Iterates of Maps Which are Non-Expansive in Hilbert's Metric

Abstract We review some results from the theory of non-expansive mappings and apply them to two metrics in particular: Thompson's part metric and Hilbert's projective metric. On symmetric cones, we generalise the notion of the cycle time defined previously for max-plus operators. We show that symmetric cones endowed with either Hilbert's or Thompson's metric satisfy Busemann's definition of a space of non-positive curvature. One result is that, on a strictly convex symmetric cone, the cycle time exists for all maps which are non-expansive in both these metrics. We also review some generalisations of the classical Denjoy-Wolff theorem.