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

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 $]1,\mathrm{\infty }[$ but non-analytic.