Semi-simple isometries of the manifold of symmetric positive definite real matrices with the trace metric

We study the semi-simple (or non-parabolic) isometries /Phi1of the manifold Pn of symmetric positive definite real matrices, endowed with the trace metric g and pay attention to the set Min (/Phi1)consisting of the points minimizing the displacement function P ↦→ d(P,/Phi1(P)). In particular, for every isometry/Phi1of (Pn, g),w eg i v et h e explicit expression and study the differential-geometric structure of the set Min (/Phi1).