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On invariant measures of the Euclidean algorithm

Published online by Cambridge University Press:  12 February 2007

S. G. DANI  and
ARNALDO NOGUEIRA
S. G. DANI
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India (e-mail: [email protected])
ARNALDO NOGUEIRA
Affiliation:
Institut de Mathématiques de Luminy, 163, avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France (e-mail: [email protected])
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Abstract

We study the ergodic properties of the additive Euclidean algorithm $f$ defined in $\mathbb{R}^2_+$. A natural extension of $f$ is obtained using the action of ${\it SL}(2, \mathbb{Z})$ on a subset of ${\it SL}(2, \mathbb{R})$. We prove that, while $f$ is an ergodic transformation with an infinite invariant measure equivalent to the Lebesgue measure, the invariant measure is not unique up to scalar multiples, and in fact there is a continuous family of such measures.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 2 , April 2007 , pp. 417 - 425
Copyright
2007 Cambridge University Press

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