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The Competive Spectral Radius of Families of Nonexpansive Mappings

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arxiv 2410.21097 v2 pith:H3HSG23R submitted 2024-10-28 math.OC math.DS

Marianne Akian , Stéphane Gaubert , Loïc Marchesini This is my paper · ORCID
classification math.OCmath.DS
keywords nonexpansivetwo-playercasegamemappingsnon-linearradiusspectral
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We consider a new class of repeated zero-sum games in which the payoff is the escape rate of a switched dynamical system, where at every stage, the transition is given by a nonexpansive operator depending on the actions of both players. This generalizes to the two-player (and non-linear) case the notion of joint spectral radius of a family of matrices. We show that the value of this game does exist, and we characterize it in terms of an infinite dimensional non-linear eigenproblem. This provides a two-player analogue of Ma\~ne's lemma from ergodic control. This also extends to the two-player case results of Kohlberg and Neyman (1981), Karlsson (2001), and Vigeral and the second author (2012), concerning the asymptotic behavior of nonexpansive mappings. We discuss two special cases of this game: order preserving and positively homogeneous self-maps of a cone equipped with Funk's and Thompson's metrics, and groups of translations.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Continuity and approximability of competitive spectral radii

    math.OC 2025-05 conditional novelty 7.0 of 10

    Under a cone-positivity condition, the competitive spectral radius is 1-Lipschitz in the matrix sets and can be approximated to arbitrary precision by solving a discretized nonlinear eigenproblem.

  2. Nonlinear Joint Spectral Radius

    math.DS 2025-07 conditional novelty 6.0 of 10

    For switched nonlinear systems whose maps are sub-homogeneous and order-preserving on a cone, a nonlinear joint spectral radius is shown to characterize asymptotic stability and bound the worst-case separation growth ...

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