### Abstract

In this paper we study a multidimensional continued fraction algorithm which is related to the Modified Jacobi-Perron algorithm considered by Podsypanin and Schweiger. We demonstrate that this algorithm has many important properties which are natural generalisations of properties of one-dimensional continued fractions. For this reason, we call the transformation associated to the algorithm the d-dimensional Gauss transformation. We construct a coordinate system for the natural extension which reveals its symmetries and allows one to give an explicit formula for the density of its invariant measure. We also discuss the ergodic properties of this invariant measure.

Original language | English |
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Pages (from-to) | 487-515 |

Number of pages | 29 |

Journal | Communications in Mathematical Physics |

Volume | 215 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jan 2001 |

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### Cite this

*Communications in Mathematical Physics*,

*215*(3), 487-515. https://doi.org/10.1007/s002200000290

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*Communications in Mathematical Physics*, vol. 215, no. 3, pp. 487-515. https://doi.org/10.1007/s002200000290

**Continued fractions and the d-dimensional Gauss transformation.** / Hardcastle, D. M.; Khanin, K.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Continued fractions and the d-dimensional Gauss transformation

AU - Hardcastle, D. M.

AU - Khanin, K.

PY - 2001/1

Y1 - 2001/1

N2 - In this paper we study a multidimensional continued fraction algorithm which is related to the Modified Jacobi-Perron algorithm considered by Podsypanin and Schweiger. We demonstrate that this algorithm has many important properties which are natural generalisations of properties of one-dimensional continued fractions. For this reason, we call the transformation associated to the algorithm the d-dimensional Gauss transformation. We construct a coordinate system for the natural extension which reveals its symmetries and allows one to give an explicit formula for the density of its invariant measure. We also discuss the ergodic properties of this invariant measure.

AB - In this paper we study a multidimensional continued fraction algorithm which is related to the Modified Jacobi-Perron algorithm considered by Podsypanin and Schweiger. We demonstrate that this algorithm has many important properties which are natural generalisations of properties of one-dimensional continued fractions. For this reason, we call the transformation associated to the algorithm the d-dimensional Gauss transformation. We construct a coordinate system for the natural extension which reveals its symmetries and allows one to give an explicit formula for the density of its invariant measure. We also discuss the ergodic properties of this invariant measure.

UR - http://www.scopus.com/inward/record.url?scp=0035530554&partnerID=8YFLogxK

U2 - 10.1007/s002200000290

DO - 10.1007/s002200000290

M3 - Article

VL - 215

SP - 487

EP - 515

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -