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

TL;DR

This paper analyzes the chaotic dynamics of the $p$-adic Potts-Bethe mapping when $q$ is divisible by $p$, using Markov partitions to fully describe its behavior and establishing the existence of Julia sets, a result not known in real cases.

Contribution

It extends previous work by characterizing the $p$-adic Potts-Bethe mapping's dynamics for divisible $q$, including the existence of Julia sets and chaos, using Markov partitions.

Findings

Full description of dynamical behavior via Markov partition

Existence of Julia set indicating chaos

Chaotic behavior not previously established in real case

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 \cite{MFKh18} 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).