On chaotic behaviour of the <i>p</i>-adic generalized Ising mapping and its application

Mukhamedov, Farrukh; Akın, Hasan; Dogan, Mutlay

doi:10.1080/10236198.2017.1340468

Cited by 4 publications

(2 citation statements)

References 44 publications

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“…We stress that the Potts–Ising mapping is a particular case of the Potts–Bethe mapping, which can be obtained from (1.1) by putting . Recently, in [30, 34] under some condition, a Julia set of the Potts–Ising mapping was described, and it was shown that restricted to its Julia set, the Potts–Ising mapping is conjugate to a full shift. Therefore, it is natural to consider the Potts–Bethe mapping for with and .…”

Section: Introductionmentioning

confidence: 99%

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%

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

Khakimov¹,

Mukhamedov²

2021

Ergod. Th. Dynam. Sys.

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“…We stress that the Potts-Ising mapping is a particular case of the Potts-Bethe mapping, which can be obtained from (1.1) by putting q = 2. Recently, in [17,21] under some condition, a Julia set of the Potts-Ising mapping have been described, and it was shown this mapping is conjugate to the full shift. Therefore, it is natural to consider the the Potts-Bethe mapping for q ≥ 3 with |q| p < 1 and k ≥ 2.…”

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 Dynamical Behavior of the p-adic λ-Ising Model on Cayley Tree

Dogan¹

2019

Z. mat. fiz. anal. geom.

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On chaotic behaviour of the p-adic generalized Ising mapping and its application

Cited by 4 publications

References 44 publications

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

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

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

On Dynamical Behavior of the p-adic λ-Ising Model on Cayley Tree

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