The three-dimensional Poincaré continued fraction algorithm (Q1895086)

The author proves that for Lebesgue-almost every line in the three-dimensional Euclidean space, the Poincaré continued fraction algorithm determines a unique vertex. Furthermore, the algorithm is nonergodic, although the Gauss map, defined by the algorithm, has an attractor and is ergodic. Finally, it is shown that the Euclidean algorithm and the horocycle flow are orbit equivalent.