# On a class of self-similar sets which contain finitely many common points

@inproceedings{Wang2021OnAC, title={On a class of self-similar sets which contain finitely many common points}, author={Zhiqiang Wang and Kan Jiang and Derong Kong and Wenxia Li}, year={2021} }

In this paper we consider a family of self-similar sets {Kλ : λ ∈ (0, 1/2]}, where each Kλ is generated by the iterated function system {fλ,0(x) = λx, fλ,1(x) = λx+(1−λ)}. For a given point x ∈ [0, 1] let Λ(x) := {λ ∈ (0, 1/2] : x ∈ Kλ}. Then x is a common point of Kλ for all λ ∈ Λ(x). Excluding the trivial cases with x ∈ {0, 1/2, 1} we show that for any x ∈ (0, 1) \ {1/2} the set Λ(x) is a topological Cantor set, and it has zero Lebesgue measure and full Hausdorff dimension. Furthermore, by… Expand

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