  We prove a suite of dynamical results, including exactness of the
transformation and piecewise-analyticity of the invariant measure, for a family
of continued fraction systems, including specific examples over reals, complex
numbers, quaternions, octonions, and in $R^3$. Our methods expand on the work
of Nakada and Hensley, and in particular fill some gaps in Hensley's analysis
of Hurwitz complex continued fractions. We further introduce a new
``serendipity'' condition for a continued fraction algorithm, which controls
the long-term behavior of the boundary of the fundamental domain under
iteration of the continued fraction map, and which is under reasonable
conditions equivalent to the finite range property. We also show that the
finite range condition is extremely delicate: perturbations of serendipitous
systems by non-quadratic irrationals do not remain serendipitous, and
experimental evidence suggests that serendipity may fail even for some rational
perturbations.
