A maximin characterisation of the escape rate of non-expansive mappings in metrically convex spaces
By Stephane Gaubert, Guillaume Vigeral. Mathematical Proceedings of the Cambridge Philosophical Society, 2010
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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.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Economics and Econometrics
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In Applied Mathematics
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Citations in the corpus
- Birkhoff's version of Hilbert's metric and its applications in analysis (2013, background): “Other Denjoy-Wolff type theorems for finite and infinite-dimensional Hilbert geometries can be found in [23, 40, 44, 54].”