Natural Extensions and Entropy of Α-continued Fractions

We construct a natural extension for each of Nakada’s α-continued fraction transformations 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). For 0 < α ≤ 1, we show that the product of the entropy with the measure of the domain equals π/6. We show that the interval (3 − √ 5)/2 ≤ α ≤ (1 + √ 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 α-expansion of α− 1 and of α.