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On some symmetric multidimensional continued fraction algorithms

Published online by Cambridge University Press:  24 January 2017

PIERRE ARNOUX  and
SÉBASTIEN LABBÉ
PIERRE ARNOUX
Affiliation:
Equipe Groupes, Dynamique, Arithmétique et Combinatoire, Institut de Mathématique de Marseille, CNRS UMR 7373, Campus de Luminy, Avenue de Luminy, Case 907, 13288 Marseille Cedex 09, France email pierre@pierrearnoux.fr
SÉBASTIEN LABBÉ
Affiliation:
Bât. B37 Institut de Mathématiques, Grande Traverse 12, 4000 Liège, Belgium email slabbe@ulg.ac.be
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Abstract

We compute explicitly the density of the invariant measure for the reverse algorithm which is absolutely continuous with respect to Lebesgue measure, using a method proposed by Arnoux and Nogueira. We also apply the same method on the unsorted version of the Brun algorithm and Cassaigne algorithm. We illustrate some experimentations on the domain of the natural extension of those algorithms. For some other algorithms, which are known to have a unique invariant measure absolutely continuous with respect to Lebesgue measure, the invariant domain found by this method seems to have a fractal boundary, and it is unclear whether it is of positive measure.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 5 , August 2018 , pp. 1601 - 1626
Copyright
© Cambridge University Press, 2017 

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