Asymptotics and Sequential Closures of Continued Fractions and Generalizations

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 Poincaré type and compared with closely related literature.

Revised Version: intermediate version loaded by mistake earlier. 52 pages, 3 figures, final version may be slightly different. Keywords: Limit Periodic Continued Fractions, $q$-Continued Fractions, Continued Fractions, Poincaré-type Recurrences, $q$-series, Infinite Products, Asymptotics, sequential closures, Ramanujan, Cauchy Distribution