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 $\Phi$ of the manifold $\mathcal{P}_n$ of symmetric positive definite real matrices, endowed with the trace metric $g$ and pay attention to the set $Min(\Phi)$ consisting of the points minimizing the displacement function $P \mapsto d(P, \Phi(P))$. In particular, for every isometry $\Phi$ of $(\mathcal{P}_n, g)$, we give the explicit expression and study the differential-geometric structure of the set $Min(\Phi)$.