Invariant measures for continued fraction algorithms with finitely many digits

Kraaikamp, C.; Langeveld, N.D.S. (2017)

Article / Letter to editor

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.

• Continued fraction expansions
• Gauss–Kuzmin–Lévy
• Entropy
• Natural extension
• Invariant measure

• All authors: Kraaikamp, C.; Langeveld, N.D.S.
• Date: 2017-10-01
• Journal: Journal of Mathematical Analysis and Applications
• Volume: 454
• Issue: 1
• Pages: 106 - 126
• DOI: doi:10.1016/j.jmaa.2017.04.067
• Link: https://reader.elsevier.com/reader/sd/pii/S0022247X17304304?token=41F252...