The -adic Ising model in an external field on a Cayley tree: periodic Gibbs measures

We consider the generalized Gibbs measures corresponding to the -adic Ising model in an external field on the Cayley tree of order two. It is established that if , then there exist three translation-invariant and two -periodic non-translation-invariant -adic generalized Gibbs measures. It becomes clear that if , , then one can find only one translation-invariant -adic generalized Gibbs measure. Moreover, the considered model also exhibits chaotic behavior if and . It turns out that even without , one could establish the existence of -periodic renormalization-group solutions when . This allows us to show the existence of a phase transition.