Natural extensions and entropy of $\alpha$-continued fractions

Abstract

We construct a natural extension for each of Nakada's $\alpha$-continued fractions and show the continuity as a function of $\alpha$ of both the entropy and the measure of the natural extension domain with respect to the density function $(1+xy{)}^{-2}$. In particular, we show that, for all $0<\alpha \le 1$, the product of the entropy with the measure of the domain equals ${\pi }^2/6$. As a key step, we give the explicit relationship between the $\alpha$-expansion of $\alpha -1$ and of $\alpha$.

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