Correlation dimension for self-similar Cantor sets with overlaps
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In this paper we consider self-similar Cantor sets ae R which are either homogeneous and \Gamma is an interval, or not homogeneous but having thickness greater than one. We have a natural labeling of the points of which comes from its construction. In case of overlaps among the cylinders of , there are some `bad' pairs of labels (ø; !) such that ø and ! label the same point of . We express, how much the correlation dimension of is smaller than the similarity dimension, in terms of the size of the set of `bad' pairs of labels. 1 Introduction In the literature there are some results (see [Fa1], [PS] or [Si] for a survey) which show that for a family of Cantor sets of overlapping construction on R, the dimension (Hausdorff or box counting) is almost surely equal to the similarity dimension, that is, the overlap between the cylinders typically does not lead to dimension drop. However, we do not understand the cause of the decrease of dimension in the exceptional cases. In this paper we...