The dynamics of doubly stochastic operators on finite dimensional simplex

TL;DR

This paper investigates the behavior of trajectories under doubly stochastic operators on finite-dimensional simplexes, showing that most trajectories tend to the simplex's center, with some exceptions.

Contribution

It introduces the concept of doubly stochastic operators on finite simplexes and analyzes their long-term trajectory behavior, highlighting convergence properties.

Findings

Trajectories generally tend to the center of the simplex.

Certain points are exceptions to the convergence.

The paper characterizes the limit behavior of these trajectories.

Abstract

We define a doubly stochastic operator on a finite dimensional simplex and study the limit behavior of the trajectories under doubly stochastic operators. We prove that except for certain points, the trajectory of a point, under the doubly stochastic operator, tends to the center of the simplex.