The Dynamics of The Potts-Bethe Mapping over ${\mathbb{Q}}_p:$ The Case $p\equiv 2(mod3)$

In this paper, we study the dynamics of the Potts-Bethe mapping associated with the $p-$adic $q-$state Potts model over the Cayley tree of order three. Namely, we establish the regularity of the Potts-Bethe mapping for the case $p\equiv 2(mod3)$ with $p\ge 5$.