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

TL;DR

This paper studies $b$-bistochastic quadratic stochastic operators, analyzing their fixed points, contraction properties, and associated Markov measures, revealing conditions for uniqueness, mixing, and absolute continuity.

Contribution

It provides a detailed analysis of fixed point uniqueness, contraction conditions, and Markov measure properties for $b$-bistochastic quadratic stochastic operators.

Findings

Fixed points are unique under certain conditions.

Strict contraction does not always imply fixed point uniqueness.

Associated Markov measures exhibit mixing and absolute continuity properties.

Abstract

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.