  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).
