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Abstract. The Rosen fractions form an infinite family which generalizes the nearest-integer continued fractions. We find planar natural extensions for the associated interval maps. This allows us to easily prove that the interval maps are weak Bernoulli, as well as to unify and generalize results of Diophantine approximation from the literature.

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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) −2 . For 0 < α ≤ 1, we show that the product of the entropy with the measure of the domain equals π 2 /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.