Dynamics of low dimensional orthogonality preserving cubic stochastic operators

Abstract

Cubic stochastic operator (CSO) was first introduced in 2004 by Rozikov and Khamraev. Since then, few studies had been done to study the dynamics of trajectory of some classes of CSOs. In this thesis, we consider the cubic stochastic operator (CSO) defined on 1 and 2-dimensional simplex. We provide a full description of orthogonal preserving (OP) cubic stochastic operators on the 1 and 2-dimensional simplex. We provide full description of the fixed points subject to two different parameters for the Volterra OP CSO on both simplex. In the last part of each case we described the behaviour of the fixed points. A concrete example of a non-ergodic orthogonal preserving (OP) Volterra cubic stochastic operator is given.