On non-ergodic volterra cubic stochastic operators

• Main Authors: Mukhamedov, Farrukh, Pah, Chin Hee, Rosli, Azizi
• Format: Article
• Language: English; English; English
• Published: Springer 2019
• Subjects: QA Mathematics
• Online Access: http://irep.iium.edu.my/73972/1/73972_On%20Non-ergodic%20Volterra%20Cubic%20Stochastic_article.pdf; http://irep.iium.edu.my/73972/2/73972_On%20Non-ergodic%20Volterra%20Cubic%20Stochastic_scopus.pdf; http://irep.iium.edu.my/73972/3/73972_On%20Non-ergodic%20Volterra%20Cubic%20Stochastic_wos.pdf; http://irep.iium.edu.my/73972/; https://link.springer.com/article/10.1007/s12346-019-00334-8
• Institution: Universiti Islam Antarabangsa Malaysia

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.