HAL: hal-00581694, version 1

arXiv: 1103.6181

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Dynamics of lambda-continued fractions and beta-shifts

Elise Janvresse 1, Benoît Rittaud 2, Thierry de la Rue 1

(2011)

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.

1:	Laboratoire de Mathématiques Raphaël Salem (LMRS)
	CNRS : UMR6085 – Université de Rouen
2:	Laboratoire d'Analyse, Géométrie et Applications (LAGA)
	CNRS : UMR7539 – Université Paris XIII - Paris Nord – Université Paris VIII - Vincennes Saint-Denis

Subject

:

Mathematics/Dynamical Systems Mathematics/Number Theory

Keyword(s): continued fractions – $\beta$-expansion

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From: Elise Janvresse <>

Submitted on: Thursday, 31 March 2011 15:37:15

Updated on: Friday, 1 April 2011 13:10:29