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Exactness of the Euclidean algorithm and of the Rauzy induction on the space of interval exchange transformations

Published online by Cambridge University Press:  30 November 2011

TOMASZ MIERNOWSKI  and
ARNALDO NOGUEIRA
TOMASZ MIERNOWSKI
Affiliation:
Institut de Mathématiques de Luminy, Aix-Marseille Université, 163, avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France (email: [email protected], [email protected])
ARNALDO NOGUEIRA
Affiliation:
Institut de Mathématiques de Luminy, Aix-Marseille Université, 163, avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France (email: [email protected], [email protected])
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Abstract

The two-dimensional homogeneous Euclidean algorithm is the central motivation for the definition of the classical multidimensional continued fraction algorithms, such as Jacobi–Perron, Poincaré, Brun and Selmer algorithms. The Rauzy induction, a generalization of the Euclidean algorithm, is a key tool in the study of interval exchange transformations. Both maps are known to be dissipative and ergodic with respect to Lebesgue measure. Here we prove that they are exact.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 1 , February 2013 , pp. 221 - 246
Copyright
Copyright © Cambridge University Press 2011

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