Bijectivity and trapping regions for complex continued fraction transformation

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.