Finite partitions for several complex continued fraction algorithms

TL;DR

This paper introduces the finite building property for complex continued fraction algorithms, enabling the representation of bijectivity domains as finite unions of Cartesian products, combining explicit and experimental methods.

Contribution

It establishes the finite building property for various algorithms and characterizes their bijectivity domains as finite unions of Cartesian products.

Findings

Bijectivity domains can be expressed as finite unions of Cartesian products.

The property applies to a large class of complex continued fraction algorithms.

Explicit and experimental methods are used to determine the sets in different coordinates.

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 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.