论文标题
关于基质产品的边际增长率
On marginal growth rates of matrix products
论文作者
论文摘要
在本文中，我们考虑了$ d \ times d $矩阵的最大序列序列的最大可能增长率，所有这些均来自某些指定的紧凑型集合，该套件已归一化，以使联合光谱半径等于$ 1 $。我们将与该集合相关的边际不稳定率序列定义为真实数字的序列，其$ n^{th} $ entry是长度$ n $的最大产品的规范，并研究了这种形式的序列的一般属性。我们描述了如何从旧的边际不稳定率序列构建新的边际不稳定序列，扩展了Protasov和Jungers的较早示例，以获得边际不稳定性率序列，其极限的生长速率与$ n $的各种非全力以赴的限制相匹配，并研究了由Matrices和Matrices of Matrices of Matrices of Matrices of Matrices $ 2 $ 2 $ 2 $ 2 $ 2 $ 2 $的边际不稳定速率序列之间的关系。我们还给出了一个有限集的第一个示例，其边际不稳定性序列渐近地类似于多项式（非全能指数）。以前的示例仅在子序列上具有此属性。
In this article we consider the maximum possible growth rate of sequences of long products of $d \times d$ matrices all of which are drawn from some specified compact set which has been normalised so as to have joint spectral radius equal to $1$. We define the marginal instability rate sequence associated to such a set to be the sequence of real numbers whose $n^{th}$ entry is the norm of the largest product of length $n$, and study the general properties of sequences of this form. We describe how new marginal instability rate sequences can be constructed from old ones, extend an earlier example of Protasov and Jungers to obtain marginal instability rate sequences whose limit superior rate of growth matches various non-integer powers of $n$, and investigate the relationship between marginal instability rate sequences arising from finite sets of matrices and those arising from sets of matrices with cardinality $2$. We also give the first example of a finite set whose marginal instability rate sequence is asymptotically similar to a polynomial with non-integer exponent. Previous examples had this property only along a subsequence.
