### 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 |

### Fingerprint

### Keywords

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

### Cite this

*Experimental Mathematics*,

*11*(1), 131-141.

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*Experimental Mathematics*, vol. 11, no. 1, pp. 131-141.

**The three-dimensional Gauss algorithm is strongly convergent almost everywhere.** / Hardcastle, D M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The three-dimensional Gauss algorithm is strongly convergent almost everywhere

AU - Hardcastle, D M

PY - 2002

Y1 - 2002

N2 - 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.

AB - 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.

KW - Brun's algorithm

KW - Jacobi-Perron algorithm

KW - Lyapunov exponents

KW - Multidimensional continued fractions

KW - Strong convergence

UR - http://www.scopus.com/inward/record.url?scp=0036015951&partnerID=8YFLogxK

M3 - Article

VL - 11

SP - 131

EP - 141

JO - Experimental Mathematics

JF - Experimental Mathematics

SN - 1058-6458

IS - 1

ER -