Invariant measures for continued fraction algorithms with finitely many digits

Abstract

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. Convergence of this method follows from [32] but the speed of convergence remains unknown. For a lot of known densities the method gives a very good approximation in a low number of iterations. 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.