Continued fractions and the d-dimensional Gauss transformation

D. M. Hardcastle, K. Khanin

Research output: Contribution to journalArticlepeer-review


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
JournalHP Laboratories Technical Report
Issue number15
Publication statusPublished - Jun 2000


  • Invariant measure
  • Multi-dimensional gauss transformation
  • Natural extension


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