On infinite dimensional orthogonal preserving and subjective quadratic stochastic operators

Abstract

In this thesis we consider a class of quadratic stochastic operators (QSOs) namely, orthogonal preserving QSOs (OP QSOs). First chapter is devoted to the literature review, problem statements, objectives, preliminaries and overviews of the whole thesis. We recall some achievements on permutation Volterra QSOs, OP QSOs and surjective QSOs defined over a finite dimensional simplex (i.e., the set of all probability distributions on a finite subset of natural numbers) in the second chapter. Indeed, permutation Volterra QSOs, OP QSOs and surjective QSOs are equivalent classes. Since finite dimensional Volterra QSOs were well-studied, therefore we continue the investigation over infinite dimensional simplex in Chapter 3. We provide a concrete form for any OP QSOs could take. Several examples are given and some properties of such mapping are described. Note that, every Volterra QSO is an OP QSO. Due to the complexness to study the dynamics for the whole set of OP QSOs on infinite dimensional simplex, so we restrict ourselves to some classes of Volterra QSOs. In particular, Chapter 4 is focusing on the study of ω−limit set (i.e., the set contains all the limiting points) of some classes of Volterra QSOs. The technique of Lypunov functions is employed here to estimate such set. The significant difference between finite and infinite case is that the ω−limit set (wrt ℓ1−norm) could be empty. Moreover, we show that there is a class of Volterra QSOs which satisfies weak ergodic but fail to be ergodic (see Defintion 4.2.4). In Chapter 5, we investigate surjectivity of infinite dimensional simplex. Unlike finite case, OP QSOs and surjective QSOs are different classes. Thus, we provide necessary and sufficient conditions for infinite dimensional surjective QSOs. Moreover, if one takes a QSO which is OP and surjective, then the operator must be a permutation Volterra.