The competitive spectral radius of families of nonexpansive mappings (Stéphane Gaubert, INRIA and CMAP, Ecole polytechnique)

04.03.2025 10:30

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 invertible 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 Mañe’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 author, concerning the asymptotic behavior of nonexpansive mappings (generalizations of the Wolff-Denjoy theorem). We also show that the value of the game admits a dual characterization when the nonexpansive maps are quasi-isometries, or when they are order preserving and positively homogeneous self-maps of a cone equipped with Funk’s and Thompson’s metrics.
This is based on joint work with Marianne Akian and Loic Marchesini,
arXiv:2410.21097

Lieu

Bâtiment: Conseil Général 7-9

Room 1-05, Tuesday 04.03.2025, Séminaire "Groupes et géométrie"

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