Abstract
Let [Formula: see text] be a [Formula: see text] self-conformal set. It is well known that [Formula: see text] satisfies the strong open set condition provided it satisfies the open set condition. A point [Formula: see text] is called a trivial point of [Formula: see text] if [Formula: see text] is a connected component of [Formula: see text]. Let [Formula: see text] be a strong open set for [Formula: see text]. In this paper, we show that whether [Formula: see text] contains a trivial point of [Formula: see text] determines several metrical and topological properties of [Formula: see text], including the maximal power law property and the dimension drop of the connected part of [Formula: see text].
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