The existence of weakly periodic Gibbs measures for the Potts model on a Cayley tree

We study the $q$-state Potts model on a Cayley tree of order $k\ge2$. In the group representation of the Cayley tree for the ferromagnetic Potts model, we single out a set of index-$2$ subgroups under which each weakly periodic Gibbs measure is translation invariant. For the anti-ferromagnetic Potts model with $k\ge2$ and $q\ge 2$, we show that a weakly periodic Gibbs measure that is not translation invariant is not unique.