One of the main problems of statistical physics is that of describing all Gibbs measures corresponding to a given Hamiltonian. It is well known that such measures form a nonempty convex compact subset in the set of all probability measures. The purpose of this paper is to investigate extreme Gibbs measures of the Vannimenus model.

Different types of the lattice spin systems with the competing interactions have rich and interesting phase diagrams. In this study a system with competing nearest-neighbor interaction J1, prolonged next-nearest-neighbor interaction Jp and ternary prolonged interaction Jtp is considered on a Cayley tree of arbitrary order k. To perform this study, an iterative scheme is developed for the corresponding Hamiltonian model. At finite temperatures several interesting properties are presented for typical values of α = T/J1, β = −Jp/J1 and γ = -Jtp/J1. This study recovers as particular cases, previous work by Vannimenus1 with γ = 0 for k = 2 and Ganikhodjaev et al.2 in the presence J1, Jp, Jtp with k = 2. The variation of the wavevector q with temperature in the modulated phase and the Lyapunov exponent associated with the trajectory of our iterative system are studied in detail.

In this paper we consider the set of all extremal Volterra quadratic stochastic operators defined on a unit simplex $S^{4}$ and show that such operators can be reinterpreted in terms of zero-sum games. We show that an extremal Volterra operator is non-ergodic and an appropriate zero-sum game is a rock-paper-scissors game if either the Volterra operator is a uniform operator or for a non-uniform Volterra operator $V$ there exists a subset $I\subset \{1,2,3,4,5\}$ with $|I|\leq 2$ such that $\sum _{i\in I}(V^{n}\mathbf{x})_{i}\rightarrow 0,$ and the restriction of $V$ on an invariant face ${\rm\Gamma}_{I}=\{\mathbf{x}\in S^{m-1}:x_{i}=0,i\in I\}$ is a uniform Volterra operator.

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