Extremal Norms for Fiber Bunched Cocycles

TL;DR

This paper establishes conditions for the existence of extremal norms in fiber-bunched cocycles, extending classical concepts like Barabanov norms to the setting of Lyapunov exponent maximization.

Contribution

It proves a non-commutative Mañé Lemma and introduces extremal norms for fiber-bunched cocycles, extending the theory of joint spectral radius.

Findings

Existence of extremal norms under irreducibility and fiber bunching.

Extension of Barabanov norms to vector bundle automorphisms.

Conditions for Lyapunov maximizing sets.

Abstract

In traditional Ergodic Optimization, one seeks to maximize Birkhoff averages. The most useful tool in this area is the celebrated Ma\~n\'e Lemma, in its various forms. In this paper, we prove a non-commutative Ma\~n\'e 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.