关于日本连分数的熵
On the entropy of Japanese continued fractions
摘要：
本文考虑一类单参数的区间扩张映射$\{T_α\}_{α\in [0,1]}$（日本连分数），它包括高斯映射（$α=1$）和最近整数及超数连分数映射（$α={1/2},α=0$）.证明了这些映射的Kolmogorov-Sinai熵$h(α)$连续依赖于参数，并且$h(α) \to 0$为$α\to 0$ .数值结果表明，这种收敛不是单调的，熵函数具有无穷多个相变和自相似结构。最后，我们找到了映射$T_α$对于$α=\frac{1}{n}$的自然扩张和不变密度。
英文摘要：
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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