矩阵序列的Mather集及其在联合谱半径研究中的应用
Mather sets for sequences of matrices and applications to the study of joint spectral radii
摘要：
定义了d次d矩阵紧集的联合谱半径。是从该集合中提取的矩阵的乘积的最大可能的指数增长性速率。在这篇文章中，我们研究遍历理论结构的矩阵序列从一个给定的集合，其产品增长的最大可能的速度。这导致了一个概念的马瑟集矩阵序列是类似于马瑟集拉格朗日动力学。我们证明了一个结构定理，建立了这些马瑟集的一般性质，并描述了在何种程度上，他们的矩阵序列的最大增长。我们给出了这个定理的应用研究联合谱半径和离散线性包含的稳定性理论。这些结果建立在关于动力系统次可加观测的最大增长轨道结构的一些一般性定理的基础上，其中包括Schreiber，Sturman和Stark的半一致次可加遍历定理的推广和Y.佩雷斯这些定理在附录中给出。
英文摘要：
The joint spectral radius of a compact set of d-times-d matrices is defined ?to be the maximum possible exponential growth rate of products of matrices drawn from that set. In this article we investigate the ergodic-theoretic structure of those sequences of matrices drawn from a given set whose products grow at the maximum possible rate. This leads to a notion of Mather set for matrix sequences which is analogous to the Mather set in Lagrangian dynamics. We prove a structure theorem establishing the general properties of these Mather sets and describing the extent to which they characterise matrix sequences of maximum growth. We give applications of this theorem to the study of joint spectral radii and to the stability theory of discrete linear inclusions. These results rest on some general theorems on the structure of orbits of maximum growth for subadditive observations of dynamical systems, including an extension of the semi-uniform subadditive ergodic theorem of Schreiber, Sturman and Stark, and an extension of a noted lemma of Y. Peres. These theorems are presented in the appendix.
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