Self-similarity, positive Lebesgue measure and nonempty interior. (English) Zbl 1396.28015
Summary: In this paper, we introduce BBI spaces (“big balls of itself”), which based on the notion of BPI spaces (“big pieces of itself”) used by David and Semmes to study self-similarity. We prove that the “self-similar” construction described by BBI spaces ensures the equivalence of positive Lebesgue measure and nonempty interior. We apply this result to self-conformal sets satisfying the WSC and prove that positive Lebesgue measure implies nonempty interior for such sets. This generalizes Zerner’s corresponding result for self-similar sets.
MSC:
| 28A80 | Fractals |
References:
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