Dynamics of $\lambda$-continued fractions and $\beta$-shifts

Élise Janvresse; Benoît Rittaud; Thierry de la Rue

doi:10.3934/dcds.2013.33.1477

Article Contents

2013, Volume 33, Issue 4: 1477-1498. Doi: 10.3934/dcds.2013.33.1477

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Dynamics of $\lambda$-continued fractions and $\beta$-shifts

1.
Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, CNRS, Avenue de l’Université, 76801 Saint Étienne du Rouvray
2.
Laboratoire Analyse, Géométrie et Applications, Université Paris 13 Institut Galilée, CNRS, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse
3.
Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, CNRS, Avenue de l'Université, Avenue de l'Université, 76801 Saint Étienne du Rouvray

Received: September 30, 2011

Revised: April 30, 2012

Published: March 2013

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Abstract

For a real number $0<\lambda<2$, we introduce a transformation $T_\lambda$ naturally associated to expansion in $\lambda$-continued fraction, for which we also give a geometrical interpretation. The symbolic coding of the orbits of $T_\lambda$ provides an algorithm to expand any positive real number in $\lambda$-continued fraction. We prove the conjugacy between $T_\lambda$ and some $\beta$-shift, $\beta>1$. Some properties of the map $\lambda\mapsto\beta(\lambda)$ are established: It is increasing and continuous from $]0, 2[$ onto $]1,\infty[$ but non-analytic.

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Mathematics Subject Classification: 511K16, 11J70, 37A45, 37B10.

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References

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