  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.
