Properties of limit sets and convergence of continued fractions (Q1336143)

Let \(E\) be a set \(\subseteq \mathbb{C}\), the complex plane. We consider continued fractions \(K(a_ n/1)\) from \(E\); i.e., \(0\neq a_ n\in E\) for all \(n\). A useful tool to study the convergence properties of these continued fractions is the notion of value sets \(V\); i.e., sets \(0\neq V\subseteq \widehat {\mathbb{C}}= \mathbb{C}\cup \infty\) with the property that \[ a/ (1+V) \subseteq V \quad \text{for all } a\in E. \] It has been shown that if \(V\) contains at least two points, then its closure \(\overline {V}\) contains the best limit set \({\mathcal V}\) of \(E\); i.e., \[ {\mathcal V}= \{w= K(a_ n/1)\in \widehat {\mathbb{C}}: \text{ all } a_ n\in E\setminus \{0\} \text{ and } K(a_ n/1) \text{ converges}\}. \] In the present paper properties of such value sets and best limit sets are related to properties of the continued fractions \(K(a_ n/ 1)\) from \(E\).