Invariance Property on Group Representations of the Cayley Tree and Its Applications

In the present paper, we give a certain condition on (not normal) subgroups of the group representation of the Cayley tree such that an invariance property holds. Except for the given condition, we show that the invariance property does not hold. In other words, we open new direction to the theory of periodic and weakly periodic Gibbs measures corresponding to subgroups of the group representation of Cayley trees. We apply our results for the Ising model.