Weakly periodic p-adic generalized Gibbs measures for the one model on a Cayley tree

In this paper, we examine the [Formula: see text]-adic Ising model with an external field on a Cayley tree. Specifically, we consider weakly periodic [Formula: see text]-adic generalized Gibbs measures, which generalize the concept of periodic Gibbs measures. In particular, we investigate weakly periodic Gibbs measures associated with normal subgroups of the group representation of a Cayley tree. We characterize weakly periodic [Formula: see text]-adic generalized Gibbs measures corresponding to normal divisors of index two for the considered model on a Cayley tree of order two, where the prime [Formula: see text]. Our results demonstrate that if [Formula: see text], there exists five unbounded weakly periodic [Formula: see text]-adic generalized Gibbs measures. Conversely, if [Formula: see text], there exists three unbounded weakly periodic [Formula: see text]-adic generalized Gibbs measures (GGMs). This confirms the occurrence of both a phase transition and a quasi-phase transition in the model for all [Formula: see text]. Furthermore, we analyze the dynamical system associated with these weakly periodic measures on [Formula: see text]. We show that the dynamical system possesses three attractive and two repelling fixed points when [Formula: see text], whereas it has three attractive fixed points when [Formula: see text].