abstract: We discuss some ergodic properties of continued fraction maps introduced by A. and. J. Hurwitz, R.B. Lakein etc. All maps discussed here are sofic systems and it is easy to see that there exists an absolutely continuous invariant ergodic probability measure for each map. Main aim of this talk is that the construction of the natural extension of each map as a map on a set of geodesics on the 3-dimensional upper-half space.