A maximin characterization of the escape rate of non-expansive mappings in metrically convex spaces

Stéphane Gaubert (1, 2) , Guillaume Vigeral (3)

1 MAXPLUS - Max-plus algebras and mathematics of decision
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Domaines: Géométrie métrique [math.MG]; Systèmes dynamiques [math.DS]

Stéphane Gaubert, Guillaume Vigeral. A maximin characterization of the escape rate of non-expansive mappings in metrically convex spaces. Mathematical Proceedings of the Cambridge Philosophical Society, 2012, 152 (2), pp.341-363. ⟨10.1017/S0305004111000673⟩. ⟨hal-00661863⟩