Invariant measures for continued fraction algorithms with finitely many digits

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:454

Article

Kraaikamp, Cor¹  Langeveld, Niels² 

[1] Delft Univ Technol, EWI DIAM, Mekelweg 4, NL-2628 CD Delft, Netherlands

[2] Leiden Univ, Math Inst, POB 9512, NL-2300 RA Leiden, Netherlands

关键词: Continued fraction expansions;    Gauss-Kuzmin-Levy;    Entropy;    Natural extension;    Invariant measure;   

DOI  :  10.1016/j.jmaa.2017.04.067

来源: Elsevier

【 摘 要 】

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-Levy 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 alpha is estimated for N = 2 and N = 36. Interesting behavior can be observed from numerical results. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

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