# On invariant measures of the Euclidean algorithm

Dani, S. G. ; Nogueira, Arnaldo (2007) On invariant measures of the Euclidean algorithm Ergodic Theory & Dynamical Systems, 27 (2). pp. 417-425. ISSN 0143-3857

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Related URL: http://dx.doi.org/10.1017/S0143385706000514

## 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.

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