TY - JOUR

T1 - Chaotic behavior of the p-adic Potts-Bethe mapping II

AU - Khakimov, Otabek

AU - Mukhamedov, Farrukh

N1 - Publisher Copyright:
© The Author(s), 2021. Published by Cambridge University Press.

PY - 2022/11/30

Y1 - 2022/11/30

N2 - 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 Kp symbols (here Kp 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).

AB - 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 Kp symbols (here Kp 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).

KW - Potts-Bethe mapping

KW - chaos

KW - p-adic numbers

KW - shift

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U2 - 10.1017/etds.2021.96

DO - 10.1017/etds.2021.96

M3 - Article

AN - SCOPUS:85116638314

SN - 0143-3857

VL - 42

SP - 3433

EP - 3457

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 11

ER -