Periodic Gibbs measures for the Potts model in translation-invariant and periodic external fields on the Cayley tree

We study the Potts model in translation-invariant and periodic external fields on the Cayley tree of order . For the Potts model in a translation-invariant external field for , the nonuniqueness of the translation-invariant and periodic Gibbs measure is shown. It is proved that for the Potts model in an external field that is not translation-invariant, translation-invariant Gibbs measures do not exist on the Cayley tree of order . Periodic Gibbs measures are also studied for the Potts model in a periodic external field. We prove that under certain conditions, the number of such measures can be at least three.