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The three-dimensional Poincaré continued fraction algorithm

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Abstract

It is proved here that for Lebesgue-almost every line in the three-dimensional Euclidean space, the Poincaré 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.

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Author notes

  1. A. Nogueira

    Present address: Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68.530, 21.945-970, Rio de Janeiro, RJ, Brazil

Authors and Affiliations

  1. Unité de Mathématiques Pures et Appliqués, École Normale Supérieure de Lyon, 46, Allée d’Italie, 69.364, Lyon Cedex 07, France

    A. Nogueira

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  1. A. Nogueira
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Correspondence to A. Nogueira.

Additional information

Partially supported by CNPq(Brazil) grant no. 30.1456-80.

An erratum to this article is available at http://dx.doi.org/10.1007/BFb0120366.

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Nogueira, A. The three-dimensional Poincaré continued fraction algorithm. Israel J. Math. 90, 373–401 (1995). https://doi.org/10.1007/BF02783221

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  • DOI: https://doi.org/10.1007/BF02783221

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