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

Invariant measures for continued fraction algorithms with finitely many digits