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We establish a maximin characterisation of the linear escape rate of theorbits of a non-expansive mapping on a complete (hemi-)metric space, under amild form of Busemann's non-positive curvature condition (we require adistinguished family of geodesics with a common origin to satisfy a convexityinequality). This characterisation, which involves horofunctions, generalisesthe Collatz-Wielandt characterisation of the spectral radius of a non-negativematrix. It yields as corollaries a theorem of Kohlberg and Neyman (1981),concerning non-expansive maps in Banach spaces, a variant of a Denjoy-Wolfftype theorem of Karlsson (2001), together with a refinement of a theorem ofGunawardena and Walsh (2003), concerning order-preserving positivelyhomogeneous self-maps of symmetric cones. An application to zero-sum stochasticgames is also given.

A maximin characterization of the escape rate of nonexpansive mappings in metrically convex spaces