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arXiv · 1011.4283

Natural extensions and entropy of $α$-continued fractions

Cor Kraaikamp·Thomas A. Schmidt·Wolfgang Steiner
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Abstract

We construct a natural extension for each of Nakada's $α$-continued fractions and show the continuity as a function of $α$ 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 $0 < α\le 1$, the product of the entropy with the measure of the domain equals $π^2/6$. As a key step, we give the explicit relationship between the $α$-expansion of $α-1$ and of $α$.

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Cor Kraaikamp, Thomas A. Schmidt, Wolfgang Steiner. 2012-06-26. Natural extensions and entropy of $α$-continued fractions. https://doi.org/10.1088/0951-7715%2F25%2F8%2F2207

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