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 $T_{\mathrm{c}}<T_{\mathrm{cr}}$ of the temperature $T>0$ such that there exist five weakly periodic Gibbs measures for $0<T<T_{\mathrm{c}}$ or $T>T_{\mathrm{cr}}$, three weakly periodic Gibbs measures for $T=T_{\mathrm{c}}$, and one weakly periodic Gibbs measure for $T_{\mathrm{c}}<T\le T_{\mathrm{cr}}$.