On the entropy of Japanese continued fractions

Abstract

We consider a one-parameter family of expanding interval maps $\{T_α\}_{α\in [0,1]}$ (japanese continued fractions) which include the Gauss map ($α=1$) and the nearest integer and by-excess continued fraction maps ($α={1/2},α=0$). We prove that the Kolmogorov-Sinai entropy $h(α)$ of these maps depends continuously on the parameter and that $h(α) \to 0$ as $α\to 0$. Numerical results suggest that this convergence is not monotone and that the entropy function has infinitely many phase transitions and a self-similar structure. Finally, we find the natural extension and the invariant densities of the maps $T_α$ for $α=\frac{1}{n}$.