On stable b-bistochastic quadratic stochastic operators and associated non-homogenous Markov chains

Abstract

In the present paper, we consider a class of quadratic stochastic operators (q.s.o.) called b-bistochastic q.s.o. defined on a finite dimensional simplex. We include several properties of b-bistochastic q.s.o. and their dynamical behaviour. One of the main findings in this paper is the description on the uniqueness of the fixed points. Besides, we list the conditions on strict contractive b-bistochastic q.s.o. on low-dimensional simplices, and it turns out that, the uniqueness of the fixed point does not imply its strict contractivity. Finally, we associate non-homogeneous Markov measures with b-bistochastic q.s.o. The defined measures were proven to satisfy the mixing property for regular b-bistochastic q.s.o. Moreover, we show that non-homogeneous Markov measures associated with a class of b-bistochastic q.s.o on one-dimensional simplex, meet the absolute continuity property.