Commensurable continued fractions

Pierre Arnoux; Thomas A. Schmidt

doi:10.3934/dcds.2014.34.4389

Article Contents

2014, Volume 34, Issue 11: 4389-4418. Doi: 10.3934/dcds.2014.34.4389

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Commensurable continued fractions

Pierre Arnoux^1, and
Thomas A. Schmidt^2,

1.
Institut de Mathématiques de Luminy (UMR6206 CNRS), 163 Avenue de Luminy, case 907, 13288 Marseille cedex 09
2.
Department of Mathematics, Oregon State University, Corvallis, OR 97331

Received: August 31, 2013

Revised: January 31, 2014

Published: October 2014

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Abstract

We compare two families of continued fractions algorithms, the symmetrized Rosen algorithm and the Veech algorithm. Each of these algorithms expands real numbers in terms of certain algebraic integers. We give explicit models of the natural extension of the maps associated with these algorithms; prove that these natural extensions are in fact conjugate to the first return map of the geodesic flow on a related surface; and, deduce that, up to a conjugacy, almost every real number has an infinite number of common approximants for both algorithms.

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Mathematics Subject Classification: Primary: 37E05, 11K50; Secondary: 37D40, 30B70.

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