Dynamics of lambda-continued fractions and beta-shifts
Résumé
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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