Dynamics of $λ$-continued fractions and $β$-shifts

For a real number $0<λ<2$, we introduce a transformation $T_λ$ naturally associated to expansion in $λ$-continued fraction, for which we also give a geometrical interpretation. The symbolic coding of the orbits of $T_λ$ provides an algorithm to expand any positive real number in $λ$-continued fraction. We prove the conjugacy between $T_λ$ and some $β$-shift, $β>1$. Some properties of the map $λ\mapstoβ(λ)$ are established: It is increasing and continuous from $]0, 2[$ onto $]1,\infty[$ but non-analytic.

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