  We study the dynamics of a family of continued fraction maps parametrized by
the unit interval. This family contains as special instances the Gauss
continued fraction map and the Fibonacci map. We determine the transfer
operators of these dynamical maps and make a preliminary study of them. We show
that their analytic invariant measures obeys a common functional equation
generalizing Lewis' functional equation and we find invariant measures for some
members of the family. We also discuss a certain involution of this family
which sends the Gauss map to the Fibonacci map.
