Open Access
April 2022 An entropy problem of the $\alpha$-continued fraction maps
Hitoshi Nakada
Author Affiliations +
Osaka J. Math. 59(2): 453-464 (April 2022).


We show that the entropy of the $\alpha$-continued fraction map w.r.t the absolutely continuous invariant probability measure is strictly less than that of the nearest integer continued fraction map when $0 \lt \alpha \lt \frac{3 - \sqrt{5}}{2}$. This answers a question by C. Kraaikamp, T. A. Schmidt, and W. Steiner (2012). To prove this result we make use of the notion of the geodesic continued fractions introduced by A. F. Beardon, M. Hockman, and I. Short (2012).


This research was partially supported by JSPS Grant-Aid for Scientific Research (C) 20K03661.


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Hitoshi Nakada. "An entropy problem of the $\alpha$-continued fraction maps." Osaka J. Math. 59 (2) 453 - 464, April 2022.


Received: 14 October 2020; Revised: 22 February 2021; Published: April 2022
First available in Project Euclid: 7 April 2022

MathSciNet: MR4405988
zbMATH: 1492.11116

Primary: 11K50 , 37A10
Secondary: 11J70

Rights: Copyright © 2022 Osaka University and Osaka City University, Departments of Mathematics

Vol.59 • No. 2 • April 2022
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