Natural extensions and entropy of $α$-continued fractions

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 $α$.

No reviews yet.

No comments yet.