On Non-Ergodic Transformations on<i>S</i>³

Abstract

In this paper we consider a set of all extremal Volterra quadratic stochastic operators on three dimensional simplex S3, show that this set is parted into four equivalence classes with respect to group of transformations generated by permutations and describe the behaviour of trajectories extremal Volterra quadratic stochastic operators for each class. It is proved that the operators in some classes are non-ergodic transformation.