连分数的渐近性和序列闭包及其推广
Asymptotics and Sequential Closures of Continued Fractions and Generalizations
摘要：
给定一个复方阵序列$a_n$，考虑它们的部分乘积序列，定义为$p_n=p_{n-1}a_{n}$。关于序列$f(p_n)$的渐近性$n\to\infty$，可以说些什么，其中$f$是连续函数？我们最一般的结果的一个特殊情况下解决这个问题的假设下，矩阵$a_n$是一个$l_1$扰动的矩阵序列有界部分产品。我们应用我们的理论来研究连分数逼近的渐近性。特别地，当连分数是椭圆型或斜驶型的$l_1$极限1-周期时，我们证明了它的逼近序列趋向于$\hat{\mathbb{C}}$中的一个圆，或趋向于一个圆上的有限点集.我们的主要定理，这样的连分数统一的斜驶和椭圆的情况下，这是收敛和发散，分别治疗。当逼近序列趋于圆时，我们得到了逼近序列极限分布的统计信息。当圆是真正的线，点被证明有一个柯西分布的参数给出的修改的原始连分数。作为一般理论的一个例子，详细研究了五个复变量的$q$ -连分式。本文中最一般的定理在Banach代数中成立。该理论也适用于$(r,s)$ -矩阵连分数和Poincaré型递归序列，并与密切相关的文献进行了比较。
英文摘要：
Given a sequence of complex square matrices, $a_n$, consider the sequence of their partial products, defined by $p_n=p_{n-1}a_{n}$. What can be said about the asymptotics as $n\to\infty$ of the sequence $f(p_n)$, where $f$ is a continuous function? A special case of our most general result addresses this question under the assumption that the matrices $a_n$ are an $l_1$ perturbation of a sequence of matrices with bounded partial products. We apply our theory to investigate the asymptotics of the approximants of continued fractions. In particular, when a continued fraction is $l_1$ limit 1-periodic of elliptic or loxodromic type, we show that its sequence of approximants tends to a circle in $\hat{\mathbb{C}}$, or to a finite set of points lying on a circle. Our main theorem on such continued fractions unifies the treatment of the loxodromic and elliptic cases, which are convergent and divergent, respectively. When an approximating sequence tends to a circle, we obtain statistical information about the limiting distribution of the approximants. When the circle is the real line, the points are shown to have a Cauchy distribution with parameters given in terms of modifications of the original continued fraction. As an example of the general theory, a detailed study of a $q$-continued fraction in five complex variables is provided. The most general theorem in the paper holds in the context of Banach algebras. The theory is also applied to $(r,s)$-matrix continued fractions and recurrence sequences of Poincaré type and compared with closely related literature.
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