Abstract
Given , let be a class of self-similar sets with each . In this paper we investigate the likelyhood of a point in the self-similar sets of . More precisely, for a given point x ∈ (0, 1) we consider the parameter set, and show that Λ(x) is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, by constructing a sequence of Cantor subsets of Λ(x) with large thickness we show that for any x, y ∈ (0, 1) the intersection Λ(x) ∩ Λ(y) also has full Hausdorff dimension.
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