正线性包含的极值范数

Extremal Norms for Fiber Bunched Cocycles

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 conditio…

极值遍历测度与矩阵半群的有限性

Extremal ergodic measures and the finiteness property of matrix semigroups

设$\bS=\{S_1,...,S_K\}$是复$d\times d$矩阵的有限集，$\varSigma_{K}^+$是所有单边无穷序列$i_{\bcdot}\colon\mathbb{N}\rightarrow\{1,...,K\}$的紧空间.马尔可夫位移$θ\colon\varSigma_{K}^+\rightarrow\varSigma_{K}^+;\ i_{\bcdot}\mapsto i_{\bcdot+1}$的遍历概率$μ_*$称为$\bS$的“极值”，如果$ρ(\bS)=\lim_{n\to\infty}\sqrt[n]{\norm{S_{i_1}...S_{i_n}}}$对$μ_*$成立-a. e. $i_{\bcdot}\in\varSigma_{K}^+$，其中$ρ(\bS)$表示$\bS$的广义/联合谱半径。利用极值范数和Kingman次可加遍历定理，证明了$\bS$具有谱有限性（即对有限长字$(i_1,...,i_n)$具有$ρ(\bS)=\sqrt[n]{ρ(S_{i_1}...S_{i_n})}$）当且仅当对$\bS$的某个极值测度$μ_*$，它至少有一个周期密度点$i_{\bcdot}\in\varSigma_{K}^+$ .