THE 3-DIMENSIONAL POINCARE CONTINUED-FRACTION ALGORITHM

It is proved here that for Lebesgue-almost every line in the three-dimensional Euclidean space, the Poincare continued fraction algorithm fixes a vertex. Besides, the algorithm is nonergodic, although the Gauss map, defined by the algorithm, has an attractor and is ergodic. It is also shown that the Euclidean algorithm and the horocycle flow are orbit equivalent.