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On the Hausdorff dimension of a set of complex continued fractions

One can ask whether there are some natural measures on T with respect to which S is ergodic or mixing. From our point of view the answer depends on the geometric structure of a representation of this space. For example, if T N, then there are, of course, 2 probability measures with respect to which S is mixing. This can be seen by noting that the permutations r of N induce distinct mixing measures /o h, where , is Gauss’ measure and h is the natural homeomorphism of N induced by r. However, if one considers the extremely natural representation of Nv via the canonical continued fraction expansion of the irrational numbers in [0, 1], then there is only one ergodi measure which is connected to the geometric structure of this set, Gauss’ measure. (See, for example [1, p. 40].) Gauss’ measure is the only ergodic measure which is absolutely continuous with respect to Lebesgue measure; this is proved in [4, p. 114]. Let us consider T N Z. Again, there are 2 measures with respect to which the shift is ergodic. There is a natural geometric representation of (N Z)N. As is shown here, the map