On Non-ergodic Volterra Cubic Stochastic Operators

Abstract

Let $$S^{m-1}$$ be the simplex in $${{\mathbb {R}}}^m$$ , and $$V:S^{m-1}\rightarrow S^{m-1}$$ be a nonlinear mapping then this operator satisfies an ergodic theorem if the limit $$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\sum _{k=1}^n V^k(x) \end{aligned}$$ exists for every $$x\in S^{m-1}$$ . It is a well known fact that this ergodicity may fail for Volterra quadratic operators, so it is natural to characterize all non-ergodic operators. However, there is an ongoing problem even in the low dimensional simplexes. In this paper, we solve the mentioned problem within Volterra cubic stochastic operators acting on two-dimensional simplex.