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Periodic Points of a \(p\)-Adic Operator and their \(p\)-Adic Gibbs Measures

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Abstract

In this paper we investigate generalized Gibbs measure (GGM) for \(p\)-adic Hard-Core (HC) model with a countable set of spin values on a Cayley tree of order \(k\geq 2\). This model is defined by \(p\)-adic parameters \(\lambda_i\), \(i\in \mathbb N\). We analyze \(p\)-adic functional equation which provides the consistency condition for the finite-dimensional generalized Gibbs distributions. Each solutions of the functional equation defines a GGM by \(p\)-adic version of Kolmogorov’s theorem. We define \(p\)-adic Gibbs distributions as limit of the consistent family of finite-dimensional generalized Gibbs distributions and show that, for our \(p\)-adic HC model on a Cayley tree, such a Gibbs distribution does not exist. Under some conditions on parameters \(p\), \(k\) and \(\lambda_i\) we find the number of translation-invariant and two-periodic GGMs for the \(p\)-adic HC model on the Cayley tree of order two.

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Notes

  1. https://en.wikipedia.org/wiki/Archimedean\(_-\)property

References

  1. V. Anashin and A. Khrennikov, Applied Algebraic Dynamics, de Gruyter Expositions in Mathematics 49 (Walter de Gruyter, Berlin-New York, 2009).

    Book  MATH  Google Scholar 

  2. L. V. Bogachev and U. A. Rozikov, “On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external field,” J. Stat. Mech.: Theory and Exp. 073205, 76 pp. (2019).

    Article  MathSciNet  MATH  Google Scholar 

  3. G. Brightwell and P. Winkler, “Graph homomorphisms and phase transitions,” J. Comb. Theory Ser. B. 77, 221–262 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Gandolfo, U. A. Rozikov and J. Ruiz, “On \(p\)-adic Gibbs measures for hard core model on a Cayley tree,” Markov Proc. Rel. Fields 18 (4), 701–720 (2012).

    MathSciNet  MATH  Google Scholar 

  5. N. N. Ganikhodzhaev, F. M. Mukhamedov and U. A. Rozikov, “Phase transitions of the Ising model on \(Z\) in the \(p\)-adic number field,” Uzbek. Mat. J. 4, 23–29 (1998).

    MathSciNet  Google Scholar 

  6. S. Katok, \(p\)-Adic Analysis Compared with Real, Student Mathematical Library 37 (AMS, Providence, RI, USA, 2007).

    MATH  Google Scholar 

  7. R. M. Khakimov, M. T. Makhammadaliev and U. A. Rozikov, “Gibbs measures for HC-model with a countable set of spin values on a Cayley tree,” [arXiv:2205.02025] (2022).

  8. A. Yu. Khrennikov, F. M. Mukhamedov and J. F. F. Mendes, “On \(p\)-adic Gibbs measures of countable state Potts model on the Cayley tree,” Nonlinearity 20, 2923–2937 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Yu. Khrennikov, K. Oleschko and M. de Jesús Correa López, “Applications of \(p\)-adic numbers: from physics to geology,” in Advances in Non-Archimedean Analysis, Contemp. Math. 665, 121–131 (AMS, Providence, RI, 2016).

    Google Scholar 

  10. A. Yu. Khrennikov, “Probability distributions on the field of \(p\)-adic numbers,” Theory Prob. Appl. 40 (1), 159–162 (1995).

    Article  MathSciNet  Google Scholar 

  11. H. K. Rosen, Elementary Number Theory and Its Applications (Addison-Westley, Canada, 1986).

    Google Scholar 

  12. N. Koblitz, \(p\)-Adic Numbers, \(p\)-Adic Analysis and Zeta-Function (Springer, Berlin, 1977).

    Book  MATH  Google Scholar 

  13. A. Le Ny, L. Liao and U. A. Rozikov, “\(p\)-Adic boundary laws and Markov chains on trees,” Lett. Math. Phys. 110 (10), 2725–2741 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  14. F. Mukhamedov, B. Omirov and M. Saburov, “On cubic equations over \(p\)-adic fields,” Int. J. Numb. Theory 10, 1171–1190 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  15. F. Mukhamedov and O. Khakimov, “On equation \(x^k=a\) over \(Q_p\) and its applications,” Izvest. Math. 84, 348–360 (2020).

    Article  MATH  Google Scholar 

  16. F. Mukhamedov and O. Khakimov, “Translation-invariant generalized \(p\)-adic Gibbs measures for the Ising model on Cayley trees,” Math. Meth. Appl. Sci. 44 (16), 12302–12316 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  17. F. Mukhamedov, O. Khakimov and A. Souissi, “A few remarks on supercyclicity of non-Archimedean linear operators on \(c_0(N)\),” \(p\)-Adic Num. Ultrametr. Anal. Appl. 14 (1), 63–75 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  18. F. Mukhamedov, “On existence of generalized Gibbs measures for one dimentional \(p\)-adic countable state Potts model,” Proc. Steklov Inst. Math. 265, 165–176 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  19. F. Mukhamedov and O. Khakimov, “Hypercyclic and supercyclic linear operators on non-Archimedean vector spaces,” Bull. Belg. Math. 25 (1), 85–105 (2018).

    MathSciNet  MATH  Google Scholar 

  20. F. Mukhamedov, M. Saburov and O. Khakimov, “On \(p\)-adic Ising-Vannimenus model on an arbitrary order Cayley tree,” J. Stat. Mech. P05032 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  21. F. Mukhamedov, “On the strong phase transition for the one-dimensional countable state \(p\)-adic Potts model,” J. Stat. Mech. P01007 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  22. U. A. Rozikov, Gibbs Measures on a Cayley Tree (World Sci. Pub., Singapore, 2013).

    Book  MATH  Google Scholar 

  23. U. A. Rozikov, “Structures of partitions of the group representation of the Cayley tree into cosets by finite-index normal subgroups, and their applications to the description of periodic Gibbs distributions,” Theor. Math. Phys. 112 (1), 929–933 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  24. U. A. Rozikov and F. H. Haydarov, “A HC model with countable set of spin values: uncountable set of Gibbs measures,” Rev. Math. Phys. , https://doi.org/10.1142/S0129055X22500398 (2022).

    Article  MathSciNet  Google Scholar 

  25. U. A. Rozikov, “What are \(p\)-adic numbers? What are they used for?” Asia Pac. Math. Newsl. 3 (4), 1–6 (2013).

    MathSciNet  Google Scholar 

  26. U. A. Rozikov, I. A. Sattarov and S. Yam, “\(p\)-Adic dynamical systems of the function \(\frac{ax}{x^2+a}\),” \(p\)-Adic Num. Ultrametr. Anal. Appl. 11 (1), 77–87 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  27. U. A. Rozikov and I. A. Sattarov, “\(p\)-Adic dynamical systems of \((2,2)\)-rational functions with unique fixed point,” Chaos Solit. Fract. 105, 260–270 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  28. U. A. Rozikov and I. A. Sattarov, “Dynamical systems of the \(p\)-adic \((2,2)\)-rational functions with two fixed points,” Resul. Math. 75 (3), 37 pp. (2020).

    MathSciNet  Google Scholar 

  29. U. A. Rozikov, “Construction of an uncountable number of limit Gibbs measures in the inhomogeneous Ising model,” Theor. Math. Phys. 118 (1), 77–84 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  30. M. Saburov and Mohd Ali K. Ahmad, “Solvability of cubic equations over \(\mathbb Q_3\),” Sains Malays. 44 (4), 635–641 (2015).

    Article  MATH  Google Scholar 

  31. V. S. Vladimirov, I. V. Volovich and E.I. Zelenov, \(p\)-Adic Analysis and Mathematical Physics (World Sci., Singapoure, 1994).

    Book  MATH  Google Scholar 

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Acknowledgments

We thank the referee for very helpful and useful comments.

Funding

The work is supported by the fundamental project (number: F-FA-2021-425) of The Ministry of Innovative Development of the Republic of Uzbekistan.

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Authors and Affiliations

  1. Institute of Mathematics, 9, University str., Tashkent, 100174, Uzbekistan

    U. A. Rozikov & I. A. Sattarov

  2. AKFA University, 264, Milliy Bog street, Yangiobod QFY, Kibray District, Tashkent Region, 111221, Uzbekistan

    U. A. Rozikov

  3. National University of Uzbekistan, University str 4, Olmazor District, Tashkent, 100174, Uzbekistan

    U. A. Rozikov

  4. Namangan State University, 316, Uychi str., Namangan, 160100, Uzbekistan

    I. A. Sattarov & A. M. Tukhtabaev

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  1. U. A. Rozikov
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  2. I. A. Sattarov
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  3. A. M. Tukhtabaev
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Correspondence to U. A. Rozikov, I. A. Sattarov or A. M. Tukhtabaev.

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Rozikov, U.A., Sattarov, I.A. & Tukhtabaev, A.M. Periodic Points of a \(p\)-Adic Operator and their \(p\)-Adic Gibbs Measures. P-Adic Num Ultrametr Anal Appl 14 (Suppl 1), S30–S44 (2022). https://doi.org/10.1134/S207004662205003X

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