Prevalence problem in the set of quadratic stochastic operators acting on L1

TL;DR

This paper investigates the prevalence of quadratic stochastic operators on L1 space, showing that norm quasi-mixing operators are dense and open, indicating typical long-term dynamics.

Contribution

It establishes that the set of norm quasi-mixing quadratic stochastic operators is dense and open in the natural metric topology on L1.

Findings

Norm quasi-mixing operators form a dense and open set

Typical long-term behavior of quadratic stochastic operators

Provides insight into stability and dynamics in L1 space

Abstract

This paper is devoted to the study of the problem of prevalence in the class of quadratic stochastic operators acting on the L1 space for the uniform topology. We obtain that the set of norm quasi-mixing quadratic stochastic operators is a dense and open set in the topology induced by a very natural metric. This shows the typical long-term behaviour of iterates of quadratic stochastic operators.