### Abstract

A proof that the three-dimensional Gauss algorithm is strongly convergent almost everywhere is given. This algorithm is equivalent to Brun's algorithm and to the modified Jacobi-Perron algorithm considered by Podsypanin and Schweiger. The proof involves the rigorous computer assisted estimation of the largest Lyapunov exponent of a cocycle associated to the algorithm. To the best of my knowledge, this is the first proof of almost everywhere strong convergence of a Jacobi-Perron type algorithm in dimension greater than two.

Original language | English |
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Pages (from-to) | 131-141 |

Number of pages | 11 |

Journal | Experimental Mathematics |

Volume | 11 |

Issue number | 1 |

Publication status | Published - 2002 |

### Keywords

- Brun's algorithm
- Jacobi-Perron algorithm
- Lyapunov exponents
- Multidimensional continued fractions
- Strong convergence

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## Cite this

Hardcastle, D. M. (2002). The three-dimensional Gauss algorithm is strongly convergent almost everywhere.

*Experimental Mathematics*,*11*(1), 131-141.