Exact Solution of the Ising Model on the Cayley Tree with Competing Ternary and Binary Interactions

The exact solution is found for the problem of phase transitions in the Ising model with competing ternary and binary interactions. For the pair of parameters $\theta =\theta (J)$ and $\theta _1=\theta _1(J_1)$ in the plane $(\theta _1,\theta )$, we find two critical curves such that a phase transition occurs for all pairs $(\theta _1,\theta )$ lying between the curves.