On the entropy of Japanese continued fractions

Details On the entropy of Japanese continued fractions On the entropy of Japanese continued fractions

2006-01-24

arxiv.org/abs/math/0601576v2

Mathematics

Dynamical Systems

42 pages, 12 figures; v2: minor changes

Scientific paper

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 $\alpha=\frac{1}{n}$.

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