Let Sm - 1 be the simplex in Rm, and V: Sm - 1→ Sm - 1 be a nonlinear mapping then this operator satisfies an ergodic theorem if the limit limn→∞1n∑k=1nVk(x)exists for every x∈ Sm - 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.
- Cubic stochastic operator
- Volterra operator
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Applied Mathematics