Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 1 / 3F (Geometry and Groups)

Under the open set condition there is a nonempty bounded open set $O$ such that the sets $S_i(O)$ are pairwise disjoint and all contained in $O$. The Hausdorff dimension of the nonempty compact invariant set is the unique $s\ge 0$ satisfyingFor the carpet here the 24 retained sub-squares give Euclidean similarities $S_{ij}(x,y)=((x+i)/5,(y+j)/5)$ for $0\le i,j\le 4$, $(i,j)\ne (2,2)$. Their images of $O=(0,1{)}^2$ are disjoint and lie in $O$, so the condition holds even though some closed images share boundary points. All contraction factors are $1/5$. Thus