January  2018, 38(1): 231-245. doi: 10.3934/dcds.2018011

Chaotic behavior of the P-adic Potts-Bethe mapping

1. 

Department of Mathematical Sciences, College of Science, The United Arab Emirates University, P.O. Box, 15551, Al Ain Abu Dhabi, UAE

2. 

Institute of Mathematics of Academy of Science of Uzbekistan, 29, Do'rmon Yo'li str., 100125, Tashkent, Uzbekistan

* Corresponding author: far75m@gmail.com

Received  December 2016 Revised  July 2017 Published  September 2017

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

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).

Citation: Farrukh Mukhamedov, Otabek Khakimov. Chaotic behavior of the P-adic Potts-Bethe mapping. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 231-245. doi: 10.3934/dcds.2018011
References:
[1]

S. AlbeverioU. 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]

N. S. AnanikianS. 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. 

[3]

V. Anashin and A. Khrennikov, Applied Algebraic Dynamics Walter de Gruyter, Berlin, New York, 2009. doi: 10.1515/9783110203011.

[4]

R. Benedetto, Reduction, dynamics, and Julia sets of rational functions, J. Number Theory, 86 (2001), 175-195.  doi: 10.1006/jnth.2000.2577.

[5]

R. Benedetto, Hyperbolic maps in p-adic dynamics, Ergod. Th. & Dynam. Sys., 21 (2001), 1-11.  doi: 10.1017/S0143385701001043.

[6]

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.

[7]

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.

[8]

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.

[9]

A. H. FanL. M. LiaoY. 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.

[10]

A. H. FanS. L. FanL. M. Liao and Y. F. Wang, On minimal deecomposition of p-adic homographic dynamical systems, Adv. Math., 257 (2014), 92-135. 

[11]

A. H. FanS. L. FanL. 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.

[12]

G. Gyorgyi, I. Kondor, L. Sasvari and T. Tel, From Phase Transitions to Chaos World Scientific, Singapore, 1992. doi: 10.1142/1633.

[13]

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.

[14]

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.

[15]

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.

[16]

N. Koblitz, P-adic Numbers, P-adic Analysis and Zeta-function Berlin, Springer, 1977.

[17]

J. Lubin, Nonarchimedean dynamical systems, Composito Math., 94 (1994), 321-346. 

[18]

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. 

[19]

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.

[20]

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.

[21]

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.

[22]

F. Mukhamedov, Renormalization method in p-adic λ-model on the Cayley tree, Int. J. Theor. Phys., 54 (2015), 3577-3595. 

[23]

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.

[24]

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.

[25]

F. Mukhamedov and O. Khakimov, On generalized self-similarity in p-adic field Fractals 24(2016), 1650041, 11pp. doi: 10.1142/S0218348X16500419.

[26]

F. Mukhamedov and O. Khakimov, On Julia set and chaos in p-adic Ising model on the Cayley tree, (submitted).

[27]

F. Mukhamedov and M. Saburov, On equation xq=a over ${\mathbb{Q}}_p$, J. Number Theor., 133 (2013), 55-58. 

[28]

F. M. Mukhamedov and U. A. Rozikov, On rational p-adic dynamical systems, Methods of Funct. Anal. and Topology, 10 (2004), 21-31. 

[29]

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.

[30]

W. Y. QiuY. F. WangJ. 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.

[31]

J. Rivera-Letelier, Dynamics of rational functions over local fields, Astérisque, 287 (2003), 147-230. 

[32]

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. 

[33]

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.

[34]

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.

[35]

J. H. Silverman, The Arithmetic of Dynamical Systems New York, Springer, 2007. doi: 10.1007/978-0-387-69904-2.

[36]

E. ThiranD. Verstegen and J. Weters, p-adic dynamics, J. Stat. Phys., 54 (1989), 893-913.  doi: 10.1007/BF01019780.

[37]

V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-adic Analysis and Mathematical Physics World Scientific, Singapour, 1994. doi: 10.1142/1581.

[38]

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.

[39]

A. Yu. Khrennikov, p-adic valued probability measures, Indag. Mathem. N.S., 7 (1996), 311-330.  doi: 10.1016/0019-3577(96)83723-2.

[40]

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.

[41]

A. Yu. Khrennikov and M. Nilsson, P-Adic Deterministic and Random Dynamical Systems Kluwer, Dordreht, 2004. doi: 10.1007/978-1-4020-2660-7.

show all references

References:
[1]

S. AlbeverioU. 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]

N. S. AnanikianS. 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. 

[3]

V. Anashin and A. Khrennikov, Applied Algebraic Dynamics Walter de Gruyter, Berlin, New York, 2009. doi: 10.1515/9783110203011.

[4]

R. Benedetto, Reduction, dynamics, and Julia sets of rational functions, J. Number Theory, 86 (2001), 175-195.  doi: 10.1006/jnth.2000.2577.

[5]

R. Benedetto, Hyperbolic maps in p-adic dynamics, Ergod. Th. & Dynam. Sys., 21 (2001), 1-11.  doi: 10.1017/S0143385701001043.

[6]

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.

[7]

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.

[8]

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.

[9]

A. H. FanL. M. LiaoY. 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.

[10]

A. H. FanS. L. FanL. M. Liao and Y. F. Wang, On minimal deecomposition of p-adic homographic dynamical systems, Adv. Math., 257 (2014), 92-135. 

[11]

A. H. FanS. L. FanL. 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.

[12]

G. Gyorgyi, I. Kondor, L. Sasvari and T. Tel, From Phase Transitions to Chaos World Scientific, Singapore, 1992. doi: 10.1142/1633.

[13]

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.

[14]

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.

[15]

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.

[16]

N. Koblitz, P-adic Numbers, P-adic Analysis and Zeta-function Berlin, Springer, 1977.

[17]

J. Lubin, Nonarchimedean dynamical systems, Composito Math., 94 (1994), 321-346. 

[18]

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. 

[19]

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.

[20]

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.

[21]

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.

[22]

F. Mukhamedov, Renormalization method in p-adic λ-model on the Cayley tree, Int. J. Theor. Phys., 54 (2015), 3577-3595. 

[23]

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.

[24]

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.

[25]

F. Mukhamedov and O. Khakimov, On generalized self-similarity in p-adic field Fractals 24(2016), 1650041, 11pp. doi: 10.1142/S0218348X16500419.

[26]

F. Mukhamedov and O. Khakimov, On Julia set and chaos in p-adic Ising model on the Cayley tree, (submitted).

[27]

F. Mukhamedov and M. Saburov, On equation xq=a over ${\mathbb{Q}}_p$, J. Number Theor., 133 (2013), 55-58. 

[28]

F. M. Mukhamedov and U. A. Rozikov, On rational p-adic dynamical systems, Methods of Funct. Anal. and Topology, 10 (2004), 21-31. 

[29]

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.

[30]

W. Y. QiuY. F. WangJ. 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.

[31]

J. Rivera-Letelier, Dynamics of rational functions over local fields, Astérisque, 287 (2003), 147-230. 

[32]

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. 

[33]

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.

[34]

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.

[35]

J. H. Silverman, The Arithmetic of Dynamical Systems New York, Springer, 2007. doi: 10.1007/978-0-387-69904-2.

[36]

E. ThiranD. Verstegen and J. Weters, p-adic dynamics, J. Stat. Phys., 54 (1989), 893-913.  doi: 10.1007/BF01019780.

[37]

V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-adic Analysis and Mathematical Physics World Scientific, Singapour, 1994. doi: 10.1142/1581.

[38]

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.

[39]

A. Yu. Khrennikov, p-adic valued probability measures, Indag. Mathem. N.S., 7 (1996), 311-330.  doi: 10.1016/0019-3577(96)83723-2.

[40]

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.

[41]

A. Yu. Khrennikov and M. Nilsson, P-Adic Deterministic and Random Dynamical Systems Kluwer, Dordreht, 2004. doi: 10.1007/978-1-4020-2660-7.

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