Limit behavior of dynamic systems corresponding to Lattice models with competing prolonged and one-level binary interactions

We study the phase diagram of the Ising model on a Cayley tree with competing prolonged next-nearest neighbour Jpand one-level next-nearest neighbour interactions J. Vannimenus proved that the phase diagram of Ising model with competing neareast-neighbour interaction J1and prolonged next-nearest neighbour interactions Jpcontains a modulated phase, as found for similar models on periodic lattices. Later Mariz et al generalized this result for Ising model with J ≠ 0. For given lattice model on a Cayley tree, i.e., Jp≠ 0; J ≠ 0 with J1= 0 we describe phase diagram and clarify the role of nearest-neighbour interaction J1and show that the class of modulated phases consists of so-called antiphase with period 4 only