Fixed points of conformal compressions of symmetric cones

Let be a Euclidean Jordan algebra and let be the associated symmetric cone. In this paper, by the contraction property of conformal compressions of the symmetric cone for the natural Riemannian and Finsler metrics on it, we represent the unique fixed point of a strict conformal compression as a limit of continued fractions on the Euclidean Jordan algebra and as the geometric mean of its images at the origin and the infinity point according to the classical triple and the Ol'shanskii polar decompositions of the compression.