Hilbert's metric on symmetric cones

Let $\mathrm{\Omega }$ be a symmetric cone. In this note, we introduce the Hilbert projective metric on $\mathrm{\Omega }$ in terms of Jordan algebras and we apply it to prove that given a linear transformation $g$ such that $g(\Omega)\subset \Omega$andarealnumber$p$,$|p|>1$,thenthereexistsauniqueelement$x\in\Omega$satisfying$g(x)=x^p$.