Qualitative Theory of Dynamical Systems

, Volume 18, Issue 3, pp 1225–1235 | Cite as

On Non-ergodic Volterra Cubic Stochastic Operators

  • Farrukh MukhamedovEmail author
  • Chin Hee Pah
  • Azizi Rosli


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.


Cubic stochastic operator Volterra operator Non-ergodic Dynamics 



The first author (FM) thanks the research grant by the UAEU, No. 31S259.

Compliance with ethical standard

Conflict of interest

The authors declare that they have no conflict of interest.


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Authors and Affiliations

  1. 1.Department of Mathematical Sciences, College of ScienceThe United Arab Emirates UniversityAl Ain, Abu DhabiUAE
  2. 2.Department of Computational and Theoretical Sciences, Faculty of ScienceIIUMKuantanMalaysia

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