Metrical Theory for a Class of Continued Fraction Transformations and Their Natural Extensions

Many results concerning the metrical theory for the simple continued fractions had been given by Gauss, L\’evy, Khintchine, etc., (see Billingsley [1]). On the other hand, the metrical theory of continued fractions to the nearest integer or of singular continued fractions has been discussed by Rieger [7], [8] and [9], in which he obtained among other things the invariant measures for these transformations. In contrast with $\{f_{\alpha}\}$ , recently Ito and Tanaka [3] considered the class of transformations $\{S_{a}\}$ including those associated with the restriction to the real axis of Hurwitz’ complex continued fractions and of simple continued fractions. Here $S_{\alpha},$ $1/2\leqq\alpha\leqq 1$ , is defined by