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Invariant measures for continued fraction algorithms with finitely many digits

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-Lévy based approximation method is used. Finally, a subfamily of the $N$-expansions is studied. In particular, the entropy as a function of a parameter $α$ is estimated for $N=2$ and $N=36$. Interesting behavior can be observed from numerical results.