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This paper provides some preliminary results on the dynamics of certaincomplex continued fractions. After establishing some general number theoreticresults, we explore the dynamics of the natural extension map associated to aspecific complex continued fraction algorithm (the "diamond" algorithm). Weprove that this map has a bijectivity domain that is a subset of a trappingregion for the map and, moreover, that both these sets have a "finite productstructure" arising from a finite partition specific to the particularalgorithm.

Bijectivity and trapping regions for complex continued fraction transformation