Dynamics of lambda-continued fractions and beta-shifts

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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Submitted on : Thursday, March 31, 2011 - 3:37:15 PM
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  • HAL Id : hal-00581694, version 1
  • ARXIV : 1103.6181

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Elise Janvresse, Benoît Rittaud, Thierry de La Rue. Dynamics of lambda-continued fractions and beta-shifts. Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2013, 33 (4), pp.1477-1498. ⟨http://aimsciences.org/journal/1078-0947⟩. ⟨hal-00581694⟩

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