Weakly periodic Gibbs measures for two and three state HC models on a Cayley tree

In this paper we study two and three state HC-model on a Cayley tree. Under some conditions on parameters of the HC-model, in the case of normal divisor of the index four, we prove the existence of the weakly periodic (non periodic) Gibbs measures which are different from the known weakly periodic measures. Moreover, we find some regions for the $\lambda$ parameter ensuring that a extreme measure is not unique.