Continued fractions and the d-dimensional Gauss transformation

D. M. Hardcastle, K. Khanin

Research output: Contribution to journalArticle

9 Citations (Scopus)


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 languageEnglish
Pages (from-to)487-515
Number of pages29
JournalCommunications in Mathematical Physics
Issue number3
Publication statusPublished - Jan 2001

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