纤维丛余圈的极值范数
Extremal Norms for Fiber Bunched Cocycles
摘要：
在传统的遍历优化中，人们寻求最大化Birkhoff平均值。在这一领域最有用的工具是著名的Mañé引理，以其各种形式。本文证明了一个非交换的Mañé引理，该引理适用于线性上循环或更一般的向量丛自同构的李雅普诺夫指数的最大化问题.更确切地说，我们提供的条件，确保存在一个极值规范，也就是说，Finsler规范，就没有向量可以扩大在一个单一的扩张因子大于最大的渐近展开率。这些条件基本上是不可约的和足够强的纤维聚束。因此，我们推广了经典的Barabanov范数的概念，用于研究联合谱半径。我们得到了几个结果，包括李雅普诺夫最大化集存在的充分条件。
英文摘要：
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.
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