On periodic Gibbs measures of p-adic Potts model on a Cayley tree

Mukhamedov, Farrukh; Khakimov, Otabek

doi:10.1134/s2070046616030043

Cited by 18 publications

(9 citation statements)

References 37 publications

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“…In [43], all fixed points of were found when and . Then, using these fixed points, the dynamics of (1.1) whenever and was investigated in [11, 31, 32]. Recently in [1, 44], the Potts–Bethe mapping was studied for the case and .…”

Section: Introductionmentioning

confidence: 99%

“…In [43], all fixed points of f θ ,q,k were found when k = 2 and |q| p < 1. Then, using these fixed points, the dynamics of (1.1) whenever k = 2 and |q| p < 1 was investigated in [11,31,32].…”

mentioning

confidence: 99%

“…Hence, to establish the existence of the phase transition, when k = 2, we showed in [41] that (A.9) has three non-trivial solutions if q is divisible by p. Note that the full description of all solutions of the last equation has been carried out in [43] when k = 2. Certain periodic points of f θ ,q,k have been carried out in [1,28,31].…”

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confidence: 99%

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Chaotic behavior of the p-adic Potts–Bethe mapping II

Khakimov¹,

Mukhamedov²

2021

Ergod. Th. Dynam. Sys.

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The renormalization group method has been developed to investigate p-adic q-state Potts models on the Cayley tree of order k. This method is closely related to the examination of dynamical behavior of the p-adic Potts–Bethe mapping which depends on the parameters q, k. In Mukhamedov and Khakimov [Chaotic behavior of the p-adic Potts–Behte mapping. Discrete Contin. Dyn. Syst.38 (2018), 231–245], we have considered the case when q is not divisible by p and, under some conditions, it was established that the mapping is conjugate to the full shift on $\kappa _p$ symbols (here $\kappa _p$ is the greatest common factor of k and $p-1$ ). The present paper is a continuation of the forementioned paper, but here we investigate the case when q is divisible by p and k is arbitrary. We are able to fully describe the dynamical behavior of the p-adic Potts–Bethe mapping by means of a Markov partition. Moreover, the existence of a Julia set is established, over which the mapping exhibits a chaotic behavior. We point out that a similar result is not known in the case of real numbers (with rigorous proofs).

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Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Chaotic behavior of the p-adic Potts–Bethe mapping II

Khakimov¹,

Mukhamedov²

2021

Ergod. Th. Dynam. Sys.

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“…Therefore, it is natural to study the dynamical systems generated by these functions in the field of p-adic numbers. Moreover, these padic dynamical systems appear in the study of non Archimedean models of physics and biology (see for example [1]- [5], [7]).…”

Section: Introductionmentioning

confidence: 99%

p-Adic (3; 2)-rational dynamical systems with three fixed points

Sattarov¹

2019

UzMJ

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In this paper we consider dynamical systems generated by (3, 2)-rational functions on the field of p-adic complex numbers. Each such function has three fixed points. We show that Siegel disks of the dynamical system may either coincide or be disjoint for different fixed points. Also, we find the basin of each attractor of the dynamical system. We show that, for some values of the parameters, there are trajectories which go arbitrary far from the fixed points.

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“…In [28] all fixed points of f θ,q,k have been found when k = 2 and |q| p < 1. Then, using these fixed points in [18,19] the dynamics of (1.1) whenever k = 2 and |q| p < 1 has been investigated. Recently in [1,29] the Potts-Bethe mapping has been studied at k = 3 with |q| p < 1.…”

Section: Introductionmentioning

confidence: 99%

Chaotic behavior of the $p$-adic Potts-Bethe mapping II

Mukhamedov¹,

Khakimov²

2019

Preprint

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In our previous investigations, we have developed the renormalization group method to p-adic q-state Potts model on the Cayley tree of order k. This method is closely related to the examination of dynamical behavior of the p-adic Potts-Bethe mapping which depends on parameters q, k. In [22] we have considered the case when q is not divisible by p, and under some conditions it was established that the mapping is conjugate to the full shift. The present paper is a continuation of the mentioned paper, but here we investigate the case when q is divisible by p and k is arbitrary. We are able to fully describe the dynamical behavior of the p-adic Potts-Bethe mapping by means of Markov partition. Moreover, the existence of Julia set is established, over which the mapping enables a chaotic behavior. We point out that a similar result is not known in the case of real numbers (with rigorous proofs).

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On periodic Gibbs measures of p-adic Potts model on a Cayley tree

Cited by 18 publications

References 37 publications

Chaotic behavior of the p-adic Potts–Bethe mapping II

Chaotic behavior of the p-adic Potts–Bethe mapping II

p-Adic (3; 2)-rational dynamical systems with three fixed points

Chaotic behavior of the $p$-adic Potts-Bethe mapping II

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