Description of weakly periodic Gibbs measures for the Ising model on a Cayley tree

We introduce the concept of a weakly periodic Gibbs measure. For the Ising model, we describe a set of such measures corresponding to normal subgroups of indices two and four in the group representation of a Cayley tree. In particular, we prove that for a Cayley tree of order four, there exist critical values T-c < T-cr of the temperature T > 0 such that there exist five weakly periodic Gibbs measures for 0 < T < T-c or T > T-cr , three weakly periodic Gibbs measures for T = T-c , and one weakly periodic Gibbs measure for T-c < T <= T-cr .