  We establish a maximin characterisation of the linear escape rate of the
orbits of a non-expansive mapping on a complete (hemi-)metric space, under a
mild form of Busemann's non-positive curvature condition (we require a
distinguished family of geodesics with a common origin to satisfy a convexity
inequality). This characterisation, which involves horofunctions, generalises
the Collatz-Wielandt characterisation of the spectral radius of a non-negative
matrix. It yields as corollaries a theorem of Kohlberg and Neyman (1981),
concerning non-expansive maps in Banach spaces, a variant of a Denjoy-Wolff
type theorem of Karlsson (2001), together with a refinement of a theorem of
Gunawardena and Walsh (2003), concerning order-preserving positively
homogeneous self-maps of symmetric cones. An application to zero-sum stochastic
games is also given.
