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

Published online by Cambridge University Press:  19 October 2011

STÉPHANE GAUBERT  and
GUILLAUME VIGERAL
STÉPHANE GAUBERT
Affiliation:
INRIA Saclay & Centre de Mathématiques Appliquées (CMAP), École Polytechnique, 91128 Palaiseau, France. e-mail: Stephane.Gaubert@inria.fr
GUILLAUME VIGERAL
Affiliation:
Université Paris-Dauphine, CEREMADE, Place du Maréchal De Lattre de Tassigny, 75775 Paris cedex 16, France. e-mail: guillaumevigeral@gmail.com
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Abstract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 152 , Issue 2 , March 2012 , pp. 341 - 363
Copyright
Copyright © Cambridge Philosophical Society 2011

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