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Communications in Mathematical Physics

, Volume 215, Issue 3, pp 487–515 | Cite as

Continued Fractions and the d-Dimensional Gauss Transformation

  • D. M. Hardcastle
  • K. Khanin

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.

Keywords

Coordinate System Invariant Measure Explicit Formula Natural Extension Continue Fraction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • D. M. Hardcastle
    • 1
  • K. Khanin
    • 1
  1. 1.Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, UK.¶E-mail: D.M.Hardcastle@ma.hw.ac.ukUK

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