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
UR - http://www.scopus.com/inward/record.url?scp=85116638314&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85116638314&partnerID=8YFLogxK
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 -