A fractal dimension estimate for a graph-directed IFS of non-similarities

Suppose a graph-directed iterated function system consists of maps f_e with upper estimates of the form d(f_e(x),f_e(y)) <= r_e d(x,y). Then the fractal dimension of the attractor K_v of the IFS is bounded above by the dimension associated to the Mauldin--Williams graph with ratios r_e. Suppose the maps f_e also have lower estimates of the form d(f_e(x),f_e(y)) >= r'_e d(x,y) and that the IFS also satisfies the strong open set condition. Then the fractal dimension of the attractor K_v of the IFS is bounded below by the dimension associated to the Mauldin--Williams graph with ratios r'_e. When r_e = r'_e, then the maps are similarities and this reduces to the dimension computation of Mauldin & Williams for that case.