  In this paper we consider continued fraction (CF) expansions on intervals
different from $[0,1]$. For every $x$ in such interval we find a CF expansion
with a finite number of possible digits. Using the natural extension, the
density of the invariant measure is obtained in a number of examples. In case
this method does not work, a Gauss-Kuzmin-L\'evy based approximation method is
used. Finally, a subfamily of the $N$-expansions is studied. In particular, the
entropy as a function of a parameter $\alpha$ is estimated for $N=2$ and
$N=36$. Interesting behavior can be observed from numerical results.
