Elementary fractal geometry. 5. Weak separation is strong separation
This work addresses a foundational issue in fractal geometry for mathematicians, providing a bridge between separation properties in self-similar sets, but it appears incremental as it builds on existing conditions and properties.
The paper tackled the problem of representing finite type self-similar sets under the weak separation condition by showing they can be expressed as graph-directed constructions that satisfy the open set condition, using a combinatorial algorithm validated in computer experiments.
For self-similar sets, there are two important separation properties: the open set condition and the weak separation condition introduced by Zerner, which may be replaced by the formally stronger finite type property of Ngai and Wang. We show that any finite type self-similar set can be represented as a graph-directed construction obeying the open set condition. The proof is based on a combinatorial algorithm which performed well in computer experiments.