Abstract
Recently, many authors have been trying to obtain Hausdorff dimension of several fractal sets. However, it is only the self-similar set that Hausdorff dimension is exactly obtained. A self-similar set is constructed inductively by using a number of similarities, and then the contraction ratios are invariant at each stage of the construction. The middle third Cantor set and the von Koch curve are examples of self-similar sets. In this paper, we extend the middle third Cantor set, which we denote C, and examine Hausdorff dimension of C.C is constructed inductively by removing an open interval from each of the closed interval at the previous stage. The length of the removed open interval is arbitrary, except that the length is less than that of the closed interval at the previous stage from which the open interval is removed. Then the contraction ratios are variant at each stage of the construction of C. It is explained in Section 1 how to construct C. In Section 3, using Billingsley's Theorem, we give the lower bound for Hausdorff dimension of C under some conditions. In Section 4, we give Hausdorff dimension of C under conditions stronger than those of Section 3. Furthermore, we give a simple example of which Hausdorff dimension is 1 and Lebegue measure is 0.
