Chaotic behavior of the <i>p</i>-adic Potts–Bethe mapping II
Abstract
Abstract 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).