Asymptotics and Sequential Closures of Continued Fractions and Generalizations

TL;DR

This paper develops a comprehensive theory for the asymptotic behavior of sequences derived from matrix products and continued fractions, unifying various cases and providing statistical and distributional insights, including for complex variables and Banach algebras.

Contribution

It introduces a general framework for analyzing asymptotics of matrix product sequences and continued fractions, encompassing elliptic and loxodromic types, and extends to Banach algebras and multivariable cases.

Findings

Approximants of certain continued fractions tend to a circle or finite set of points.

When approximants tend to a circle, their distribution follows a Cauchy distribution.

The theory applies to complex-variable $q$-continued fractions and matrix continued fractions.

Abstract

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 Poincar\'e type and compared with closely related literature.