On Non-ergodic Volterra Cubic Stochastic Operators

• Autores: Farrukh Mukhamedov, Hee Pah Chin, Azizi Rosli
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 18, Nº 3, 2019, págs. 1225-1235
• Idioma: inglés
• DOI: 10.1007/s12346-019-00334-8
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• Resumen

  • 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.