On invariant measures of the Euclidean algorithm

• Autores: S. G. Dani, Arnaldo Nogueira
• Localización: Ergodic theory and dynamical systems, ISSN 0143-3857, Vol. 27, Nº 2, 2007, págs. 417-425
• Idioma: inglés
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• Resumen

  • 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.