Uniqueness of fixed points of b-bistochastic quadratic stochastic operators and associated nonhomogenous Markov chains

In the present paper, we consider a class of quadratic stochastic operators (q.s.o.) called $ b- $bistochastic q.s.o. We include several properties of $ b- $bistochastic q.s.o. and their dynamical behavior. 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 strict contraction. Finally, we associated Markov measures with $b$-bistochastic q.s.o. On a class of $b$-bistochastic q.s.o. on finite dimensional simplex, the defined measures were proven to satisfy the mixing property. Moreover, we show that Markov measures associated with a class of $ b- $bistochastic q.s.o on one dimensional simplex meets the absolute continuity property.