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 $lambdamapstobeta(lambda)$ are established: It is increasing and continuous from $]0, 2[$ onto $]1,infty[$ but non-analytic.