The three-dimensional Gauss algorithm is strongly convergent almost everywhere

D M Hardcastle

Research output: Contribution to journalArticle

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 languageEnglish
Pages (from-to)131-141
Number of pages11
JournalExperimental Mathematics
Volume11
Issue number1
Publication statusPublished - 2002

Fingerprint

Gauss
Three-dimensional
Jacobi
Largest Lyapunov Exponent
Cocycle
Strong Convergence

Keywords

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

Cite this

@article{9ca5a54dee5e42799852381e2426d13f,
title = "The three-dimensional Gauss algorithm is strongly convergent almost everywhere",
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.",
keywords = "Brun's algorithm, Jacobi-Perron algorithm, Lyapunov exponents, Multidimensional continued fractions, Strong convergence",
author = "Hardcastle, {D M}",
year = "2002",
language = "English",
volume = "11",
pages = "131--141",
journal = "Experimental Mathematics",
issn = "1058-6458",
publisher = "A K Peters",
number = "1",

}

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

In: Experimental Mathematics, Vol. 11, No. 1, 2002, p. 131-141.

Research output: Contribution to journalArticle

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 -