Hausdorff dimension of self-similar sets with overlaps

By Qi-Rong Deng, J. Harding, Tianyou Hu. 2008

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Dimension theory of iterated function systems (2009, background): “Partial results have only been obtained for conformal IFS that satisfy the finite type condition (see [45] for the definition).”
Self-similar Sets as Hyperbolic Boundaries (2009, background): “Another possible extension of the previous consideration is to assume the {Sj}j=1 of similitudes to satisfy the weak separation condition ([18], [19]) or the finite type condition [22] instead of the open set condition.”
On the Assouad dimension of self-similar sets with overlaps (2014, background): “An example of this dimension drop is given by the modified Sierpiński triangle of Ngai and Wang [22, Figure 2, page 670].”
Multifractal formalism for self-similar measures with weak separation condition (2009, background): “We remark that all the IFS in the above examples satisfy the finite type condition, a notion introduced by Ngai and Wang [26] which is stronger than the WSC [27].”
A Generalized Finite Type Condition for Iterated Function Systems (2007, methodology): “The following is the sequence of nested index sets used in the original finite type condition [20].”