Advanced Search
Article Contents
Article Contents

Chaotic behavior of the P-adic Potts-Bethe mapping

The authors would like to thank an anonymous referee for his useful suggestions which allowed to improve the content of the paper.

Abstract Full Text(HTML) Related Papers Cited by
  • In the previous investigations of the authors the renormalization group method to p-adic models on Cayley trees has been developed. This method is closely related to the investigation of p-adic dynamical systems associated with a given model. In this paper, we study chaotic behavior of the Potts-Bethe mapping. We point out that a similar kind of result is not known in the case of real numbers (with rigorous proofs).

    Mathematics Subject Classification: Primary:37B05, 37B10;Secondary:12J12, 39A70.


    \begin{equation} \\ \end{equation}
  • 加载中
  •   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.
      N. S. Ananikian , S. K. Dallakian  and  B. Hu , Chaotic Properties of the Q-state Potts Model on the Bethe Lattice: Q <2, Complex Systems, 11 (1997) , 213-222. 
      V. Anashin and A. Khrennikov, Applied Algebraic Dynamics Walter de Gruyter, Berlin, New York, 2009. doi: 10.1515/9783110203011.
      R. Benedetto , Reduction, dynamics, and Julia sets of rational functions, J. Number Theory, 86 (2001) , 175-195.  doi: 10.1006/jnth.2000.2577.
      R. Benedetto , Hyperbolic maps in p-adic dynamics, Ergod. Th. & Dynam. Sys., 21 (2001) , 1-11.  doi: 10.1017/S0143385701001043.
      F. A. Bosco  and  R. S. Jr Goulart , Fractal dimension of the Julia set associated with the Yang-Lee zeros of the ising model on the Cayley tree, Europhys. Let., 4 (1987) , 1103-1108.  doi: 10.1209/0295-5075/4/10/004.
      H. Diao  and  C. E. Silva , Digraph representations of rational functions over the p-adic numbers, p-Adic Numbers, Ultametric Anal. Appl., 3 (2011) , 23-38.  doi: 10.1134/S2070046611010031.
      T. P. Eggarter , Cayley trees, the Ising problem, and the thermodynamic limit, Phys. Rev. B, 9 (1974) , 2989-2992.  doi: 10.1103/PhysRevB.9.2989.
      A. H. Fan , L. M. Liao , Y. F. Wang  and  D. Zhou , p-adic repellers in Qp are subshifts of finite type, C. R. Math. Acad. Sci Paris, 344 (2007) , 219-224.  doi: 10.1016/j.crma.2006.12.007.
      A. H. Fan , S. L. Fan , L. M. Liao  and  Y. F. Wang , On minimal deecomposition of p-adic homographic dynamical systems, Adv. Math., 257 (2014) , 92-135. 
      A. H. Fan , S. L. Fan , L. M. Liao  and  Y. F. Wang , Minimality of p-adic rational maps with good reduction, Discrete Cont. Dyn. Sys., 37 (2017) , 3161-3182.  doi: 10.3934/dcds.2017135.
      G. Gyorgyi, I. Kondor, L. Sasvari and T. Tel, From Phase Transitions to Chaos World Scientific, Singapore, 1992. doi: 10.1142/1633.
      M. Herman and J. -C. Yoccoz, Generalizations of some theorems of small divisors to nonArchimedean fields, In: Geometric Dynamics (Rio de Janeiro 1981), Lec. Notes in Math. , Springer, Berlin, 1007 (1983), 408–447 doi: 10.1007/BFb0061427.
      S. Kaplan , A survey of symbolic dynamics and celestial mechanics, Qualitative Theor. Dyn. Sys., 7 (2008) , 181-193.  doi: 10.1007/s12346-008-0010-5.
      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.
      N. Koblitz, P-adic Numbers, P-adic Analysis and Zeta-function Berlin, Springer, 1977.
      J. Lubin , Nonarchimedean dynamical systems, Composito Math., 94 (1994) , 321-346. 
      S. Ludkovsky  and  A. Yu. Khrennikov , Stochastic processes on non-Archimedean spaces with values in non-Archimedean fields, Markov Process. Related Fields, 9 (2003) , 131-162. 
      J. L. Monroe , Julia sets associated with the Potts model on the Bethe lattice and other recursively solved systems, J. Phys. A: Math. Gen., 34 (2001) , 6405-6412.  doi: 10.1088/0305-4470/34/33/305.
      F. A. Mukhamedov , Dynamical system appoach to phase transitions p-adic Potts model on the Cayley tree of order two, Rep. Math. Phys., 70 (2012) , 385-406.  doi: 10.1016/S0034-4877(12)60053-6.
      F. Mukhamedov , On dynamical systems and phase transitions for q+1-state p-adic Potts model on the Cayley tree, Math. Phys. Anal. Geom., 53 (2013) , 49-87.  doi: 10.1007/s11040-012-9120-z.
      F. Mukhamedov , Renormalization method in p-adic λ-model on the Cayley tree, Int. J. Theor. Phys., 54 (2015) , 3577-3595. 
      F. Mukhamedov and H. Akin, Phase transitions for p-adic Potts model on the Cayley tree of order three, J. Stat. Mech. (2013), P07014, 30pp.
      F. Mukhamedov  and  O. Khakimov , Phase transition and chaos: $P$-adic Potts model on a Cayley tree, Chaos, Solitons & Fractals, 87 (2016) , 190-196.  doi: 10.1016/j.chaos.2016.04.003.
      F. Mukhamedov and O. Khakimov, On generalized self-similarity in p-adic field Fractals 24(2016), 1650041, 11pp. doi: 10.1142/S0218348X16500419.
      F. Mukhamedov and O. Khakimov, On Julia set and chaos in p-adic Ising model on the Cayley tree, (submitted).
      F. Mukhamedov  and  M. Saburov , On equation xq=a over ${\mathbb{Q}}_p$, J. Number Theor., 133 (2013) , 55-58. 
      F. M. Mukhamedov  and  U. A. Rozikov , On rational p-adic dynamical systems, Methods of Funct. Anal. and Topology, 10 (2004) , 21-31. 
      F. M. Mukhamedov  and  U. A. Rozikov , On Gibbs measures of p-adic Potts model on the Cayley tree, Indag. Math. N.S., 15 (2004) , 85-99.  doi: 10.1016/S0019-3577(04)90007-9.
      W. Y. Qiu , Y. F. Wang , J. H. Yang  and  Y. C. Yin , On metric properties of limiting sets of contractive analytic non-Archimedean dynamical systems, J. Math. Anal. App., 414 (2014) , 386-401.  doi: 10.1016/j.jmaa.2014.01.015.
      J. Rivera-Letelier , Dynamics of rational functions over local fields, Astérisque, 287 (2003) , 147-230. 
      U. A. Rozikov  and  O. N. Khakimov , Description of all translation-invariant $p$-dic Gibbs measures for the Potts model on a Cayley tree, Markov Proces. Rel. Fields, 21 (2015) , 177-204. 
      M. Saburov and M. A. Kh. Ahmad, On descriptions of all translation invariant p-adic Gibbs measures for the Potts model on the Cayley tree of order three Math. Phys. Anal. Geom. 18(2015), Art. 26, 33 pp. doi: 10.1007/s11040-015-9194-5.
      M. Saburov and M. A. Kh. Ahmad, The dynamics of the potts-bethe mapping over $\mathbb Q_p$: The case $p\equiv 2 (mod 3)$, J. Phys. : Conf. Ser. 819(2017), 012017.
      J. H. Silverman, The Arithmetic of Dynamical Systems New York, Springer, 2007. doi: 10.1007/978-0-387-69904-2.
      E. Thiran , D. Verstegen  and  J. Weters , p-adic dynamics, J. Stat. Phys., 54 (1989) , 893-913.  doi: 10.1007/BF01019780.
      V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-adic Analysis and Mathematical Physics World Scientific, Singapour, 1994. doi: 10.1142/1581.
      C. F. Woodcock  and  N. P. Smart , p-adic chaos and random number generation, Experiment Math., 7 (1998) , 333-342.  doi: 10.1080/10586458.1998.10504379.
      A. Yu. Khrennikov , p-adic valued probability measures, Indag. Mathem. N.S., 7 (1996) , 311-330.  doi: 10.1016/0019-3577(96)83723-2.
      A. Yu. Khrennikov, p-adic description of chaos, In: Nonlinear Physics: Theory and Experiment. Editors E. Alfinito, M. Boti. , World Scientific, Singapore, (1996), 177–184.
      A. Yu. Khrennikov and M. Nilsson, P-Adic Deterministic and Random Dynamical Systems Kluwer, Dordreht, 2004. doi: 10.1007/978-1-4020-2660-7.
  • 加载中

Article Metrics

HTML views(331) PDF downloads(169) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint