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American Institute of Mathematical Sciences (AIMS), Discrete and Continuous Dynamical Systems - Series A, 11(34), p. 4389-4418, 2014

DOI: 10.3934/dcds.2014.34.4389

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Commensurable continued fractions

Journal article published in 2014 by Pierre Arnoux, Thomas A. Schmidt
This paper is made freely available by the publisher.
This paper is made freely available by the publisher.

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Abstract

41 pages, 10 figures ; International audience ; 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.