TY - JOUR

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

AU - Khakimov, Otabek

AU - Mukhamedov, Farrukh

N1 - Funding Information:
The present work is supported by the UAEU UPAR Grant No. G00003247 (Fund No. 31S391). We are grateful to an anonymous referee whose useful suggestions and comments allowed us to improve the presentation of this paper.
Publisher Copyright:
© The Author(s), 2021. Published by Cambridge University Press.

PY - 2021

Y1 - 2021

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 κp symbols (here κp 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 κp symbols (here κp 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 - chaos

KW - p-adic numbers

KW - Potts-Bethe mapping

KW - shift

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U2 - 10.1017/etds.2021.96

DO - 10.1017/etds.2021.96

M3 - Article

AN - SCOPUS:85116638314

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

ER -