Serendipitous decompositions of higher-dimensional continued fractions

Abstract: 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.