On the chaotic Thompson metric of matrices

Abstract

In this article, we introduce and study properties of the Thompson metric induced by the chaotic order on the cone of positive definite matrices. Furthermore, we introduce an approach to characterize geodesic curves associated to Thompson metrics and study in-betweenness type results for the metric induced by the chaotic order. Among other results, we obtain that the map A↦d≪(A,Qα,z(B,A));z>0, satisfies Audenaert’s in-betweenness for 0≤α≤1.