For real-valued continued fractions, the natural extension
of the Gauss map has a global attractor with a simple structure coming
from a "cycle property". Because this cycle structure is strictly
one-dimensional, the "finite building property" was developed as an
alternative to analyze /complex/ continued fraction algorithms. For
algorithms with this property, the domain in $\mathbb C\times\mathbb C$
of the natural
extension of the continued fraction map can be described 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.