Open set condition

A finite family of contracting similarities $S_1,\dots ,S_k$ satisfies the open set condition if there is a nonempty bounded open set $O$ with $S_i(O)\subseteq O$ and $S_i(O)\cap S_j(O)=\varnothing$ for $i\ne j$. This permits the closures to meet. For the invariant compact set, this separation makes its Hausdorff dimension equal to the similarity exponent $s$ defined by ${\sum }_i{c}_i^s=1$, where $c_i$ are the contraction ratios. The level-$n$ cylinder sets supply the upper dimension bound; the separated cylinder masses with weights $c_i^s$ supply the matching mass-distribution lower bound. For the 24 maps of ratio $1/5$ retaining all but the central square, $O=(0,1{)}^2$ works and $s=\log 24/\log 5$.