Some Strange Properties of Quadratic Stochastic Volterra Operators

One of the fascinating results in the one dimensional nonlinear dynamical system is that a mapping which maps a compact connected subset of the real line into itself is regular if and only if it does not have any order periodic points except fixed points. However, in general, this result does not hold true in the high dimensional case. In this paper, we provide a counter example for such kind of mappings among quadratic stochastic Volterra operators. Moreover, we showed an equivalence of notions of regularity, transitivity and Ergodic principle for quadratic stochastic Volterra operators acting on the finite dimensional simplex. Apart from these, we study the fixed point set of the composition of two quadratic stochastic Volterra operators.