On the entropy of Japanese continued fractions

Abstract

We consider a one-parameter family of expanding interval maps $\{T_{\alpha }{\}}_{\alpha \in [0,1]}$ (japanese continued fractions) which include the Gauss map ($\alpha =1$) and the nearest integer and by-excess continued fraction maps ($\alpha =1/2,\alpha =0$). We prove that the Kolmogorov-Sinai entropy $h(\alpha )$ of these maps depends continuously on the parameter and that $h(\alpha )\to 0$ as $\alpha \to 0$. 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 $T_{\alpha }$ for $\alpha =\frac{1}{n}$.

42 pages, 12 figures; v2: minor changes