Bijectivity and trapping regions for complex continued fraction transformation

TL;DR

This paper investigates the dynamics of a specific complex continued fraction algorithm, establishing bijectivity and trapping regions with a finite product structure, advancing understanding of its mathematical properties.

Contribution

It introduces new results on the bijectivity domain and trapping regions for the diamond complex continued fraction map, with a finite partition structure.

Findings

The natural extension map has a bijectivity domain within a trapping region.

Both sets exhibit a finite product structure from a finite partition.

Preliminary number theoretic results support the dynamical analysis.

Abstract

This paper provides some preliminary results on the dynamics of certain complex continued fractions. After establishing some general number theoretic results, we explore the dynamics of the natural extension map associated to a specific complex continued fraction algorithm (the "diamond" algorithm). We prove that this map has a bijectivity domain that is a subset of a trapping region for the map and, moreover, that both these sets have a "finite product structure" arising from a finite partition specific to the particular algorithm.