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On the entropy of Japanese continued fractions

Laura Luzzi·Stefano Marmi
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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 $α=\frac{1}{n}$.

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Laura Luzzi, Stefano Marmi. 2006-06-01. On the entropy of Japanese continued fractions. https://arxiv.org/abs/math/0601576

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