p-Adic Numbers and Their Functions, 2nd ed. Cambridge Tracts in Mathematics, vol. 76. Cambridge: Cambridge University Press, 1981

    Google Scholar 

  19. Mukhamedov F, Khakimov O. Phase transition and chaos: p-adic Potts model on a Cayley tree. Chaos Solitons Fractals, 2016, 87: 190–196

    Article  MathSciNet  MATH  Google Scholar 

  20. Mukhamedov F, Khakimov O. Chaotic behavior of the p-adic Potts-Bethe mapping. Discrete Contin Dyn Syst, 2018, 38: 231–245

    Article  MathSciNet  MATH  Google Scholar 

  21. Mukhamedov F, Rozikov U A. On rational p-adic dynamical systems. Methods Funct Anal Topology, 2004, 10: 21–31

    MathSciNet  MATH  Google Scholar 

  22. Rivera-Letelier J. Dynamique des fonctions rationnelles sur des corps locaux. Astérisque, 2003, 287: 147–230

    MathSciNet  MATH  Google Scholar 

  23. Sattarov I A. p-Adic (3; 2)-rational dynamical systems. p-Adic Numbers Ultrametric Anal Appl, 2015, 7: 39–55

    Article  MathSciNet  MATH  Google Scholar 

  24. Silverman J. The Arithmetic of Dynamical Systems. Graduate Texts in Mathematics, vol. 241. New York: Springer, 2007

    Book  MATH  Google Scholar 

  25. Thiran E, Verstegen D, Weyers J. p-Adic dynamics. J Stat Phys, 1989, 54: 893–913

    Article  MathSciNet  MATH  Google Scholar 

  26. Woodcock C F, Smart N P. p-Adic chaos and random number generation. Experiment Math, 1998, 7: 333–342

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11401236 and 11471132) and Self-Determined Research Funds of Central China Normal University (Grant No. CCNU17QN0009).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, China

    Shilei Fan

  2. Laboratoire d’Analyse et de Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val de Marne, Créteil, 94010, France

    Lingmin Liao

Authors
  1. Shilei Fan
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  2. Lingmin Liao
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Correspondence to Shilei Fan.

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Dedicated to the Memory of Professor Lei Tan

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Fan, S., Liao, L. Rational map ax + 1/x on the projective line over ℚ2. Sci. China Math. 61, 2221–2236 (2018). https://doi.org/10.1007/s11425-017-9229-3

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