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

Abstract

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).