Natural extensions and entropy of α-continued fractions

We construct a natural extension for each of Nakada's alpha-continued fraction transformations 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). For 0 < alpha <= 1, we show that the product of the entropy with the measure of the domain equals pi(2)/6. We show that the interval (3 - root 5)/2 <= alpha <= (1 + root 5)/2 is a maximal interval upon which the entropy is constant. As a key step for all this, we give the explicit relationship between the a-expansion of alpha - 1 and of alpha.