Abstract

We present a property satisfied by a large variety of complex continued fraction algorithms (the “finite building property”) and use it to explore the structure of bijectivity domains for natural extensions of Gauss maps. Specifically, we show that these domains can each be given as a finite union of Cartesian products in $\mathbb{C}\times \mathbb{C}.$ In one complex coordinate, the sets come from explicit manipulation of the continued fraction algorithm, while in the other coordinate the sets are determined by experimental means.

中文翻译：

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几种复连分数算法的有限分区

摘要

我们提出了一个由大量复杂连分数算法满足的性质（“有限建筑性质”），并用它来探索双射域的结构，以实现高斯图的自然扩展。具体来说，我们展示了这些域中的每一个都可以作为笛卡尔积的有限联合给出$\mathbb{C}\times \mathbb{C}.$在一个复杂坐标中，集合来自连分数算法的显式操作，而在另一个坐标中，集合由实验方法确定。