On the Hausdorff dimension of a set of complex continued fractions

R. J. Gardner, R. D. Mauldin
1983 Illinois Journal of Mathematics
This note arose from some general considerations concerning geometric representations of the shift operator. Specifically, consider an infinite set T, the product space TN, and shift operator S T N --> T defined by S(<tl, t2, t3, ...) <t2, t3 . One can ask whether there are some natural measures on T N 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
more » ... , 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 N v 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 h((bl, b2 )) is a homeomorphism of (N Z) onto a subset of the open disc in the plane with center (1/2, 0) and radius 1/2. Our question is, is there an