On $(k_0)$-translation-invariant and $(k_0)$-periodic Gibbs measures for Potts model on Cayley tree

As a rule, the solving of problem arising while studying the thermodynamical properties of physical and biological system is made in the framework of the theory of Gibbs measure. The Gibbs measure is a fundamental notion defining the probability of a microscopic state of a given physical system defined by a given Hamiltonian. It is known that to each Gibbs measure one phase of a physical system is associated to, and if this Gibbs measure is not unique then one says that a phase transition is present. In view of this the study of the Gibbs measure is of a special interest. In this paper we study $(k_0)$-translation-invariant $(k_0)$-periodic Gibbs measures for the Potts model on the Cayley tree. Such measures are constructed by means of translation-invariant and periodic Gibbs measures. For the ferromagnetic Potts model, in the case $k_0=3$ we prove the existence of $(k_0)$-translation-invariant, that is, $(3)$-translation-invariant Gibbs measures. For antiferromagnetic Potts model and also in the case $k_0=3$ we prove the existence of $(k_0)$-periodic ($(3)$-periodic) Gibbs measures on the Cayley tree.