Dynamics of linear maps of idempotent measures

TL;DR

This paper characterizes all linear operators acting on the simplex of idempotent measures, classifies them into two types, and analyzes their fixed points and dynamical systems behavior.

Contribution

It provides a complete description of linear maps on idempotent measures, including fixed points and dynamics, distinguishing two main classes of matrices.

Findings

Identified two classes of linear operators on idempotent measures.

Derived fixed points for each class of operators.

Analyzed the dynamical systems generated by these operators.

Abstract

We describe all linear operators which maps $n-1$-dimensional simplex of idempotent measures to itself. Such operators divided to two classes: the first class contains all $n\times n$-matrices with non-negative entries which has at least one zero-row; the second class contains all $n\times n$-matrices with non-negative entries which in each row and in each column has exactly one non-zero entry. These matrices play a role of the stochastic matrices in case of idempotent measures. For both classes of linear maps we find fixed points. We also study the dynamical systems generated by the linear maps of the set of idempotent measures.