Extremal norms for fiber-bunched cocycles

Bochi, Jairo; Garibaldi, Eduardo

Journal de l’École polytechnique - Mathématiques, Volume 6 (2019), pp. 947-1004.

In traditional Ergodic Optimization, one seeks to maximize Birkhoff averages. The most useful tool in this area is the celebrated Mañé Lemma, in its various forms. In this paper, we prove a non-commutative Mañé Lemma, suited to the problem of maximization of Lyapunov exponents of linear cocycles or, more generally, vector bundle automorphisms. More precisely, we provide conditions that ensure the existence of an extremal norm, that is, a Finsler norm with respect to which no vector can be expanded in a single iterate by a factor bigger than the maximal asymptotic expansion rate. These conditions are essentially irreducibility and sufficiently strong fiber-bunching. Therefore we extend the classic concept of Barabanov norm, which is used in the study of the joint spectral radius. We obtain several consequences, including sufficient conditions for the existence of Lyapunov maximizing sets.

En optimisation ergodique traditionnelle, on cherche à maximiser des moyennes de Birkhoff. L’outil le plus utile dans ce domaine est le célèbre lemme de Mañé, sous ses diverses formes. Dans cet article, nous montrons un lemme de Mañé non commutatif, adapté au problème de la maximisation des exposants de Lyapunov de cocycles linéaires ou, plus généralement, des automorphismes de fibrés vectoriels. Plus précisément, nous fournissons des conditions qui garantissent l’existence d’une norme extrémale, c’est-à-dire une norme de Finsler pour laquelle aucun vecteur ne peut être dilaté en une seule itération par un facteur plus grand que le taux de croissance asymptotique maximal. Ces conditions sont essentiellement l’irréductibilité et un resserrement des fibres suffisamment fort. Nous étendons donc le concept classique de norme de Barabanov, utilisé dans l’étude du rayon spectral joint. Nous obtenons plusieurs conséquences, notamment des conditions suffisantes pour l’existence des ensembles maximisants de Lyapunov.

Classification: 37H15, 37D20, 37D30, 15A60, 93D30
Keywords: Linear cocycle, extremal norm, Lyapunov exponent, ergodic optimization, joint spectral radius
Mot clés : Cocycle linéaire, norme extrémale, exposant de Lyapunov, optimisation ergodique, rayon spectral joint

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