An entropy problem of the $\alpha$-continued fraction maps

Abstract

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).